前言
之前写过一篇维纳滤波在图像复原中的作用,讲述了图像退化模型以及维纳滤波的作用。维纳滤波使用的前提是知道信号和噪声的功率谱,但在实际应用中较难得到,只能根据先验知识进行估计。
本文介绍盲去卷积复原算法,并在MATLAB中进行实验,和维纳滤波的复原效果进行一个对比。盲去卷积的方法有多种,本文主要介绍由fish提出的基于露西-理查德森(Richardson-Lucy)的盲去卷积算法。
盲去卷积原理
露西-理查德森算法属于图像复原中的非线性算法,与维纳滤波这种较为直接的算法不同,该算法使用非线性迭代技术,在计算量、性能方面都有了一定提升。
露西-理查德森算法是由贝叶斯公式推导而来,因为使用了条件概率(即算法考虑了信号的固有波动,因此具有复原噪声图像的能力)。贝叶斯公式如下:
结合图像退化/复原模型,可以得到迭代函数:
其中 fi 就是第i轮迭代复原图像,对应贝叶斯公式中的p(x),g是退化函数,对应贝叶斯公式的p(y|x),c为退化图像(c(y)dy意为在退化图像上积分),如果满足等晕条件,即图像各区域的模糊函数相同,则迭代公式可化简如下:
这就是路西-理查德森迭代公式,其中c是退化图像,g是退化函数,f是第k轮复原图像。如果系统的退化函数PSF(g(x))已知,只要有一个初始估计f就可以进行迭代求解了。在开始迭代后,由于算法的形式,估计值会与真实值的差距迅速减小,从而后续迭代过程f的更新速度会逐渐变慢,直至收敛。算法的另一优点就是初始值f>0,后续迭代值均会保持非负性,并且能量不会发散。
盲去卷积需要两步进行复原,原因是我们既不知道原始图像f,也不知道退化函数g。求解过程示意图如下:
即在第k轮迭代,我们假a设原始图像已知,即k-1轮得到的fk-1,再通过R-L公式求解gk,随后,再用gk求解fk,反复迭代,最后求得最终f和g。因此,在求解最初,我们需要同时假设一个复原图像f0和一个退化函数g0。迭代公式如下:
此外,有人采用这种盲去卷积方法进行了相关实验,下图为实验原图:
下图(a)左、右分别为附加标准差1.5%、10%泊松噪声的退化图像,图(b)左、右分别为1.5%的复原图像和PSF,图(c)为10%对应结果
盲去卷积MATLAB实验
将一幅原始图像,进行模糊处理(模拟大气湍流),分别使用维纳滤波(由于没加噪声,就是逆滤波)和盲去卷积进行复原,复原结果如下:
%% Deblurring Images Using the Blind Deconvolution Algorithm %%盲反卷积算法复原图像 % 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 %在不知道图像失真信息(模糊和噪声)信息情况下,盲反卷积算法可以有效地加以利用。该算法 %对图像和点扩展函数(PSF)的同时进行复原。每次迭代都使用加速收敛Richardson-Lucy %算法。额外的光学系统(如照相机)的特性可作为输入参数,帮助改善图像复原质量。可以通 %过用户指定的函数对PSF进行限制 % Copyright 2004-2005 The MathWorks, Inc. %% Step 1: Read Image %%第一步:读取图像 % The example reads in an intensity image. The |deconvblind| function can % handle arrays of any dimension. %该示例读取一个灰度图像。| deconvblind |函数可以处理任何维数组。 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:模拟一个模糊 % 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|. %模拟一个现实中存在的模糊图像(例如,由于相机抖动或对焦不足)。这个例子通过对真实 %图像进行高斯滤波器模拟图像模糊(使用|imfilter|)。高斯滤波器是一个点扩展函数, %|PSF|。 PSF=fspecial('gaussian',7,10); Blurred=imfilter(I,PSF,'symmetric','conv'); %对图像I进行滤波处理; figure;imshow(Blurred);title('Blurred Image'); %% Step 3: Restore the Blurred Image Using PSFs of Various Sizes %%第三步:使用不同的点扩展函数复原模糊图像 % 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. %为了说明知道真实PSF的大小的重要性,这个例子执行三个修复。PSF函数重建每次都是从统一 %的全一数组开始。 %% % 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. %第一次复原,|J1|和|P1|,使用一个较小数组,| UNDERPSF |,来对PSF的初步猜测。该 %UNDERPSF数组每维比真实PSF少4个元素。 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. %第二次复原,|J2|和|P2|,使用一个元素全为1的数组,| OVERPSF|,初始PSF每维比真 %实PSF多4个元素。 