cvpr cvpr2013 cvpr2013-90 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Dohyung Seo, Jeffrey Ho, Baba C. Vemuri
Abstract: In this paper, we introduce a novel framework for computing a path of diffeomorphisms between a pair of input diffeomorphisms. Direct computation of a geodesic path on the space of diffeomorphisms Diff(Ω) is difficult, and it can be attributed mainly to the infinite dimensionality of Diff(Ω). Our proposed framework, to some degree, bypasses this difficulty using the quotient map of Diff(Ω) to the quotient space Diff(M)/Diff(M)μ obtained by quotienting out the subgroup of volume-preserving diffeomorphisms Diff(M)μ. This quotient space was recently identified as the unit sphere in a Hilbert space in mathematics literature, a space with well-known geometric properties. Our framework leverages this recent result by computing the diffeomorphic path in two stages. First, we project the given diffeomorphism pair onto this sphere and then compute the geodesic path between these projected points. Sec- ond, we lift the geodesic on the sphere back to the space of diffeomerphisms, by solving a quadratic programming problem with bilinear constraints using the augmented Lagrangian technique with penalty terms. In this way, we can estimate the path of diffeomorphisms, first, staying in the space of diffeomorphisms, and second, preserving shapes/volumes in the deformed images along the path as much as possible. We have applied our framework to interpolate intermediate frames of frame-sub-sampled video sequences. In the reported experiments, our approach compares favorably with the popular Large Deformation Diffeomorphic Metric Mapping framework (LDDMM).
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract In this paper, we introduce a novel framework for computing a path of diffeomorphisms between a pair of input diffeomorphisms. [sent-2, score-0.67]
2 Direct computation of a geodesic path on the space of diffeomorphisms Diff(Ω) is difficult, and it can be attributed mainly to the infinite dimensionality of Diff(Ω). [sent-3, score-0.823]
3 Our proposed framework, to some degree, bypasses this difficulty using the quotient map of Diff(Ω) to the quotient space Diff(M)/Diff(M)μ obtained by quotienting out the subgroup of volume-preserving diffeomorphisms Diff(M)μ. [sent-4, score-0.781]
4 This quotient space was recently identified as the unit sphere in a Hilbert space in mathematics literature, a space with well-known geometric properties. [sent-5, score-0.281]
5 Our framework leverages this recent result by computing the diffeomorphic path in two stages. [sent-6, score-0.547]
6 First, we project the given diffeomorphism pair onto this sphere and then compute the geodesic path between these projected points. [sent-7, score-0.547]
7 Sec- ond, we lift the geodesic on the sphere back to the space of diffeomerphisms, by solving a quadratic programming problem with bilinear constraints using the augmented Lagrangian technique with penalty terms. [sent-8, score-0.418]
8 In this way, we can estimate the path of diffeomorphisms, first, staying in the space of diffeomorphisms, and second, preserving shapes/volumes in the deformed images along the path as much as possible. [sent-9, score-0.373]
9 We have applied our framework to interpolate intermediate frames of frame-sub-sampled video sequences. [sent-10, score-0.1]
10 Introduction Image and shape matching using 2D diffeomorphisms (diffeomorphic matching) is a popular technique in many computer vision and medical imaging applications such as ∗This research was in part supported by the NIH grant NS066340 to BCV Jeffrey Ho Baba C. [sent-13, score-0.496]
11 Recall that a diffeomorphism is a smooth bijective mapping between two (image) domains, and many important types of transformations and deformations used in computer vision can be taken as 2D/3D diffeomorphisms between appropriate domains. [sent-17, score-0.671]
12 A computational problem that has a fundamental importance in diffeomorphic matching is to determine a smooth path of diffeomorphisms that connects/interpolates a given pair of diffeomorphisms. [sent-18, score-1.043]
13 Given two diffeomorphisms ρ0, ρ1, a smooth path Ωin → → DifΩ f(. [sent-20, score-0.67]
14 ) ∈ Diff(Ω) is a diffeomorphism oanr dea φc(h0 t, . [sent-22, score-0.153]
15 ΩIn) itsh ias paper, we present a novel framework for computing such a smooth path for a given pair of diffeomorphisms ρ0 and ρ1 . [sent-26, score-0.67]
16 However, evaluating diffeomorphic path between two given diffeomorphisms is a challenging problem. [sent-27, score-1.043]
17 This is mainly due to first, the infinite dimensionality of the space of diffeomorphisms and second, the lack of knowledge of the metric on this space. [sent-28, score-0.574]
