cvpr cvpr2013 cvpr2013-393 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Maria Zontak, Inbar Mosseri, Michal Irani
Abstract: Recurrence of small clean image patches across different scales of a natural image has been successfully used for solving ill-posed problems in clean images (e.g., superresolution from a single image). In this paper we show how this multi-scale property can be extended to solve ill-posed problems under noisy conditions, such as image denoising. While clean patches are obscured by severe noise in the original scale of a noisy image, noise levels drop dramatically at coarser image scales. This allows for the unknown hidden clean patches to “naturally emerge ” in some coarser scale of the noisy image. We further show that patch recurrence across scales is strengthened when using directional pyramids (that blur and subsample only in one direction). Our statistical experiments show that for almost any noisy image patch (more than 99%), there exists a “good” clean version of itself at the same relative image coordinates in some coarser scale of the image.This is a strong phenomenon of noise-contaminated natural images, which can serve as a strong prior for separating the signal from the noise. Finally, incorporating this multi-scale prior into a simple denoising algorithm yields state-of-the-art denois- ing results.
Reference: text
sentIndex sentText sentNum sentScore
1 of Computer Science and Applied Mathematics The Weizmann Institute of Science, ISRAEL Abstract Recurrence of small clean image patches across different scales of a natural image has been successfully used for solving ill-posed problems in clean images (e. [sent-2, score-1.241]
2 In this paper we show how this multi-scale property can be extended to solve ill-posed problems under noisy conditions, such as image denoising. [sent-5, score-0.274]
3 While clean patches are obscured by severe noise in the original scale of a noisy image, noise levels drop dramatically at coarser image scales. [sent-6, score-1.619]
4 This allows for the unknown hidden clean patches to “naturally emerge ” in some coarser scale of the noisy image. [sent-7, score-1.298]
5 We further show that patch recurrence across scales is strengthened when using directional pyramids (that blur and subsample only in one direction). [sent-8, score-1.262]
6 Our statistical experiments show that for almost any noisy image patch (more than 99%), there exists a “good” clean version of itself at the same relative image coordinates in some coarser scale of the image. [sent-9, score-1.315]
7 This is a strong phenomenon of noise-contaminated natural images, which can serve as a strong prior for separating the signal from the noise. [sent-10, score-0.236]
8 Introduction The goal of natural image denoising is to recover a clean image I from its noise contaminated version IN. [sent-13, score-0.709]
9 A leap improvement in denoising has been obtained by [3], who introduced a strong natural-image prior, which exploits the recurrence of small image patches internally, within a natural image. [sent-18, score-0.793]
10 External statistics of natural image patches learned from large collections of clean images, has also been used as priors for denoising. [sent-20, score-0.721]
11 While the external methods benefit from a large space of clean patches, the internal methods benefit from an imagespecific space of patches, which was shown by [16] to be very powerful. [sent-22, score-0.524]
12 In contrast a big challenge for internal methods is to aggregate enough noisy patches which share the same underlying signal. [sent-24, score-0.572]
13 This is particularly problematic when large patches are used (often necessary in the presence of severe noise [4, 9, 17]). [sent-25, score-0.455]
14 In this paper we propose × a new internal search space, which on one hand is imagespecific, yet, contains clean patches – the space of coarse patches of the noisy image. [sent-26, score-1.3]
15 We present a strong multi-scale prior for solving illposed problems under severe noise, which is based on the recurrence of small patches across different scales of a natural image (where a coarser image scale is generated by blurring & subsampling the image). [sent-27, score-1.233]
16 Most patch-based denoising methods perform deniosing by exploiting patch repetitions within the same scale, whereas our prior suggests to use patch repetitions across different scales. [sent-28, score-0.899]
17 It was previously shown [7, 16] that small patches in clean natural images tend to recur across image scales. [sent-30, score-0.84]
