集成电路X射线图像的多正则化图像复原

集成电路X射线图像的多正则化图像复原 Multi-regularization image restoration method for X-ray images of integrated circuit first-author 马 鸽（1988-），女，山东济宁人，博士，讲师，2016年于华南理工大学获得博士学位，现为广州大学机器人工程专业教师，主要从事机器视觉、图像处理等方面的研究。Email：m_ge@gzhu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227280&type= 22.01333237 32.00399017 马 鸽 （1988-），女，山东济宁人，博士，讲师，2016年于华南理工大学获得博士学位，现为广州大学机器人工程专业教师，主要从事机器视觉、图像处理等方面的研究。Email： m_ge@gzhu.edu.cn m_ge@gzhu.edu.cn corresp E-mail： tzou@gzhu.edu.cn E-mail： tzou@gzhu.edu.cn 邹 涛（1975-），男，辽宁营口人，博士后，研究员，2005年于上海交通大学获得博士学位，现为广州大学机械与电气工程学院院长和智能制造工程研究院院长，主要从事智能制造、工业软件研发与工程应用等方面的研究。Email：tzou@gzhu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227295&type= 22.01333618 32.00399780 邹 涛 （1975-），男，辽宁营口人，博士后，研究员，2005年于上海交通大学获得博士学位，现为广州大学机械与电气工程学院院长和智能制造工程研究院院长，主要从事智能制造、工业软件研发与工程应用等方面的研究。Email： tzou@gzhu.edu.cn tzou@gzhu.edu.cn 广州大学 机械与电气工程学院，广东 广州 510006 广州大学 机械与电气工程学院，广东 广州 510006 School of Mechanical and Electrical Engineering， Guangzhou University， Guangzhou 510006， China School of Mechanical and Electrical Engineering， Guangzhou University， Guangzhou 510006， China 针对集成电路X射线图像噪声强烈、对比度低的特点，本文充分考虑图像边缘细节和平滑区域在细节保持和噪声去除上的不同需求，提出一种多正则化图像复原方法。该方法基于快速傅里叶变换下的高斯高通滤波和高斯低通滤波获取集成电路X射线图像的边缘细节结果和平滑滤波结果作为正则化图像复原的观测图像，充分利用 <math id="M1"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227307&type= 2.03200006 3.21733332 正则项在细节保持以及全变分（Total Variation， <math id="M2"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227319&type= 4.82600021 2.45533323 ）正则项在噪声去除上的优越性设计了一种 <math id="M3"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227331&type= 4.82600021 2.45533323 - <math id="M4"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227340&type= 2.03200006 3.21733332 混合正则化模型，解决整幅图像采用单一正则化项造成的细节过度平滑缺失或去噪效果差的问题。标准自然光图像和集成电路的X射线图像两个系列的实验表明了本文方法能够在有效去除噪声的同时保留更多的细节信息，为后续集成电路的缺陷检测奠定基础。 The X-ray images of Integrated Circuits （IC） generally have high noise and low contrast characteristics. Considering the different needs in detail preservation and noise removal of edge details and smooth regions， this paper proposes a multi-regularization image restoration method. Firstly， by employing a Fourier transform based Gauss high-pass filter and Gauss low-pass filter， the edge detail and smoothing filter results were obtained as new observed images for image restoration. Then， a TV- l 1 mixed regularization model， which takes full use of the advantages of l 1 regularization term in detail preservation and Total Variation （TV） regularization term in noise removal， was designed. The model is capable of addressing the problem of excessive smoothing or defective denoising caused by a single regularization term. Experiments on standard and IC X-ray images show that the proposed method can effectively remove noise while retaining more details， laying a foundation for subsequent defect detection of IC. 图像复原 多正则化 傅里叶变换 image restoration regularization Fourier transform 1 引 言 1 X射线成像是通过发射X射线穿透被检测元器件，依据待检测元器件内部产生的射线能量衰减情况及其衰减强度的不同，由平板探测器将衰减数据转换成数字图像传递给计算机的过程。X射线成像检测是通过不同材料对X射线的吸收差异，对物体内部结构成像然后进行内部缺陷检测的技术。因其成像具有分辨率高、无损、非接触等优点，在集成电路精密制造业、航空航天、特种装备检测等领域取得了广泛关注 ［ 1 1 - 3 3 1-3 ］ 。然而，集成电路的X射线图像成像过程中存在多个噪声来源，如电子随机热运动产生的热噪声、X射线源光子累积产生的噪声等，因此表现出噪声强烈的特点。同时，集成电路多层封装间隔微小，X射线出射距离短，因此呈现出对比度低的特点，给后续识别检测工作带来很大困难。因此，研究X射线图像复原技术具有理论意义和实际应用价值。 文献［ 4 4 ］将热噪声和光子累积噪声分别建模为加性噪声和乘性噪声，得到集成电路X射线成像的混合噪声退化模型。但是由于工业生产的快速性要求，实际过程中常采用加大电压的方式实现快速累积成像，大大减少了泊松特性，因此文献［ 5 5 ］将其建模为高斯加性噪声退化模型。 图像复原是从采集的退化图像获得原始干净图像的逆过程。复原方法包括滤波方法和正则化方法，滤波方法通常具有确定性的滤波函数，如高斯滤波、中值滤波、逆滤波等，但往往受限于噪声去除和细节保持的平衡问题，而正则化方法因其灵活多变的图像先验知识模型和求解算法，被广泛地应用于工业检测、机器视觉、医学诊断等领域 ［ 6 6 - 9 9 6-9 ］ 。 正则化方法将图像复原问题归结为由数据保真项和正则化项组成的目标函数模型求解问题。常用的正则化模型可分为基于 <math id="M5"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227415&type= 2.37066650 3.72533321 范数的模型和基于全变分的模型两大类。 <math id="M6"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227357&type= 2.37066650 3.72533321 范数刻画图像的边缘细节信息，具有较好的细节保持能力。尤其地，通过小波变换、字典分解等 ［ 10 10 - 12 12 10-12 ］ 可充分利用图像的稀疏特性来降低数据冗余带来的采样数据存储和硬件设备困难的方法，被广泛地应用于核磁共振成像、医学CT图像重建、图像稀疏重建等领域 ［ 13 13 - 14 14 13-14 ］ 。 全变分（Total Variation， <math id="M7"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 ）正则化模型 ［ 15 15 ］ 最初于1992年由Rudin，Osher和Fatemi提出，但该模型易产生“阶跃”现象，呈现“分片常数”效应。文献［ 16 16 ］则将保真项等效变换到微分空间，并采用ADMM（Alternating Direction Method of Multipliers）分裂算法进行求解获得了快速准确的复原效果。 