Non-Iterative Restoration For Weakly Blurred And Strongly .
CVPRCVPR#****#****CVPR 2010 Submission #****. CONFIDENTIAL REVIEW COPY. DO NOT DISTRIBUTE.000054001002055056003004Non-Iterative Restoration for Weakly Blurred and Strongly Noisy Images059005006007060061Anonymous CVPR submission062008Paper ID 27028063064065066Abstract013014015067068069We present a non-iterative algorithm for blind restoration of an image which has been corrupted by my mild blur,and strong noise. The most successful deblurring algorithms to date address the problem in a setting where thecorrupting noise is assumed so small as to be essentiallynegligible. Unfortunately, in practical applications, muchstronger noise is often present, which renders most blinddeblurring algorithms useless. Furthermore, it is often thecase that most images we wish to enhance in practice sufferfrom mild, rather than severe blur. This motivates our study,where we develop a locally adaptive and non-parametricmethod for high fidelity enhancement of such distorted images. We illustrate that our algorithm can produce state ofthe art results under practical imaging conditions.070071072073074075(a) Original image0790800810820830300311. Introduction032Blur and noise are the two common problems that exist in digital imaging. In recent years, several blinddeconvolution algorithms [3, 11, 5] have been proposedto restore images degraded by blur (usually motion blur).These algorithms are generally designed under the assumption that the point spread function (PSF) of blur is spatiallyinvariant, and that noise is very weak or virtually absent.Currently the most practical way for avoiding motion bluris perhaps still by shortening the exposure time of the camera. However, this would inevitably reduce the number ofincoming photons, and result in weakly blurred but highlynoisy images [13, 4]. At the same time, in many cases limited accuracy of auto-focus systems and low light conditionmay also produce images with weak (defocus) blur and relatively strong noise (including luminance and chrominancenoise). Such situations are very common in consumer imaging applications.Unfortunately, even when dealing with weakly blurredimages, the presence of noise can be a significant problemfor the state of the art deblurring algorithms. Even moderate amounts of noise can strongly influence the PSF estimation process in blind deconvolution. As a case in 9050051052053076077078029033034057058084085(b) Shan et al. [11], SIGGRAPH 2008086087088089090091092093094095096(c) The proposed methodFigure 1. Results of deblurring a weakly blurred and strongly noisyimage.consider the method of Fergus et al. [3], whose impressiveperformance relies on a model of the expected sharp imagegradients. The inherent ambiguity between the high frequency components of noise and those of real structure willconfuse the algorithm, rendering the overall performancefar from that demonstrated in the original paper. A similar situation happens in methods that focus on small blur1097098099100101102103104105106107
CVPRCVPR#****#****CVPR 2010 Submission #****. CONFIDENTIAL REVIEW COPY. DO NOT 541551561571581591601611622. Problem analysiskernels. Joshi et al. [5] estimate the PSF by first predicting sharp edges from the blurred image. Next, under theassumption that the underlying edges are ideal step ones, aBayesian non-blind deconvolution technique is applied tothe observed and predicted image pair. Difference of Gaussians (DoG) edge detector is used to find blurred edges,but the locations of zero-crossing pixels can easily be influenced by strong noise. Besides, since the intensities of thetwo sides of edges are predicted pixel by pixel accordingto the maximum and minimum values along each edge profile, the intensity variation caused by noise will inevitablyproduce ”strip artifacts” on the sharp edges, and thereforereduce the estimation accuracy.We model the process that degrades an ideal sharp imagef (denoted in a lexicographically ordered vector) into theobserved blurry and noisy data g as:163164165166167g Hf n(1)where H represents a blurring matrix, and n denotes additive noise. Assuming that H is known (for now), the goalof non-blind deblurring is to reconstruct an image f̂ withits high-frequency components (edges, texture, etc.) fairlyrestored, and meanwhile being as clean as possible. Let usfirst take a look at how a classical MAP-based deconvolution method attempts to solve this problem.In general, the MAP framework seeks an estimate f̂ byminimizing the following type of cost function:In general, even if the PSF can be accurately estimated,current non-blind deconvolution approaches still do nothandle both the deblurring and denoising tasks simultaneously very well. The Richardson-Lucy (RL) algorithm[10, 8] is favored by many deblurring approaches becauseof its low computational cost. However, RL is also a sourcefor ringing artifacts since it is difficult to incorporate withinit sophisticated prior constraints for latent images [7]. Amore popular class of algorithms belongs to maximum aposteriori or MAP based estimation, which usually includesa data fitting term and a regularization term in the cost function [11, 6, 1]. As we will illustrate in Section. 