iccv iccv2013 iccv2013-394 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Ruixuan Wang, Emanuele Trucco
Abstract: This paper introduces a ‘low-rank prior’ for small oriented noise-free image patches: considering an oriented patch as a matrix, a low-rank matrix approximation is enough to preserve the texture details in the properly oriented patch. Based on this prior, we propose a single-patch method within a generalized joint low-rank and sparse matrix recovery framework to simultaneously detect and remove non-pointwise random-valued impulse noise (e.g., very small blobs). A weighting matrix is incorporated in the framework to encode an initial estimate of the spatial noise distribution. An accelerated proximal gradient method is adapted to estimate the optimal noise-free image patches. Experiments show the effectiveness of our framework in removing non-pointwise random-valued impulse noise.
Reference: text
sentIndex sentText sentNum sentScore
1 Single-patch low-rank prior for non-pointwise impulse noise removal Ruixuan Wang Emanuele Trucco School of Computing, University of Dundee, UK {ruixuanwang , manue lt rucco} @ comput ing . [sent-1, score-0.769]
2 uk Abstract This paper introduces a ‘low-rank prior’ for small oriented noise-free image patches: considering an oriented patch as a matrix, a low-rank matrix approximation is enough to preserve the texture details in the properly oriented patch. [sent-4, score-0.654]
3 Based on this prior, we propose a single-patch method within a generalized joint low-rank and sparse matrix recovery framework to simultaneously detect and remove non-pointwise random-valued impulse noise (e. [sent-5, score-0.917]
4 Experiments show the effectiveness of our framework in removing non-pointwise random-valued impulse noise. [sent-10, score-0.576]
5 Introduction This paper aims to remove random-valued impulse noise (RVIN) with varying sizes and irregular shapes (so called ‘non-pointwise’ RVIN, e. [sent-12, score-0.784]
6 While the original matrix recovery framework has been recently used for image and video denoising [11], it requires multiple similar image patches, with each patch vectorized as a column in the matrix. [sent-17, score-0.564]
7 patches wpixithelisn) a single image a lnikde ltoy collaboratively and effectively remove traditional (singlepixel) RVIN. [sent-21, score-0.281]
8 Related work We briefly discuss related work on impulse noise removal and low-rank matrix recovery; see [2] and [12] for recent, comprehensive reviews on denoising. [sent-29, score-0.801]
9 There are mainly two types of impulse noise: salt-andpepper noise (black or white), and RVIN (any gray value). [sent-30, score-0.665]
10 , adaptive center weighted median filter (ACWMF) [4], rankordered absolute difference (ROAD) noise detector followed by a trilateral filtering [9], and a logarithmic version of the ROAD followed by edge-preserving regularization (EPR) for pixel restoration (ROLD-EPR) [7]. [sent-34, score-0.261]
11 , when noise is structured rather than single-pixel, the overall noisy removal will be limited. [sent-39, score-0.254]
12 Given the excellent performance of non-local methods [2, 5], learned sparse models [8, 18], and the combination of both [6, 17] for random Gaussian noise, they were explored for impulse noise as well [20, 22]. [sent-40, score-0.699]
13 , self-similarity) to group similar image patches together, followed by collaborative filtering [2, 5]. [sent-43, score-0.247]
14 Sparse methods also use redundant information by assuming each patch can be well approximated by a linear combination of a small subset of patches (‘words’) within a large dictionary. [sent-44, score-0.473]
15 , 8 8 pixels) as it may become datifcfihcu silzt eto is f liinmdi multiple 8sim ×il 8ar p larger-size patches within an image for non-local methods, and to represent a ooppyyrriigghhtt 11007733 larger-size patch by a linear combination of other patches for sparse methods. [sent-48, score-0.688]
16 Crucially, impulse noise often needs to be detected first to reduce the effect of noisy pixels on patch matching and dictionary learning. [sent-49, score-1.05]
17 Similar to twostage methods, the overall accuracy of impulse noise removal is largely limited by the performance of the initial impulse noise location. [sent-50, score-1.427]
18 A joint low-rank and sparse matrix recovery framework was recently used to detect and remove impulse noise simultaneously [11], remove background, and remove shadows and specularities from face images [3]. [sent-51, score-1.051]
19 , 10 10 to 40 40) image patch from images of natural or mg. [sent-56, score-0.265]
