引言

圖像模糊是一種拍攝常見的現象，我曾在圖像去模糊（維納濾波）?介紹過。這里不再詳述，只給出物理模型，這里我們仍在頻率域表示?

G(u,v)=H(u,v)F(u,v)+N(u,v)(1)
其中提到最簡單的復原方法是直接做逆濾波（Inverse filter）。?
F??(u,v)=G(u,v)H(u,v)(2)
該除法是陣列操作，即按位除。?
在含有噪聲情況下，將(1)式兩端除以H(u,v)?
F??(u,v)=F(u,v)+N(u,v)H(u,v)(3)
這里N(u,v)未知，式子表明，即使知道退化函數也不能準備復原圖像。還有當退化函數H(u,v)是零或者非常小的值，則N(u,v)H(u,v)很容易支配整個式子。?
下面我將用代碼說明一下逆濾波，這里我采用直接編碼形式。對了，前面我提到過，當噪聲信息比NSR等于0時，此時維娜濾波等同于逆濾波。因此可以直接使用matlab自帶deconvwnr函數，將第三個參數NSR設置成0即可，省事的同學可以試一下。

代碼

<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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這里使用了經典的lena圖是灰度圖像，分別對圖像進行運動模糊，逆濾波，運動模糊+高斯噪聲，假定噪聲未知直接逆濾波，噪聲已知逆濾波。

效果

說明

逆濾波對噪聲非常敏感，除非我們知道噪聲的分布情況（事實上，這也很難知道），逆濾波幾乎不可用，可以從二排中間看出，恢復圖像效果極差。但若知道噪聲分布，也是可以完全復原信息的。可以從二排最后一張圖可以看出。寫作本文的目的也僅是在數學角度上對圖像模糊現象進行分析，后續會介紹更加有效的圖像復原方法，敬請關注。

相關閱讀及參考文獻

圖像去模糊（維納濾波）?http://blog.csdn.net/bluecol/article/details/46242355?
圖像去模糊（約束最小二乘濾波）?http://blog.csdn.net/bluecol/article/details/47359421?
數字圖像處理（第三版） 岡薩雷斯著 chapter 5，圖像復原與重建

轉載請保留以下信息

作者 日期 聯系方式

風吹夏天	2015年8月8日	wincoder@qq.com

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