Framelet变换高光谱图像光谱加权稀疏解混

Framelet变换高光谱图像光谱加权稀疏解混 Spectral weighted sparse unmixing of hyperspectral images based on framelet transform first-author 徐晨光（1985-），男，江西抚州人，现为南昌工程学院信息工程学院教师，主要从事遥感影像处理方面的研究。E-mail： xcg@nit.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440872&type= 22.01332855 32.00399780 徐晨光 （1985-），男，江西抚州人，现为南昌工程学院信息工程学院教师，主要从事遥感影像处理方面的研究。E-mail： xcg@nit.edu.cn xcg@nit.edu.cn corresp E-mail： dengcz@nit.edu.cn E-mail： dengcz@nit.edu.cn 邓承志（1980-），男，江西兴国人，教授，2008年于华中科技大学获得博士学位，现为南昌工程学院教务处处长，兼任江西省水信息协同感知与智能处理重点实验室副主任、江西省电子学会秘书长、中国高等学校电工学研究会江西分会副理事长、江西省人工智能学会理事，主要从事遥感影像处理、图像处理方面的研究。E-mail： dengcz@nit.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440879&type= 22.01333618 32.00399780 邓承志 （1980-），男，江西兴国人，教授，2008年于华中科技大学获得博士学位，现为南昌工程学院教务处处长，兼任江西省水信息协同感知与智能处理重点实验室副主任、江西省电子学会秘书长、中国高等学校电工学研究会江西分会副理事长、江西省人工智能学会理事，主要从事遥感影像处理、图像处理方面的研究。E-mail： dengcz@nit.edu.cn dengcz@nit.edu.cn 南昌工程学院 江西省水信息协同感知与智能处理重点实验室，江西 南昌 330099 南昌工程学院 江西省水信息协同感知与智能处理重点实验室，江西 南昌 330099 Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing，Nanchang Institute of Technology， Nanchang 330099， China Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing，Nanchang Institute of Technology， Nanchang 330099， China 空域中高光谱数据由于信息过于分散，冗余过多，且易受噪声的影响，其特征提取难度较大。为了提高高光谱图像解混的鲁棒性和稀疏性，提出了一种framelet变换高光谱图像光谱加权稀疏解混方法。介绍了高光谱稀疏解混和framelet变换方法的理论知识，接着利用framelet变换对高光谱图像解混建模，并且在该模型上加入变换域光谱加权稀疏正则项，提出framelet变换的高光谱图像光谱加权稀疏解混模型。最后，利用交替方向乘子法对模型进行求解。实验结果表明：信号与重建误差比（SRE）提高12.4%~1 045%，丰度重构正确率（ P s ）保持在16%的误差内。与其他相关稀疏解混方法相比，本文提出的算法具有良好的抗噪性和稀疏性能，获得了更好的解混结果。 Hyperspectral sparse unmixing methods have attracted considerable attention， and most current sparse unmixing methods are implemented in the spatial domain； however， the hyperspectral data used by these methods complicate feature extraction owing to scattered information， redundancy， and noisy spatial signals. To improve the robustness and sparsity of the unmixing results of hyperspectral images， a spectral-weighted sparse unmixing method of hyperspectral images based on the framelet transform （SFSU） is proposed. First， we introduce the theoretical knowledge of hyperspectral sparse unmixing and the framelet transform. Following this， we develop a hyperspectral image unmixing model based on the framelet transform using this theory. In this model， a spectral-weighted sparse regularization term is added to construct the SFSU. Finally， to solve the SFSU model， an alternating direction method of multipliers is presented. According to the experimental results， the signal-to-reconstruction error ratio is found to increase by 12.4%-1 045%， and the probability of success （ P s ） remains within 16% error. The proposed model demonstrates better anti-noise and sparse performance compared with other related sparse unmixing methods and yields better unmixing results. 高光谱遥感 Framelet变换 光谱加权 稀疏解混 交替方向乘子法 hyper spectral remote sensing framelet transform sectral weighted sparse unmixing alternating direction method of multipliers（ADMM） 1 引 言 1 高光谱遥感图像不但具有地物的空间信息，而且含有丰富的光谱信息 ［ 1 1 - 2 2 1-2 ］ 。但是由于高光谱传感设备空间分辨率的限制，接收到的复杂地物的高光谱图像中单个像元内包含的地物并不单一，呈现出混合像元的现象 ［ 3 3 ］ 。混合像元的精确分解对后期数据处理具有重要的作用。 高光谱图像解混是目前解决混合像元分解问题的有效方法之一，它的目标是把混合像元分解成为端元和与之对应的丰度 ［ 4 4 - 5 5 4-5 ］ 。其解混模型是一种光谱混合模型，模型一般分为线性模型和非线性模型两种。线性光谱混合模型是假设入射光通过一次反射就被传感器接收成像，而非线性光谱混合模型则是通过多次反射和折射之后才被传感器接收成像 ［ 6 6 - 7 7 6-7 ］ 。虽然非线性混合模型能够比较真实地描述入射光传播的自然现象，但是由于非线性参数过多，很难求解，往往用线性模型来近似表示 ［ 8 8 - 9 9 8-9 ］ 。 稀疏解混方法是高光谱图像解混中比较常用的一种半监督解混方法，由于它利用预先已知的大型光谱库 ［ 10 10 ］ 作为先验知识，致使光谱库中端元数目远大于实际构成混合像元的端元数目，因此其求解模型是一个欠定问题 ［ 11 11 ］ 。为了解决该问题，Iordache ［ 12 12 ］ 在1985年提出一种经典的L 1 稀疏解混求解方法SUnSAL。Bioucas ［ 13 13 ］ 在此基础上提出了一种基于L 21 的稀疏解混算法CLSUnSAL。而Zheng ［ 14 14 ］ 为了更好地表征像元丰度系数的稀疏性，在文献［ 12 12 ］模型的稀疏项中增加一个惩罚因子（光谱加权），使得算法能更充分地利用高光谱数据中的光谱信息，从而有效地改善解混结果。虽然这些算法都是通过改进空间域中各种不同的正则项来不断地优化解混的效果，但是空间域的高光谱数据存在信息过于分散、冗余过多、易受噪声影响等缺点 ［ 15 15 ］ ，如果把高光谱数据通过一定形式的变换来减少这些不利因素，然后再进行解混，可以较好地提高解混效果。 Framelet变换方法具有变换前后信息不丢失和去相关性的特征，并且变换后的图像中大部分信息集中在低频部分，而细节和噪声都集中在高频部分 ［ 16 16 ］ 。该变换方法具有更好的稳定性，且更容易对图像进行特征提取 ［ 17 17 ］ 。Zhang ［ 18 18 ］ 等提出了一种基于framelet的高光谱稀疏解混模型，把变换分成了低频近似分量和中高频细节分量，通过调节两个分量的比例和限制项参数对高光谱图像进行更有效的稀疏解混。 虽然变换域算法在抗噪性能上比空间域算法有了很大的改善，但是对丰度系数的稀疏性先验表征不充分，没有很好地利用高光谱信息，解混效果不佳。为了更充分地利用高光谱的光谱特性，更精确地对高光谱图像进行稀疏解混，更好地提升解混结果和算法的稳定性，本文提出了一种framelet变换高光谱图像光谱加权稀疏解混方法（Spectral-weighted Framelet Sparse Unmixing， SFSU）。该算法充分利用framelet变换的优势，在对高光谱图像进行去噪的同时，增加了变换域的光谱加权约束以促进高光谱图像端元之间的稀疏，提升解混结果的抗噪性和稀疏性。 2 高光谱图像光谱加权稀疏解混 1 2.1　高光谱稀疏解混 2 假设一个光谱波段数为 <math id="M1"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440884&type= 2.45533323 2.79399991 ，端元个数为 <math id="M2"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441020&type= 2.62466669 2.79399991 的高光谱端元库定义为 <math id="M3"><mi mathvariant="bold-italic">S</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440889&type= 27.26266861 4.48733330 ，它表示 <math id="M4"><mi>L</mi><mo>×</mo><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440906&type= 9.82133293 2.79399991 的端元字典集，其中 <math id="M5"><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msubsup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>L</mi></mrow></msubsup><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440931&type= 31.32666588 4.40266657 <math id="M6"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440919&type= 2.62466669 2.79399991 ，是光谱库中第 <math id="M7"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440938&type= 1.10066664 2.79399991 个端元的光谱特征。并且假设一个包含 <math id="M8"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440943&type= 3.30200005 2.87866688 个像素的高光谱图像，其中每个像素的观测高光谱数据表示为 <math id="M9"><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440926&type= 28.19400024 4.57200003 ，则高光谱观测矩阵 <math id="M10"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440948&type= 30.31066895 4.91066647 表示为 <math id="M11"><mi>L</mi><mo>×</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440965&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440963&type= 10.49866676 2.87866688 的矩阵。高光谱线性混合模型表示为： <math display="block" id="M12"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo>+</mo><mi mathvariant="bold-italic">E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440952&type= 20.23533440 2.96333337 ， 其中： <math id="M13"><mi mathvariant="bold-italic">X</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>P</mi><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440972&type= 14.81666565 3.38666677 表示光谱库中各端元的比例丰度； <math id="M14"><mi mathvariant="bold-italic">E</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>L</mi><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440977&type= 14.73200035 3.38666677 为误差噪声。 利用线性模型解混进行逼近，通常会添加一些先验条件，例如非负约束 <math id="M15"><mi mathvariant="bold-italic">X</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440955&type= 9.56733322 3.30200005 和稀疏性约束 <math id="M16"><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440991&type= 14.64733315 4.57200003 。线性解混模型如下： <math display="block" id="M17"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Y</mi><mo>-</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">X</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440983&type= 62.73799896 7.87400007 其中 <math id="M18"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441848&type= 1.77800000 2.79399991 表示为正则化参数。 2.2　光谱加权稀疏解混 2 为了提高模型（2）中稀疏项的稀疏性，更好地表征像元丰度系数的稀疏性，并且充分地利用高光谱数据中的光谱信息，在模型（2）的稀疏项中增加一个光谱加权因子 <math id="M19"><mi mathvariant="bold-italic">Ω</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>P</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440986&type= 45.29666519 3.97933316 ，其中 <math id="M20"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441020&type= 2.62466669 2.79399991 为矩阵 <math id="M21"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441025&type= 3.13266683 2.79399991 的行数（光谱库中端元的个数）， <math id="M22"><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>ε</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441029&type= 31.24200058 10.41399956 。光谱加权因子的添加，使迭代求解时对非零系数施加更大的惩罚，从而改善解混结果。