20 a,n 1-0m×a1d0e objects or scenes p(waticthhin fr our experiments, see Section 6), if rotated by a characteristic orientation defined later, has a low-rank approximation with texture details (including edges) well preserved (see Section 6. [sent-57, score-0.381]
21 L∗ can be considered a low-rank matrix due to the lowrank prior for single patches (Section 6. [sent-61, score-0.349]
22 Also, since the number of pixels corrupted by impulse noise is generally much smaller than the total number of pixels, S∗ is a sparse matrix. [sent-63, score-0.822]
23 While the above optimization framework has been used for image and video denoising [11], each image patch was ∗ ×× × considered as a column in a matrix. [sent-87, score-0.391]
24 (a) A synthetic clean image patch of size 40 40 pixels with (raa)nk A 2 sy0. [sent-90, score-0.347]
25 n (hbe)t Ac c synthetic noisy patch by adding a eslmsa wliltehr (3 3 pixels) non-pointwise RVIN at the top-left corner and a larger x(9e ×s) 9 n pixels) one aer RouVnIdN th ate t cheen ttoepr. [sent-91, score-0.324]
26 Now more specifically, using larger-size patches in the multi-patch method will generally make multiple patches less similar to each other and hence lead to over smoothing of the current patch (Figures 9 and 10). [sent-94, score-0.631]
27 Compared to the multi-patch method, our method requires no search as it considers a single patch as the matrix P and the patch size can be larger (e. [sent-95, score-0.61]
28 ×M 41o)re w importantly, using largersize patches allows us to remove non-pointwise RVIN. [sent-98, score-0.25]
29 , with much higher intensity value than the signal) non-pointwise impulse noise exists, the true solutions S∗ and L∗ often correspond to a much larger λ? [sent-109, score-0.686]
30 1 (due to the set of higher impulse noise mvaulcuehs) la arngedr modestly smaller ? [sent-111, score-0.697]
31 , E1(L∗ , S∗) > E1 and the non-pointwise impulse noise will remain, to some extent, in the estimated optimal signal (Figure 1c). [sent-115, score-0.73]
32 Hence we propose a generalized version of the optimization framework to denoise an image patch effectively in the presence of non-pointwise (multi-pixel) RVIN: ∗ dˆSe sˆLm. [sent-116, score-0.296]
33 W can encode the initially estimated spatial distribution of impulse noise in the image patch. [sent-127, score-0.718]
34 Initial estimates, obtained by any impulse noise detector, correspond to entries in W with values close to 0. [sent-128, score-0.751]
35 1, bweilclamu soer (ea lti lkeealyst part ospf)o tnhde higher impulse enor iλse? [sent-131, score-0.547]
36 First, the candidate impulse Algorithm 1 APG method to minimize Equation Algorithm 1APG method to minimize Equation (2) (2) Input: P, W, λ. [sent-171, score-0.587]
37 Output: L = max(ρμk,μ); ← L,S ← S noise locations in an image patch are estimated by any of the methods suggested below to obtain a binary weighting matrix W0, in which each entry is set to 1 at the initially estimated impulse noise pixels and 0 elsewhere. [sent-174, score-1.366]
38 Consequently, eth 1e entries of W at or near the initially estimated impulse noise locations will have smaller values than elsewhere. [sent-177, score-0.848]
39 In practice, the binary matrix W0 can be generated by any existing impulse noise detectors (e. [sent-178, score-0.724]
40 Characteristic orientation for each patch One potential issue in denoising methods is edge blurring and loss of sharpness. [sent-185, score-0.486]
41 For example, even a patch with a simple pattern may have a high rank (Figure 2, patch in blue rectangle). [sent-187, score-0.611]
42 In this case, the low-rank approximation of the patch will blur the sharp edge (Figure 2b). [sent-188, score-0.326]
43 Instead, if we can find a low-rank patch (Figure 2, patch in yellow rectangle) by rotating around the target image point, 11007755 (a) (b) ×× (c) Figure 2: Effect of characteristic orientation on low-rank patch approximation. [sent-189, score-1.021]
44 (b) A low-rank (rank 15) approximation of a 41 41 image patch around t(hraen image acpepnrteoxr. [sent-191, score-0.326]
45 m(ca)t iTohne o olfow a- 4ra1n ×k (4r1an ikm a1)g approximation of an oriented image patch around the same point. [sent-192, score-0.387]
46 the low-rank approximation of this patch will more likely preserve the sharpness of the edges (Figure 2c). [sent-193, score-0.465]
47 Similar observations apply for patches with other texture patterns like corners, and experiments (Figure 8) show the importance of characteristic orientation in denoising. [sent-194, score-0.434]
