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盲去卷积原理及在图像复原的应用
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前言
?????? 之前寫過一篇維納濾波在圖像復(fù)原中的作用，講述了圖像退化模型以及維納濾波的作用。維納濾波使用的前提是知道信號和噪聲的功率譜，但在實際應(yīng)用中較難得到，只能根據(jù)先驗知識進行估計。
?????? 本文介紹盲去卷積復(fù)原算法，并在MATLAB中進行實驗，和維納濾波的復(fù)原效果進行一個對比。盲去卷積的方法有多種，本文主要介紹由fish提出的基于露西-理查德森(Richardson-Lucy)的盲去卷積算法。
盲去卷積原理
?????露西-理查德森算法屬于圖像復(fù)原中的非線性算法，與維納濾波這種較為直接的算法不同，該算法使用非線性迭代技術(shù)，在計算量、性能方面都有了一定提升。
?????露西-理查德森算法是由貝葉斯公式推導(dǎo)而來，因為使用了條件概率（即算法考慮了信號的固有波動，因此具有復(fù)原噪聲圖像的能力）。貝葉斯公式如下：
????? 結(jié)合圖像退化/復(fù)原模型，可以得到迭代函數(shù)：
??
????其中 fi 就是第i輪迭代復(fù)原圖像，對應(yīng)貝葉斯公式中的p(x)，g是退化函數(shù)，對應(yīng)貝葉斯公式的p(y|x)，c為退化圖像（c(y)dy意為在退化圖像上積分），如果滿足等暈條件，即圖像各區(qū)域的模糊函數(shù)相同，則迭代公式可化簡如下：
???? 這就是路西-理查德森迭代公式，其中c是退化圖像，g是退化函數(shù)，f是第k輪復(fù)原圖像。如果系統(tǒng)的退化函數(shù)PSF(g(x))已知，只要有一個初始估計f就可以進行迭代求解了。在開始迭代后，由于算法的形式，估計值會與真實值的差距迅速減小，從而后續(xù)迭代過程f的更新速度會逐漸變慢，直至收斂。算法的另一優(yōu)點就是初始值f>0，后續(xù)迭代值均會保持非負(fù)性，并且能量不會發(fā)散。
?????盲去卷積需要兩步進行復(fù)原，原因是我們既不知道原始圖像f，也不知道退化函數(shù)g。求解過程示意圖如下：
????? 即在第k輪迭代，我們假a設(shè)原始圖像已知，即k-1輪得到的fk-1，再通過R-L公式求解gk，隨后，再用gk求解fk，反復(fù)迭代，最后求得最終f和g。因此，在求解最初，我們需要同時假設(shè)一個復(fù)原圖像f0和一個退化函數(shù)g0。迭代公式如下：
????? 此外，有人采用這種盲去卷積方法進行了相關(guān)實驗，下圖為實驗原圖：
下圖(a)左、右分別為附加標(biāo)準(zhǔn)差1.5%、10%泊松噪聲的退化圖像，圖(b)左、右分別為1.5%的復(fù)原圖像和PSF,圖(c)為10%對應(yīng)結(jié)果
?????
盲去卷積MATLAB實驗
?????將一幅原始圖像，進行模糊處理（模擬大氣湍流），分別使用維納濾波（由于沒加噪聲，就是逆濾波）和盲去卷積進行復(fù)原，復(fù)原結(jié)果如下：
下面是網(wǎng)上找到的有關(guān)盲去卷積的MATLAB程序，可以加深理解。
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%%?Deblurring?Images?Using?the?Blind?Deconvolution?Algorithm??? %%盲反卷積算法復(fù)原圖像?? %?The?Blind?Deconvolution?Algorithm?can?be?used?effectively?when?no?? %?information?about?the?distortion?(blurring?and?noise)?is?known.?The?? %?algorithm?restores?the?image?and?the?point-spread?function?(PSF)?? %?simultaneously.?The?accelerated,?damped?Richardson-Lucy?algorithm?is?used?? %?in?each?iteration.?Additional?optical?system?(e.g.?camera)?? %?characteristics?can?be?used?as?input?parameters?that?could?help?to?? %?improve?the?quality?of?the?image?restoration.?PSF?constraints?can?be?? %?passed?in?through?a?user-specified?function?? %在不知道圖像失真信息(模糊和噪聲)信息情況下，盲反卷積算法可以有效地加以利用。該算法?? %對圖像和點擴展函數(shù)（PSF）的同時進行復(fù)原。每次迭代都使用加速收斂Richardson-Lucy??? %算法。額外的光學(xué)系統(tǒng)（如照相機）的特性可作為輸入?yún)?shù)，幫助改善圖像復(fù)原質(zhì)量。可以通?? %過用戶指定的函數(shù)對PSF進行限制?? %?Copyright?2004-2005?The?MathWorks,?Inc.?? ??? %%?Step?1:?Read?Image?? %%第一步：讀取圖像?? %?The?example?reads?in?an?intensity?image.?The?|deconvblind|?function?can?? %?handle?arrays?of?any?dimension.?? %該示例讀取一個灰度圖像。|?deconvblind?|函數(shù)可以處理任何維數(shù)組。?? I?=?imread('view.tif');?? figure;imshow(I);title('Original?Image');?? %text(size(I,2),size(I,1)+15,?...?? %????'Image?courtesy?of?Massachusetts?Institute?of?Technology',?...?? %'FontSize',7,'HorizontalAlignment','right');???? ????? ?? ??? %%?Step?2:?Simulate?a?Blur?? %%第二步：模擬一個模糊?? %?Simulate?a?real-life?image?that?could?be?blurred?