iccv iccv2013 iccv2013-421 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Jan Lellmann, Evgeny Strekalovskiy, Sabrina Koetter, Daniel Cremers
Abstract: While total variation is among the most popular regularizers for variational problems, its extension to functions with values in a manifold is an open problem. In this paper, we propose the first algorithm to solve such problems which applies to arbitrary Riemannian manifolds. The key idea is to reformulate the variational problem as a multilabel optimization problem with an infinite number of labels. This leads to a hard optimization problem which can be approximately solved using convex relaxation techniques. The framework can be easily adapted to different manifolds including spheres and three-dimensional rotations, and allows to obtain accurate solutions even with a relatively coarse discretization. With numerous examples we demonstrate that the proposed framework can be applied to variational models that incorporate chromaticity values, normal fields, or camera trajectories.
Reference: text
sentIndex sentText sentNum sentScore
1 The key idea is to reformulate the variational problem as a multilabel optimization problem with an infinite number of labels. [sent-3, score-0.273]
2 This leads to a hard optimization problem which can be approximately solved using convex relaxation techniques. [sent-4, score-0.161]
3 With numerous examples we demonstrate that the proposed framework can be applied to variational models that incorporate chromaticity values, normal fields, or camera trajectories. [sent-6, score-0.436]
4 Total Variation for Manifold-Valued Functions For functions u : Ω → Rl, Ω ⊆ Rd, the total variation TV (u) =p∈C∞c(Ω,Rsul×pd),? [sent-10, score-0.232]
5 dx, (1) plays a central role in variational image processing because of its numerous favorable properties: it preserves discontinuities, as a sharp transition from 0 to 1has the same cost as a smooth monotone transition. [sent-16, score-0.177]
6 In many applications of computer vision, however, the functions of interest take on values which do not lie in a Euclidean space such as Rl, but rather on a manifold see Figure 1. [sent-19, score-0.393]
7 We propose an algorithm for total variation (TV)-regularization of functions with values on arbitrary Riemannian manifolds. [sent-22, score-0.296]
8 The key idea is to represent the manifold by a set of grid points (labels) and to solve a convex optimization problem that admits sub-label accuracy: in the above example, the blue points may lie between grid-points. [sent-26, score-0.535]
9 Processing the normal field of a geometric structure, such as a surface or three-dimensional shape, leads to variational problems in terms of functions that assume values on the two-dimensional sphere S2. [sent-28, score-0.529]
10 While the latter can usually be dealt with by a suitable modification of the differential operators, constraining the range of u to a manifold is generally a nonconvex constraint, which makes optimization much harder. [sent-30, score-0.257]
11 Another major challenge in extending the concept of total variation to such manifold-valued data is that we need to assure algorithmically that discontinuities in the values are properly handled. [sent-31, score-0.262]
12 In particular, jumps in the values should be measured with respect to the geodesic distance on that manifold. [sent-32, score-0.211]
13 For general manifolds the estimation of the geodesic distance itself may also be a challenging computational problem. [sent-33, score-0.219]
14 Related Work Giaquinta and Mucci [8, 9] studied the notion of total variation for functions u : Ω → M with values in general vmaarniaitfiooldns f oMr f. [sent-36, score-0.296]
15 u It provides a separation of the differentiable part and the jump part where the jump set Su is penalized with the geodesic distance dM (u−, u+) between the two values u and on either side of the jump. [sent-42, score-0.254]
16 The authors theoretically u+ study the functions with bounded total variation and prove several properties such as lower-semicontinuity and structure theorems. [sent-43, score-0.287]
17 However, they do not provide an algorithm for implementing this regularizer in a variational setting. [sent-44, score-0.185]
18 In fact, their theoretical analysis is based on an embedding of the given manifold M in a higher-dimensional Eudcilindgea onf space vwenhic mha nisi fionlfdea Msib lien tao implement: Firstly, Ethuehigher-dimensional embedding space increases the computational complexity. [sent-45, score-0.257]
19 Secondly, one cannot numerically constrain the values of the estimated solution to the generally non-convex manifold M. [sent-46, score-0.383]
20 o [r7 t]h prove tiahle eaxsieste ofnc teh eo fc imrcilneim Mize =rs f Sor certain energies in the space of functions with bounded total cyclic variation, again using an embedding in the Euclidean plane. [sent-49, score-0.246]
