iccv iccv2013 iccv2013-408 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Carlos Fernandez-Granda, Emmanuel J. Candès
Abstract: We present a framework to super-resolve planar regions found in urban scenes and other man-made environments by taking into account their 3D geometry. Such regions have highly structured straight edges, but this prior is challenging to exploit due to deformations induced by the projection onto the imaging plane. Our method factors out such deformations by using recently developed tools based on convex optimization to learn a transform that maps the image to a domain where its gradient has a simple group-sparse structure. This allows to obtain a novel convex regularizer that enforces global consistency constraints between the edges of the image. Computational experiments with real images show that this data-driven approach to the design of regularizers promoting transform-invariant group sparsity is very effective at high super-resolution factors. We view our approach as complementary to most recent superresolution methods, which tend to focus on hallucinating high-frequency textures.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract We present a framework to super-resolve planar regions found in urban scenes and other man-made environments by taking into account their 3D geometry. [sent-2, score-0.19]
2 Such regions have highly structured straight edges, but this prior is challenging to exploit due to deformations induced by the projection onto the imaging plane. [sent-3, score-0.314]
3 Our method factors out such deformations by using recently developed tools based on convex optimization to learn a transform that maps the image to a domain where its gradient has a simple group-sparse structure. [sent-4, score-0.398]
4 This allows to obtain a novel convex regularizer that enforces global consistency constraints between the edges of the image. [sent-5, score-0.436]
5 Computational experiments with real images show that this data-driven approach to the design of regularizers promoting transform-invariant group sparsity is very effective at high super-resolution factors. [sent-6, score-0.143]
6 The aim is to obtain a higher-resolution image by upsampling a single image. [sent-12, score-0.289]
7 We can only hope to reconstruct certain very specific structures (see [4] for theoretical results on the super-resolution of pointwise objects) or to hallucinate high-frequency textures that are visually pleasing [1]. [sent-14, score-0.214]
8 In this work we consider planar regions taken from 3D scenes that have straight edges aligned in a few main directions, such as the one in Figure 1. [sent-15, score-0.518]
9 They are ubiquitous in urban environments and recent large-scale urban 3D mapping efforts (such as the Apple 3D map) make such data readily availEmmanuel J. [sent-17, score-0.096]
10 Existing super-resolution techniques can be applied to this class of textures to obtain reasonably good upsampling results up to factors of three or four. [sent-22, score-0.496]
11 In this work, we explore the possibility of attaining higher upsampling factors by harnessing such prior knowledge for images with structured edges. [sent-24, score-0.414]
12 Unfortunately, the non-uniform blur and deformations induced by the projection of 3D surfaces onto the imaging plane make it very challenging to exploit prior knowledge about the structure of the data directly. [sent-25, score-0.163]
13 In fact, in our quest to super-resolve regions with structured edges we face two fundamental questions that are at the core ofmany problems in computer vision: 1. [sent-26, score-0.301]
14 A solution in the case of highly structured 3D scenes is to use recent advances in the recovery of low-rank textures [19], defined as low-rank structures deformed by affine or projective transformations. [sent-36, score-0.335]
15 In the case of images with highly structured edges, the sparsity pattern of the gradient tends to follow a lowrank pattern, as illustrated by Figure 1. [sent-38, score-0.184]
16 Learning the domain transform that reveals the lowrank structure of the data. [sent-40, score-0.238]
17 Under the assumptions that the edge structure of the 3D scene is approximately low rank, this data-driven procedure produces a convex regularizer that allows to super-resolve the image very effectively. [sent-43, score-0.37]
18 These example-based methods enforce local consistency to produce sharp-looking edges and are able to hallucinate high-frequency textures very effectively at moderate upsampling factors, especially if prior knowledge about these textures is available [9, 7, 14]. [sent-47, score-0.849]