(e.g.,?due?to?camera?? %?motion?or?lack?of?focus).?The?example?simulates?the?blur?by?convolving?a?? %?Gaussian?filter?with?the?true?image?(using?|imfilter|).?The?Gaussian?filter?? %?then?represents?a?point-spread?function,?|PSF|.?? ?%模擬一個現(xiàn)實中存在的模糊圖像（例如，由于相機抖動或?qū)共蛔?#xff09;。這個例子通過對真實?? %圖像進行高斯濾波器模擬圖像模糊（使用|imfilter|）。高斯濾波器是一個點擴展函數(shù)，?? %|PSF|。?? PSF=fspecial('gaussian',7,10);?? Blurred=imfilter(I,PSF,'symmetric','conv');??%對圖像I進行濾波處理；?? figure;imshow(Blurred);title('Blurred?Image');???? ?? ????? ?? ??? %%?Step?3:?Restore?the?Blurred?Image?Using?PSFs?of?Various?Sizes?? %%第三步：使用不同的點擴展函數(shù)復(fù)原模糊圖像?? %?To?illustrate?the?importance?of?knowing?the?size?of?the?true?PSF,?this?? %?example?performs?three?restorations.?Each?time?the?PSF?reconstruction?? %?starts?from?a?uniform?array--an?array?of?ones.?? %為了說明知道真實PSF的大小的重要性，這個例子執(zhí)行三個修復(fù)。PSF函數(shù)重建每次都是從統(tǒng)一?? %的全一數(shù)組開始。?? %%?? %?The?first?restoration,?|J1|?and?|P1|,?uses?an?undersized?array,?|UNDERPSF|,?for?? %?an?initial?guess?of?the?PSF.?The?size?of?the?UNDERPSF?array?is?4?pixels?? %?shorter?in?each?dimension?than?the?true?PSF.??? %第一次復(fù)原，|J1|和|P1|，使用一個較小數(shù)組，|?UNDERPSF?|，來對PSF的初步猜測。該?? %UNDERPSF數(shù)組每維比真實PSF少4個元素。?? UNDERPSF?=?ones(size(PSF)-4);?? [J1?P1]?=?deconvblind(Blurred,UNDERPSF);?? figure;imshow(J1);title('Deblurring?with?Undersized?PSF');??? ?? ????? ?? %%?? %?The?second?restoration,?|J2|?and?|P2|,?uses?an?array?of?ones,?|OVERPSF|,?for?an?? %?initial?PSF?that?is?4?pixels?longer?in?each?dimension?than?the?true?PSF.?? %第二次復(fù)原，|J2|和|P2|，使用一個元素全為1的數(shù)組，|?OVERPSF|，初始PSF每維比真?? %實PSF多4個元素。?? OVERPSF?=?padarray(UNDERPSF,[4?4],'replicate','both');?? [J2?P2]?=?deconvblind(Blurred,OVERPSF);?? figure;imshow(J2);title('Deblurring?with?Oversized?PSF');???? ????? ?? ??? %%?? %?The?third?restoration,?|J3|?and?|P3|,?uses?an?array?of?ones,?|INITPSF|,?for?an?? %?initial?PSF?that?is?exactly?of?the?same?size?as?the?true?PSF.?? %第三次復(fù)原，|J3|和|P3|，使用一個全為一的數(shù)組|?INITPSF?|作為初次PSF，每維與真正?? %的PSF相同。?? INITPSF?=?padarray(UNDERPSF,[2?2],'replicate','both');?? [J3?P3]?=?deconvblind(Blurred,INITPSF);?? figure;imshow(J3);title('Deblurring?with?INITPSF');???? ?? ????? ?? ??? %%?Step?4:?Analyzing?the?Restored?PSF?? %%第四步：分析復(fù)原函數(shù)PSF?? %?All?three?restorations?also?produce?a?PSF.?The?following?pictures?show?? %?how?the?analysis?of?the?reconstructed?PSF?might?help?in?guessing?the?? %?right?size?for?the?initial?PSF.?In?the?true?PSF,?a?Gaussian?filter,?the?? %?maximum?values?are?at?the?center?(white)?and?diminish?at?the?borders?(black).?? %所有這三個復(fù)原也產(chǎn)生PSF。以下圖片顯示對PSF重建分析的如何可能有助于猜測最初PSF的大?? %小。