但是，以上方法均对整幅图像进行统一处理，采用单一的正则化约束条件或者组合的多正则化约束进行求解，容易造成去噪不彻底或过度平滑。文献［ 4 4 ］对集成电路X射线图像的加性噪声和乘性噪声分别采用 <math id="M8"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 正则化项和 <math id="M9"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227415&type= 2.37066650 3.72533321 正则项进行约束，并采用显式差分算法和梯度投影（Gradient Projection， GP）算法进行求解获得了很好的去噪效果。 本文基于图像边缘细节、平滑特征和不同正则化项的复原特点，提出集成电路X射线图像的多正则化图像复原算法。该方法采用高斯高通滤波和高斯低通滤波分别获取图像的边缘细节结果和平滑滤波结果，充分利用 <math id="M10"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227415&type= 2.37066650 3.72533321 正则项在细节保持以及 <math id="M11"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 正则项在噪声去除上的优越性，对边缘细节结果和平滑滤波结果分别采用 <math id="M12"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227415&type= 2.37066650 3.72533321 正则化项和 <math id="M13"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 正则化项约束进行求解，进而获得完整的集成电路X射线图像复原结果，解决整幅图像采用单一正则化项可能造成的细节过度平滑缺失和去噪效果差等问题。 2 基于快速傅里叶变换的图像滤波方法 1 集成电路的X射线图像退化模型可表示为： <math display="block" id="M14"><mi mathvariant="bold-italic">f</mi><mo>=</mo><mi mathvariant="bold-italic">H</mi><mi mathvariant="bold-italic">x</mi><mo>+</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227434&type= 18.54199982 3.72533321 ， 其中： <math id="M15"><mi mathvariant="bold-italic">f</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">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=37227447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227442&type= 14.39333248 3.89466691 是观测到的退化图像， <math id="M16"><mi mathvariant="bold-italic">H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227452&type= 3.47133350 2.79399991 为观测阵， <math id="M17"><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227461&type= 2.28600001 2.79399991 为待求解的干净图像， <math id="M18"><mi mathvariant="bold-italic">n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227469&type= 2.11666679 2.79399991 是符合均值为 μ 、方差为 ξ 的高斯分布 <math id="M19"><mi mathvariant="bold-italic">n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227478&type= 2.11666679 2.79399991 ∼N（μ ， ξ ）的加性噪声。 图像的傅里叶变换是将图像由空间域变换至频率域，其中低频对应集成电路X射线图像内变化较为缓慢的灰度分量，简记为平滑特征；高频对应图像内变化较快的灰度分量，多表现为边缘细节部分，因此，本节基于快速傅里叶变换（Fast Fourier Transform， FFT）计算速度快、效率高的特点，分别采用高斯低通滤波和高斯高通滤波获得低频对应的平滑滤波结果和高频对应的边缘细节结果。 具体步骤如下： 第一步，对观测图像 <math id="M20"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227798&type= 2.45533323 3.64066648 进行快速傅里叶变换得到其频谱 <math id="M21"><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227497&type= 12.10733318 3.97933316 ，其中 <math id="M22"><mi>u</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>M</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227504&type= 29.97200012 3.21733332 ， <math id="M23"><mi>v</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227515&type= 30.14133072 3.21733332 ， <math id="M24"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227528&type= 7.11199999 3.55599999 是傅里叶变换； 第二步，对频谱 <math id="M25"><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227538&type= 12.10733318 3.97933316 作能量估计，设定滤波器截止频率 <math id="M26"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227583&type= 4.14866638 3.72533321 。 选定初始截止频率，获取高斯低通滤波后的 <math id="M27"><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227559&type= 12.27666664 3.97933316 。逐步增大截止频率并不断计算当前截止频率条件下的 <math id="M28"><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227570&type= 12.27666664 3.97933316 ，当满足条件（2）时，当前截止频率记为滤波器的截止频率 <math id="M29"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227583&type= 4.14866638 3.72533321 。 <math display="block" id="M30"><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>></mo><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227592&type= 22.09799957 10.92199993 ， <math id="M31"><mi mathvariant="normal">其</mi><mi mathvariant="normal">中</mi><mo>:</mo><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227600&type= 23.28333473 4.74133301 表示整个图像频谱的能量， | <math id="M32"><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>'</mi></mrow></msup><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227608&type= 12.27666664 3.97933316 | 表示高斯低通滤波后频谱的能量， <math id="M33"><mi>β</mi><mo>∈</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227617&type= 14.39333248 3.97933316 为经验常数，用于选取滤波器的截止频率； 第三步，基于高斯低通滤波和高斯高通滤波原理对图像进行频域滤波，获取低频部分 <math id="M34"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227624&type= 17.69533348 3.97933316 和高频部分 <math id="M35"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227631&type= 16.17133331 3.97933316 。 <math display="block" id="M36"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">P</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>·</mo><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227641&type= 50.71533585 3.97933316 ， <math display="block" id="M37"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">P</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>·</mo><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227649&type= 49.44533539 3.97933316 ， 其中： <math display="block" id="M38"><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">P</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227659&type= 30.56466866 7.02733374 ， <math display="block" id="M39"><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">P</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">P</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227669&type= 41.31733322 3.97933316 . 第四步，对低频和高频部分进行反傅里叶变换，获得低频对应的图像平滑滤波结果 <math id="M40"><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227676&type= 8.04333401 4.57200003 和高频对应的图像边缘细节结果 <math id="M41"><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227684&type= 6.43466663 4.57200003 。 <math display="block" id="M42"><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227691&type= 30.73399925 4.06400013 ， <math display="block" id="M43"><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227697&type= 27.68600082 4.06400013 ， 其中 <math id="M44"><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi mathvariant="bold-italic">F</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi><mi>N</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>u</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>M</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msubsup><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mfenced separators="|"><mrow><mfrac><mrow><mi>u</mi><mi>x</mi></mrow><mrow><mi>M</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>v</mi><mi>y</mi></mrow><mrow><mi>N</mi></mrow></mfrac></mrow></mfenced></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227704&type= 67.73333740 8.55133343 。 3 基于分区的多正则化图像复原算法 1 正则化图像复原的目标函数模型求解问题可描述为： <math display="block" id="M45"><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">H</mi><mi mathvariant="bold-italic">x</mi><mo>-</mo><mi mathvariant="bold-italic">f</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>δ</mi><mi>Φ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227712&type= 43.01066971 7.87400007 ， 其中： <math id="M46"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">H</mi><mi mathvariant="bold-italic">x</mi><mo>-</mo><mi mathvariant="bold-italic">f</mi></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=37227725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227722&type= 17.01799965 6.09600019 是数据保真项， <math id="M47"><mi>Φ</mi><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=37227732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227729&type= 8.89000034 3.55599999 为正则化项。正则化参数 <math id="M48"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227736&type= 1.94733346 2.79399991 为平衡数据保真项和正则化项之间的权重。 针对集成电路X射线图像噪声强烈、对比度低的特点，本文提出一种多正则化图像复原方法。该方法充分考虑图像边缘细节和图像平滑区域在细节保持和噪声去除上的不同需求，设计了一种 <math id="M49"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227742&type= 12.86933327 3.80999994 范数的混合正则化模型，如 式（10） 式（10） 所示： <math display="block" id="M50"><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>x</mi></mrow></munder><mtext> </mtext><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">H</mi><mi mathvariant="bold-italic">x</mi><mo>-</mo><mi mathvariant="bold-italic">f</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><mo>·</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227751&type= 62.99200058 7.87400007 <math display="block" id="M51"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>·</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><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=37227765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227758&type= 26.24666595 4.57200003 ， 其中： <math id="M52"><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">和</mi><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227769&type= 10.07533360 4.23333359 为正则化参数， <math id="M53"><mi mathvariant="bold-italic">D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227775&type= 3.04800010 2.79399991 为梯度算子， <math id="M54"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><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=37227784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227781&type= 10.24466610 4.57200003 为 <math id="M55"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 正则化项，如式（ 11 11 ）~（ 12 12 ）所示。 <math id="M56"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227792&type= 1.77800000 2.79399991 为观测图像选择因子，当 <math id="M57"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227798&type= 2.45533323 3.64066648 是高频对应的图像边缘细节结果时， <math id="M58"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227804&type= 8.46666718 2.79399991 ；当 <math id="M59"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227810&type= 2.45533323 3.64066648 是低频对应的图像平滑滤波结果时， <math id="M60"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227817&type= 8.46666718 2.79399991 。 若 <math id="M61"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><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=37227845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227842&type= 10.24466610 4.57200003 为各向同性 <math id="M62"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227830&type= 5.67266655 2.87866688 ，则： <math display="block" id="M63"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>M</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mroot><mrow><msubsup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227837&type= 68.83399963 7.02733374 . 若 <math id="M64"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><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=37227845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227842&type= 10.24466610 4.57200003 为各向异性 <math id="M65"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 ，则： <math display="block" id="M66"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>M</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mfenced><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227853&type= 73.06733704 7.53533268 ， 其中： <math id="M67"><msub><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227861&type= 34.45933533 4.99533367 ， <math id="M68"><msub><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227867&type= 34.45933533 4.99533367 ， <math id="M69"><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227873&type= 4.23333359 3.72533321 表示水平梯度（Horizontal gradient）， <math id="M70"><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227878&type= 4.23333359 3.72533321 表示垂直梯度（Vertical gradient）， <math id="M71"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227883&type= 21.08200073 3.72533321 ， <math id="M72"><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>M</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227887&type= 20.57399940 3.72533321 ，（ <math id="M73"><mi>i</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>M</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227891&type= 32.93532944 3.21733332 ， <math id="M74"><mi>j</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227895&type= 17.52599907 3.64066648 <math id="M75"><mo>⋯</mo><mo>,</mo><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227899&type= 15.32466698 3.21733332 ）。 当 <math id="M76"><mi mathvariant="bold-italic">f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227902&type= 2.62466669 3.64066648 是高频对应的图像边缘细节结果时， <math id="M77"><mi mathvariant="bold-italic">f</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub><mo>,</mo><mi>λ</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227908&type= 22.86000061 4.57200003 ，问题（10）转化为求解 <math id="M78"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227915&type= 9.56733322 3.72533321 问题： <math display="block" id="M79"><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>*</mi></mrow></msubsup><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><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">H</mi><mi mathvariant="bold-italic">x</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>λ</mi><mo>·</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><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=37227921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227920&type= 63.07666779 7.87400007 . 