小波多重分形遥感图像混合去噪

小波多重分形遥感图像混合去噪 Removing mixed noise from remote sensing images by wavelet multifractal method first-author 成丽波（1971-），女，吉林公主岭人，教授，博士生导师，1995年于吉林师范大学获得学士学位，2004年、2009年于吉林大学分别获得硕士和博士学位，现为长春理工大学数学与统计学院院长，主要从事小波分析、数学建模、遥感图像处理等方面的研究。E-mail： clbyy@126.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315039&type= 22.01332855 32.00400162 成丽波 （1971-），女，吉林公主岭人，教授，博士生导师，1995年于吉林师范大学获得学士学位，2004年、2009年于吉林大学分别获得硕士和博士学位，现为长春理工大学数学与统计学院院长，主要从事小波分析、数学建模、遥感图像处理等方面的研究。E-mail： clbyy@126.com clbyy@126.com corresp E-mail： liyuhaoxy@sina.com E-mail： liyuhaoxy@sina.com 李昂臻（1995-），男，吉林长春人，硕士研究生，2019年于吉林师范大学获得学士学位，主要研究方向为遥感图像去噪。E-mail：liyuhaoxy@sina.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315055&type= 22.01332855 32.00399399 李昂臻 （1995-），男，吉林长春人，硕士研究生，2019年于吉林师范大学获得学士学位，主要研究方向为遥感图像去噪。E-mail： liyuhaoxy@sina.com liyuhaoxy@sina.com 长春理工大学 数学与统计学院，吉林 长春 130022 长春理工大学 数学与统计学院，吉林 长春 130022 School of Mathematics and Statistics， Changchun University of Science and Technology， Changchun 130022， China School of Mathematics and Statistics， Changchun University of Science and Technology， Changchun 130022， China 为了去除遥感图像的混合噪声，建立了小波多重分形去噪算法。该算法主要利用小波分析进行信号分解，多重分形提取图像特征。通过小波分解进行图像分解时，利用小波半软阈值指数衰减阈值法进行加性噪声初步处理。采用多重分形理论，找出含噪图像的多重分形谱，并构造偏移算子 <math id="M1"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315653&type= 2.53999996 2.45533323 对加性噪声进行二次处理。利用方向梯度与二维掩码层进行逐像素乘法获得稀疏梯度集，重建去噪图像。最后，计算出去噪图像的评价指标数值，根据数值分析评价去噪效果。实验结果表明：对6幅随机添加噪声的图片进行去噪，去噪图像的峰值信噪比最高为26.700 <math id="M2"><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315077&type= 3.72533321 2.37066650 ，边缘保持指数最高为0.449。该方法能够有效去除遥感图像的混合噪声，基本满足遥感图像混合去噪的可视性，细节留存等要求。 To remove the mixed noise from remote sensing images， a wavelet multifractal denoising algorithm was developed. The algorithm mainly uses wavelet analysis for signal decomposition and multifractal to extract image features. First， image decomposition was performed by wavelet decomposition， and additive noise was preliminarily processed using the exponential decay threshold method of the wavelet semi-soft threshold. Second， using the multifractal theory， the multifractal spectrum of the noisy image was found， and an offset operator is constructed to process the additive noise twice. Then， the sparse gradient set was obtained by multiplying the direction gradient with the two-dimensional mask layer pixel by pixel， and the denoised image is reconstructed. Finally， the evaluation index value of the denoised image was calculated， and the denoising effect was evaluated according to the numerical analysis. The experimental results show that the method can effectively remove the mixed noise of remote sensing images. The maximum peak signal-to-noise ratio of the denoised images is 26.700 dB by denoising six randomly added noise images. Moreover， the highest edge preservation index is 0.449. It can meet the requirements of the visibility and detail preservation of mixed denoising of remote sensing images and provide a reliable basis for subsequent analysis. 图像处理 遥感图像去噪 多重分形 半软阈值 小波分析 image processing remote sensing image denoising multifractal semi-soft threshold wavelet analysis 1 引 言 1 随着卫星遥感技术的发展及成熟，遥感图像在车载GPS导航、手机地图软件的实时地图中得到了大量的应用。在资源分析 ［ 1 1 - 2 2 1-2 ］ ，河流水源研究 ［ 3 3 ］ ，地质地貌勘探 ［ 4 4 ］ 以及气象灾害 ［ 5 5 - 7 7 5-7 ］ 等领域也都需要卫星遥感图像的身影。随着图像传感器功能的提升，目前遥感图像的空间分辨率已经到了米级 ［ 8 8 - 9 9 8-9 ］ 。然而，受传感器的灵敏度及成像环境的影响，遥感图像会被噪声污染。噪声会影响图像的视觉效果，甚至会掩盖数字图像的真实信息，因此去噪是卫星遥感图像预处理中的重要环节。 传统去噪算法主要利用滤波分解，再通过滤波器进行噪声去除 ［ 10 10 - 15 15 10-15 ］ 。李辉 ［ 13 13 ］ 通过改进的双树复小波进行图像去噪。Huang ［ 14 14 ］ 采用基于核主成分分析（Kernel Principal Components，KPCA）希尔伯特-黄变换（Hilbert-Huang Transform，HHT）进行去噪，该方法改进了传统的主成分分析只能处理线性信号的缺点，可以对非线性遥感图像信号进行处理。 常见的乘性噪声去除方法利用全变差（Total Variation，TV）模型 ［ 16 16 - 17 17 16-17 ］ 或 <math id="M3"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315088&type= 3.55599999 3.72533321 最小化决策树 ［ 18 18 ］ 进行去噪，这两种方法旨在去除Gamma乘性噪声及斑点噪声。