引言

图像模糊是一种拍摄常见的现象，我曾在图像去模糊（维纳滤波） 介绍过。这里不再详述，只给出物理模型，这里我们仍在频率域表示

其中提到最简单的复原方法是直接做逆滤波（Inverse filter）。

该除法是阵列操作，即按位除。 
在含有噪声情况下，将(1)式两端除以H(u,v)

这里N(u,v)未知，式子表明，即使知道退化函数也不能准备复原图像。还有当退化函数H(u,v)是零或者非常小的值，则N(u,v)H(u,v)很容易支配整个式子。 
下面我将用代码说明一下逆滤波，这里我采用直接编码形式。对了，前面我提到过，当噪声信息比NSR等于0时，此时维娜滤波等同于逆滤波。因此可以直接使用matlab自带deconvwnr函数，将第三个参数NSR设置成0即可，省事的同学可以试一下。

代码

<code class="hljs vhdl has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;">close <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">all</span>;
clear <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">all</span>;
clc;
% Display the original image.
I = im2double(imread(<span class="hljs-attribute" style="box-sizing: border-box;">'lena</span>.jpg'));
[hei,wid,~] = size(I);
subplot(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>),imshow(I);
title(<span class="hljs-attribute" style="box-sizing: border-box;">'Original</span> Image (courtesy <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">of</span> MIT)');% Simulate a motion blur.
LEN = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">21</span>;
THETA = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">11</span>;
PSF = fspecial(<span class="hljs-attribute" style="box-sizing: border-box;">'motion</span>', LEN, THETA);
blurred = imfilter(I, PSF, <span class="hljs-attribute" style="box-sizing: border-box;">'conv</span>', <span class="hljs-attribute" style="box-sizing: border-box;">'circular</span>');
subplot(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>), imshow(blurred); title(<span class="hljs-attribute" style="box-sizing: border-box;">'Blurred</span> Image');% Inverse filter
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">If</span> = fft2(blurred);
Pf = fft2(PSF,hei,wid);
deblurred = ifft2(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">If</span>./Pf);
subplot(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>), imshow(deblurred); title(<span class="hljs-attribute" style="box-sizing: border-box;">'Restore</span> Image')% Simulate additive noise.
noise_mean = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;
noise_var = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0.0001</span>;
blurred_noisy = imnoise(blurred, <span class="hljs-attribute" style="box-sizing: border-box;">'gaussian</span>', ...noise_mean, noise_var);
subplot(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">4</span>), imshow(blurred_noisy)
title(<span class="hljs-attribute" style="box-sizing: border-box;">'Simulate</span> Blur <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">and</span> Noise')% Try restoration assuming no noise.
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">If</span> = fft2(blurred_noisy);
deblurred2 = ifft2(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">If</span>./Pf);
subplot(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">5</span>), imshow(deblurred2)
title(<span class="hljs-attribute" style="box-sizing: border-box;">'Restoration</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">of</span> Blurred Assuming No Noise');% Try restoration <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">with</span> noise <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">is</span> known.
noisy = blurred_noisy - blurred;
Nf = fft2(noisy);
deblurred2 = ifft2(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">If</span>./Pf - Nf./Pf);
subplot(<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">6</span>), imshow(deblurred2)
title(<span class="hljs-attribute" style="box-sizing: border-box;">'Restoration</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">of</span> Blurred <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">with</span> Noise <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">Is</span> Known')</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; 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padding: 0px 5px;">40</li><li style="box-sizing: border-box; padding: 0px 5px;">41</li><li style="box-sizing: border-box; padding: 0px 5px;">42</li><li style="box-sizing: border-box; padding: 0px 5px;">43</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; 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padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li><li style="box-sizing: border-box; padding: 0px 5px;">29</li><li style="box-sizing: border-box; padding: 0px 5px;">30</li><li style="box-sizing: border-box; padding: 0px 5px;">31</li><li style="box-sizing: border-box; padding: 0px 5px;">32</li><li style="box-sizing: border-box; padding: 0px 5px;">33</li><li style="box-sizing: border-box; padding: 0px 5px;">34</li><li style="box-sizing: border-box; padding: 0px 5px;">35</li><li style="box-sizing: border-box; padding: 0px 5px;">36</li><li style="box-sizing: border-box; padding: 0px 5px;">37</li><li style="box-sizing: border-box; padding: 0px 5px;">38</li><li style="box-sizing: border-box; padding: 0px 5px;">39</li><li style="box-sizing: border-box; padding: 0px 5px;">40</li><li style="box-sizing: border-box; padding: 0px 5px;">41</li><li style="box-sizing: border-box; padding: 0px 5px;">42</li><li style="box-sizing: border-box; padding: 0px 5px;">43</li></ul>

这里使用了经典的lena图是灰度图像，分别对图像进行运动模糊，逆滤波，运动模糊+高斯噪声，假定噪声未知直接逆滤波，噪声已知逆滤波。

效果

说明

逆滤波对噪声非常敏感，除非我们知道噪声的分布情况（事实上，这也很难知道），逆滤波几乎不可用，可以从二排中间看出，恢复图像效果极差。但若知道噪声分布，也是可以完全复原信息的。可以从二排最后一张图可以看出。写作本文的目的也仅是在数学角度上对图像模糊现象进行分析，后续会介绍更加有效的图像复原方法，敬请关注。

相关阅读及参考文献

图像去模糊（维纳滤波） http://blog.csdn.net/bluecol/article/details/46242355 
图像去模糊（约束最小二乘滤波） http://blog.csdn.net/bluecol/article/details/47359421 
数字图像处理（第三版） 冈萨雷斯著 chapter 5，图像复原与重建

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