本文采用GPSR-Basic方法 ［ 17 17 ］ 进行求解，步骤如下： 第一步，令 <math id="M80"><mi mathvariant="bold-italic">x</mi><mo>=</mo><mi mathvariant="bold-italic">a</mi><mo>-</mo><mi mathvariant="bold-italic">b</mi><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="bold-italic">b</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227924&type= 36.66066742 3.47133350 ，将问题（13）转化为无约束优化问题。 <math display="block" id="M81"><mi mathvariant="bold-italic">s</mi><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="bold-italic">s</mi></mrow></munder><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">s</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">s</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227928&type= 51.81600189 7.87400007 ， 其中： <math id="M82"><mi mathvariant="bold-italic">s</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">a</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">b</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mi mathvariant="bold-italic">c</mi><mo>=</mo><mi mathvariant="bold-italic">α</mi><msub><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>N</mi></mrow></msub><mo>+</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">f</mi></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">f</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><msub><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>N</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227931&type= 52.23933029 10.24466610 是包含 2N 个1的列向量， <math id="M83"><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">H</mi></mtd><mtd><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">H</mi></mtd></mtr><mtr><mtd><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">H</mi></mtd><mtd><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">H</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227935&type= 38.26933289 8.55133343 。 第二步，梯度投影迭代更新。 <math display="block" id="M84"><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>-</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>∇</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227939&type= 36.23733139 5.24933338 ， 其中 <math id="M85"><msub><mrow><mfenced separators="|"><mrow><mi>*</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mtext> </mtext><mfenced separators="|"><mrow><mi>*</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227943&type= 27.00866890 4.65666676 ， <math id="M86"><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227945&type= 9.56733322 3.04800010 。 第三步，回溯线搜索 <math id="M87"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>∈</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227947&type= 15.07066536 4.48733330 。 若满足 <math id="M88"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>∇</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227949&type= 60.11333084 4.65666676 ，则令 <math id="M89"><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227951&type= 14.81666565 3.04800010 。 最后，判断截止条件，若： <math display="block" id="M90"><mfrac><mrow><mo stretchy="false">|</mo><mo stretchy="false">|</mo><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">|</mo><msubsup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mo stretchy="false">|</mo><mo stretchy="false">|</mo><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">|</mo><msubsup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>≤</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227953&type= 24.04533195 9.05933285 . 则 <math id="M91"><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">l</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227955&type= 37.42266846 4.40266657 ，否则返回第二步。 若 <math id="M92"><mi mathvariant="bold-italic">f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227959&type= 2.62466669 3.64066648 是低频对应的图像平滑滤波结果时， <math id="M93"><mi mathvariant="bold-italic">f</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mo>,</mo><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227965&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227963&type= 24.38400078 4.57200003 ，则问题（10）转化为求解 <math id="M94"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227966&type= 12.86933327 3.80999994 问题（15）， <math display="block" id="M95"><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>x</mi></mrow></munder><mtext> </mtext><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">H</mi><mi mathvariant="bold-italic">x</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><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=37227970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227968&type= 64.85466003 7.87400007 . 