增加光谱加权因子后的稀疏解混模型如下： <math display="block" id="M23"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>X</mi></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Y</mi><mo>-</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Ω</mi><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">X</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441033&type= 65.61666107 7.87400007 由于 <math id="M24"><mi mathvariant="bold-italic">Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441037&type= 3.13266683 2.79399991 为对角矩阵，为了便于计算，稀疏项转化为 <math id="M25"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Ω</mi><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441039&type= 27.34733391 4.57200003 ，其中 <math id="M26"><mo>•</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441042&type= 1.26999998 2.79399991 表示矩阵的点乘，令 <math id="M27"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>/</mo><mfenced separators="|"><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ε</mi></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441055&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441053&type= 39.03133011 6.60400009 则 <math id="M28"><mi mathvariant="bold-italic">Κ</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo>…</mo></mtd><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mo>…</mo></mtd><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>P</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>…</mo></mtd><mtd><mo>…</mo></mtd><mtd><mo>…</mo></mtd><mtd><mo>…</mo></mtd><mtd><mo>…</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo>…</mo></mtd><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mo>…</mo></mtd><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>P</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441047&type= 47.41333389 14.22399998 。 3 Framelet变换高光谱图像光谱加权稀疏解混 1 3.1　Framelet变换方法 2 Framelet变换 ［ 19 19 - 20 20 19-20 ］ 是在紧框架基础上构建的。紧框架定义为：如果 <math id="M29"><mi>χ</mi><mo>⊂</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441066&type= 16.67933464 4.82600021 满足： <math display="block" id="M30"><mtable><mtr><mtd><mi>f</mi><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>χ</mi></mrow></munder><mrow><mfenced open="<" close=">" separators="|"><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></mfenced></mrow><mi>g</mi><mo>,</mo><mo>∀</mo><mi>f</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">或</mi><mtext> </mtext><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><msup><mrow><munder><mo movablelimits="false">∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>χ</mi></mrow></munder><mrow><mfenced open="|" close="|" separators="|"><mrow><mfenced open="<" close=">" separators="|"><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></mfenced></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo><mo>∀</mo><mi>f</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441062&type= 55.71067047 17.44133186 其中 <math id="M31"><mfenced open="<" close=">" separators="|"><mrow><mo>⋅</mo><mo>,</mo><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441073&type= 7.78933382 3.97933316 是 <math id="M32"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441078&type= 10.15999985 4.48733330 空间的内积， <math id="M33"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mo>⋅</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441080&type= 5.84200001 5.24933338 表示 <math id="M34"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441135&type= 10.15999985 4.48733330 空间的二范数，那么称 <math id="M35"><mi>χ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441085&type= 2.03200006 3.64066648 是在 <math id="M36"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441088&type= 10.15999985 4.48733330 空间的紧框架。 对于给定的 <math id="M37"><mi mathvariant="bold-italic">Φ</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mi>ϕ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></mfenced><mo>⊂</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441109&type= 39.79333496 4.74133301 ，对 <math id="M38"><mi mathvariant="bold-italic">Φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441124&type= 3.13266683 2.79399991 进行扩张和平移可以得到小波系统 <math id="M39"><mi>χ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441120&type= 8.80533314 3.97933316 ，定义为： <math display="block" id="M40"><mi>χ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>ϕ</mi></mrow><mrow><mi>j</mi></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi></mrow></msup><mi>x</mi><mo>-</mo><mi>l</mi></mrow></mfenced><mo>:</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>r</mi><mo>;</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>∈</mo><mi mathvariant="bold">Z</mi></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441116&type= 70.86599731 5.24933338 当 <math id="M41"><mi>χ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Φ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441120&type= 8.80533314 3.97933316 形成的 <math id="M42"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441135&type= 10.15999985 4.48733330 空间的紧框架时，称之为小波紧框架， <math id="M43"><msup><mrow><mi>ϕ</mi></mrow><mrow><mi>j</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441154&type= 2.79399991 3.89466691 称为一个紧framelet。 为了构建小波框架，定义一个紧支撑缩放函数 <math id="M44"><mi>ψ</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441142&type= 16.00200081 4.74133301 ，该函数满足： <math display="block" id="M45"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>l</mi><mo>∈</mo><mi mathvariant="bold">Z</mi></mrow></munder><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi>x</mi><mo>-</mo><mi>l</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441143&type= 44.11133575 6.77333355 其中 <math id="M46"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441165&type= 16.76399994 4.82600021 ，是缩放函数掩膜（低通滤波器）。 通过framelet基 <math id="M47"><mi>ψ</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>r</mi></mrow></mfenced><mo>⊂</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441167&type= 39.62400055 4.82600021 <math id="M48"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441152&type= 10.15999985 4.48733330 和缩放函数，构建出framelet系统。该系统定义为：令 <math id="M49"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>r</mi></mrow></mfenced><mo>⊂</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441186&type= 42.58732986 4.82600021 为一个framelet掩膜（高通滤波器）集合，则framelet系统为： <math display="block" id="M50"><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>l</mi><mo>∈</mo><mi mathvariant="bold">Z</mi></mrow></munder><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi>x</mi><mo>-</mo><mi>l</mi><mo stretchy="false">)</mo><mo>,</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mi>r</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441175&type= 62.22999954 6.77333355 因此，系统framelet <math id="M51"><mi>ψ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441195&type= 2.11666679 3.55599999 ，相当于定义一组滤波器 <math id="M52"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441198&type= 25.90800095 4.57200003 满足： <math display="block" id="M53"><mtable><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ω</mi><mo>+</mo><mi>γ</mi><mi mathvariant="normal">π</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></mrow><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ω</mi><mo>+</mo><mi>γ</mi><mi mathvariant="normal">π</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>γ</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441202&type= 63.66933060 15.40933418 其中 <math id="M54"><mi>ω</mi><mo>∈</mo><mi mathvariant="fraktur">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441217&type= 9.14400005 3.13266683 ，并且 <math id="M55"><msub><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>l</mi></mrow></msub><mrow><mi>ξ</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>i</mi><mi>l</mi><mi>ω</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441209&type= 31.07266808 4.74133301 和 <math id="M56"><msub><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>l</mi></mrow></msub><mrow><mi>ξ</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>i</mi><mi>l</mi><mi>ω</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441228&type= 31.07266808 4.74133301 是一个增量函数。 在实际应用中，通常用分段线性B样条作为framelet系统，定义的缩放函数掩膜为： <math id="M57"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441225&type= 23.02933502 7.87400007 ，相应的两个高通滤波器为： <math id="M58"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1,0</mn><mo>,</mo><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441243&type= 31.24200058 8.29733276 ， <math id="M59"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441231&type= 31.58066559 7.87400007 。 