2 the data fitting term simulates the imaging model, and basically sharpens the input image during the optimization process, whilethe regularization term plays a role as a denoising filter. Although the combination of these two terms can result in agood balance between high frequency content restorationand noise suppression, the noise effect may still remain inthe output data, and corrupt the smoothness of latent objectstructure (see Fig. 1(b)).f̂ arg min k Hf g k2 λR(Gx f , Gy f )2170171172173174175176177178(2)179180The first term on the right-hand side is called the likelihood, or data-fitting, term. R(Gx f , Gy f ) is the regularization term designed to stabilize noise effects, where Gxand Gy are matrix forms of horizontal and vertical gradient operators. The regularization parameter λ can be tunedto adjust the contributions of the data fitting term and theregularization term.Many studies have been focused on developing a smartregularization function so that it can filter noise meanwhilepreserving high frequency image content, such as edges[11, 6, 1, 3]. These MAP-based methods, regardless of theirlevel of sophistication, are generally reliant on global priorassumptions (such as sparsity) on the statistics of the latent image gradients. This, however, does not guarantee thatthese algorithms will capture the local characteristics of theimage well, particularly in the presence of significant noise.To illustrate this problem, a simulated non-blind deblurringexample is given in Fig. 2, where an ideal step edge isconvolved with a known PSF, and the results are estimatedby a MAP-based deconvolution method called bilateral total variation (BTV), which is very good at preserving sharpedges [2]. Both noise-free and Gaussian noise (whose standard deviation σn 5) deconvolution cases were studied asreported in Fig. 2. It can be observed from the results thatfor the noiseless case, the latent edge was perfectly recovered. Meanwhile, for the noisy image, though the BTV termsuccessfully removed noise in most areas, protected highvalue gradients in the edge region, thus yielding a sparsegradient distribution, the restored edge still looks unnatural.Similar phenomena can also be observed in the real example results of another leading method [11] illustrated in Fig.1(b).To further understand the behavior of MAP-based deconvolution, it is instructive to analyze the optimization problem in (2), which is often solved using a gradient-following181182fIn this paper, we analyze the shortcomings of currentMAP-based deconvolution algorithms, and show that theintrinsic contradiction between the data fitting term andthe regularization term limits their performance in deconvolution methods. To alleviate this problem, we proposea new approach called geometric locally adaptive deblurring (GLAD). The key idea behind the proposed approach isthat instead of approaching deconvolution with global image priors (as done in MAP approaches), we consider deblurring according to the local image structure. In this way,we illustrate that deblurring and denoising can be efficientlydone together. Based on this idea, a non-iterative restorationalgorithm is presented in this paper for weakly blurred andstrongly noisy images. This algorithm is derived from local adaptive regression [12], and does not require any PSFestimation. The proposed method is computationally muchcheaper than iterative MAP-based deblurring approaches,and can effectively remove both blur and noise artifacts (including chrominance artifacts, as in Fig. 12213214215
CVPRCVPR#****#****CVPR 2010 Submission #****. CONFIDENTIAL REVIEW COPY. DO NOT 59260261262263264265266267268269(d)(e)Figure 2. MAP deconvolution using BTV regularization [2]. (a)original edge; (b) noise-free blurred edge; (c) reconstructed resultof (b); (d) noisy blurred edge; (e) reconstructed result of (d)(5)284285TThe noise n, though smoothed by H , would be added tof (k) in each iteration, and be inevitably amplified by thehigh-pass sharpening process without regularization.A general form of regularization is given by:R(Gx f , Gy f ) k Gx f kνν k Gy f kνν(4)Let us first put the regularization term aside and assume theadditive noise n to be negligible so that g Hf . The blurmatrix H plays a role as a low-pass filter, and therefore sodoes HT H. The term f (k) HT Hf (k) , which extracts lowfrequency components from image f (k) , can thus be viewedas high-pass filtering. Although some components of thisterm will be counteracted by f (k) HT g f (k) HT Hf ,there will still be some high-frequency components left,scaled by the factor µ, and added back to f (k) , as long asf (k) is more blurry than the underlying ideal image1 f . Thisoperation can thus be viewed as a high-pass sharpening filter. Ideally, through the iterations this sharpening processcontinues