结合微分空间在梯度细节保持上的优越性，本文采用文献［ 16 16 ］的D-ADMM（Derivative Alternating Direction Method of Multipliers）进行求解，力求平滑区域在平滑去噪的同时保留更多的微小细节信息，避免过渡平滑带来的集成电路X射线图像的细节损失。 通过微分空间和图像空间的等效条件，问题（15）转化为： <math display="block" id="M96"><mi mathvariant="bold-italic">d</mi><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>d</mi></mrow></munder><mtext> </mtext><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><mi mathvariant="bold-italic">H</mi><mi mathvariant="bold-italic">d</mi><mo>-</mo><mi mathvariant="bold-italic">D</mi><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">d</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227971&type= 63.92333221 7.87400007 ， <math display="block" id="M97"><mi>s</mi><mo>.</mo><mi>t</mi><mo>.</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">d</mi><mo>=</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227973&type= 42.41799927 4.40266657 令 <math id="M98"><mi mathvariant="bold-italic">z</mi><mo>=</mo><mi mathvariant="bold-italic">d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227975&type= 9.14400005 2.87866688 ，由增广拉格朗日方法，问题（16）转化为无约束优化问题， <math display="block" id="M99"><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">d</mi><mo>,</mo><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">q</mi><mo>,</mo><mi mathvariant="bold-italic">p</mi></mrow></mfenced><mo>=</mo><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">H</mi><mi mathvariant="bold-italic">d</mi><mo>-</mo><mi mathvariant="bold-italic">D</mi><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227978&type= 72.30532837 7.87400007 <math display="block" id="M100"><mfrac><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">d</mi><mo>-</mo><mi mathvariant="bold-italic">z</mi><mo>+</mo><mi mathvariant="bold-italic">q</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>+</mo><mi mathvariant="bold-italic">p</mi></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=37227984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227982&type= 65.44733429 7.87400007 ， 其中： <math id="M101"><mi mathvariant="bold-italic">d</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow></msub><mi mathvariant="bold-italic">x</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow></msub><mi>x</mi><mo>,</mo><mi mathvariant="bold-italic">z</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">q</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227986&type= 112.26799774 5.50333309 是增广拉格朗日的惩罚参数。 在ADMM框架下（17）可以转化为多个简单的子问题进行求解，具体如下： 第一步，求 <math id="M102"><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227989&type= 7.19666624 3.97933316 ， <math id="M103"><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227993&type= 7.19666624 3.97933316 ： <math display="block" id="M104"><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced open="{" close="}" separators="|"><mrow><mo stretchy="false">[</mo><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mfenced><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfenced><mo stretchy="false">]</mo><mi mathvariant="normal">∅</mi><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37227997&type= 73.57534027 5.50333309 ， 其中： <math id="M105"><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">H</mi><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">I</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228001&type= 106.68000031 5.16466665 ， <math id="M106"><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228005&type= 3.72533321 3.64066648 表示按像素相除（entry-wise division）。 <math display="block" id="M107"><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfenced><mo stretchy="false">)</mo><mi mathvariant="normal">∅</mi><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>v</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228008&type= 69.25733185 4.82600021 ， 其中： <math id="M108"><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">H</mi><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228012&type= 45.38133240 4.91066647 ， <math id="M109"><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228016&type= 59.43600082 5.16466665 。 