通常在算法中framelet变换可以用矩阵 <math id="M60"><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441362&type= 3.97933316 2.96333337 表示，具体的推导过程详见文献［ 19 19 ］。 对于滤波器 <math id="M61"><mi>h</mi><mo>=</mo><msubsup><mrow><mfenced open="{" close="}" separators="|"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>j</mi><mo>=</mo><mo>-</mo><mi>J</mi></mrow><mrow><mi>J</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441238&type= 23.45266724 6.51933336 ，令矩阵 <math id="M62"><mi mathvariant="bold-italic">S</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441240&type= 7.78933382 3.55599999 为Neumann（对称）边界条件下滤波器 <math id="M63"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441258&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441250&type= 1.94733346 2.79399991 的卷积算子： <math display="block" id="M64"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">S</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mtd><mtd><mtable><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mi>J</mi><mo stretchy="false">)</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr></mtable></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr></mtable></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mtable><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mtable><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr></mtable></mtd><mtd><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd><mtd><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mi>J</mi><mo stretchy="false">)</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mtd><mtd><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441260&type= 70.61199951 54.18666840 在给定的过滤器 <math id="M65"><mi>h</mi><mo>=</mo><msubsup><mrow><mfenced open="{" close="}" separators="|"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>j</mi><mo>=</mo><mo>-</mo><mi>J</mi></mrow><mrow><mi>J</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441274&type= 23.45266724 6.51933336 ， <math id="M66"><mi>ς</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441279&type= 1.69333339 3.64066648 层的滤波器 <math id="M67"><msup><mrow><mi>h</mi></mrow><mrow><mfenced separators="|"><mrow><mi>ς</mi></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441283&type= 4.65666676 3.64066648 对应没有下采样的分解，为： <math display="block" id="M68"><msup><mrow><mi>h</mi></mrow><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mi>J</mi><mo stretchy="false">)</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mi>J</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441267&type= 63.07666779 23.79133224 . 则： <math display="block" id="M69"><mi mathvariant="bold-italic">W</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mo>∏</mo><mrow><mi>ς</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd></mtr><mtr><mtd><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mo>∏</mo><mrow><mi>ς</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mi>r</mi></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mo>∏</mo><mrow><mi>ς</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mi>r</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>L</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441271&type= 54.18666840 47.07466888 ， 其中： <math id="M70"><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ς</mi></mrow></msubsup><mo>≡</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441290&type= 20.06599998 4.65666676 ， <math id="M71"><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441362&type= 3.97933316 2.96333337 为多层分解算子， <math id="M72"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>ς</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msubsup><mrow><mi mathvariant="bold-italic">Η</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441296&type= 30.47999954 5.16466665 为低通滤波器， <math id="M73"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441388&type= 7.19666624 3.80999994 等于 <math id="M74"><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441362&type= 3.97933316 2.96333337 中剩余的值. 因此，离散图像（矢量形式） <math id="M75"><mi mathvariant="bold-italic">f</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441321&type= 13.54666710 3.89466691 的变换形式可以写成 <math id="M76"><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441338&type= 5.50333309 3.64066648 。根据酉扩展原理（UEP） ［ 21 21 ］ ，变换算子 <math id="M77"><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441326&type= 3.97933316 2.96333337 必须满足 <math id="M78"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">W</mi><mo>=</mo><mi mathvariant="bold">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441343&type= 15.40933418 3.13266683 ，其中 <math id="M79"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441332&type= 5.67266655 3.13266683 是 <math id="M80"><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441362&type= 3.97933316 2.96333337 的逆变换矩阵算符， <math id="M81"><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441355&type= 1.86266661 2.79399991 是单位矩阵。 3.2　Framelet变换稀疏解混 2 空域稀疏解混方法的高光谱图像信息过于分散，冗余过多，且易受噪声影响，致使其特征提取不便 ［ 22 22 ］ 。通过framelet变换可以使图像问题简单化，并且变换后绝大部分信息集中在低频部分，而中高频部分只有很少一部分细节和噪声信息，有利于特征的提取。通过控制低频和中高频的比例，可以达到增强抗噪性能的目的。Framelet变换的高光谱稀疏解混就是利用这种特性，通过把framelet变换方法融入到稀疏解混中完成高光谱解混。 因此，把 式（2） 式（2） 添加framelet变换方法得到framelet变换高光谱图像稀疏解混方法： <math display="block" id="M82"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">Y</mi><mo>-</mo><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">X</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441369&type= 76.62332916 7.87400007 其中 <math id="M83"><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441362&type= 3.97933316 2.96333337 表示变换算子矩阵。 为了方便对中高频信息进行滤波，Zhang ［ 18 18 ］ 等把 <math id="M84"><mi mathvariant="bold-italic">W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441362&type= 3.97933316 2.96333337 变换算子矩阵分成两部分 <math id="M85"><mi mathvariant="bold-italic">W</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441377&type= 25.73866653 3.89466691 ，其中 <math id="M86"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441384&type= 5.16466665 3.80999994 表示低频变换算子矩阵， <math id="M87"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441388&type= 7.19666624 3.80999994 表示中高频变换算子矩阵。 因此 式（11） 式（11） 可以写成： <math display="block" id="M88"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold">X</mi></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mtext> </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mtext> </mtext><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">X</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441410&type= 125.05266571 7.87400007 其中 <math id="M89"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442073&type= 1.94733346 2.79399991 是低频项和中高频项的平衡参数。当 <math id="M90"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442073&type= 1.94733346 2.79399991 =1时， 式（13） 式（13） 和 式（12） 式（12） 等效，对抗噪不具备优势；当 <math id="M91"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442073&type= 1.94733346 2.79399991 在（0，1）区间变化时，可以发挥出一定的抗噪性能。 3.3　Framelet变换的光谱加权稀疏解混 2 高光谱数据包含了大量光谱信息。在基于framelet变换的高光谱稀疏解混算法中，通常都是做简单的变换，而对高光谱的光谱特性利用不足，导致先验信息表示不充分，解混结果不优的问题。Framelet变换的光谱加权稀疏解混是把framelet变换稀疏解混算法（ 式（13） 式（13） ）与光谱加权稀疏解混算法（ 式（3） 式（3） ）进行融合，使新的模型在抗噪性和稀疏性上都有很大的提升。基于此，本文把光谱加权约束引入到变换域高光谱稀疏解混中，提出一种framelet变换光谱加权稀疏解混方法。 3 . 3 . 1　解混模型 3 解混模型分为两部分：第一部分为观测矩阵与近似值的变换的逼近项（包含了低频逼近项和中高频逼近项），第二部分为变换域光谱权重约束项。 <math display="block" id="M92"><mtable><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="‖" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><msubsup><mrow><mfenced open="" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext> </mtext><mi mathvariant="bold-italic">X</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441435&type= 65.70133209 15.57866764 其中 <math id="M93"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 表示光谱加权因子。 式（14） 式（14） 中第一部分的第一项是低频逼近项，为观测数据的近似逼近项。第二项是中高频逼近项，为观测数据边缘轮廓细节逼近项，大部分噪声都在该项中。由于变换后，大部分信息都集中在第一项低频逼近项中，而绝大多数噪声都集中在第二项中高频逼近项中，通过调节 <math id="M94"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441466&type= 1.94733346 2.79399991 参数来控制低频逼近项与中高频逼近项的平衡比例，从而达到抑制高光谱解混数据中噪声的效果。在噪声大的情况下，通过减小 <math id="M95"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441460&type= 1.94733346 2.79399991 的值来抑制噪声。因此， <math id="M96"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441466&type= 1.94733346 2.79399991 参数的设置对高光谱解混去噪起着关键作用。 