48 Based on the assumption that the optimally oriented patch is low-rank, we expect that the difference between the oriented patch and its low-rank approximation will be minimum at the optimal (‘characteristic’) orientation. [sent-195, score-0.733]
49 To compute the latter, let P(θ) denote an oriented m n image patch haet a given image position, n ro otariteendt eadnt miclo ×c knw iimseby an orientation angle θ with respect to the image row direction, and the low-rank approximation (to a fixed quality level) of P(θ). [sent-196, score-0.482]
50 Then the characteristic orientation at the current image position can be estimated as P˜(θ) ˆθ = argθ∈ m[i0n,π]d(θ) = argθ m∈[0i,nπ]m1n? [sent-197, score-0.253]
51 Such a simple, threshold-free approximation proved effective enough to find reliably the characteristic orientation for each image patch (see Section 6. [sent-214, score-0.552]
52 z Feo 4r1 a ×n i4m1 pixels ahn sdi z5e0 %64 overlapped by neighboring patches along 4b1ot phi dxeirlesc atniodn 5s,0 totally lita ptapkedes approximately 3 minutes to generate the denoised image. [sent-240, score-0.416]
53 Since each image pixel is often covered by multiple patches, the final denoised value at each pixel in the image is averaged from the corresponding denoised pixels in these multiple patches. [sent-241, score-0.453]
54 Low-rank prior in single patches We illustrate the experimental foundation of the lowrank prior for small noise-free image patches. [sent-244, score-0.317]
55 In practice, due to noise, an image patch as a matrix is seldom lowrank. [sent-245, score-0.324]
56 However, if the assumption of low-rank prior is true, the column-wise signal variation in an image patch should be mostly preserved in a much lower-dimensional space, and the low-rank approximation should preserve meaningful textural details. [sent-246, score-0.539]
57 , rotated to their characteristic orientations) patches with sizes 21 21 and 41 41pixels were generated fpraotmch eeasc whi dtha tsaizseets by ×un2i1fo arnmd sampling xine lesa wche image. [sent-250, score-0.39]
58 Every patch siisg fniarlst v decomposed by mSV ×D nan idm atghee minimum number lˆ of singular values necessary to preserve the predefined level of signal variation is determined by lˆ = argminJ∈[1,min(m,n)]{β < ? [sent-252, score-0.408]
59 histogram can be easily generated recording the frequency of patches with a particular rank value. [sent-261, score-0.264]
60 The cumulative rank histogram in Figure (3a) (thinner blue line) shows that when preserving 95% of the column-wise signal variations, about 90% image patches have low-rank approximations with rank equal or smaller than 11. [sent-262, score-0.415]
61 9, more than 98% image patches have low-rank approximations with rank equal or smaller than 10. [sent-264, score-0.296]
62 This shows that most oriented patches can be approximated by their low-rank versions which keep most signal variations. [sent-266, score-0.282]
63 The second test shows that low-rank patch approxima11007766 × ×× (a) (b) Figure 3: Low-rank prior in image patches with size (a) 21 21 (from Caltech256) and (b) 41 41 pixels (from SceneCategory15). [sent-267, score-0.557]
64 (a) (b) Figure 4: Average rank value of each cluster for image patches with size (a) 21 21 (from Caltech256) and (b) p4a1 c×h e4s1 pixels i(zfero (ma) SceneCategory15). [sent-268, score-0.346]
65 95, most clusters have average rank values less than 10 for 21 21 patches a hnadv ele assv ethraagne e2 r0a nfokr v4a1l u×e 4s1 l patches. [sent-274, score-0.288]
66 Within each such cluster, the highest-rank (lˆ) image patch was chosen to represent the most complex visual pattern. [sent-277, score-0.265]
67 It can be observed that, even for the patches with most complex texture patterns, the textural details have all been preserved in the lowrank approximations. [sent-279, score-0.375]
68 In addition, when adding RVIN to the image patches by corrupting 5% pixels in each patch, Figure 4 (dashed curves) also shows that the average rank values increased. [sent-282, score-0.349]
69 The proposed low-rank matrix recovery framework just makes use of this observation for removing noise from image patches. [sent-285, score-0.298]
70 Determination of characteristic orientation The first test here checks whether the determination of characteristic orientation is invariant to changes in patch sizes. [sent-288, score-0.808]
71 As an example, we use an image patch (Figure 6a top) from a noisy and low-contrast underwater hydrocolonoscopy image (Figure 8a). [sent-289, score-0.515]
72 Figure (6) shows that for a large range of patch sizes (e. [sent-290, score-0.317]
73 This invariance to patch sizes is especially beneficial to denoising images with textures at different scales. [sent-295, score-0.443]