然而，遥感图像中的乘性噪声多为Gaussian噪声，针对此类噪声目前主流的去除方法有LEE滤波、BM3D（Block-Matching and 3D Filtering）算法和多重分形算法。Zhong ［ 19 19 ］ 等通过改进LEE滤波对遥感图像中的乘性噪声进行了去除。 遥感图像去噪时除了要关注其噪声的去除程度之外，还要避免图像的细节与结构遭到破坏。因此，在去噪的同时还要考虑边缘保持。Zhang ［ 20 20 ］ 利用Contorlet在进行遥感图像去噪的同时保留了边缘细节。Pham等 ［ 21 21 ］ 对去噪算法进行改进，构造边缘保护模型。Deepa等 ［ 22 22 ］ 提出了一种最小二乘法加权正则化的去噪技术，旨在保护图像边缘。2020年，Suman等 ［ 23 23 - 24 24 23-24 ］ 通过多重分形理论，融合小波分析以及TV模型构造出针对遥感图像中乘性噪声去除的算法，该算法可以有效去除遥感图像中的乘性噪声，并且可以高效保持图像的边缘细节和图像结构。 本文在此基础上提出了一种卫星遥感图像混合噪声去除算法，该算法利用小波半软阈值指数衰减阈值法对低频噪声进行平滑处理，之后采用多重分形算法进行二次处理估计去噪图像。该方法弥补了小波半软阈值指数衰减法去噪能力薄弱，仅能进行去噪预处理，而多重分形对于加性噪声去噪效果不明显的不足。 2 混合噪声模型 1 在图像处理中，混合噪声图像通常表示为： <math display="block" id="M4"><mi>f</mi><mo>=</mo><mi>η</mi><mo>×</mo><mi>g</mi><mo>+</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315099&type= 21.25133324 3.72533321 ， 其中： <math id="M5"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315110&type= 1.94733346 3.64066648 为原始图像， <math id="M6"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315123&type= 1.86266661 3.64066648 为乘性噪声， <math id="M7"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315730&type= 1.86266661 2.79399991 为加性噪声， <math id="M8"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315723&type= 1.10066664 3.64066648 为被噪声污染的图像。 根据混合噪声模型，混合噪声可以分两步进行去除，首先对加性噪声进行去噪处理，之后再对乘性噪声进行去噪处理，同时还要保持图像细节及结构的完整。 3 小波多重分形去噪算法 1 本文在Suman构造的多重分形去除乘性噪声算法的基础上，融合小波半软阈值指数衰减模型 ［ 25 25 ］ 来去除低频信号中的加性噪声，并利用偏移算子 <math id="M9"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315470&type= 2.96333337 2.87866688 ［ 26 26 ］ 去除剩余加性噪声。具体流程如 图1 图1 所示。 3.1　小波半软阈值 2 首先对遥感图像进行小波变换： <math display="block" id="M10"><mfenced separators="|"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mi>ω</mi></mrow></msup><mi>f</mi></mrow></mfenced><mfenced separators="|"><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315189&type= 23.36800003 4.82600021 <math display="block" id="M11"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>a</mi></mrow></mfenced></mrow><mrow/></mroot></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mover><mrow><mi>ψ</mi><mfenced separators="|"><mrow><mfrac><mrow><mi>t</mi><mo>-</mo><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mrow></mfenced></mrow><mo>¯</mo></mover></mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo>,</mo><mi>a</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315202&type= 51.13866425 11.43000031 ， 其中： a 为尺度因子， b 为平移因子， f （ t ）为图像信号。利用 <math id="M12"><mi>ψ</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315211&type= 7.11199999 3.97933316 对图像进行分解，获得小波系数集 <math id="M13"><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315221&type= 6.01133299 4.48733330 。 常规的小波去噪采用小波硬阈值或小波软阈值，这两种方法对于信号的衰减太强，会使后续的图像产生不可逆的细节损失。因此为了避免细节破坏，本文采用小波半软阈值指数衰减阈值法。该方法可以有效去除小波分解后的低频滤波中平滑且不可忽视的加性噪声。选取两个阈值 <math id="M14"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315230&type= 20.23533440 4.14866638 。当 <math id="M15"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315238&type= 20.48933220 4.48733330 时更接近软阈值；当 <math id="M16"><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>></mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315251&type= 12.78466606 4.48733330 时更接近硬阈值。因此半软阈值函数如下： <math display="block" id="M17"><msubsup><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn 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mathvariant="normal">1</mn></mrow></msub><mo><</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315271&type= 20.48933220 4.48733330 的线性变换改为指数形式，即有： <math display="block" id="M19"><msubsup><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mo>/</mo><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mfenced></mrow></msup><mo>,</mo></mtd><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup><mo>,</mo></mtd><mtd><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315280&type= 74.67600250 19.98133469 . 