第三项为丰度矩阵稀疏正则化项，其中光谱加权因子 <math id="M97"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 表示对不同光谱间丰度稀疏性进行测量，在求解时对不稀疏的光谱施加更大的惩罚，从而提高解混的稀疏效果。 <math id="M98"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 在 t +1次迭代中表示为： <math display="block" id="M99"><msup><mrow><mi mathvariant="bold-italic">Κ</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow/><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>t</mi></mrow></msubsup></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>t</mi></mrow></msubsup></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow/><mrow><mi>P</mi></mrow></msub></mrow><mrow><mi>t</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow/><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>t</mi></mrow></msubsup></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>t</mi></mrow></msubsup></mtd><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd><mtd><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow/><mrow><mi>P</mi></mrow></msub></mrow><mrow><mi>t</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441479&type= 53.08599854 16.08666801 其中 <math id="M100"><msubsup><mrow><mi>k</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>t</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ε</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441498&type= 30.56466866 10.41399956 ， <math id="M101"><msubsup><mrow><mi mathvariant="bold-italic">Κ</mi></mrow><mrow/><mrow><mi>t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441496&type= 4.14866638 3.97933316 表示 <math id="M102"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 第 t 次迭代结果； <math id="M103"><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441615&type= 11.09133339 3.97933316 表示丰度矩阵 <math id="M104"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441591&type= 3.13266683 2.79399991 在第 t 次迭代结果中第 i 行的值； <math id="M105"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441534&type= 2.28600001 3.64066648 表示 <math id="M106"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441591&type= 3.13266683 2.79399991 矩阵的行数； <math id="M107"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441540&type= 1.43933344 2.79399991 表示最小值，以防 <math id="M108"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441553&type= 16.17133331 5.33400011 得到零的值，T表示矩阵的转置。 式（15） 式（15） 中矩阵的每行元素值都相等，且每行的值都依赖于丰度矩阵对应行的 <math id="M109"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441565&type= 16.17133331 5.33400011 值。而丰度矩阵 <math id="M110"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441591&type= 3.13266683 2.79399991 中每一行表示一个端元光谱对应的丰度值。 <math id="M111"><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441576&type= 7.28133297 2.79399991 表示 <math id="M112"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441591&type= 3.13266683 2.79399991 矩阵中各个元素乘上该元素对应行的 <math id="M113"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ε</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441595&type= 23.11400032 10.41399956 的加权值。该加权值与其对应行端元的 <math id="M114"><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441597&type= 11.09133339 3.97933316 有关。端元对应的丰度 <math id="M115"><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441600&type= 11.09133339 3.97933316 越小， <math id="M116"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441587&type= 16.17133331 5.33400011 值越小，对应的 <math id="M117"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ε</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441622&type= 22.60600090 10.41399956 就越大，进而在下一次迭代过程中对该行施加更大的约束；而如果端元对应的丰度比较大， <math id="M118"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441609&type= 16.17133331 5.33400011 值就越大，则对应的 <math id="M119"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ε</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441612&type= 23.11400032 10.41399956 就越小，进而在下一次迭代过程中对 <math id="M120"><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>t</mi></mrow></msup><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mo>:</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441615&type= 11.09133339 3.97933316 施加的约束就越小。由此促进了高光谱图像端元之间的稀疏性。 根据 式（11） 式（11） 可以得到 <math id="M121"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441649&type= 73.32133484 5.16466665 ，该公式是低频逼近项和中高频逼近项之和。再根据低通滤波的特点 ［ 18 18 ］ ，低频逼近项 <math id="M122"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≈</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441639&type= 38.18466568 5.16466665 ，从而 <math id="M123"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≈</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441655&type= 36.83000183 5.16466665 <math id="M124"><mo>+</mo><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441646&type= 26.50066566 5.16466665 。因此， 式（14） 式（14） 中的光谱权重项 <math id="M125"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441685&type= 19.64266586 5.16466665 项包含了 <math id="M126"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441667&type= 12.36133385 4.57200003 。在参考文献［ 13 13 ］和［ 21 21 ］中，光谱加权因子 <math id="M127"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441672&type= 3.21733332 2.79399991 得到成功使用，并且实验结果也证明增加光谱加权因子 <math id="M128"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 的稀疏项，对丰度的稀疏性有很大的增强作用。因此，本文提出的模型（14）中也增加了变换域的光谱权重项 <math id="M129"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441685&type= 19.64266586 5.16466665 以替代 <math id="M130"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">Κ</mi><mo>•</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441690&type= 12.36133385 4.57200003 项，来达到增强丰度的稀疏功能。 3 . 3 . 2　求解方法 3 本文利用交替方向乘子法ADMM ［ 23 23 ］ 对模型（14）进行求解。在求解算法中，采用内外两层循环来加快求解的收敛速度。其中，内层循环作用是利用ADMM算法对算法中相关变量和拉格朗日算子进行迭代更新，而外层循环则是对光谱加权因子 <math id="M131"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 进行更新。具体来说，光谱加权因子 <math id="M132"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 是由外层循环决定的，而在每次内层循环中，光谱加权因子 <math id="M133"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441716&type= 3.21733332 2.79399991 的值都是保持不变的，内层循环只是利用ADMM对相关变量和拉格朗日算子进行交替迭代工作。求解思路如下： 第一步：公式化简。 根据卷积定理，矩阵在空间域的点乘的变换等价于矩阵在变换域的卷积，如下： <math display="block" id="M134"><mi>f</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>•</mo><mi>g</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>↔</mo><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mi mathvariant="normal">*</mi><mi>G</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441720&type= 59.09733200 4.82600021 因此 式（14） 式（14） 简化为： <math display="block" id="M135"><mtable columnalign="left"><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi></mrow></munder><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="‖" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><msubsup><mrow><mfenced open="" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">Κ</mi><mi mathvariant="normal">*</mi><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>+</mo><msub><mrow><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441724&type= 67.81800079 15.57866764 其中 <math id="M136"><mi mathvariant="normal">*</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441728&type= 1.77800000 2.79399991 代表卷积， <math id="M137"><msub><mrow><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441732&type= 14.73200035 4.48733330 代表 <math id="M138"><mi mathvariant="bold-italic">X</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441738&type= 9.56733322 3.30200005 。 第二部：ADMM求解。 