18 , 5 5) re-appears “as is” (without any spatial transforma5tio ×n) i)n coarser rvse r“saison iss” ”o (fw tihteh oimuta agney (s speea iFailg. [sent-33, score-0.273]
19 It was further shown by [6] that due to local scale invariance in natural images, patches tend to recur at coarser image scales at similar relative image coordinates as in the fine scale. [sent-37, score-0.92]
20 We show that these natural-image properties can be exploited for “pulling out” cleaner versions of the signal from recurrence” × × ×× coarser scales of the noisy image (see Fig. [sent-38, score-0.755]
21 It is further noted that for some patches (especially edge patches), the cross-scale recurrence property is strengthened when using directional pyramids (that blur and subsample only in one direction). [sent-41, score-1.137]
22 Surprisingly, our experiments indicate that with very 1 1 1 111119999955333 The local scale-invariance property of small patches in clean natural images scale. [sent-42, score-0.755]
23 We display the 5 5 patches at the same relative coordinates in all 4 images (i. [sent-43, score-0.374]
24 , the 5 5 patch centered at (x0, y0) in the fine ( thx0e/ s3a,m mye0/ r3el)a itinv teh ceo coarse tsecsal iens a)l. [sent-45, score-0.367]
25 lT 4he imrea igse strong similarity b peatwtcehe nce corresponding 5 5 patches ssccaallee. [sent-46, score-0.345]
26 s, Waned d tihspel 5a y× t h5e patch 5ce pnattecrheeds a att saccarolesss,-s acnadle t Since the noise drops dramatically with scale, the clean patches naturally “emerge” at coarser pyramid levels of the noisy image hine 5th×e 5cle paant image. [sent-47, score-1.835]
27 Nreodte a tth (axt the patch a int th thee coarse s sccaalele of the noisy image isism maillsaor very stwimeielanr c otor trhesep colnedanin patch. [sent-48, score-0.629]
28 high probability, for every noisy patch, a very good representative of its unknown clean patch resides within a small set of 60 patches (all its 5 5 descendant patches at the same sreelta otifv 6e0 0i pmaatcghe ecso (oarldli intsa5t e×s)5. [sent-49, score-1.814]
29 Finally, incorporating this multi-scale prior into a simple denoising algorithm, yields results better than state-of-theart methods, especially for high noise levels. [sent-51, score-0.272]
30 N faoctte hthatat w tehe u term y“m smualtlil-s 5ca ×le 5 denoising” of [10, 2] refers to a different notion using patches of different – × sizes from various locations within the image. [sent-53, score-0.303]
31 The multiscale property we exploit here is the “fractal-like” scale invariance property of natural images, namely, recurrence of small patches of a fixed size (e. [sent-54, score-0.697]
32 , edge-detection in noisy images, super-resolution under noisy conditions, etc. [sent-61, score-0.48]
33 2 we analyze and quantify the local cross-scale emergence of clean patches in noisy images, and show that this is a very strong phenomenon. [sent-64, score-0.958]
34 We further show that patch recurrence across scales is strengthened when using directional pyramids. [sent-65, score-0.978]
35 3 proposes an approach for finding the “best” coarser patch for any noisy patch in a noisy image, and discusses its limitations. [sent-67, score-1.361]
36 4 proposes a very simple denoising algorithm which exploits this multi-scale prior, providing state-of-the-art denoising results (especially for high noise levels). [sent-69, score-0.403]
37 Statistics of Clean Patches in Noisy Images We demonstrate how the local scale-invariance property of small patches in natural images (Fig. [sent-73, score-0.382]
38 a) can be used for “pulling out” cleaner versions of the signal from coarser scales of the noisy image (Fig. [sent-75, score-0.755]
39 We first extend the × notion of ‘pyramids’ from the standard isotropic pyramid to directional pyramids. [sent-78, score-0.362]
40 We then show that patch recurrence across scale is further strengthened when using directional pyramids. [sent-79, score-0.902]
41 We generate a directional pyramid by blurring and subsampling the image only in one direction. [sent-82, score-0.336]
42 This is different from the commonly used isotropic image pyramid, which preserves the aspect ratio, as well as from the Steerable Pyramid [15], which applies 1D directional filtering, but subsamples the image in both directions. [sent-83, score-0.249]
43 For some patches (especially edge patches), the cross-scale recurrence property is strengthened when using directional pyramids. [sent-84, score-0.853]