该方法可以平缓信号改变量，防止细节信号丢失。 3.2　多重分形 2 自然系统存在自相似性，具有多重分形性质。遥感图像作为传感器获取的自然图像，其中的观测信号显然也存在自相似性，因此遥感图像也具有多重分形性质。多重分形能够定量地刻画复杂几何对象在不同层次的分形特征，并通过多重分形谱表达出来。 由于乘性噪声是附加在信号上的，所以本文采用多重分形方法 ［ 21 21 - 22 22 21-22 ］ 进行噪声去除，该方法不同于传统滤波法，其去噪主要依据图像本身的信息特征进行估计，即寻找 <math id="M20"><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mo>≃</mo><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315289&type= 8.63599968 3.72533321 。首先，根据多重分形理论从 <math id="M21"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315723&type= 1.10066664 3.64066648 中提取无噪信息，再从无噪信息重建去噪图像。 3 . 2 . 1　信息提取 3 遥感图像 <math id="M22"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315308&type= 6.85799980 4.06400013 在像素点 <math id="M23"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315368&type= 1.10066664 2.79399991 处的奇异值指数 <math id="M24"><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315326&type= 6.77333355 3.55599999 为： <math display="block" id="M25"><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfrac><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>μ</mi></mrow></msub><mfenced separators="|"><mrow><mi>B</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>r</mi></mrow></mfenced></mrow></mfenced><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mi>Φ</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mi>r</mi></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315332&type= 60.36733627 9.82133293 其中 <math id="M26"><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315358&type= 6.94266701 3.97933316 为Hausdorff指数。定义集合 <math id="M27"><mi>E</mi><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315347&type= 8.38199997 3.97933316 为具有相同 <math id="M28"><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315358&type= 6.94266701 3.97933316 值的所有点 <math id="M29"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315368&type= 1.10066664 2.79399991 构成的集合，即： <math display="block" id="M30"><mi>E</mi><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>t</mi><mo>:</mo><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>h</mi></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315375&type= 35.30599976 4.74133301 为了分析退化的奇异过程，定义半径 <math id="M31"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315381&type= 1.43933344 2.79399991 的 <math id="M32"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315387&type= 1.94733346 2.79399991 邻域局部奇异点集合为： <math display="block" id="M33"><msub><mrow><mi>E</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi>t</mi><mo>:</mo><mi>h</mi><mo>-</mo><mi>ε</mi><mo>≤</mo><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>≤</mo><mi>h</mi><mo>+</mo><mi>ε</mi></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315395&type= 55.03333664 4.74133301 根据 式（7） 式（7） ，引入Hausdorff维度 <math id="M34"><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mi>H</mi></mrow></msub><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315402&type= 10.49866676 3.72533321 ： <math display="block" id="M35"><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mi>H</mi></mrow></msub><mi>E</mi><mo>=</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mfenced open="{" close="}" separators="|"><mrow><mi>s</mi><mo>:</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>r</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">0</mn></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315408&type= 54.94866943 9.65200043 ， 其中 <math id="M36"><msub><mrow><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo><</mo><mi>i</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315417&type= 15.74800014 4.57200003 是 <math id="M37"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315425&type= 2.62466669 2.79399991 的 <math id="M38"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315431&type= 1.52400005 2.79399991 覆盖，即： <math display="block" id="M39"><mi>E</mi><mo>⊂</mo><msubsup><mo>∪</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow/><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo><</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi><mo>⊂</mo><mi>f</mi></mrow></msub><mo>,</mo><mo>∀</mo><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315439&type= 48.85266495 4.40266657 . 