根据拉格朗日扩展公式： <math id="M139"><mi>ζ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo>,</mo><mi mathvariant="bold-italic">D</mi></mrow></mfenced><mo>=</mo><mi>g</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">V</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441753&type= 36.23733139 4.06400013 <math id="M140"><mo>+</mo><mfrac><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">G</mi><mi mathvariant="bold-italic">U</mi><mo>-</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">V</mi><mo>-</mo><mi mathvariant="bold-italic">D</mi></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441741&type= 35.72933197 7.87400007 ，其中 <math id="M141"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441860&type= 2.11666679 3.64066648 表示归一化拉格朗日乘子，通常为正数； <math id="M142"><mi>g</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">V</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mtext> </mtext><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mtext> </mtext><mo>+</mo><mtext> </mtext><mfrac><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mtext> </mtext><mfenced open="‖" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mtext> </mtext><mi mathvariant="bold-italic">Y</mi><mtext> </mtext><mo>-</mo></mrow></mfenced><mtext> </mtext><msubsup><mrow><mfenced open="" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mtext> </mtext><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441748&type= 93.81066895 7.87400007 <math id="M143"><mi>λ</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">Κ</mi><mi mathvariant="normal">*</mi><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>+</mo><msub><mrow><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441763&type= 41.48666382 5.16466665 <math id="M144"><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo>,</mo><mi mathvariant="bold-italic">D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441767&type= 12.44600010 3.21733332 为方程待求解的未知矩阵，其中 <math id="M145"><mi mathvariant="bold-italic">U</mi><mo>=</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441752&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441771&type= 12.27666664 3.21733332 <math id="M146"><mi mathvariant="bold-italic">V</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="bold-italic">D</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441773&type= 55.96466827 3.97933316 。 <math id="M147"><mi mathvariant="bold-italic">G</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="bold-italic">S</mi><mo>,</mo><mi mathvariant="bold-italic">W</mi><mo>,</mo><mi mathvariant="bold-italic">I</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441786&type= 23.87599945 4.74133301 和 <math id="M148"><mi mathvariant="bold-italic">B</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mo stretchy="false">[</mo><mi mathvariant="bold-italic">I</mi><mo>,</mo><mi mathvariant="bold-italic">I</mi><mo>,</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441775&type= 27.00866890 3.97933316 为关系矩阵，并且满足 <math id="M149"><mi mathvariant="bold-italic">G</mi><mi mathvariant="bold-italic">U</mi><mo>=</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441791&type= 16.25600052 2.96333337 。可以得到： <math display="block" id="M150"><mtable><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo>,</mo><mi mathvariant="bold-italic">D</mi></mrow></munder><mi>ζ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo>,</mo><mi mathvariant="bold-italic">D</mi></mrow></mfenced><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo>,</mo><mi mathvariant="bold-italic">D</mi></mrow></munder><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="‖" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><mi mathvariant="bold-italic">Y</mi><mo>-</mo></mrow></mfenced><msubsup><mrow><mfenced open="" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">M</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">Κ</mi><mi mathvariant="normal">*</mi><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfrac><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">U</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">W</mi><mi mathvariant="bold-italic">U</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">U</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441793&type= 138.00666809 17.61066628 其中： <math id="M151"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">W</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441782&type= 92.87934113 4.57200003 <math id="M152"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441796&type= 46.48199844 4.57200003 上标表示迭代次数， t 或是 t +1表示内层循环迭代次数。以下公式也同样适用。根据ADMM交替迭代进行求解，可以得到： <math display="block" id="M153"><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow/><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">S</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">I</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced open="[" close="" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo></mrow></mfenced><mfenced open="" close="]" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441797&type= 105.74867249 5.24933338 <math display="block" id="M154"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced open="[" close="" separators="|"><mrow><mfenced close="" separators="|"><mrow><mfenced close="" separators="|"><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced><mi>F</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mi mathvariant="bold-italic">W</mi></mrow></mfenced><mfenced open="" separators="|"><mrow><mo>+</mo><mi>α</mi></mrow></mfenced><mi>F</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">Y</mi></mrow></mfenced><mo>+</mo></mrow></mfenced><mfenced open="" close="]" separators="|"><mrow><mfenced open="" separators="|"><mrow><mfenced open="" separators="|"><mrow><mi>μ</mi><mi>F</mi><mfenced close="" separators="|"><mrow><mi mathvariant="bold-italic">S</mi><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow/><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>-</mo></mrow></mfenced><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup></mrow></mfenced></mrow></mfenced><mo>/</mo><mfenced separators="|"><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>F</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mi mathvariant="bold-italic">W</mi></mrow></mfenced><mo>+</mo><mi>α</mi><mo>+</mo><mi>μ</mi></mrow></mfenced></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441815&type= 153.41600037 6.77333355 其中 <math id="M155"><mi>F</mi><mfenced separators="|"><mrow><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441807&type= 7.36600018 3.97933316 和 <math id="M156"><msup><mrow><mi>F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441809&type= 9.90600014 3.97933316 分别为傅里叶的正反变换， <math display="block" id="M157"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mi mathvariant="bold-italic">W</mi><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow/><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>,</mo><mfrac><mrow><mi>λ</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><msubsup><mrow><mi mathvariant="bold-italic">Κ</mi></mrow><mrow/><mrow><mi>k</mi></mrow></msubsup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441812&type= 62.73799896 10.58333302 ， 其中 <math id="M158"><msubsup><mrow><mi mathvariant="bold-italic">Κ</mi></mrow><mrow/><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441831&type= 4.40266657 3.97933316 的上标 k 表示外层循环迭代次数。以下公式也同样适用。 <math display="block" id="M159"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441834&type= 42.75666428 4.57200003 . 3 . 3 . 3　算法描述 3 当在运用framelet变换的高光谱图像光谱加权稀疏解混前，需要对算法进行初始化和参数设置。 3.3.3.1　初始化 4 算法进行了内外两层循环，内层循环利用ADMM方法得到相关待求变量和拉格朗日算子的迭代值，外层循环更新内层循环中光谱加权因子的值，这样的做法可以使算法在光谱加权因子稳定的情况下加快收敛速度。因此在初始状态，外循环次数 k 和内循环次数 t 都必须初始设置为0。初始化 <math id="M160"><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="bold">T</mi></mrow></msup><mi mathvariant="bold-italic">S</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">I</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="bold">T</mi></mrow></msup><mi mathvariant="bold-italic">Y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441838&type= 40.97866440 5.16466665 ， <math id="M161"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mo>=</mo><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441842&type= 16.17133331 4.06400013 ， <math id="M162"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mo>=</mo><mi mathvariant="bold-italic">W</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441844&type= 17.52599907 4.06400013 ， <math id="M163"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441825&type= 13.88533401 4.06400013 ，并且设置拉格朗日算子初始化值为0。 3.3.3.2　参数设计 4 求解 式（17） 式（17） 时，需要设置3个参数。第一个是稀疏项正则化参数 <math id="M164"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441848&type= 1.77800000 2.79399991 ，第二个是变换平衡参数 <math id="M165"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442073&type= 1.94733346 2.79399991 ，第三个是拉格朗日算子平衡参数 <math id="M166"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441860&type= 2.11666679 3.64066648 ，通常这3个值都必须大于0。