19 However, they are not well adapted to deal with global features, such as the straight edges in Figure 1. [sent-48, score-0.39]
20 To recapitulate, our main contribution is a principled methodology for the super-resolution of planar regions with regular structures, which achieves high-quality results at upsampling factors that are problematic for other methods. [sent-51, score-0.538]
21 Directional total variation Consider the problem of designing a regularizer adapted to the problem of super-resolving images with sharp edges oriented in a few main directions. [sent-59, score-0.582]
22 1 norm of the gradient [11], also known as the total variation (TV) of the image, or related non-convex penalties [10] in order to obtain an estimate with a sparse gradient (see also [15] for a recent approach that takes discretization into account). [sent-61, score-0.19]
23 Unfortunately, minimizing the total variation often fails to superresolve two-dimensional edges, even in the case of very simple piecewise-constant images such as the checkerboard shown in Figure 3. [sent-62, score-0.13]
24 This failure is largely due to the fact that the regularizer is agnostic to the orientation of the edges in the image, and in particular to the correlation between the orientation of nearby edges. [sent-63, score-0.377]
25 This suggests resorting to a regularizer that is better adapted to the high-level structure of the image gradient. [sent-64, score-0.257]
26 Let us assume that, as is the case for the checkerboard in Figure 3, we happen to know the directions of most edges in the image. [sent-65, score-0.333]
27 In this case, the gradient in the image is not only sparse, but group sparse [18], since its nonzero elements are grouped along horizontal and vertical lines. [sent-66, score-0.246]
28 As a result, a more suitable regularizer is the directional total variation (DTV) of the image, defined as DTV(I) =x? [sent-67, score-0.296]
29 It is designed to favor edges that are aligned horizontally and vertically. [sent-83, score-0.337]
30 A similar regularizer has been proposed for multiple change-point detection in time-series analysis [3]. [sent-84, score-0.181]
31 The top of Figure 3 compares the results of minimizing the TV and DTV cost functions to perform nonblind deblurring of a checkerboard image. [sent-85, score-0.251]
32 Transform-invariant regularization 21 Gaussian The DTV regularizer proposed in Section 2. [sent-93, score-0.276]
33 We seldom encounter images where the edges are perfectly aligned horizontally and vertically. [sent-95, score-0.337]
34 We consider images such that there exists an affine or projective transform τ for which most of the edges of I◦τ are aligned vertically and horizontally. [sent-97, score-0.536]
35 eInd general yiitn igs not straightforward to design a regularizer adapted to such a model. [sent-100, score-0.257]
36 The reason is that the gradient is no longer group sparse along a few main directions. [sent-101, score-0.142]
37 However, it is group sparse modulo the transform τ. [sent-102, score-0.208]
38 2) × a cost function that promotes straight edges in the transformed image, where TI-DTV stands for transforminvariant directional total variation. [sent-104, score-0.47]
39 2) is invariant to affine or projective transforms of I, as long as we are able to estimate them a priori. [sent-106, score-0.117]
40 As we will see, this allows to factor out significant deformations induced by the camera projection. [sent-107, score-0.148]
41 At the bottom of Figure 4 we can see the results of minimizing the TI-DTV cost function to perform nonblind deblurring of a tilted checkerboard. [sent-108, score-0.158]
42 Transform-invariant low-rank textures 21 In order to use the cost function proposed in (2. [sent-116, score-0.195]
43 2), it is necessary to learn a transform “τ” mapping the image to a domain where its edges are mostly aligned vertically and horizontally. [sent-117, score-0.464]
44 We propose doing this by exploiting the fact that images with vertical and horizontal edges tend to be approximately low rank when viewed as a matrix. [sent-118, score-0.35]
45 This is obviously the case for the checkerboard in Figure 4, but holds much more broadly. [sent-119, score-0.093]
46 The main edges are indeed aligned horizontally and vertically by the transformation associated to the low-rank texture. [sent-121, score-0.416]
47 An image with low-rank structured edges might lose its sharpness at low resolutions, but it remains an approximately low-rank texture. [sent-122, score-0.314]