在真正的PSF中，高斯濾波器的最高值在中心（白），到邊界消失（黑）。?? figure;?? subplot(221);imshow(PSF,[],'InitialMagnification','fit');?? title('True?PSF');?? subplot(222);imshow(P1,[],'InitialMagnification','fit');?? title('Reconstructed?Undersized?PSF');?? subplot(223);imshow(P2,[],'InitialMagnification','fit');?? title('Reconstructed?Oversized?PSF');?? subplot(224);imshow(P3,[],'InitialMagnification','fit');?? title('Reconstructed?true?PSF');???? ?? ????? ?? ??? %%??? %?The?PSF?reconstructed?in?the?first?restoration,?|P1|,?obviously?does?not?? %?fit?into?the?constrained?size.?It?has?a?strong?signal?variation?at?the?? %?borders.?The?corresponding?image,?|J1|,?does?not?show?any?improved?clarity?? %?vs.?the?blurred?image,.?? ?%第一次復(fù)原的PSF，|P1|，顯然不適合大小的限制。它在邊界有一個強烈的變化信號。?? %相應(yīng)的圖片|J1|，與模糊圖像|Blurred|比沒有表現(xiàn)出清晰度提高。?? %%?? %?The?PSF?reconstructed?in?the?second?restoration,?|P2|,?becomes?very?smooth?? %?at?the?edges.?This?implies?that?the?restoration?can?handle?a?PSF?of?a?? %?smaller?size.?The?corresponding?image,?|J2|,?shows?some?deblurring?but?it?? %?is?strongly?corrupted?by?the?ringing.?? ?%第二次復(fù)原的PSF，|P2|，邊緣變得非常平滑。這意味著復(fù)原可以處理一個更細(xì)致的?? %PSF。相應(yīng)的圖片|J2|，顯得清晰了，但被一些“振鈴”強烈破壞。?? %%?? %?Finally,?the?PSF?reconstructed?in?the?third?restoration,?|P3|,?is?somewhat?? %?intermediate?between?|P1|?and?|P2|.?The?array,?|P3|,?resembles?the?true?PSF?? %?very?well.?The?corresponding?image,?|J3|,?shows?significant?improvement;?? %?however?it?is?still?corrupted?by?the?ringing.?? ?%最后，第三次復(fù)原的PSF，|P3|，介于|P1|和|P2|之間。該陣列|P3|，非常接近真?? %正的PSF。相應(yīng)的圖片，|J3|，顯示了顯著改善，但它仍然被一些“振鈴”破壞。?? ?? ?? %%?Step?5:?Improving?the?Restoration?? %%第五步：改善圖像復(fù)原?? %?The?ringing?in?the?restored?image,?|J3|,?occurs?along?the?areas?of?sharp?? %?intensity?contrast?in?the?image?and?along?the?image?borders.?This?example?? %?shows?how?to?reduce?the?ringing?effect?by?specifying?a?weighting?? %?function.?The?algorithm?weights?each?pixel?according?to?the?|WEIGHT|?array?? %?while?restoring?the?image?and?the?PSF.?In?our?example,?we?start?by?? %?finding?the?"sharp"?pixels?using?the?edge?function.?By?trial?and?error,?? %?we?determine?that?a?desirable?threshold?level?is?0.3.?? %在復(fù)原圖像|J3|內(nèi)部灰度對比鮮明的地方和圖像邊界都出現(xiàn)了“振鈴”。這個例子說明了如何?? %通過定義一個加權(quán)函數(shù)來減少圖像中的“振鈴”。該算法是在對圖像和PSF進行復(fù)原時，對每個?? %像元根據(jù)|WEIGHT|數(shù)組進行加權(quán)計算。在我們的例子，我們從用邊緣函數(shù)查找“鮮明”像元?? %開始。通過反復(fù)試驗，我們確定理想的閾值為0.3。?? ?? %WEIGHT?=?edge(I,'sobel',.3);???? ?WEIGHT?=?edge(Blurred,'sobel',.3);??? %%?? %?To?widen?the?area,?we?use?|imdilate|?and?pass?in?a?structuring?element,?|se|.?? %為了拓寬領(lǐng)域，我們使用|imdilate|并傳遞一個結(jié)構(gòu)元素|se|。?? se?=?strel('disk',2);?? WEIGHT?=?1-double(imdilate(WEIGHT,se));???? ??? %%?? %?The?pixels?close?to?the?borders?are?also?assigned?the?value?0.?? %在邊界附近像素的值也被分配為0。?? WEIGHT([1:3?end-[0:2]],:)?=?0;?? WEIGHT(:,[1:3?end-[0:2]])?