最后可得： <math display="block" id="M40"><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mi>H</mi></mrow></msub><mi>E</mi><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315445&type= 28.70199966 4.06400013 . 多重分形测度 <math id="M41"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315454&type= 1.94733346 3.64066648 的Hausdorff谱为 <math id="M42"><mfenced separators="|"><mrow><mi>h</mi><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315462&type= 14.47799969 4.82600021 ，也称为局部奇异谱。 3 . 2 . 2　图像去噪与重构 3 构造偏移算子 <math id="M43"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315470&type= 2.96333337 2.87866688 ： <math display="block" id="M44"><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced><mo>=</mo><mi>O</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>ε</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315477&type= 55.45666885 4.99533367 . 通过变换多重分形谱，将含有加性噪声的不规则信号 <math id="M45"><mi>f</mi><mfenced separators="|"><mrow><mi>h</mi><mo><</mo><mn mathvariant="normal">2</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315483&type= 20.82799911 4.06400013 和 <math id="M46"><mi>f</mi><mfenced separators="|"><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow></mfenced><mo><</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315490&type= 20.82799911 4.06400013 平滑处理，由多重分形理论可知，该变换不会改变谱的相对强度，因此不会影响图像的细节。 设限制于 <math id="M47"><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315499&type= 7.61999989 4.57200003 的标准密度测度为 <math id="M48"><msub><mrow><mi>δ</mi></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315506&type= 4.06400013 3.64066648 ，利用水平垂直两方向的方向梯度 <math id="M49"><mo>∇</mo><mi>x</mi><mo>,</mo><mo>∇</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315513&type= 11.26066685 3.72533321 逐像素乘法得出稀疏梯度集： <math display="block" id="M50"><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mo>∇</mo><mi>x</mi><mo>⋅</mo><mi>f</mi></mrow></mfenced><mo>⋅</mo><msub><mrow><mi>δ</mi></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mo>∇</mo><mi>y</mi><mo>⋅</mo><mi>f</mi></mrow></mfenced><mo>⋅</mo><msub><mrow><mi>δ</mi></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315519&type= 54.52533722 4.99533367 利用 <math id="M51"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315526&type= 10.41399956 5.92666674 重建去噪图像 <math id="M52"><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>≃</mo><mi>g</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315533&type= 19.04999924 3.97933316 ，其中： <math display="block" id="M53"><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><munder><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>g</mi></mrow></munder><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315539&type= 63.33066940 7.78933382 采用欧拉-拉格朗日公式法简化函数，得到： <math display="block" id="M54"><mi>d</mi><mi>i</mi><mi>v</mi><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi>d</mi><mi>i</mi><mi>v</mi><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315549&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315545&type= 41.48666382 4.91066647 其中“ <math id="M55"><mi>d</mi><mi>i</mi><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315551&type= 5.07999992 2.79399991 ”为发散运算符。