由于高光谱像元分解时，在不同的端元、不同噪声，或不同像元尺寸的情况下，较优结果的3个参数值都可能不一样。因此，在每次高光谱数据解混时，需要多次动态调整3个参数，使分解结果更为精确。 3.3.3.3　停止条件 4 算法分为内外两层，内层循环停止和跳出条件为：当内循环达到设定数值（设定为5）或是前后两次迭代误差值小于设定数（设定为0.000 1）时，跳出内循环。外循环停止条件为：当外循环达到设定数值（设定为60）或是前后两次迭代误差值小于设定数（设定为0.000 1）时，跳出外循环结束计算。SFSU算法伪代码如 表1 表1 所示。 1. Input： spectrum library <math id="M167"><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441854&type= 2.03200006 2.37066650 and observed data <math id="M168"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441873&type= 2.70933342 2.37066650 . 2. Initialize： <math id="M169"><mi>k</mi><mo>=</mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441864&type= 7.19666624 2.37066650 ， <math id="M170"><mi>t</mi><mo>=</mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441878&type= 6.68866634 2.37066650 ， <math id="M171"><mi>α</mi><mo>></mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441867&type= 7.45066643 2.45533323 ， <math id="M172"><mi>λ</mi><mo>></mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441883&type= 7.28133297 2.45533323 ， <math id="M173"><mi>μ</mi><mo>></mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441869&type= 7.61999989 3.13266683 ， <math id="M174"><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="bold">T</mi></mrow></msup><mi mathvariant="bold-italic">S</mi><mo>+</mo><mn>2</mn><mi mathvariant="bold-italic">I</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi mathvariant="bold">T</mi></mrow></msup><mi mathvariant="bold-italic">Y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441889&type= 35.39066696 4.57200003 ， <math id="M175"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn>0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441893&type= 13.88533401 3.47133350 ， <math id="M176"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mi mathvariant="bold-italic">W</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn>0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441897&type= 15.07066536 3.47133350 ， <math id="M177"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn>0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441872&type= 11.93799973 3.47133350 ， <math id="M178"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441904&type= 9.39799976 3.38666677 ， <math id="M179"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441906&type= 9.39799976 3.38666677 ， <math id="M180"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mn>0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441921&type= 9.39799976 3.38666677 。 3. Repeat until the stop condition is reached （1） Update <math id="M181"><mi mathvariant="bold-italic">Κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441909&type= 2.70933342 2.37066650 with formula （15） formula （15） ， （2） Repeat until the stop condition is reached （a） Update variables <math id="M182"><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow/><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441913&type= 8.46666718 3.80999994 <math id="M183"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441917&type= 8.46666718 3.80999994 <math id="M184"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441936&type= 8.46666718 3.80999994 <math id="M185"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441944&type= 6.85799980 3.47133350 with formulas （ 19 19 ）-（ 22 22 ） respectively. （b） Update Lagrangian operators： <math id="M186"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441946&type= 42.50266647 3.97933316 <math id="M187"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">W</mi><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441949&type= 43.34933472 3.97933316 <math id="M188"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441931&type= 40.30133438 3.97933316 （c） Update inner iteration times： <math id="M189"><mi>t</mi><mo>=</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441964&type= 13.46199989 2.53999996 （d） variable replacement： <math id="M190"><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow/><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow/><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441969&type= 17.94933319 3.38666677 ， <math id="M191"><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow/><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow/><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441953&type= 17.10266685 3.38666677 . （3） Update outer iteration times： <math id="M192"><mi>k</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441955&type= 14.39333248 2.53999996 4. Output ： <math id="M193"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441958&type= 2.70933342 2.37066650 . 4 实验与讨论 1 本节通过模拟和真实数据实验来验证所提出算法的先进性，并且与SUnSAL，CLSUnSAL，RWCLSUnSAL及FSU算法进行比较。采用 <math id="M194"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">10</mn><mo>⋅</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><mo>/</mo><mi>E</mi><mfenced separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>-</mo><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441993&type= 73.23667145 7.02733374 表示信号与重建误差比，并且利用丰度重构正确率 <math id="M195"><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mi>P</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>-</mo><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≤</mo><mi>t</mi><mi>h</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441981&type= 46.39733505 6.26533365 进行定量分析，来衡量混合像元分解的精度。公式中SRE的单位为dB， <math id="M196"><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441998&type= 2.28600001 2.79399991 为正确的丰度值， <math id="M197"><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441985&type= 2.28600001 2.87866688 为丰度的估计值， <math id="M198"><mi mathvariant="bold-italic">E</mi><mfenced separators="|"><mrow><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441987&type= 7.61999989 3.97933316 表示期望， <math id="M199"><mi>P</mi><mfenced separators="|"><mrow><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441989&type= 7.28133297 3.97933316 表示概率。SRE值越大或是 <math id="M200"><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43441991&type= 3.13266683 4.57200003 值越大时，算法越接近真实值，其稀疏性能也较好。 4.1　模拟高光谱数据实验 2 在模拟高光谱数据实验中，选择两组高光谱数据（DC1和DC2）进行模拟实验。这两组数据都采用同一种光谱库 <math id="M201"><mi mathvariant="bold">A</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">224</mn><mo>×</mo><mn mathvariant="normal">240</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442008&type= 17.78000069 3.38666677 进行解混。光谱库 <math id="M202"><mi mathvariant="bold">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442092&type= 2.79399991 2.79399991 是在美国地质勘探局数据库中随机选择240个端元光谱数据作为实验中的光谱字典，该字典具有224个不同波段，并且均匀分布在0.4~2.5 μm内。这两组数据设定不同的端元丰度，并根据线性混合模型（1），在3种不同的信噪比（SNR）情况下（20，30，40 dB）生成3种不同的带噪高光谱模拟数据。 图1 图1 表示构成高光谱模拟数据1（DC1）的5个不同端元的丰度，每个丰度矩阵大小为75×75，其背景由这5种端元按不同比例（0.114 9∶0.074 1∶0.200 3∶0.205 5∶0.405 1）混合而成。 图2 图2 是高光谱模拟数据2（DC2）的9个不同端元的丰度值，它们同样是通过线性混合而成，每个丰度矩阵大小为100×100。 表2 表2 和 表3 表3 分别展示了不同SNR下，利用SUnSAL，CLSUnSAL，RWCLSUnSAL，FSU和SFSU 5种算法对DC1和DC2解混的最优结果和对应的参数值，粗体表示同一SNR下解混结果最好的数值。 2 Algorithms 4 SNR=20 4 SNR=30 4 SNR=40 <math id="M203"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi><mo>/</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442048&type= 10.24466610 2.45533323 <math id="M204"><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442050&type= 2.45533323 3.47133350 <math id="M205"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442052&type= 1.35466671 2.11666679 <math id="M206"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442039&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442062&type= 1.52400005 2.11666679 <math id="M207"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi><mo>/</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442048&type= 10.24466610 2.45533323 <math id="M208"><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442050&type= 2.45533323 3.47133350 <math id="M209"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442052&type= 1.35466671 2.11666679 <math id="M210"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442046&type= 1.52400005 2.11666679 <math id="M211"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi><mo>/</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442048&type= 10.24466610 2.45533323 <math id="M212"><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442050&type= 