21 Unfortunately, this solution does not extend to general Riemannian manifolds, as unwrapping the manifold to a simple Euclidean one is typically not feasible. [sent-52, score-0.257]
22 [15, 17] proposed relaxations of the labeling problem on continuous domains that allow to find good – and often globally optimal solutions using convex optimization, see also [20, 1]. [sent-57, score-0.167]
23 This constitutes a convex relaxation of the finite labeling problem similar to Linear Programming (LP) relaxation [19, 21]: each of the values in J is asso(cLiaPte)d r ewlaitxha one o[1f9 9th, e2 1u]n:it e vaechcto ofrs he1e , . [sent-83, score-0.472]
24 , eesl i∈n RJl, i s bu ats sion-termediate values are allowed in order to obt∈ain R a convex problem. [sent-86, score-0.155]
25 The regularizer in (3) can be seen as total variation for functions with values in the finite set J, where the mtioentri foc rd f cuoncnttrioonlss twhiet weighting othfe a jump ferto mJ, ,l wabheelr ie1 t htoe label i2. [sent-87, score-0.485]
26 A straightforward approach to apply this idea to manifolds is to choose a finite set of points z1, . [sent-89, score-0.28]
27 as two major drawbacks: Firstly, it generally requires a quadratic number of constraints, which severely restricts the resolution with which the manifold can be discretized. [sent-98, score-0.257]
28 Together, these features ensure that the minimization problem is computationally feasible, and allow to obtain accurate solutions with values on the manifold that are not restricted to a finite set. [sent-115, score-0.422]
29 We validate our method on a variety of inverse problems, including the denoising of chromaticity values, the inpainting and denoising of normal fields and the denoising of camera trajectories. [sent-116, score-1.092]
30 Rudin-Osher-Fatemi (ROF) denoising (blue) of a vector-valued signal u : [0, 1] → R2 (red), visualized as a curve in R2 . [sent-134, score-0.248]
31 o,n [−d f1r,o1m] left), the pairwise formulation exhibits a strong bias towards grid points (second from right). [sent-141, score-0.221]
32 The proposed method has greatly reduced bias and allows the minimizer to assume values between grid points, which can be considered as a form of sub-label accuracy in the range of u (right). [sent-142, score-0.215]
33 Proposed Model In order to extend (3) to the case where J consists of all points on a o(c eoxntneencdte (d3,) R toie thmean cnasiean w) hse-rdeim Je ncsoinosniaslts m ofan alilfold M, s ≥ 1, we replace the unit simplex formulation fuo? [sent-145, score-0.291]
34 , choosing a finite set of labels J, in cwrehitcizhe case (r5o)b rleemdu,c ie. [sent-182, score-0.14]
35 , However, the Riemannian manifold structure allows to reformulate the dual constraints (6) in a more elaborate way. [sent-184, score-0.337]
36 T tahneg key ipdaecae i tso t Mo replace tohien pairwise constraints (6) (TzM)d, by a local condition on the gradient in terms of the spectral norm ? [sent-189, score-0.279]
37 We provide a short proof of the equivalency in the appendix, as it is instructive to verify that in fact the spectral norm appears in the constraint, rather than a numerically more convenient – and more commonly used – matrix norm such as the Frobenius norm. [sent-202, score-0.207]
38 The gradient-based formulation (7) has two major benefits: • • It can be accurately discretized using O(l) local terms Iint ccaonnt braes atc ctcou Ora(tel2ly) t deirsmcsre required nfogr Oth(el )fu l olc pairwise formulation (6). [sent-203, score-0.217]
39 It allows to incorporate knowledge about the local manifold s ttoruc intcuroer ionr ateterm ksn oowf ltehdeg edif afeboreunttia thl operator Dz. [sent-204, score-0.257]
40 While it is possible to reduce the computational cost for the pairwise constraints by restricting the interaction to local neighborhoods, the approximation of the geodesic distance dM is more accurate when using the local terms obtained through the gradient-based formulation. [sent-205, score-0.201]
41 2, where we compare the result of Rudin-Osher-Fatemi (ROF) denoising of a func- tion u : [0, 1] → [−1, 1]2 using a discretization of the set [t−io1n, u1]2 : [w0,ith1] ]l →= [2−819, equally spaces points and gradients e[−va1lu,1at]ed at the center of each cell. [sent-207, score-0.385]
42 For such a moderate number of points the full pairwise model (6) already requires 83521 constraints at each image point, and shows a 2946 pronounced bias towards grid points. [sent-208, score-0.261]
43 We assume that the manifold M is discretized using l ∈ N points ez t1h , . [sent-218, score-0.485]
44 age domain ≈Ω ⊆ Rd is discretized using n ∈ N points x im1 , . [sent-234, score-0.228]
45 , uedli) ∈s aP vMec ftoorr 1u ≤∈ i( ≤ n, where PM is the discretized space ∈o fP probfoabri 1lit ≤y measures on rthee P manifold M: PM := {y ∈ Rl | y ? [sent-244, score-0.419]
46 For computation of the gradients on the s-dimensional manifold M we allow to specify m ∈ N evaluation points ym1a , . [sent-267, score-0.323]