18 However, in this framework (applied to image registration), the diffeomorphic path φ is not explicitly computed; instead, a timedependent vector field v(t, . [sent-30, score-0.547]
19 )||2L2, (1) where I1, I2 are the two input images, and the diffeomorphic path φ is obtained by integrating the time-dependent 1 1 12 2 2 2 2 57 5 vector field v via the following ODE ddtφ(t,. [sent-34, score-0.547]
20 [2] have shown that the diffeomorphic paths determined by LDDMM stay in the space of diffeomorphisms. [sent-38, score-0.435]
21 The LDDMM framework has been further elaborated in the context of evolution equations on groups of diffeomorphisms and momentum conservation [3]. [sent-39, score-0.548]
22 introduced geodesic equations on the group of diffeomorphisms based on the LDDMM in [3] and Sommer et al. [sent-41, score-0.628]
23 These frameworks involve two different optimizations: image matching and smoothing of the diffeomorphic paths in the space of diffeomorphisms. [sent-44, score-0.452]
24 Second, the smoothness constraint over a diffeomorphic path may lead to unsatisfactory correspondences between input images. [sent-49, score-0.547]
25 One way to overcome these issues is to separate registration stage and the stage of estimating diffeomorphic paths. [sent-51, score-0.414]
26 The key contribution in this paper is that we design a framework for interpolating the two given differomorphisms, which is formulated separately from the registration task. [sent-52, score-0.093]
27 Instead of interpolating the diffeomorphisms in their space directly, we project diffeomorphisms on to the space of densities, whose geodesics are readily computed, and then lift the geodesics back to the space of diffeomorphisms efficiently. [sent-53, score-1.839]
28 More specifically, let M denote a compact ndimensional Riemannian manifold, and Diff(M) and Diffμ (M) respectively denote the infinite-dimensional group ofdiffeomorphisms of M and its infinite-dimensional subgroup of volume-preserving diffeomorphisms. [sent-54, score-0.076]
29 The quotient space Diff(M)/Diff(M)μ was studied in depth and identified with the space of densities functions Dens(M) on M in [6]. [sent-55, score-0.193]
30 The latter space can then be canonically embedded into a sphere in the Hilbert space (See [6] for details); therefore, the geodesics on Dens(M) can be computed readily. [sent-56, score-0.239]
31 In LDDMM, the diffeomorphic path is computed directly in Diff(M) without referencing the quotient space Dens(M) = Diff(M)/Diff(M)μ. [sent-57, score-0.7]
32 Third row: the path of images warped by the diffeomorphisms along the path estimated using our method. [sent-60, score-0.844]
33 We remark that it is precisely because the geometry of the quotient space is available (and simple), the geodesic path can first be determined on the quotient space. [sent-63, score-0.544]
34 The second step of lifting the path in Dens(M) back to Diff(M) is then formulated as a quadratic programming problem with bilinear constraints that minimizes point-wise L2-norm of deformation fields along the diffeomorphic path. [sent-64, score-0.82]
35 We validate the proposed framework by applying it to the problem of filling-in missing frames in a video footage. [sent-68, score-0.145]
36 Model formulation In this section, we present the novel framework for computing the path of diffeomorphisms. [sent-71, score-0.174]
37 The estimation of geodesics on the space of diffeomorphisms requires the knowledge of its metric. [sent-73, score-0.594]
38 However, in this paper, instead of working on the manifold of diffeomorphisms whose metric is unknown, we map diffeomorphisms to a known space, the space of densities, and take advantage of the fact that geodesics for the canonical metric on the sphere are the great circles. [sent-74, score-1.245]
39 Our algorithm consists of the following two steps: 1 1 12 2 2 2 2 68 6 • • Map a given pair of diffeomorphisms to the space of dMenapsit aie gs,iv eD nen psa(irM o)f adnifdf computing tsh teo geodesic e b oe-f tween the two projected points. [sent-75, score-0.644]
40 The Space of Densities and Diffeomorphisms Given a pair of diffeomorphisms mapping Ω to Ω, and without loss of generality, we can assume one of them is the identity map id and denote the other diffeomorphism by ρ. [sent-79, score-0.746]
41 The goal is to compute the one-parameter family of diffeomorphisms, φ(t), between id and ρ, such that φ(0) = id and φ(t = tf) = ρ where, 0 ≤ t ≤ tf. [sent-80, score-0.116]
42 Furthermore, since φ is a diffeomorphism of M, the integral transform μ(M) =? [sent-83, score-0.153]