第二步，更新 <math id="M110"><msup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228020&type= 6.51933336 3.04800010 ： <math display="block" id="M111"><msup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>z</mi></mrow></munder><mtext> </mtext><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">z</mi><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228024&type= 70.18866730 8.80533314 ， 对于各向同性 <math id="M112"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 ，有： <math display="block" id="M113"><msup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>-</mo><mfrac><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228031&type= 66.29399872 9.99066734 ， 其中 <math id="M114"><msup><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mroot><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228035&type= 54.18666840 5.24933338 。 对于各向异性 <math id="M115"><mi mathvariant="normal">T</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228039&type= 5.67266655 2.87866688 ，有： <math display="block" id="M116"><msup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced><mo>-</mo><mfrac><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228042&type= 72.64399719 9.99066734 . 第三步，更新 <math id="M117"><msup><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228046&type= 6.94266701 3.89466691 ， <math id="M118"><msup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228051&type= 6.51933336 3.89466691 ： <math display="block" id="M119"><msup><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228054&type= 48.25999832 4.82600021 ， <math display="block" id="M120"><msup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228058&type= 37.33800125 3.89466691 . 最后，判断截止条件。 若满足如下条件： <math display="block" id="M121"><mfrac><mrow><mo stretchy="false">|</mo><mo stretchy="false">|</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">|</mo><msubsup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mfenced open="|" close="|" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>≤</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228063&type= 26.24666595 11.85333347 . 则得到复原图像 <math id="M122"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">U</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced><mo>=</mo><mi mathvariant="bold-italic">U</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced><mi mathvariant="normal">。</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228066&type= 46.73600006 4.57200003 若条件不满足则跳转至第一步， <math id="M123"><mi>k</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228070&type= 14.81666565 2.96333337 。 综合高频对应的图像边缘细节复原结果和低频对应的图像平滑部分复原结果得到期望的集成电路X射线图像复原结果，获得噪声去除和细节保持的有效平衡。 4 实验与结果分析 1 为了验证本文算法的有效性，采用标准自然光图像和集成电路的X射线图像进行两个系列的实验验证。实验的软硬件实验环境为： Matlab（R2016b）， Inter（R） Core（TM） i5-8250U CPU @1.60 GHz ，8 GB内存。X射线成像示意图及本文采用的X射线设备如 图1 图1 所示。 4.1　标准灰度图像实验验证 2 该实验采用Boat，Cameraman，Lena，Peppers 4幅经典的自然光图像进行实验验证，原图如 图2 图2 所示。实验参数如下：噪声方差为 <math id="M124"><mi>μ</mi><mo>=</mo><mn mathvariant="normal">5</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228080&type= 20.23533440 3.89466691 ，经验系数 <math id="M125"><mi>β</mi><mo>=</mo><mn mathvariant="normal">0.72</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228105&type= 14.64733315 3.64066648 ，正则化参数 <math id="M126"><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">9</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">9</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228087&type= 44.36533356 4.40266657 ，截止条件 <math id="M127"><mi>ε</mi><mo>=</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228113&type= 12.86933327 3.04800010 。 本文采用均方误差（MSE），峰值信噪比（PSNR），结构相似性（SSIM）对4幅图像的复原结果进行分析，如 表1 表1 所示。显然，本文方法的去噪指标MSE、PSNR和细节保持指标SSIM均优于对比算法，表明本文方法的复原结果得到了噪声去除和细节保持的有效平衡，因此可将其用于集成电路的X射线图像复原。 Methods MSE PSNR SSIM 4 Boat Isotropic D-ADMM 45.85 31.55 0.836 7 Anisotropic D-ADMM 40.49 32.10 0.857 7 Isotropic proposed method 31.03 33.25 0.886 2 Anisotropic proposed method 31.25 33.22 0.885 8 4 Cameraman Isotropic D-ADMM 30.92 33.26 0.844 2 Anisotropic D-ADMM 24.90 34.20 0.885 4 Isotropic proposed method 12.69 37.13 0.948 8 Anisotropic proposed method 12.87 37.07 0.948 8 4 Lena Isotropic D-ADMM 34.52 32.78 0.835 6 Anisotropic D-ADMM 29.04 33.53 0.867 5 Isotropic proposed method 20.42 35.06 0.909 5 Anisotropic proposed method 20.32 35.09 0.910 0 4 Peppers Isotropic D-ADMM 36.56 32.54 0.818 8 Anisotropic D-ADMM 31.44 33.19 0.844 7 Isotropic proposed method 26.39 33.95 0.876 7 Anisotropic proposed method 26.42 33.95 0.876 5 4.2　集成电路X射线图像实验验证 2 集成电路的X射线图像选择电容元件、焊点等3幅图像进行验证。