47 , ym M∈ wMe aallt owwh itcoh s tpheec gradient c Non estvraaliunatt (io7)n s phooiunltds be enforced;∈ t Mhesea tcwohulidc hftohr example bceo nthster acineltl 7ce)nstheorsu lidf M = Rs and the zk form a regular mesh. [sent-270, score-0.247]
48 Gradient Discretization on the Manifold In every point yj ∈ M we compute, for all points zk ∈ M, k ∈ Nj in the neighborhood, pthutee ,in fvoerr asell exponential map kvj ∈,k := expy−j1(zk) ∈ TyjM. [sent-282, score-0.353]
49 The extra variable c in the minimization receives the estimate for the value pt (xi, yj), which is unknown as p is discretized only on the points zk. [sent-295, score-0.228]
50 By choosing a suitable parametrization of the tangent space at each point yj, problem (10) can be written in terms of matrices Mj ∈ Rr×s (where s is the dimension of the manifold and the∈ref Rore of the tangent space) and a sparse indexing matrix Pj ∈ Rr×l (both independent of i) as g∈Rmsi,cn∈R? [sent-296, score-0.522]
51 Discretized Model With the above remarks, and identifying matrices with their vector representation where necessary, the fully assembled discretized problem takes the following form: u∈mRinn×l p∈Rn×l×dm,g∈axRn×m×s×d? [sent-328, score-0.162]
52 The scaling by b can be removed by setting b = e, in which case the problem remains the same except for an element-wise scaling of ui by bi, and the constraints for u turn into the unit simplex constraints ui ∈ Δl . [sent-350, score-0.479]
53 Note that each of the gi,j is a matrix in Rs×d, wh∈ere Δ s is the dimension of the manifold and d is the dimension of the image domain Ω. [sent-351, score-0.257]
54 In order to deal with the nontrivial spectral norm constraints we use an approach based on [10], which allows to explicitly compute projections on the constraint set for image domains Ω with dimension 1 and 2. [sent-373, score-0.132]
55 Another possible way is to replace the spectral norm by the Frobenius norm, in which case (19)–(20) becomes a standard second-order cone program (SOCP). [sent-374, score-0.14]
56 Let us summarize why in our opinion the proposed scheme is well suited as a generic model for optimization problems with values on manifolds: • The solver implementation is fully independent of the aTchteua slo lsvtreurct imurpel eomf tehnet mtiaonni ifsol fdu. [sent-375, score-0.196]
57 l Implementing a new manifold only requires to supply a set of points zk, k = 1, . [sent-376, score-0.323]
58 • By choosing the yj on the midpoints between pairs of points oions tihneg m theesh y defined by the zk and choosing suitable Aj , Bj, and Pj, the discretized model also covers the pairwise formulation (6), as well as variants with a reduced number of interaction terms. [sent-385, score-0.648]
59 Means on Manifolds As (5) is essentially a convex relaxation approach, it is possible and actually desirable to obtain probability measures ui with several non-zero components. [sent-390, score-0.272]
60 1 This approach fails for manifolds other than convex subsets of Rs with the usual Euclidean distance, as otherwise z may not be a point in M. [sent-409, score-0.204]
61 (25) In (23) all points zk are mapped to the tangent space TzM by the inverse exponential map, the mean z¯? [sent-438, score-0.382]
62 is then computed in the tangent space in (24), and finally projected back onto the manifold to update z in (25). [sent-439, score-0.37]
63 On the unit sphere S2 it consistently required 5 − 10 iterations ttoh converge rtoe Salmitos cto nmsiasctehnintley precision, a−nd 1 2 i0te0r aittieorna-s tions were sufficient to obtain reliable results even in cases where u? [sent-441, score-0.22]
64 [13, 2] suggested a denoising model for RGB-valued color images I Ω → R3 : tnhoaits separates denoising -ovfa tluhee brightness |gIe| : IΩ :→ Ω R → →fro Rm denoising otefs st dhee chromaticity bCr g=h In/es|sI| : Ω : Ω→ → S R2. [sent-448, score-0.996]
65 Trohme fdoernmoeisri negnc oofd tehse mainly geometric in I/fo|Ir|m a:ti Ωon →, w Shile the latter consists of vectors in the two-dimensional unit sphere and captures the color information. [sent-449, score-0.22]
66 Denoisngofacol rimageI:Ω→R3bysepar tely denoising Ditse intensity maticity I/ |I| ∈ sSit2y mthea proposed m| ∈etho Sd. [sent-452, score-0.248]
67 added noise; denoised |I| ∈ loRr by tahgee eu Isu :al Ω ΩR O→F mRodel, and chroby t ∈he R RR bOyF t hmeo udsuela flo RrO OvFal umeosd on aSnd2 using bLye ftht teo R right, top lto fo obro vttaolumes: input image; chromaticity; combined denoising result. [sent-453, score-0.304]
68 Left to right: smoothed chromaticity; smoothed brightness and chromaticity; smoothed brightness. [sent-456, score-0.165]
69 Smoothing the brightness and chromaticity components differently gives rise to interesting visual effects (Fig. [sent-459, score-0.252]
70 Although t(hxe) range loef u eisp dinisgctr heetiz veedl using a regular grid, the sub-label accuracy of our approach allows u to assume values between grid points, giving a more natural, smooth result compared to treating the problem as a labeling problem (Fig. [sent-464, score-0.232]