43 =The Jreafocre, using the squareroot of the density function, we can define the map Φ from Diff(M) to a sphere in a Hilbert space via Φ : φ → f = ? [sent-91, score-0.122]
44 MΦ2(φ)dμ = μ(M) (4) since shows that Φ(φ) belongs to a sphere with radius ? [sent-93, score-0.097]
45 int on the unit sphere and for the two diffeomorphism id, ρ mapped on to the sphere by Φ, the unique geodesic between them can be readily computed using the formula f(t) =sin1(θ)[sin(θ − t)f1+ sin(t)f2] (5) where f1 = ? [sent-96, score-0.492]
46 id are projected to the same point on the sphere. [sent-104, score-0.077]
47 Hence, the geodesic joining the two projected points is degenerate – i. [sent-105, score-0.157]
48 , it consists of just one point– and the lifted diffeomorphic path in Diff(M) is then a path joining ρ and id consisting of only volume-preserving diffeomorphisms. [sent-107, score-0.839]
49 In this paper, diffeomorphisms are expressed as deformation vector fields. [sent-108, score-0.583]
50 In the case of 2-D, φ(t) = (x + U(r, t) , y + V (r, t)) where r = (x, y), and (U(r, t) , V (r, t)) denote the deformation vector fields. [sent-109, score-0.104]
51 Lifting the Geodesic Path to the Space of Diffeomorphisms After a geodesic on the sphere is obtained, we have to lift this path back to Diff(M). [sent-118, score-0.454]
52 This is because the equation is bilinear , and geometrically, this corresponds to the fact that Eq. [sent-121, score-0.064]
53 Therefore, the computation ofthe lifted path must be regularized, and we propose using the L2 smoothness ofthe deformation vector fields over time as the main regularization criterion. [sent-123, score-0.314]
54 The lifting problem now leads to a quadratic program with bilinear constraints for 2D domains min? [sent-125, score-0.136]
55 In our numerical scheme, we consider it as the change in area of a triangle in the mesh with four neighboring pixel points replaced by the deformation field at vertices. [sent-195, score-0.106]
56 After vectorizing all deformation vector fields, the optimization problem becomes a quadratic program with bilinear constraints given by: min21U? [sent-204, score-0.177]
57 Du(V)U ( Dv (U)V) corresponds to the bilinear terms in Eq. [sent-227, score-0.064]
58 Experimental Results We evaluate the effectiveness and efficiency of the proposed framework using the difficult problem image interpolation for missing frames in videos. [sent-254, score-0.153]
59 Specifically, for the experiment, an image sequence is extracted from a video and a few frames are selected as the key frames. [sent-255, score-0.161]
60 We then apply our method to compute a diffeomorphic path connecting two adjacent key frames, and the intermediate frames are computed and warped by the appropriate diffeomorphism along the diffeomorphic path. [sent-256, score-1.197]
61 These filled-in im- age sequences are compared with the original sequences by measuring the pSNR of the computed frames w. [sent-257, score-0.149]
62 Second, a registration map– or diffeomorphism denoted by ρ–between the two images is computed using an image registration method. [sent-263, score-0.254]
63 We then set id as the diffeomorphism which maps the source image to itself, and in this way, we have the two diffeomorphisms to be interpolated, id and ρ. [sent-264, score-0.765]
64 Next, we compute a diffeomorphic path, φ(t) joining id and ρ, using the proposed algorithm. [sent-265, score-0.465]
65 Finally, with this computed diffeomorphic path φ(t), we can simulate an image sequence I(t) between Is and It simply by warping the image I(t) = Is ◦ φ(t). [sent-266, score-0.604]
66 We have two cylinders: one is positioned upright and the other one is the 50-degree-rotated version of the upright cylinder. [sent-269, score-0.082]
67 We evaluate deformation fields that deform the upright cylinder to the rotated one keeping the boundary fixed. [sent-270, score-0.19]
68 The ground truth is generated by rotating the upright cylinder by 50/19 degree consecutively so that we have 20 sequential images. [sent-272, score-0.094]
69 Beginning at the leftmost plot, rotating cylinder experiment, bending motion experiment and dancing motion experiment. [sent-279, score-0.24]
70 The reported times for the proposed method include registration and interpolation stages. [sent-283, score-0.067]
71 For the dancing motion sequence, we were not able to find suit- able parameters for LDDMM and LDDMM does not produce convergent results. [sent-284, score-0.111]
72 MethodThe proposed methodLDDMM Rotation test24min 40sec42min 20sec Bending woman Dancing woman 16min 7sec 12min 30sec 41min 57sec No convergence pSNR scores for the sequences are plotted in Fig. [sent-285, score-0.072]