until f (k) becomes equally sharp as f . Once thishappens, the contribution of f (k) HT Hf (k) would be moreor less exactly counteracted by the term f (k) HT g, andthe sharpening stops.The above view toward the optimization process can helpinterpret the cause of ringing artifacts that frequently appear in deblurred images. It used to be commonly believedthat these artifacts are due to the Gibbs phenomenon [13].However, according to the experiments in [11], Yuan et al.1 Usually270 µHT n µλ R(Gx f , Gy f )algorithm such as steepest descent. Such iterative solutionscan be stopped after some number of iterations to effecta balance between the level of deblurring and denoisingachieved in the resulting solution. The iteration process ofthe canonical gradient descent algorithm can be denoted as:hif (k 1) f (k) µ HT (Hf (k) g) λ R(Gx f , Gy f )(3)where µ is the step size in the direction of gradient, and denotes the gradient of the regularization term. To furtheranalyze how the estimate evolves during each iteration, werewrite the above equation as follows:hif (k 1) f (k) µ (f (k) HT Hf (k) ) (f (k) HT g) µλ R(Gx f , Gy f )suggest that ringing is caused rather by erroneous estimation of the PSF instead. Here, we can provide a detailedexpression for this phenomenon. Namely, if the PSF is incorrectly estimated, the term f (k) HT g would fail to stopthe sharpening process properly during the iteration andover-sharpened components would be produced, like overshoots around edges. These overshoots provide even moreunwanted high-frequency components and finally result inringing.Now let us consider the realistic scenario where g isnoisy. We thus have:hif (k 1) f (k) µ (f (k) HT Hf (k) ) (f (k) HT Hf 290When ν 2 this is Tikhonov regularization based on Gaussian prior, and if 0 ν 1 it becomes a heavy-tailed function representing a sparse prior on the expected image gradients [6]. Although it is difficult to denote R(Gx f , Gy f )in a closed form for an arbitrary ν, it can be concludedthat during the optimization iteration, a sparse prior tendsto concentrate on smoothing pixels with median or smallgradient values, leaving high-value gradients preserved. Inother words, it performs like a smoothing filter assuming aglobal model for the locally treated pixels.Through the above analysis we can see that in MAPbased algorithms, deblurring and denoising are consideredsomewhat separately and in apparent contradiction: duringeach optimization iteration, the data fitting term sharpensthe image, while the regularization term smooths the image to stabilize the noise effect. The intrinsic contradictionbetween these two terms results in a limited deblurring performance, which leaves much to be desired.2913. Proposed algorithm310In this section, we propose an approach called geometriclocally adaptive deblurring (GLAD). This approach takeslocal image structure into consideration, so that it can efficiently combine deblurring and denoising together. Then,the steering kernel (SK) regression technique for image reconstruction [12] is briefly introduced, which is able to capture local image structure even with presence of mild blurand strong noise. Based on the SK, an approach of constructing GLAD kernels for weakly blurred gray-scale image restoration is developed. Finally we extend this approach to color images with a strategy for removing chrominance artifacts.the iteration is initialized by setting f (0) 3
CVPRCVPR#****#****CVPR 2010 Submission #****. CONFIDENTIAL REVIEW COPY. DO NOT DISTRIBUTE.324[xi , yi ], its SK is mathematically represented as:p det(Cl )(xl xi )T Cl (xl xi )K(xl xi ) exph2 2h2(8)where xl denotes a given location inside the SK windowcentered at xi , h is a global smoothing parameter, and Cl isa covariance matrix estimated from a collection of gradientswithin an analysis window wl around xl .Examples of the SK in a weakly blurred image (addedwith white Gaussian noise, whose σn 10) are given inFig. 4 (a)-(c), where we can see that the estimated kernelsreliably captured local image structures. Since the definition (8) of the SK seems like a simple Gaussian function, itmay appear mysterious that the resulting kernel values donot simply have elliptical contours. This is a subtle, butimportant point, which stems from the fact that a separatecovariance matrix Cl is estimated and used at each pixellocation. This yields a far richer set of shapes for the resulting kernel weights, as seen in Fig. 4, than would otherwisebe expected of the humble Gaussian kernel. In the flat region, the SK is wide and basically isotropic, indicating thatthere is no strong directional structure inside this area. Inthe edge region, the shape of SK depicts the edge outline,and the kernel values basically indicate the pixel intensitysimilarity with respect to the pixel of interest. In the regionthat contains small scale image details, the correspondingSK also shrinks to a small point. Besides, it can be observedthat the SK estimation is basically robust to high levels ofnoise.Let us