=?0;?? figure;imshow(WEIGHT);title('Weight?array');???? ?? ????? ?? ?%%?? %?The?image?is?restored?by?calling?deconvblind?with?the?|WEIGHT|?array?and?an?? %?increased?number?of?iterations?(30).?Almost?all?the?ringing?is?suppressed.?? %該圖像通過|WEIGHT|數(shù)組和增加重復(fù)次數(shù)（30）調(diào)用deconvblind函數(shù)來復(fù)原。幾乎所?? %有的“振鈴”被抑制。?? [J?P]?=?deconvblind(Blurred,INITPSF,30,[],WEIGHT);?? figure;imshow(J);title('Deblurred?Image');???? ????? ?? ??? %%?Step?6:?Using?Additional?Constraints?on?the?PSF?Restoration?? %第六步：使用附加約束對PSF復(fù)原?? %?The?example?shows?how?you?can?specify?additional?constraints?on?the?PSF.?? %這個例子說明了如何在PSF上指定額外的限制。?? %?The?function,?|FUN|,?below?returns?a?modified?PSF?array?which?deconvblind?? %?uses?for?the?next?iteration.??? %函數(shù)|FUN|返還一個修改了的PSF數(shù)組，用作deconvblind函數(shù)的下一次重復(fù)。?? %?In?this?example,?|FUN|?modifies?the?PSF?by?cropping?it?by?|P1|?and?|P2|?number?? %?of?pixels?in?each?dimension,?and?then?padding?the?array?back?to?its?? %?original?size?with?zeros.?This?operation?does?not?change?the?values?in?? %?the?center?of?the?PSF,?but?effectively?reduces?the?PSF?size?by?|2*P1|?and?? %?|2*P2|?pixels.??? %在這個例子中，通過對PSF數(shù)組各維數(shù)剪切|P1|和|P2|個值實現(xiàn)對PSF的修改，對數(shù)組填充?? %回零。此操作不會改變在PSF中心的值，而且有效地在各維減少了|2*P1|和|?2*P2|元?? %素。?? ??? P1?=?2;?? P2?=?2;?? FUN?=?@(PSF)?padarray(PSF(P1+1:end-P1,P2+1:end-P2),[P1?P2]);???? ??? %%?? %?The?anonymous?function,?|FUN|,?is?passed?into?|deconvblind|?last.?? %該匿名函數(shù)|FUN|，最后傳遞給|?deconvblind?|。?? %%?? %?In?this?example,?the?size?of?the?initial?PSF,?|OVERPSF|,?is?4?pixels?larger?? %?than?the?true?PSF.?Setting?P1=2?and?P2=2?as?parameters?in?|FUN|?? %?effectively?makes?the?valuable?space?in?|OVERPSF|?the?same?size?as?the?true?? %?PSF.?Therefore,?the?outcome,?|JF|?and?|PF|,?is?similar?to?the?result?of?? %?deconvolution?with?the?right?sized?PSF?and?no?|FUN|?call,?|J|?and?|P|,?from?? %?step?4.?? %在這個例子中，初始PSF，|OVERPSF|，每維比真正的PSF多4個像素，。設(shè)置P1=2和P2=2作?? %為|FUN|的參數(shù)，可有效地使|OVERPSF|與真正的PSF的大小相同。因此，得到的結(jié)果|JF|?? %和|PF|，與第四步不使用|FUN|而僅用正確尺寸PSF盲反卷積得到的結(jié)果|J|和|P|類似。?? [JF?PF]?=?deconvblind(Blurred,OVERPSF,30,[],WEIGHT,FUN);?? figure;imshow(JF);title('Deblurred?Image');???? ?? ????? ?? ??? %%?? %?If?we?had?used?the?oversized?initial?PSF,?|OVERPSF|,?without?the?? %?constraining?function,?|FUN|,?the?resulting?image?would?be?similar?to?the?? %?unsatisfactory?result,?|J2|,?achieved?in?Step?3.?? %?? %?Note,?that?any?unspecified?parameters?before?|FUN|?can?be?omitted,?such?as?? %?|DAMPAR|?and?|READOUT|?in?this?example,?without?requiring?a?place?holder,?? %?([]).?? ?%如果我們使用了沒有約束的函數(shù)|FUN|的較大的初始PSF，|?OVERPSF?|，所得圖像將類?? %似第3步得到的效果并不理想的|J2|。??? %注意，任何在|FUN|之前未指定參數(shù)都可以省略，如|DAMPAR|和|READOUT|在這個例子中，而不需要指示他們的位置，([])。?? ??? displayEndOfDemoMessage(mfilename) ?
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