73 All resulting sequences and original video footages are provides as movie files in the supplemental material. [sent-288, score-0.075]
74 Bending motion sequence In this experiment, we present a video sequence in which a person is depicting a bending motion. [sent-293, score-0.181]
75 75 fps, and we have extracted 34 frames from videos. [sent-295, score-0.082]
76 We choose the first, the last and the seventeenth frames as keyframes so that there are fifteen missing frames between keyframes. [sent-299, score-0.297]
77 3, the two groups of missing frames in each sequence are highlighted with rectangular boxes, with the three remaining key frames. [sent-301, score-0.213]
78 Next we compute two diffeomorphic deformation fields: one from the first to the the seventeenth frame, and one from the seventeenth frame to the last frame. [sent-302, score-0.594]
79 The missing frames are computed as described in the previous section. [sent-303, score-0.146]
80 The resulting frames are compared with the ground truth and also with results produced by LDDMM: visual comparison shown in in Fig. [sent-304, score-0.082]
81 Additionally, LDDMM requires more key frames (between the first and seventeenth frames) in order to ensure its convergence. [sent-311, score-0.172]
82 Dancing motion sequence In this experiment, we work on a video sequence with a more complicated motion namely, dancing. [sent-314, score-0.144]
83 Twenty-seven image frames have been sampled from the original video, and twenty-two frames have been removed, leaving five key frames. [sent-315, score-0.187]
84 These keyframes and missing frames are shown in Fig. [sent-316, score-0.148]
85 Due the amount of motion variation, we have retained more key frames (five) than the previous experiment on bending motion (three). [sent-318, score-0.234]
86 We compute the same number of missing frames using our method as in the previous experiment. [sent-319, score-0.127]
87 We remark that for this experiment we were not able to find suitable parameters for LDDMM to produce convergent results. [sent-324, score-0.068]
88 Conclusion In this paper, we have introduced a novel framework for computing a diffeomorphic path that interpolates the two given diffeomorphisms, and we have demonstrated its applicability to image interpolation problems for reconstructing missing or corrupted images in videos. [sent-326, score-0.618]
89 Computing diffeomorphic paths is often challenging, and the difficulty mainly arises from the nonlinearity and the infinite dimensionality of the space. [sent-327, score-0.434]
90 However, some of these difficulties can be circumvented by appealing to the fact that the quotient space of Diff(M) by (the subgroup of) volume-preserving diffeomorphisms can be identified with a convex subset ofthe unit sphere in a Hilbert space. [sent-329, score-0.787]
91 Using this decomposition (loosely speaking), the geometry of the sphere, which is well-known and computationally straightforward, provides us with geodesics that can be readily computed. [sent-330, score-0.095]
92 In particular, for the problems that requires treating the volume-preserving diffeomorphisms as the nuisance parameters, the path computed by our method is exactly the geodesic paths for the problems. [sent-331, score-0.83]
93 Fur1 1 12 2 23 2 2 9 1 9 Figure 3: The original missing frames are highlighted with blue bounding rectangles, and the frames computed using our method and LDDMM are highlighted with red and green bounding rectangles, respectively. [sent-333, score-0.278]
94 The three key frames are shown individually without bounding rectangles. [sent-334, score-0.105]
95 The remaining five key frames are also shown in the sequence. [sent-336, score-0.105]
96 Gerig, Unbiased diffeomorphic atlas construction for computational anatomy, NeuroImage, 23, S 15 1-S 160, (2004) [2] M. [sent-348, score-0.399]
97 Younes, Computing large deformation metric mappings vis geodesic flows of diffeomorphisms, IJCV 61(2), 139-157, (2005) [3] L. [sent-354, score-0.22]
98 Lauze, and X, Pennec, A multiscale kernel bundle for LDDMM: towards sparse deformation description across space and scales, IPMI, 624-635, (201 1) [5] S. [sent-360, score-0.15]
99 Lauze, M, Nielsen, and X, Pennec, Kernel Bundle EPDiff: Evolution Equations for Multi-scale Diffeomorphic Image Registration, SSVM, 677-688, (201 1) [6] Geometry of diffeomorphism groups, complete integrability and geometric statistics, arXiv: 1105. [sent-362, score-0.153]
100 Raiconi, On the minimization of quadratic functions with bilinear constraints via augmented Lagrangians, Journal of optimization theory and applications, 55, 1, 23-36, (Oct. [sent-365, score-0.113]
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