describe more precisely how the kernel values in(8) are computed. The local gradient matrix for the windowwl centered at xl is defined as: . (9)D Gx (xm ) Gy (xm ) , xm gure 3. Spatially adaptive deblurring example. (a) blurredand noisy edge; (b) filtered edge; (c)-(d) Locally adaptive denoise/deblur filters.3403413423433443.1. MotivationThe fundamental problem of MAP methods is that theyseek to find a global filter S that can do deblurring and denoising 73374375376377f̂ SHf Sn(7)where the first term attempts to invert H, and the secondterm ends up amplifying the noise.Another possible way to go is to design a filter adaptiveto the local image statistics. In this alternative view, it is notnecessary to globally invert the blur operator H. Instead, itsaction should be locally controlled to respond to the signalcharacteristics implied in the measured image. That is tosay, for instance, in places where the effect of blur is not felt(such as flat areas of the image) our filter should concentratemore on noise reduction. Meanwhile, in edge-containingregions, our filter should attempt to deblur the image onlyin the edge profile direction, and still effect smoothing alongthe edge direction. (See example shown in Fig. 3)We term this approach geometric locally adaptive deblurring (GLAD), to distinguish it from more standarddeconvolution-based deblurring approaches. In order to implement GLAD, a technique that can capture the local image structure even in the presence of blur and noise is required first.where [Gx (xm ), Gy (xm )] denote the image gradient at[xm , ym ]. The dominant direction v1 and its perpendiculardirection v2 within the region wl can be estimated by computing the (compact) singular value decomposition (SVD)of D: Ts1 0 Tv1 v2D USV U(10)0 s23.2. Steering kernel constructionThe key idea behind SK is to robustly obtain the localstructure of images by analyzing estimated gradients, anduse this structure information to determine the shape andsize of a canonical kernel [12]. By now, SK has been successfully utilized to address the image denoising problem[12].Assuming a pixel of interest is located at position xi Here the singular values s1 s2 0 represent the energyin the directions v1 and v2 respectively. The matrix Clcan then be stably estimated through the following formula[12]:2XCl γ̺q vq vqT(11)q 27428429430431
CVPRCVPR#****#****CVPR 2010 Submission #****. CONFIDENTIAL REVIEW COPY. DO NOT DISTRIBUTE.432is then computed as a local regression with the kernel S asfollows:Px wl S(x xi )g(x)ˆf (xi ) P(14)x wl S(x xi )433434435436where g is the measured blurry and noisy image. Fig.4 (d)-(f) illustrate examples of GLAD kernels. Considerthe edge pixel for instance. The negative values appearingalong the two sides of the edge outline indicate that the deblurring effect happens in the direction perpendicular to theedge orientation. Meanwhile, we have most positive values along the outline of the edge, which means the kernelsmooths (denoises) the edge simultaneously. On the otherhand, in the flat region, though there exist few small negative values across the region, the kernel can still be viewedlargely as a smoothing filter.One problem that frequently arises in practice, in partdue to limited depth of field of cameras, is that the degreeof blur varies in space. As a result, with a globally designed deblurring approach, some regions that happen tobe already in-focus may be overdeblurred. In such cases,as we explained in Section 2, an improper estimate of thelocal PSF would fail to stop the ”sharpening” process during the iteration, resulting in overshoots or ringing artifacts.Without additional finesse, similar problems may plague theproposed method. If the local region is already in focus,a fixed, and unnecessarily high value of κ would produceovershoots along the edge pixels. To alleviate this prob
Non-Iterative Restoration for Weakly Blurred and Strongly Noisy Images Anonymous CVPR submission Paper ID **** Abstract We present a non-iterative algorithm for blind restora-tion of an image which has been corruptedby my mild blur, and strong noise. The most successful deblurring algo-rithms to date address the problem in a setting where the
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Image restoration techniques are used to make the corrupted image as similar as that of the original image. Figure.3 shows classification of restoration techniques. Basically, restoration techniques are classified into blind restoration techniques and non-blind restoration techniques [15]. Non-blind restoration techniques
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Image restoration techniques are used to make the degraded or corrupted image as similar as that of the original image. Fig. 1 shows classification of restoration techniques. Basically, restoration techniques are classified into blind restoration techniques and non-blind restoration technique. Non-blind restoration techniques are further
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