48 This is crucial for our interests, since we can consequently use TILT to learn the transform associated to the edge structure and then apply the regularizer proposed in Section 2. [sent-123, score-0.339]
49 The authors of [19] develop robust computational tools that allow to extract low-rank textures distorted by affine or projective transformations. [sent-126, score-0.353]
50 In essence, TILT al33333381 lows to compute a transform τ from an image I such that I τ = L + E, where L is a low-rank matrix and E ac◦ cIo ◦un τts = =for L sparse d wevheiaretio Lns i sfro am lo twh-er alnowk- mraantkri xm oanddel. [sent-128, score-0.203]
51 We refer to [19] for more details on transform-invariant low-rank textures and on how to solve Problem (2. [sent-135, score-0.15]
52 3), but we would like to mention that the presence of the sparse term E is of vital importance if we apply TILT to retrieve low-rank textures from low-resolution inputs. [sent-136, score-0.24]
53 The reason is that it accounts for artifacts caused by blur and pixelation. [sent-137, score-0.105]
54 This is illustrated by Figure 6, which shows the low-rank and sparse components obtained from a blurry image. [sent-138, score-0.147]
55 Super-resolution via TI-DTV regularization We finally have all the tools to tackle the problem of super-resolving an image obtained from a 3D scene with structured edges. [sent-144, score-0.216]
56 We propose to leverage a data-driven convex regularizer adapted to the 3D geometry revealed by TILT. [sent-145, score-0.316]
57 4) for a downsampling operator D : RN1×N2 → Rn1×n2 and a blurring ksaemrnpelli nKg ∈p rRatNo1r × DN2 :. [sent-147, score-0.146]
58 We apply TILT to the low-resolution image ILR in order to obtain a transform τ which reveals the low-rank edge structure of the image. [sent-150, score-0.201]
59 In practice, we upsample ILR using bicubic interpolation before learning the transformation. [sent-151, score-0.39]
60 5) where λ and β are regularization parameters,TV represents the usual total variation operator for discrete images (i. [sent-181, score-0.186]
61 the sum of the horizontal and vertical finite differences) and Aτ is a linear operator that maps the image to the domain where we seek to penalize the directional total variation. [sent-183, score-0.281]
62 For color images we apply this procedure to the illuminance channel and upsample the chrominance components Cb and Cr using bicubic interpolation. [sent-184, score-0.362]
63 In practice, we use a Gaussian kernel with a standard deviation σ slightly greater than the upsampling factor divided by two. [sent-191, score-0.346]
64 This framework consists in casting the problem as a conic program, determining the dual problem and applying a firstorder method, such as accelerated gradient descent, to solve a smoothed version of the dual. [sent-199, score-0.133]
65 5) by implementing functions to apply the operator Aτ and to compute the dual of the mixed ? [sent-204, score-0.163]
66 This can be done by implementing Aτ with a sparse matrix that samples the transformed image on a new grid using bilinear interpolation. [sent-211, score-0.096]
67 5), another option to super-resolve our class of images of interest is to work directly in the rectified domain penalizing the DTV norm. [sent-226, score-0.1]
68 More precisely, one can compute the rectified lowresolution image ILτR = ILR ◦ τ and then solve the optimiza- tion problem mI˜HτiRn? [sent-227, score-0.135]
69 Adequate modeling of the downsampling operator is crucial to super-resolve effectively [12], so it not surprising that the results for this alternative method are not as sharp as those obtained with TI-DTV regularization, as shown in Figure 7. [sent-249, score-0.144]
70 6) to super-resolve at an upsampling factor of 8 using geometric information obtained from the low-resolution image. [sent-253, score-0.346]
71 We focus on qualitative comparisons, since there is no clear metric capable of quantifying the quality of super-resolved images (for instance, at high upsampling factors the mean-square error can be better for blurry images that do not enhance any features of interest). [sent-263, score-0.482]
72 In our first example, we take large planar regions from five images in the SUN database [16], shown in Figure 8, and compare our method with other representative super-resolution methods developed in the literature. [sent-265, score-0.142]
73 Although we apply the algorithms to the whole planar region, zoomed-in areas are shown due to space limitations. [sent-266, score-0.138]