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. %第三次复原,|J3|和|P3|,使用一个全为一的数组| INITPSF |作为初次PSF,每维与真正 %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:分析复原函数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). %所有这三个复原也产生PSF。以下图片显示对PSF重建分析的如何可能有助于猜测最初PSF的大 %小。在真正的PSF中,高斯滤波器的最高值在中心(白),到边界消失(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,. %第一次复原的PSF,|P1|,显然不适合大小的限制。它在边界有一个强烈的变化信号。 %相应的图片|J1|,与模糊图像|Blurred|比没有表现出清晰度提高。 %% % 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. %第二次复原的PSF,|P2|,边缘变得非常平滑。这意味着复原可以处理一个更细致的 %PSF。相应的图片|J2|,显得清晰了,但被一些“振铃”强烈破坏。 %% % 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,第三次复原的PSF,|P3|,介于|P1|和|P2|之间。该阵列|P3|,非常接近真 %正的PSF。相应的图片,|J3|,显示了显着改善,但它仍然被一些“振铃”破坏。 %% Step 5: Improving the Restoration %%第五步:改善图像复原 % 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. %在复原图像|J3|内部灰度对比鲜明的地方和图像边界都出现了“振铃”。这个例子说明了如何 %通过定义一个加权函数来减少图像中的“振铃”。该算法是在对图像和PSF进行复原时,对每个 %像元根据|WEIGHT|数组进行加权计算。在我们的例子,我们从用边缘函数查找“鲜明”像元 %开始。通过反复试验,我们确定理想的阈值为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|. %为了拓宽领域,我们使用|imdilate|并传递一个结构元素|se|。 se = strel('disk',2); WEIGHT = 1-double(imdilate(WEIGHT,se)); %% % The pixels close to the borders are also assigned the value 0. %在边界附近像素的值也被分配为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. %该图像通过|WEIGHT|数组和增加重复次数(30)调用deconvblind函数来复原。几乎所 %有的“振铃”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:使用附加约束对PSF复原 % The example shows how you can specify additional constraints on the PSF. %这个例子说明了如何在PSF上指定额外的限制。 % The function, |FUN|, below returns a modified PSF array which deconvblind % uses for the next iteration. %函数|FUN|返还一个修改了的PSF数组,用作deconvblind函数的下一次重复。 % 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. %在这个例子中,通过对PSF数组各维数剪切|P1|和|P2|个值实现对PSF的修改,对数组填充 %回零。此操作不会改变在PSF中心的值,而且有效地在各维减少了|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. %该匿名函数|FUN|,最后传递给| 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. %在这个例子中,初始PSF,|OVERPSF|,每维比真正的PSF多4个像素,。设置P1=2和P2=2作 %为|FUN|的参数,可有效地使|OVERPSF|与真正的PSF的大小相同。因此,得到的结果|JF| %和|PF|,与第四步不使用|FUN|而仅用正确尺寸PSF盲反卷积得到的结果|J|和|P|类似。 [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, % ([]). %如果我们使用了没有约束的函数|FUN|的较大的初始PSF,| OVERPSF |,所得图像将类 %似第3步得到的效果并不理想的|J2|。 %注意,任何在|FUN|之前未指定参数都可以省略,如|DAMPAR|和|READOUT|在这个例子中,而不需要指示他们的位置,([])。 displayEndOfDemoMessage(mfilename)