44 In particular, very thin edges which tend to disappear at low scales of the isotropic (or steerable) pyramid, are preserved well by our directional pyramids. [sent-85, score-0.402]
45 The ‘hidden’ clean patches are obscured by noise at the original input image scale. [sent-88, score-0.815]
46 However, these (unknown) clean patches recur at coarser image scales, at the same relative image coordinates. [sent-89, score-1.014]
47 Since the noise level drops dramatically at coarser image scales (see Fig. [sent-90, score-0.535]
48 b), the unknown clean patches naturally emerge at some coarser pyramid levels. [sent-92, score-1.154]
49 Note that although the directional pyramids change the aspect ratio of the image, the patches compared across scales remain ofthe same size and shape (square 5 5 paacrtcohsesss)c ianle sallr smcaaliens. [sent-93, score-0.785]
50 (b) The drop in noise variance σ2 as a function of the pyramid scale (empirically evaluated from many randomGaussian-noise images). [sent-97, score-0.31]
51 (c) Examples of noisy patches and their corresponding clean patches ‘emerging at coarse pyramid scales. [sent-98, score-1.384]
52 ’ × Patches are marked in (a) and magnified in (c): The noisy horizontal (red) and vertical (green) edge patches have corresponding clean patches at coarse scales of the respective directionalpyramid (at the same relative coordinates). [sent-99, score-1.421]
53 The noisy uniform patch (cyan) has a clean patch at coarse scales of the Iso-pyramid. [sent-100, score-1.413]
54 ve N a mgeoloyd: (ci)lea Hno w re mpraesneyn t5at ×ive 5 patch of the clean patch in a coarser scale of the noisy image? [sent-103, score-1.536]
55 ) We show that statistically this is indeed a very strong phenomenon of noisy images. [sent-108, score-0.306]
56 In fact, as will be shown, the vast majority of image patches (more than 99%) will benefit significantly from going down in scale, and will have a very good representative patch directly below them, at the same relative image coordinates (i. [sent-109, score-0.799]
57 , on the ‘needle’ of their descendant patches see Fig. [sent-111, score-0.436]
58 Each clean image I (converted to grayscale) was first contaminated by Gaussian noise with zero mean and variance σ2, resulting in a noisy image IN = I N. [sent-115, score-0.772]
59 Each noisy patch pN has a needle in all 3 pyramids (illustrated here only for the Isotropic pyramid). [sent-118, score-0.99]
60 (a) All the patches along the needle of the noisy patch are at the same relative image coordinates. [sent-119, score-1.177]
61 Initially the patches get better (cleaner), but eventually new structures enter the patch. [sent-120, score-0.303]
62 The “best” representative patch on the needle is marked in orange. [sent-121, score-0.625]
63 (b) Zooming in on the descendant patches {psNc} along the needle (sc = 1, . [sent-122, score-0.723]
64 types of pyramids: (i) Isotropic pyramid (blur and subsample1 both in x and in y), (ii) X-pyramid (blur and subsample only in the x direction), and (iii) Y-pyramid (blur and subsample only in the y direction) – see Fig. [sent-126, score-0.249]
65 Namely, if (x, y) are the coor- ×× dinate of the clean 5 5 patch p (and the noisy patch pN), tdhinena feor o ef tacheh cscleaalen sc ×= 5 50 p. [sent-137, score-1.316]
66 9ats we compare p oonislyy apgatacinhs pt the 5 5 patch psNc whose coordinates are: (i) (0. [sent-138, score-0.354]
67 |, ,| as t)h, ew “eb cehsot”s e re thpere osennet wathiviech ho mf tihneclean patch p. [sent-148, score-0.337]
68 4 visually displays the resulting image when each patch in the noisy image is replaced by its single “best” descendant 5 5 patch psNc along its needle. [sent-150, score-1.024]
69 ) It is evident that the resulting image is significantly cleaner than the noisy input. [sent-152, score-0.336]
70 In fact, it is significantly cleaner (has significantly larger PSNR2) than what can be achieved by today’s state-of-the art denoising algorithms (see PSNR comparisons in Fig. [sent-153, score-0.235]
71 Note that this is not a denoising algorithm, since we used the original clean image to guide the selection of patches. [sent-155, score-0.512]
72 However, those patches were selected from the pyramid of the noisy image; no additional processing was (Overlaps 1using Matlab “imresize” with a bicubic kernel 2PSNR= 20 log10 (255/σ), stands for “Peak Signal to Noise Ratio”. [sent-156, score-0.656]
73 Noisy images (man-made & natural scenes), and their corresponding ‘Oracle ’ images – generated by replacing each noisy patch with its ‘best’ descendant patch along its limited ’multi-scale needle ’ in the noisy image pyramid (see text). [sent-159, score-1.688]
