图像去模糊（逆滤波）

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 

G(u,v)=H(u,v)F(u,v)+N(u,v)(1)

其中提到最简单的复原方法是直接做逆滤波（Inverse filter）。 

F̂ (u,v)=G(u,v)H(u,v)(2)

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) 

F̂ (u,v)=F(u,v)+N(u,v)H(u,v)(3)

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; 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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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作者日期联系方式风吹夏天2015年8月8日[e-mail]