74 The second test checks whether the orientation determination is robust to noisy patches. [sent-296, score-0.245]
75 Given an image patch (Figure 7 top left), Figure (7) shows that the estimate of the characteristic orientation is robust to RVIN noise, even if 30% of image pixels have been damaged by RVIN. [sent-297, score-0.552]
76 (a) An original low-contrast 41 41 image patch (top) and the oArnien otreigdi nvaelrs loiown- c(boontttroamst) 4 4(1b e×tte 4r1 1v i mewag on pmatocnhit (otorp). [sent-305, score-0.287]
77 )(b an) dd t(θhe) over all possible orientation angles with three patch sizes. [sent-306, score-0.36]
78 Top row (left to right): original image patch, oriented patches with noise sparsity level at 0. [sent-308, score-0.42]
79 Bottom: estimated characteristic orientation with varying noise sparsity levels, with mean (solid curve) and standard deviation (dotted curve) values from 10 runs. [sent-316, score-0.431]
80 In the denoised result, based on the proposed matrix recovery framework, certain edges have been blurred when using SIFT-orientation method (Figure 8e). [sent-318, score-0.345]
81 While sampling error is introduced to the oriented patches due to rotation of the original patches, the image quality loss due to sampling error appears to be much smaller compared to image enhancement due to noise removal from the oriented patches. [sent-320, score-0.554]
82 This is supported by Figures 8d-8f, which shows that both orientation methods perform better than without using any orientation determination (Figure 8d). [sent-321, score-0.252]
83 Removal of impulse noise To evaluate our method quantitatively, RVIN with different sizes (e. [sent-324, score-0.717]
84 Four denoising methods were chosen for comparison: the median filtering as the baseline method, the ROLDEPR method [7], the multi-patch low-rank matrix recovery method (MPLR) [11], and the proposed method without applying the weighting matrix W (henceforth ‘Proposed \W’). [sent-336, score-0.452]
85 For MPLR, all the parameters were set as suggested in [11], except that four different patch sizes (i. [sent-342, score-0.317]
86 i,l anr patches were 8s,e1ar6c,h3e2d} ()t wo generate a nd2 1 ×0 1 ad1 dmitaitornixa)l across the whole image when denoising each image patch. [sent-346, score-0.309]
87 The highest PSNR over the different patch sizes was reported. [sent-347, score-0.338]
88 For our method, the patch size was fixed to 31 √× 31, tphoer highest PouSNr mRe was reported over ed iwffaesr feinxte λd =o 3s1/×√3311,, where s ∈ {1. [sent-348, score-0.307]
89 Particle removal in hydrocolonoscopy images The proposed denoising method is also capable of removing particles in hydrocolonoscopy images, where the particles with various sizes (maximumly 15 15 pixels) were suspended riino water, creating non-pointwise RpixVeINls). [sent-404, score-0.886]
90 Figure 10 displays an ×× exemplar hydrocolonoscopy image and the denoised result with the MPLR and the proposed method. [sent-406, score-0.363]
91 The patch size is 41 41 for the proposed method and 16 16 for the MPLR 4m1e×tho4d1 fino ro rthdeer p rtoo remove large-size particles. [sent-407, score-0.353]
92 The ‘Proposed \W’ can preserve sharpness e b 1ut0 cca annndot 1 remove large-size particles effectively (Figure 10e and 10f). [sent-409, score-0.293]
93 The proposed method can remove (a) (b) (c) Figure 9: Part of denoised images from different methods ×× with noise size (a) 1 1, (b) 2 2, and (c) 3 3 pixels. [sent-410, score-0.378]
94 Conclusions This paper has introduced a low-rank prior for small oriented (rotated by a characteristic orientation angle) noise11007799 (a)(b) (c)(d) (e)(f) (g)(h) Figure 10: Removal of particles suspended in a hydrocolonoscopy image. [sent-416, score-0.647]
95 (b) De- × noised image by MPLR with patch size 16 16 pixels. [sent-418, score-0.286]
96 The low-rank prior suggests that a single patch can be effectively denoised within a low-rank matrix recovery framework. [sent-424, score-0.646]
97 Without resorting to other similar patches, the single-patch method can effectively remove non-pointwise RVIN within a generalized low-rank matrix recovery framework, and encode the initial estimation of noise locations effectively. [sent-425, score-0.4]
98 Removing random Gaussian noise and video denoising will be explored as future work. [sent-427, score-0.244]
99 A universal noise removal algorithm with an impulse detector. [sent-485, score-0.742]
100 A restoration algorithm for images contaminated by mixed gaussian plus random-valued impulse noise. [sent-573, score-0.61]
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