74 done to improve the resulting image; only a single patch was selected, and in a very limited search space – only along the multi-scale ‘needle’ descending from the noisy patch. [sent-160, score-0.576]
75 A cleaner patch exists at a coarser pyramid level of the noisy image, somewhere directly ‘underneath ’ the noisy patch, at the same relative image coordinates. [sent-163, score-1.287]
76 Note that the Oracle selection is restricted to an extremely limited search range, namely: For each noisy patch, select one of 60 ‘candidate’ patches (20 patches on its needle 3 pyramids). [sent-165, score-1.112]
77 l space so af tainl y yp soesasribchle5 5 patches (25625, assuming 256 graylevels), or relative t5o × ×the 5 huge space o6f all natural clean image patches, or even relative to the space of all patches within the noisy image (hundreds of thousand of patches, all of which are noisy). [sent-167, score-1.328]
78 As can be seen, for most noisy patches, the good representative patches tend to be in relatively low scales. [sent-175, score-0.622]
79 This behavior is quite intuitive, as there is a tradeoff between two factors: On one hand, we introduce more inaccuracies in the clean signal as we go down the scale (increasing ‘bias’). [sent-177, score-0.487]
80 On the other hand, the noise levels drop dramatically as we go down the scales (decreasing variance). [sent-178, score-0.337]
81 The total patch error is the sum of these two factors: the signal error plus the noise (bias/variance tradeoff). [sent-179, score-0.465]
82 Extremely few patches prefer to stay at the top scale (e. [sent-188, score-0.428]
83 The majority of patches prefer the Isotropic pyramid (65% for σ = 25). [sent-192, score-0.492]
84 These are mostly smooth patches (which are a large majority of the image). [sent-193, score-0.325]
85 The remaining patches (mostly edges and smooth patches near the edges) prefer the directional pyramids (19% prefer the x-pyramid and 15% prefer the y-pyramid for σ = 25). [sent-194, score-1.103]
86 This is due to using directional pyramids only in the X and Y directions. [sent-199, score-0.335]
87 Adding directional pyramids at various other angles is bound to improve the resulting image, both visually 1 1 1 1 1 19 9 98 6 6 (a) ‘Best scale’ vs. [sent-200, score-0.335]
88 4 we explain how the directional pyramids can be extended to arbitrary angles, to provide better representatives for arbitrary patch orientations. [sent-206, score-0.687]
89 Nev- ertheless, most oriented patches still manage to find reasonable representatives in coarse levels ofthe isotropic pyramid (e. [sent-207, score-0.637]
90 Hence, the above 3 pyramids alone (with their very limited search space) already provide a good representation of the clean image, with very low errors. [sent-211, score-0.542]
91 It is important to note that the “best” Oracle patch (the “best” patch along the the multi-scale ‘needle’) is not necessarily the best ‘Nearest Neighbor’ (NN) of the clean patch. [sent-212, score-1.024]
92 In fact, it is most likely that the best NN of the clean patch resides elsewhere in the noisy pyramid. [sent-213, score-0.962]
93 Moreover, a better representative can surely be found in a huge collection of clean natural patches. [sent-214, score-0.482]
94 Nevertheless, what our experiments indicate is that there exists a very good representative of the clean patch in a tiny well-defined search space: one of 60 patches. [sent-215, score-0.757]
95 Estimating the Multi-Scale Patch Errors To mimic the oracle, we wish to estimate the mean squared error (MSE) between the clean patch p and each of the noisy patches along its corresponding ’multi-scale needle’ in the noisy pyramid. [sent-225, score-1.492]
96 The problem is that we do not have the original clean patch p, only its noisy version pN. [sent-226, score-0.951]
97 We next show how the MSE errors with respect to the unknown clean patch p can be estimated using pN. [sent-227, score-0.715]
98 Let pN = p + n be the noisy patch, where n ∼ N(0, σ2) is assumed= =to p p b+e Gna buess thiaen n nooisisye p (ait. [sent-228, score-0.24]
99 Let psNc = psc nsc denote a coarsescale patch along the ‘multi-scale needle’ of pN, where psc denotes the coarser version of the clean patch p, and nsc is a coarser version of the original noise n (at scale “sc”). [sent-232, score-1.916]
100 The exact noise realization in each patch is not known. [sent-246, score-0.415]
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