2.45533323 3.47133350 <math id="M213"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442052&type= 1.35466671 2.11666679 <math id="M214"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442069&type= 1.52400005 2.11666679 SUnSAL 2.844 5 0.178 5 0.5 - 6.117 1 0.808 5 0.05 - 11.149 8 0.999 6 0.01 - CLSUnSAL 5.911 0.927 5 20 - 7.515 0 0.967 8 1.5 - 14.853 7 1 0.5 - RWCLSUnSAL 0.965 8 0.089 1 0.007 - 5.104 9 0.581 9 0.001 - 8.901 9 0.981 7 0.000 2 - FSU 8.530 6 0.972 4 0.05 0.005 11.103 4 0.993 1 0.01 0.05 18.273 1 0.999 1 0.02 0.1 SFSU 11.018 6 0.988 6 0.05 0.001 20.871 7 1 0.05 0.01 26.979 0 1 0.07 0.08 2 Algorithms 4 SNR=20 4 SNR=30 4 SNR=40 <math id="M215"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi><mo>/</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442048&type= 10.24466610 2.45533323 <math id="M216"><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442050&type= 2.45533323 3.47133350 <math id="M217"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442052&type= 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442052&type= 1.35466671 2.11666679 <math id="M222"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442046&type= 1.52400005 2.11666679 <math id="M223"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi><mo>/</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442048&type= 10.24466610 2.45533323 <math id="M224"><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442050&type= 2.45533323 3.47133350 <math id="M225"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442052&type= 1.35466671 2.11666679 <math id="M226"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442069&type= 1.52400005 2.11666679 SUnSAL 4.216 7 0.559 7 0.2 - 8.475 8 0.798 3 0.02 - 14.271 2 0.990 2 0.005 - CLSUnSAL 3.414 8 0.468 6 2.2 - 6.619 8 0.735 8 0.33 - 11.446 8 0.971 5 0.02 - RWCLSUnSAL 3.683 2 0.533 5 0.03 - 8.185 3 0.812 7 0.01 - 14.868 4 0.973 90 0.000 1 - FSU 7.447 3 0.744 1 0.05 0.005 12.627 5 0.967 4 0.01 0.05 16.341 2 0.999 6 0.001 0.1 SFSU 8.369 5 0.842 5 0.2 0.006 16.044 6 0.963 2 0.007 0.01 29.523 7 1 0.007 0.1 通过观察 表2 表2 和 表3 表3 的实验结果数据，可以得到以下结论：（1）使用变换的解混算法（FSU和SFSU）的解混效果比没有使用变换的解混算法（SUnSAL，CLSUnSAL，RWCLSUnSAL）的解混结果好很多，特别是增加了变换域光谱权重的SFSU算法，是效果最好的；（2）在低信噪比的情况下，FSU和SFSU算法明显比其他几种算法的解混结果要好很多（20 dB时DC1数据中SRE提高比例超过29.2%，DC2数据中SRE提高比例超过12.4%），这说明变换域方法对带噪数据具有较大的改善效果；（3）变换域算法FSU和SFSU 解混的最优结果中， <math id="M227"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442073&type= 1.94733346 2.79399991 参数值随着信噪比的增高而变大（如SFSU算法DC1数据，在20 dB时 <math id="M228"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.001</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442075&type= 18.03400040 3.21733332 在30 dB时 <math id="M229"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.01</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442077&type= 16.17133331 3.21733332 在40 dB时 <math id="M230"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.08</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442079&type= 14.39333248 2.79399991 ），说明在大噪声条件下，噪声集中的中高频部分受到抑制，低频部分起主导作用；而随着信噪比的增大，中高频的比重逐渐增多，这证明了 式（14） 式（14） 的原理；（4）随着信噪比的提升，SFSU算法结果远超其他算法（30 dB时DC1数据中SRE提高比例最少90.9%，DC2数据中SRE提高比例至少27.1%；40 dB时DC1数据中SRE提高比例至少47.6%，DC2数据中SRE提高比例至少81%），这说明增加变换域光谱加权后，算法的精确度得到了更好的提升。 图3 图3 为DC1数据在信噪比为40 dB的情况下，几种算法的最优解混结果的丰度差值。在最优解混结果情况下，几种算法对应的参数可见 表2 表2 中40 dB列。由于每种算法都对应5种端元的丰度，且效果相似，因此本文只以第5个端元丰度图的比较结果为例进行展示。从差值图中可以得到与上述实验相同的结论，即SFSU算法的误差最小，性能最好。 图4 图4 为DC2数据在信噪比为40 dB的情况下，几种算法的最优解混估计丰度和估计丰度与真实丰度的差值。在最优解混结果情况下，5种算法的参数可见表4。与DC1一样，由于每种算法都对应了5种端元的丰度，且效果相似，因此只展示了第5个端元的丰度图比较结果。从 图4 图4 差值图中可以得到与DC1一样的结果，SFSU算法效果最好。 为了更好地对算法解混结果的稀疏度进行比较，本文随机选择了DC1和DC2的100个像元进行展示。 图5 图5 为DC1和DC2在20 dB的情况下，几种算法的最佳解混结果的丰度，5种算法对应参数可见 表3 表3 和表4。图中每一行表示一个端元对应的丰度。由图可知，SUnSAL，CLSUnSAL和RWCLSUnSAL 3种算法中出现了较为明显的干扰端元的丰度或噪声，而变换域算法FSU和SFSU，由于具有较好的抗噪功能，其丰度图与真实丰度图非常接近，特别是本文提出的增加光谱权重的SFSU算法，其稀疏特性有了更进一步的提升，估计丰度值与真实丰度值更为逼近，尤其是在DC2的SFSU解混效果特别明显（不但稀疏度与真实值很接近，而且端元丰度与真实丰度也很接近）。 比较以上两个模拟实验结果，可以得到一致的结论：SFSU算法在抗噪性能和稀疏性能上都有最好的效果。 4.2　真实高光谱数据实验 2 实验使用Cuprite数据集作为真实高光谱数据进行比较分析。该数据集是由1997年的机载可见红外成像光谱仪（AVIRIS）收集，覆盖了美国内华达州的Cuprite地区。原始图像有224个波段，波长从370~2 480 nm，空间分辨率为20 m，光谱分辨率为10 nm。实验图像使用原始图像的部分场景，尺寸为250×191像素，去除由于水的吸收和低信噪比会受到影响的波段（1~2，105~115，150~170和223~224 nm），最终留下188个光谱波段。 本实验中使用的光谱库与模拟实验中使用的光谱库 <math id="M231"><mi mathvariant="bold">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442092&type= 2.79399991 2.79399991 相同，并且同样把受噪声影响的光谱带从 <math id="M232"><mi mathvariant="bold">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442092&type= 2.79399991 2.79399991 中剔除。 图6 图6 为利用Tricorder软件生成的矿物质材料的分布。由于无法获取该高光谱数据的真实丰度，因此只能把这几种不同的稀疏解混算法的估计丰度与矿物图进行比较，对其性能进行定性分析。 图7 图7 为将SUnSAL，CLSUnSAL，RWCLSUnSAL，FSU和SFSU算法应用于Cuprite矿区高光谱数据解混，估算出的几种主要矿物（明矾、白云石和玉髓）的分布情况，即它们的端元丰度图，并进行了可视化（定性）比较。为了便于比较，从 图6 图6 中的Tricorder软件产品中提取这些材料的空间分布，并显示在 图7 图7 中。几种解混算法经过优化后使用的参数分别是SUnSAL（ <math id="M233"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.01</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442105&type= 14.22399998 2.79399991 ），CLSUnSAL（ <math id="M234"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442097&type= 12.36133385 2.79399991 ），RWCLSUnSAL（ <math id="M235"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.001</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442099&type= 16.08666801 2.79399991 ），FSU（ <math id="M236"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442101&type= 12.53066730 2.79399991 ， <math id="M237"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.005</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442066&type= 16.08666801 2.79399991 ）和SFSU（ <math id="M238"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442103&type= 12.53066730 2.79399991 ， <math id="M239"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.01</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442105&type= 14.22399998 2.79399991 ）。 1;2; 由 图7 图7 的实验结果可以得到：（1）使用变换域的解混算法（FSU，SFSU）获得的丰度图表现出良好的稳定性，比其他几种算法显示出更少的异常值。在白云石丰度图中，FSU和SFSU算法得到的丰度估算结果中，异常值明显少于其他算法。这说明在真实高光谱数据图像解混中，变换域方法有助于去除高光谱数据中的噪声。（2）具有变换域光谱加权约束的解混算法SFSU是几种算法中效果最好的。在玉髓丰度图中，SFSU算法得到的丰度结果与Tricorder软件生成的玉髓丰度图最为接近，而SUnSAL，CLSUnSAL，RWCLSUnSA丰度图在左上角与Tricorder图有明显的差别，FSU丰度图过于平滑，稀疏性没有SFSU好。这说明增加变换域光谱加权约束的高光谱解混算法SFSU，不管是在抗噪性能上，还是在稀疏性能上，都优于其他算法。 5 结 论 1 本文提出了一种framelet变换高光谱图像光谱加权稀疏解混方法，该算法充分利用变换域的优势，同时考虑高光谱的光谱特性，使得解混结果同时具有光谱的稀疏性和变换域方法的抗噪性，解混效果得到很好的提升。在模拟数据实验和真实数据实验中，将SUnSAL，CLSUnSAL，RWCLSUnSA，FSU和SFSU算法进行详细的分析和对比。实验结果证明，SFSU算法的 <math id="M240"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442119&type= 7.28133297 2.79399991 和 <math id="M241"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442117&type= 3.55599999 3.72533321 值都比其他解混方法好， <math id="M242"><mi mathvariant="normal">S</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442119&type= 7.28133297 2.79399991 提高了12.4%~1 045%， <math id="M243"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43442111&type= 3.55599999 3.72533321 保持在16%的误差内，表现出更好的解混效果。 参考文献 KESHAVA N ， MUSTARD J F . Spectral unmixing ［J］. IEEE Signal Processing Magazine ， 2002 ， 19 （ 1 ）： 44 - 57 . doi: 10.1109/79.974727 http://dx.doi.org/10.1109/79.974727 Spectral unmixing 徐晨光 ， 张绍泉 ， 李璠 ， 等 . 高光谱遥感图像光谱加权稀疏解混算法分析和比较 ［J］. 南昌工程学院学报 ， 2020 ， 39 （ 6 ）： 87 - 93 . 高光谱遥感图像光谱加权稀疏解混算法分析和比较 XU CH G ， ZHANG SH Q ， LI F ， et al . Analysis and comparison of spectral weighted sparse unmixing algorithms of hyperspectral remote sensing images ［J］. Journal of Nanchang Institute of Technology ， 2020 ， 39 （ 6 ）： 87 - 93 . （in Chinese） Analysis and comparison of spectral weighted sparse unmixing algorithms of hyperspectral remote sensing images BOARDMAN J W . Automating spectral unmixing of AVIRIS data using convex geometry concepts ［C］. Proc. of AVIRIS ， 1993 ： 11 - 14 . Automating spectral unmixing of AVIRIS data using convex geometry concepts 张少磊 ， 付光远 ， 汪洪桥 ， 等 . 基于向量总变差约束局部光谱解混的高光谱图像超分辨 ［J］. 光学 精密工程 ， 2019 ， 27 （ 12 ）： 2683 - 2692 . doi: 10.3788/ope.20192712.2683 http://dx.doi.org/10.3788/ope.20192712.2683 基于向量总变差约束局部光谱解混的高光谱图像超分辨 ZHANG SH L ， FU G Y ， WANG H Q ， et al . Total-variation-regularized local spectral unmixing for hyperspectral image super-resolution ［J］. Opt. Precision Eng. ， 2019 ， 27 （ 12 ）： 2683 - 2692 . （in Chinese） . doi: 10.3788/ope.20192712.2683 http://dx.doi.org/10.3788/ope.20192712.2683 Total-variation-regularized local spectral unmixing for hyperspectral image super-resolution 张绍泉 ， 黄志浩 ， 邓承志 ， 等 . 