将 式（13） 式（13） 投影到傅里叶基底上得到： <math display="block" id="M56"><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>ω</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mi>i</mi><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>x</mi></mrow></msub><mfenced separators="|"><mrow><mi>ω</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>y</mi></mrow></msub><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub><mfenced separators="|"><mrow><mi>ω</mi></mrow></mfenced></mrow><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315555&type= 51.90066528 10.07533360 利用傅里叶逆变换获得去噪图像信号： <math display="block" id="M57"><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced open="{" close="}" separators="|"><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>ω</mi></mrow></mfenced></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315561&type= 31.66533470 4.74133301 最后利用小波逆变换将信号 <math id="M58"><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315567&type= 7.11199999 3.97933316 重构为去噪图像。 4 实验与结果 1 4.1　小波多重分形算法 2 本文在处理器为Intel（R） Core（TM） i7-10750H CPU @ 2.60 GHz 2.59 GHz，运行内存为16 G的64位Win11操作系统上进行实验。使用的编程语言是2020版MATLAB。 遥感图像取自武汉多时相场景变化检测数据集 （MtS-WH）。通过MATLAB进行噪声的随机添加，以此模拟实际情况。其中，加性噪声标准差 <math id="M59"><msub><mrow><mi>δ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,40</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315574&type= 17.78000069 3.97933316 ，乘性噪声标准差 <math id="M60"><msub><mrow><mi>δ</mi></mrow><mrow><mo>×</mo></mrow></msub><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,25</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315577&type= 17.44133186 3.97933316 。MATLAB随机取值如 表1 表1 所示，填噪效果如 图2 图2 所示。针对6幅随机噪声图像，采用小波多重分形去噪算法进行去噪，具体流程如 表2 表2 所示。 序 号 <math id="M61"><msub><mrow><mi>δ</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315581&type= 3.21733332 3.21733332 <math id="M62"><msub><mrow><mi>δ</mi></mrow><mrow><mo>×</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315586&type= 3.04800010 3.21733332 N1 27 10 N2 35 23 N3 17 24 N4 30 15 N5 1 17 N6 6 6 步 骤 流 程 Step 1 输入含噪图像 <math id="M63"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315605&type= 0.93133330 3.13266683 Step 2 利用 <math id="M64"><mi>d</mi><mi>b</mi><mn>4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315610&type= 4.99533367 2.37066650 小波分解图像信号，获得小波系数集 <math id="M65"><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315615&type= 5.24933338 3.97933316 Step 3 利用小波半软阈值指数衰减阈值法对低频信号进行去噪处理，获得修正小波系数集 <math id="M66"><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315628&type= 5.24933338 3.97933316 Step 4 通过 <math id="M67"><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315628&type= 5.24933338 3.97933316 与多重分形理论获得多重分形谱 <math id="M68"><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315636&type= 6.60400009 3.47133350 与 <math id="M69"><mi>x</mi><mo>,</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315642&type= 4.82600021 3.13266683 方向的方向梯度 <math id="M70"><mo>∇</mo><mi>x</mi><mo>,</mo><mo>∇</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315669&type= 9.65200043 3.13266683 Step 5 通过偏移算子 <math id="M71"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315653&type= 2.53999996 2.45533323 ，获得去除加性噪声后的多重分形谱 <math id="M72"><msub><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315659&type= 6.60400009 3.89466691 Step 6 通过 <math id="M73"><msub><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315664&type= 6.60400009 3.89466691 与 <math id="M74"><mo>∇</mo><mi>x</mi><mo>,</mo><mo>∇</mo><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315669&type= 9.65200043 3.13266683 进行逐像素乘法获得稀疏梯度集 <math id="M75"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>f</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315675&type= 8.97466660 5.07999992 Step 7 使用欧拉-拉格朗日法与傅里叶逆变换获得去噪图像信号 Step 8 利用小波重构获得去噪图像 <math id="M76"><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mo>≃</mo><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315680&type= 7.45066643 3.21733332 4.2　对比实验 2 图像去噪效果的评价分为主观定性分析与客观定量分析两种。