71 Variational processing of such data requires optimization over the two-dimensional unit sphere S2. [sent-469, score-0.22]
72 Figure 5m sehnoswions an application Sof our method to ROF denoising of the normal field for terrain data obtained from [6] in order to compute a shaded model. [sent-470, score-0.347]
73 Denoising the normal field considerably improves the visual quality, preserving sharp transitions in the normal field such as along the mountain ridges due to the total variation-based regularizer. [sent-471, score-0.298]
74 Solving the problem as a labeling problem on a finite grid leads to an artificial piecewise constant solution (bottom left). [sent-478, score-0.232]
75 Left to right: Input contours with outer normal field; normal field estimated using a total variation regularizer with values on the unit sphere (x- and y-components of the normals colorcoded as hue); shaded object based on the reconstructed normals. [sent-482, score-0.739]
76 The sphere was discretized using l = 162 points obtained by subdividing the edges of an icosahedron twice, and the gradients evaluated at the centers of all (triangular) faces. [sent-483, score-0.446]
77 The same approach can be used to inpaint normal fields based on given contour lines with outer normals by setting the data term to zero outside of the contours. [sent-484, score-0.201]
78 The proposed approach can be applied to this setting by representing the rotations as unit quaternions. [sent-490, score-0.173]
79 These can be viewed as points on the three-dimensional unit sphere S3, provided th poatin antipodal points are iiodneanlti ufineidt, whietrhe Sthe geodesic distance d(a, b) = arccos( | ? [sent-491, score-0.458]
80 Optimization on the two-dimensional unit sphere applied to denoising of normals in S2 for three-dimensional visualization. [sent-495, score-0.519]
81 Optimization over the space SO(3) of three-dimensional rotations applied to denoising of camera orientations. [sent-501, score-0.315]
82 The rotations are discretized using a quaternion representation, and optimization is subsequently carried out over the three-dimensional manifold of unit quaternions. [sent-504, score-0.655]
83 Discretization of the set of unit quaternions using 720 points as used in Fig. [sent-506, score-0.221]
84 We generate the points zk from the vertices of the hexacosichoron, which is a regular polytope in R4 akin to the icosahedron in R3, by subdividing the faces and eliminating opposite points. [sent-509, score-0.373]
85 Using the same numerical solver as for the previous experiments, this allows to apply Rudin-Osher-Fatemi denoising to the task of smoothing the rotation component of a camera trajectory (Fig. [sent-513, score-0.293]
86 Conclusion We proposed a framework for TV-regularization of functions with values in an arbitrary Riemannian manifold. [sent-516, score-0.136]
87 To this end, we formulated the TV-regularization as a multilabel optimization problem with an infinite number of labels, and used a specialized discretization that allows to better incorporate knowledge about the manifold structure. [sent-517, score-0.462]
88 Using this approach, it becomes possible to solve variational problems for manifold-valued functions that consist of a possibly non-convex data term and a total variation regularizer. [sent-518, score-0.412]
89 Suitable constraints on the dual variables ensure that the TV-regularizer correctly penalizes jumps according to the geodesic distance on the manifold. [sent-519, score-0.227]
90 We experimentally validated the proposed method on a variety of inverse problems, including the denoising of chromaticity values and camera trajectories, optical flow computation, as well as inpainting and denoising of normal fields. [sent-520, score-0.857]
91 The proposed approach allows to obtain more accurate solutions compared to treating the problem as a finite labeling problem, while reducing the computational effort. [sent-521, score-0.214]
92 Many properties of minimizers of total variationregularized models in manifolds are still not fully understood. [sent-522, score-0.217]
93 A first-order primal-dual algorithm for convex problems with applications to imaging. [sent-625, score-0.132]
94 Variational problems for maps of bounded variation with values in S1. [sent-652, score-0.262]
95 Maps of bounded variation with values into a manifold: Total variation and relaxed energy. [sent-670, score-0.323]
96 The natural vectorial total variation which arises from geometric measure theory. [sent-679, score-0.16]
97 Exact optimization for Markov random fields with convex priors. [sent-696, score-0.142]
98 Variational models for image colorization via chromaticity and brightness decomposition. [sent-703, score-0.252]
99 Convex optimization for multi-class image labeling with a novel family of total variation based regularizers. [sent-719, score-0.236]
100 Optimality bounds for a variational relaxation of the image partitioning problem. [sent-729, score-0.209]
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