光谱加权协同稀疏和全变差正则化高光谱图像解混 ［J］. 电子学报 ， 2020 ， 48 （ 12 ）： 2453 - 2461 . doi: 10.3969/j.issn.0372-2112.2020.12.022 http://dx.doi.org/10.3969/j.issn.0372-2112.2020.12.022 光谱加权协同稀疏和全变差正则化高光谱图像解混 ZHANG SH Q ， HUANG ZH H ， DENG CH ZH ， et al . Spectral reweighted collaborative sparsity and total variation based hyperspectral unmixing method ［J］. Acta Electronica Sinica ， 2020 ， 48 （ 12 ）： 2453 - 2461 . （in Chinese） . doi: 10.3969/j.issn.0372-2112.2020.12.022 http://dx.doi.org/10.3969/j.issn.0372-2112.2020.12.022 Spectral reweighted collaborative sparsity and total variation based hyperspectral unmixing method BHATT J S ， JOSHI M V . Deep learning in hyperspectral unmixing： a review ［C］. IGARSS 2020 - 2020 IEEEInternational Geoscience and Remote Sensing Symposium . 262，2020 ， Waikoloa ， HI， USA . IEEE ， 2021 ： 2189 - 2192 . doi: 10.1109/igarss39084.2020.9324546 http://dx.doi.org/10.1109/igarss39084.2020.9324546 Deep learning in hyperspectral unmixing： a review LI C ， LI J ， SUI C H ， et al . Spatial-spectral nonlinear hyperspectral unmixing under complex noise ［J］. IEEE Sensors Journal ， 2022 ， 22 （ 5 ）： 4338 - 4346 . doi: 10.1109/jsen.2022.3143852 http://dx.doi.org/10.1109/jsen.2022.3143852 Spatial-spectral nonlinear hyperspectral unmixing under complex noise 贾麒 ， 廖守亿 ， 张作宇 ， 等 . 重加权稀疏非负矩阵分解的高光谱解混 ［J］. 红外与激光工程 ， 2020 ， 49 （ S2 ）： 283 - 299 . 重加权稀疏非负矩阵分解的高光谱解混 JIA Q ， LIAO SH Y ， ZHANG Z Y ， et al . Reweighted sparse nonnegative matrix decomposition for hyperspectral unmixing ［J］. Infrared and Laser Engineering ， 2020 ， 49 （ S2 ）： 283 - 299 . （in Chinese） Reweighted sparse nonnegative matrix decomposition for hyperspectral unmixing 徐晨光 ， 邓承志 ， 朱华生 . 近似稀疏约束的多层非负矩阵分解高光谱解混 ［J］. 红外与激光工程 ， 2018 ， 47 （ 11 ）： 265 - 273 . doi: 10.3788/irla201847.1117010 http://dx.doi.org/10.3788/irla201847.1117010 近似稀疏约束的多层非负矩阵分解高光谱解混 XU CH G ， DENG CH ZH ， ZHU H SH . Approximate sparse regularized multilayer NMF for hyperspectral unmixing ［J］. Infrared and Laser Engineering ， 2018 ， 47 （ 11 ）： 265 - 273 . （in Chinese） . doi: 10.3788/irla201847.1117010 http://dx.doi.org/10.3788/irla201847.1117010 Approximate sparse regularized multilayer NMF for hyperspectral unmixing IORDACHE M D ， PLAZA A ， BIOUCAS-DIAS J . On the use of spectral libraries to perform sparse unmixing of hyperspectral data ［C］. 2010 2nd Workshop on Hyperspectral Image and Signal Processing： Evolution in Remote Sensing . 1416，2010 ， Reykjavik ， Iceland . IEEE ， 2010 ： 1 - 4 . doi: 10.1109/whispers.2010.5594888 http://dx.doi.org/10.1109/whispers.2010.5594888 On the use of spectral libraries to perform sparse unmixing of hyperspectral data WANG P ， SHEN X ， NI K ， et al . Hyperspectral sparse unmixing based on multiple dictionary pruning ［J］. International Journal of Remote Sensing ， 2022 ， 43 （ 7 ）： 2712 - 2734 . doi: 10.1080/01431161.2022.2068358 http://dx.doi.org/10.1080/01431161.2022.2068358 Hyperspectral sparse unmixing based on multiple dictionary pruning BIOUCAS-DIAS J M ， FIGUEIREDO M A T . Alternating direction algorithms for constrained sparse regression： application to hyperspectral unmixing ［C］. 2010 2nd Workshop on Hyperspectral Image and Signal Processing： Evolution in Remote Sensing . 1416，2010 ， Reykjavik ， Iceland . IEEE ， 2010 ： 1 - 4 . doi: 10.1109/whispers.2010.5594963 http://dx.doi.org/10.1109/whispers.2010.5594963 Alternating direction algorithms for constrained sparse regression： application to hyperspectral unmixing IORDACHE M D ， BIOUCAS-DIAS J M ， PLAZA A . Collaborative sparse regression for hyperspectral unmixing ［J］. IEEE Transactions on Geoscience and Remote Sensing ， 2014 ， 52 （ 1 ）： 341 - 354 . doi: 10.1109/tgrs.2013.2240001 http://dx.doi.org/10.1109/tgrs.2013.2240001 Collaborative sparse regression for hyperspectral unmixing ZHENG C Y ， LI H ， WANG Q ， et al . Reweighted sparse regression for hyperspectral unmixing ［J］. IEEE Transactions on Geoscience and Remote Sensing ， 2016 ， 54 （ 1 ）： 479 - 488 . doi: 10.1109/tgrs.2015.2459763 http://dx.doi.org/10.1109/tgrs.2015.2459763 Reweighted sparse regression for hyperspectral unmixing 李忠伟 ， 张浩 ， 王雷全 ， 等 . 融合空谱-梯度特征的深度高光谱图像去噪 ［J］. 光学 精密工程 ， 2022 ， 30 （ 5 ）： 615 - 629 . doi: 10.37188/ope.2021.0485 http://dx.doi.org/10.37188/ope.2021.0485 融合空谱-梯度特征的深度高光谱图像去噪 LI ZH W ， ZHANG H ， WANG L Q ， et al . Deep hyperspectral image denoising by fusing space spectrum-gradient features ［J］. Opt. Precision Eng. ， 2022 ， 30 （ 5 ）： 615 - 629 . （in Chinese） . doi: 10.37188/ope.2021.0485 http://dx.doi.org/10.37188/ope.2021.0485 Deep hyperspectral image denoising by fusing space spectrum-gradient features XU C G ， WU Z M ， LI F ， et al . Spectral-spatial joint sparsity unmixing of hyperspectral images based on framelet transform ［J］. Infrared Physics & Technology ， 2021 ， 112 ： 103564 . doi: 10.1016/j.infrared.2020.103564 http://dx.doi.org/10.1016/j.infrared.2020.103564 Spectral-spatial joint sparsity unmixing of hyperspectral images based on framelet transform RAPIN J ， BOBIN J ， LARUE A ， et al . NMF with sparse regularizations in transformed domains ［J］. SIAM Journal on Imaging Sciences ， 2014 ， 7 （ 4 ）： 2020 - 2047 . doi: 10.1137/140952314 http://dx.doi.org/10.1137/140952314 NMF with sparse regularizations in transformed domains ZHANG G X ， XU Y Y ， FANG F M . Framelet-based sparse unmixing of hyperspectral images ［J］. IEEE Transactions on Image Processing： a Publication of the IEEE Signal Processing Society ， 2016 ， 25 （ 4 ）： 1516 - 1529 . doi: 10.1109/tip.2016.2523345 http://dx.doi.org/10.1109/tip.2016.2523345 Framelet-based sparse unmixing of hyperspectral images CAI J F ， CHAN R H ， SHEN Z W . A framelet-based image inpainting algorithm ［J］. Applied and Computational Harmonic Analysis ， 2008 ， 24 （ 2 ）： 131 - 149 . doi: 10.1016/j.acha.2007.10.002 http://dx.doi.org/10.1016/j.acha.2007.10.002 A framelet-based image inpainting algorithm JIANG T X ， HUANG T Z ， ZHAO X L ， et al . Matrix factorization for low-rank tensor completion using framelet prior ［J］. Information Sciences ， 2018 ， 436/437 ： 403 - 417 . doi: 10.1016/j.ins.2018.01.035 http://dx.doi.org/10.1016/j.ins.2018.01.035 Matrix factorization for low-rank tensor completion using framelet prior WANG R ， LI H C ， LIAO W Z ， et al . Double reweighted sparse regression for hyperspectral unmixing ［C］. 2016 IEEE International Geoscience and Remote Sensing Symposium （IGARSS）. 1015，2016 ， Beijing， China. IEEE ， 2016 ： 6986 - 6989 . doi: 10.1109/igarss.2016.7730822 http://dx.doi.org/10.1109/igarss.2016.7730822 Double reweighted sparse regression for hyperspectral unmixing 刘敬 ， 李青妍 ， 刘逸 . 基于核加权类对准则的高光谱影像特征提取 ［J］. 光学 精密工程 ， 2021 ， 29 （ 6 ）： 1397 - 1405 . doi: 10.37188/OPE.20212906.1397 http://dx.doi.org/10.37188/OPE.20212906.1397 基于核加权类对准则的高光谱影像特征提取 LIU J ， LI Q Y ， LIU Y . Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion ［J］. Opt. Precision Eng. ， 2021 ， 29 （ 6 ）： 1397 - 1405 . （in Chinese） . doi: 10.37188/OPE.20212906.1397 http://dx.doi.org/10.37188/OPE.20212906.1397 Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion RASTI B ， KOIRALA B ， SCHEUNDERS P ， et al . UnDIP： hyperspectral unmixing using deep image prior ［J］. IEEE Transactions on Geoscience and Remote Sensing ， 2022 ， 60 ： 1 - 15 . doi: 10.1109/tgrs.2021.3067802 http://dx.doi.org/10.1109/tgrs.2021.3067802 UnDIP： hyperspectral unmixing using deep image prior 10.37188/OPE.20233109.1404 TP751.1 A 1004-924X（2023）09-1404-14 徐晨光,徐洪雨,郁春艳等.Framelet变换高光谱图像光谱加权稀疏解混[J].光学精密工程,2023,31(09):1404-1417. 徐晨光,徐洪雨,郁春艳等.Framelet变换高光谱图像光谱加权稀疏解混[J].光学精密工程,2023,31(09):1404-1417. XU Chenguang,XU Hongyu,YU Chunyan,et al.Spectral weighted sparse unmixing of hyperspectral images based on framelet transform[J].Optics and Precision Engineering,2023,31(09):1404-1417. XU Chenguang,XU Hongyu,YU Chunyan,et al.Spectral weighted sparse unmixing of hyperspectral images based on framelet transform[J].Optics and Precision Engineering,2023,31(09):1404-1417. 国家自然科学基金资助项目（No.61865012，No.62141105）；江西省重点研发项目（No.20202BBGL73081，No.20181ACG70022）；南昌工程学院大学生创新创业训练计划资助项目（No.2021026）；江西省大学生创新创业训练计划资助项目（No.S202211319025，No.201211319001，No.201311319034）；国家大学生创新创业训练计划资助项目（No.201911319016） 2022-05-18 2022-07-08 2023-05-10 2023-05-15 37010519 光学精密工程 9 31