实验参数选择为 <math id="M128"><mi>β</mi><mo>=</mo><mn mathvariant="normal">0.72</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228105&type= 14.64733315 3.64066648 ，正则化参数 <math id="M129"><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">5</mn></mrow></msup><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">8</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228108&type= 44.36533356 4.40266657 ，截止条件 <math id="M130"><mi>ε</mi><mo>=</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228113&type= 12.86933327 3.04800010 。 图3 图3 ~ 图5 图5 所示为三幅图像的复原结果，对比方法为文献［ 4 4 ］所述的混合去噪方法、各向同性D-ADMM和各向异性D-ADMM。显然，各方法均可获得集成电路X射线图像的有效复原，但是，本文方法的复原结果对比度高，边缘更加清晰。图中放大显示的细节部分表明本文方法能够更清晰地展示电容条状结构、焊点气泡边缘等细节信息，进一步表明了本文方法在噪声去除的同时具有更好的细节保持效果，为后续缺陷检测、提高产品可靠性奠定基础。 1;2; 为更进一步验证本文算法的优越性，本节计算出以上复原结果的拉普拉斯梯度（Laplacian Sum， LS），灰度平均梯度（Gray Mean Grads， GMG），边缘强度（Edge Intensity， EI）对X射线图像的复原结果进行客观评价，如 表2 表2 所示。实验结果表明本文算法在去除噪声的同时能够保留更多的细节信息。 2 3 Capacitance element 3 Welding spot 1 3 Welding spot 2 LS GMG EI LS GMG EI LS GMG EI Isotropic D-ADMM 0.037 0.008 8 0.085 0.025 0.005 3 0.048 0.015 0.003 4 0.032 Anisotropic D-ADMM 0.037 0.008 6 0.084 0.024 0.005 1 0.046 0.015 0.003 3 0.03 Method of reference ［ 4 4 ］ 0.029 0.006 6 0.07 0.014 0.002 7 0.028 0.013 0.002 2 0.022 Isotropic proposed method 0.049 0.010 7 0.092 0.047 0.008 8 0.059 0.032 0.006 0.040 Anisotropic proposed method 0.049 0.010 7 0.091 0.047 0.008 8 0.058 0.032 0.006 0.039 随后，本文对另10幅集成电路的X射线图像进行了实验验证，客观评价如 表3 表3 所示，再次验证了本文方法的有效性。 续表3　另10幅集成电路X射线图像实验结果 2 3 1 3 2 LS GMG EI LS GMG EI Isotropic D-ADMM 0.024 0.005 0.051 0.032 0.008 0.080 Anisotropic D-ADMM 0.024 0.005 0.048 0.033 0.008 0.077 Method of reference ［ 4 4 ］ 0.017 0.003 0.034 0.025 0.006 0.064 Isotropic proposed method 0.045 0.009 0.060 0.053 0.011 0.089 Anisotropic proposed method 0.045 0.009 0.059 0.054 0.011 0.088 2 3 3 3 4 LS GMG EI LS GMG EI Isotropic D-ADMM 0.031 0.006 0.055 0.042 0.009 0.075 Anisotropic D-ADMM 0.030 0.006 0.052 0.041 0.008 0.072 Method of reference ［ 4 4 ］ 0.016 0.003 0.030 0.018 0.004 0.038 Isotropic proposed method 0.052 0.010 0.061 0.072 0.014 0.089 Anisotropic proposed method 0.052 0.009 0.060 0.072 0.014 0.088 2 3 5 3 6 LS GMG EI LS GMG EI Isotropic D-ADMM 0.027 0.006 0.053 0.025 0.005 0.048 Anisotropic D-ADMM 0.027 0.006 0.051 0.024 0.005 0.046 Method of reference ［ 4 4 ］ 0.017 0.004 0.037 0.014 0.003 0.028 Isotropic proposed method 0.046 0.009 0.062 0.047 0.009 0.059 Anisotropic proposed method 0.046 0.009 0.061 0.047 0.009 0.058 2 3 7 3 8 LS GMG EI LS GMG EI Isotropic D-ADMM 0.036 0.008 0.069 0.028 0.006 0.055 Anisotropic D-ADMM 0.035 0.007 0.066 0.027 0.006 0.052 Method of reference ［ 4 4 ］ 0.018 0.004 0.039 0.016 0.003 0.033 Isotropic proposed method 0.063 0.012 0.081 0.050 0.010 0.066 Anisotropic proposed method 0.063 0.012 0.080 0.050 0.010 0.065 2 3 9 3 10 LS GMG EI LS GMG EI Isotropic D-ADMM 0.026 0.006 0.052 0.032 0.007 0.061 Anisotropic D-ADMM 0.025 0.005 0.050 0.031 0.007 0.058 Method of reference ［ 4 4 ］ 0.017 0.003 0.034 0.016 0.003 0.034 Isotropic proposed method 0.047 0.009 0.062 0.056 0.011 0.072 Anisotropic proposed method 0.047 0.009 0.061 0.056 0.011 0.071 5 结 论 1 本文在分析集成电路X射线成像特点的基础上，提出了一种多正则化集成电路X射线图像复原算法。首先，通过高斯高通滤波和高斯低通滤波获取图像的高频边缘细节结果和低频平滑滤波结果并以此作为图像复原的观测图；然后提出了一种 <math id="M131"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228154&type= 2.37066650 3.72533321 -TV混合正则化图像复原模型。该模型充分考虑了边缘细节和平滑部分的复原需求，解决了整幅图像采用单一正则化项造成的细节过度平滑缺失或去噪效果差的问题。实验表明本文方法的复原结果对比度高，边缘清晰，在集成电路X射线图像复原中获得了去噪和细节保持的有效平衡，可为后续缺陷检测、提高产品可靠性奠定基础。 参考文献 MIKHAYLOV A ， ZAKHAROVA M ， VLNIESKA V ， et al . 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MA Ge,LIN Sen,LI Zhifu,et al.Multi-regularization image restoration method for X-ray images of integrated circuit[J].Optics and Precision Engineering,2022,30(22):2913-2922. 广州市科技计划项目（No.202102020876）；广东省自然科学基金资助项目（No.2018A030310065）；国家自然科学基金资助项目（No.61803110，No.62173102）； 2022-04-29 2022-05-23 2022-11-25 2022-11-24 光学精密工程 22 30