定性分析主要由图像视觉效果判断图像质量，很难消除主观因素的影响。定量分析通常采用数值分析，最常用的评价标准有均方误差（Mean Squared Error，MSE）和峰值信噪比（Peak Signal to Noise Ratio， PSNR），其公式如下： <math display="block" id="M77"><mi>M</mi><mi>S</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></mfrac><msup><mrow><msub><mo>∑</mo><mrow><mi>t</mi></mrow></msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></mfenced></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315691&type= 58.33533096 7.87400007 ， <math display="block" id="M78"><mi>P</mi><mi>S</mi><mi>N</mi><mi>R</mi><mo>=</mo><mn mathvariant="normal">10</mn><mo>⋅</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mi>m</mi><mo>×</mo><mi>n</mi><mo>×</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><msub><mo>∑</mo><mrow><mi>t</mi></mrow></msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></mfenced></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315697&type= 63.50000000 14.98600006 ， 式中 <math id="M79"><msubsup><mrow><mi>g</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315703&type= 5.50333309 4.82600021 为图像灰度最大值。 边缘保持指数（Edge Protection Index， EPI）是一种定量的边缘保持方法。在图像噪声抑制中，EPI定义为： <math display="block" id="M80"><mi>E</mi><mi>P</mi><mi>I</mi><mo>=</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mfenced open="|" close="|" separators="|"><mrow><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>-</mo><mover accent="true"><mi>g</mi><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></mfenced></mrow></mfenced></mrow></mrow></mrow><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></mfenced></mrow></mfenced></mrow></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315712&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315709&type= 63.66933060 12.86933327 ， 其中： <math id="M81"><mover accent="true"><mi>g</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315720&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315715&type= 1.94733346 3.72533321 为去除噪声的SAR图像， <math id="M82"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315723&type= 1.10066664 3.64066648 为原始SAR图像， <math id="M83"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315730&type= 1.86266661 2.79399991 ， <math id="M84"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315735&type= 2.79399991 2.79399991 分别为SAR图像的列数和行数。边缘保持指数值越高，图像边缘的保存能力越好。较好的去噪SAR图像通常具有合理的EPI值。 为了验证本文构造的小波多重分形去噪算法的有效性，本文利用自适应维纳滤波，传统小波去噪算法，改进LEE滤波，TV模型进行对比实验，实验结果通过PSNR与EPI进行数值表达。结果如 图3 图3 ~ 图8 图8 所示，数值解如 表3 表3 和 表4 表4 所示。 算 法 N1 N2 N3 N4 N5 N6 Wavelet 22.752 22.785 22.883 21.764 23.285 14.273 Lee-filter 23.824 22.856 23.211 24.767 23.506 15.754 TV model 24.641 23.411 23.698 24.738 24.655 20.477 BM3D-SAR 24.533 23.617 23.851 26.492 25.947 21.874 Wavelet-multifractal 24.893 24.201 23.977 26.700 26.554 22.500 算 法 N1 N2 N3 N4 N5 N6 Wavelet 0.337 0.354 0.313 0.285 0.248 0.308 Lee filter 0.365 0.375 0.355 0.375 0.296 0.327 TV model 0.378 0.386 0.364 0.365 0.288 0.355 BM3D-SAR 0.358 0.380 0.387 0.398 0.274 0.361 Wavelet-multifractal 0.421 0.406 0.433 0.449 0.302 0.405 通过6组对比实验能够发现，在混合噪声去除上，本文方法兼具去噪效果与视觉效果。 由于图像被混合噪声污染，去噪后的PSNR与EPI普遍较低。由 表3 表3 和 表4 表4 可知，小波多重分形算法对于含有混合噪声的遥感图像去噪效果最好，PSNR值普遍高于其他方法，EPI值有着明显的提升，说明该方法在去噪的同时能够极大程度地保留边缘，有着较强的细节留存能力。 5 结 论 1 本文针对混合噪声，利用小波半软阈值指数衰减阈值法去除部分加性噪声，通过带有偏移算子 <math id="M85"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315832&type= 2.96333337 2.87866688 的多重分形进行剩余加性噪声的去除并保持图像平滑，不丢失细节信息。最后，采用稀疏梯度集估计去噪图像，该方法可以有效去除附加在信号上的乘性噪声并保证细节完整。实验结果表明：对6幅图片进行去噪，去噪图像的PSNR值最高为26.700 <math id="M86"><mi mathvariant="normal">d</mi><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=33315839&type= 4.40266657 2.79399991 ，边缘EPI值最高为0.449。该方法能够有效去除遥感图像的混合噪声。 参考文献 QUILTY J ， ADAMOWSKI J . 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