74 Interpolation algorithms are represented by bicubic interpolation, which we compute using the Matlab function imresize. [sent-267, score-0.249]
75 eter, since TV regularization is usually the method of choice to promote sharp edges in image processing. [sent-289, score-0.343]
76 It is actually noted in [17] that exemplar-based methods have difficulties dealing with highly structured textures such as building facades, because it is difficult to build a dictionary that can exhaust edges in all directions and scales. [sent-292, score-0.458]
77 Planar regions with structured edges extracted from images in the SUN database [16]. [sent-295, score-0.301]
78 33333403 online for this algorithm, which allows to apply an upsampling factor of4. [sent-297, score-0.379]
79 For the rest ofthe methods, including ours, we apply an upsampling factor of 8. [sent-298, score-0.379]
80 In all cases bicubic interpolation produces images that are very blurry. [sent-300, score-0.379]
81 The results for TV regularization are sharper, but they contain significant artifacts which make edges appear wobbly instead of straight. [sent-301, score-0.359]
82 Despite its reduced upsampling factor, the sparse-coding algorithm is also not capable of super-resolving edges effectively and its results are only slightly better than those of bicubic interpolation. [sent-302, score-0.78]
83 In contrast, TI-DTV regularization produces clear straight edges that correspond to the global geometry of the planar surface, yielding upsampled images that are significantly sharper than the rest. [sent-303, score-0.669]
84 As expected the edges align mostly horizontally and vertically following the low-rank structure. [sent-305, score-0.354]
85 For the top example in Figure 12, where the lowresolution image has size 120x136, the running time required by TI-DTV regularization is of 123. [sent-306, score-0.175]
86 Super-resolution of text Text follows the model that we consider to some extent, since most letters contain horizontal or vertical edges. [sent-313, score-0.104]
87 As a result, TI-DTV regularization is capable of effectively super-resolving letter or characters printed on distorted surfaces. [sent-314, score-0.174]
88 To demonstrate this, Figure 11 shows four such examples and compares the results obtained from bicubic interpolation and our method. [sent-315, score-0.342]
89 TI-DTV regularization is clearly superior in all cases, despite the fact that some letters have edges that are not aligned with the low-rank structure (see the following section). [sent-317, score-0.353]
90 Limitations As we have made clear throughout this paper, our method is geared to the super-resolution of planar surfaces that are approximately low-rank and have straight edges that are oriented following the low-rank structure. [sent-320, score-0.504]
91 If these conditions are not met, the algorithm might produce artifacts in regions that resemble horizontal or vertical edges. [sent-321, score-0.209]
92 In any case, the method often degrades gracefully in regions that do not have straight edges with the right orientation. [sent-324, score-0.351]
93 In Figure 10, for example, TI-DTV does not produce any artifacts around the arc and at the same time super-resolves sharply the edges in the rest of the image. [sent-325, score-0.298]
94 In comparison, TV makes the arc look sharper but generates obvious artifacts. [sent-326, score-0.099]
95 Bicubic TV TI-DTV that does not completely conform to the transform-invariant lowrank model using bicubic interpolation, TV regularization and TIDTV regularization. [sent-329, score-0.412]
96 Conclusion and extensions We believe that developing tools capable of constraining non-local image structure is a crucial step towards achieving high-quality super-resolution at large upsampling factors. [sent-332, score-0.388]
97 Our contributions are the introduction of a principled methodology in which such constraints are imposed through data-driven non-parametric regularizers and a robust implementation of this methodology for a particular class of images, which yields state-of-the-art results. [sent-333, score-0.161]
98 Templates for convex cone problems with applications to sparse signal recovery. [sent-362, score-0.116]
99 Results from super-resolving the images in Figure 8 using bicubic interpolation, total-variation regularization, sparse coding [17] and our proposed algorithm. [sent-495, score-0.306]
100 The upsampling factor was 4 for sparse coding and 8 for the rest of the methods. [sent-496, score-0.403]
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Abstract: unkown-abstract
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