iccv iccv2013 iccv2013-301 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Etienne Huot, Giuseppe Papari, Isabelle Herlin
Abstract: This paper describes modeling and numerical computation of orthogonal bases, which are used to describe images and motion fields. Motion estimation from image data is then studied on subspaces spanned by these bases. A reduced model is obtained as the Galerkin projection on these subspaces of a physical model, based on Euler and optical flow equations. A data assimilation method is studied, which assimilates coefficients of image data in the reduced model in order to estimate motion coefficients. The approach is first quantified on synthetic data: it demonstrates the interest of model reduction as a compromise between results quality and computational cost. Results obtained on real data are then displayed so as to illustrate the method.
Reference: text
sentIndex sentText sentNum sentScore
1 fr Giuseppe Papari Lithicon Norway AS NORWAY papari @ l ithi con . [sent-3, score-0.06]
2 fr s le Abstract This paper describes modeling and numerical computation of orthogonal bases, which are used to describe images and motion fields. [sent-6, score-0.28]
3 Motion estimation from image data is then studied on subspaces spanned by these bases. [sent-7, score-0.147]
4 A reduced model is obtained as the Galerkin projection on these subspaces of a physical model, based on Euler and optical flow equations. [sent-8, score-0.351]
5 A data assimilation method is studied, which assimilates coefficients of image data in the reduced model in order to estimate motion coefficients. [sent-9, score-0.777]
6 The approach is first quantified on synthetic data: it demonstrates the interest of model reduction as a compromise between results quality and computational cost. [sent-10, score-0.074]
7 Results obtained on real data are then displayed so as to illustrate the method. [sent-11, score-0.159]
8 A powerful class of methods for this task is based on data assimilation (DA), which emerged in this field less than ten years ago [1, 16, 20], after being widely used in remote sensing, geophysical and meteorological applications [4, 14, 21]. [sent-14, score-0.745]
9 DA relies on the use of numerical models obtained by discretizing and approximating highly complex and non linear geophysical models. [sent-15, score-0.138]
10 The issue of model reduction, obtained when projecting the dynamic equations on a subspace, arises in a natural way when studying the numerical analysis literature [10, 17]. [sent-16, score-0.116]
11 [5] apply model reduction for estimating flow dynamics from particle image velocimery measures with a data assimilation method. [sent-18, score-0.59]
12 However, their reduction method does not allow to either constrain properties on motion fields or to apply specific boundary conditions. [sent-22, score-0.296]
13 In this paper, a novel projection basis is proposed (Section 3), which is derived by an optimality criterion, that takes into account the shape of the basin in which the water flows, and some desirable properties of the concerned motion, such as smoothness and zero divergence. [sent-23, score-0.283]
14 These waveforms are applied, by means of a Galerkin projection, × to obtain a reduced model of the fluid dynamic system under study which is used for image assimilation (Section 2). [sent-24, score-1.082]
15 Model reduction and data assimilation In this section, the full and reduced model of the physical system under study are reviewed (Subsection 2. [sent-27, score-0.697]
16 1), and the variational approach to image assimilation is described (Subsection 2. [sent-28, score-0.539]
17 Dynamic model In the following, a numerical procedure for the solution of the fluid dynamic equations is presented, which is based on the Galerkin projection of the state vector on motion and image subspaces spanned by given families of projection functions. [sent-32, score-0.696]
18 As the concern is motion estimation from these data, the state vector X is composed of a 2D vector field w(r), which represents the horizontal velocity of the water, and a scalar field Is (r), which represents the surface temperature of the water. [sent-35, score-0.672]
19 The latter is also called pseudo-image in the image assimilation literature. [sent-36, score-0.507]
20 mparison between the state vector and the real image obse? [sent-42, score-0.095]
21 This equation is now projected i(nwto · ∇the) subspaces spanned by the orthogonal bases Φ = {φi (r)}i=1. [sent-45, score-0.317]
22 As Φ are Ψ are orthogonal bases, we obtain: age subspaces, dinegfin theed by: ? [sent-73, score-0.05]
23 ting Is (t) on the basis Ψ, B(k) the K K matrix whose (i, j) element is: B(k)i,j=? [sent-105, score-0.1]
24 K A reduced state vector is defined as XR(t) (7) = ? [sent-117, score-0.213]
25 , and System (7) is summarized by: ddXtR(t) + MR(XR(t)) = 0 (8) the reduced model MR being the Galerkin projection of the full model M on bases Φ and Ψ. [sent-119, score-0.361]
26 Data assimilation Motion estimation will be obtained by means of data assimilation in the reduced model described in the previous subsection, where the reduced state vector XR satisfies the evolution equation (8). [sent-122, score-1.458]
27 Observations, denoted by the vector Y(t), are linked to the state vector by an observation operator H: Y(t) = HXR(t) + ? [sent-123, score-0.173]
28 R(t) (9) where H is the projection operator that maps XR(t) → b(t). [sent-124, score-0.124]
29 Some heuristics are usually available on the value of the state vector at date 0. [sent-128, score-0.125]
30 This is described by the background value XRb of the state vector: XR(0) = XRb + ? [sent-129, score-0.062]
31 For a posteriori estimation of 33334536 XR(0) given the observations, the following cost functional J needs to be minimized: J(XR(0)) =? [sent-135, score-0.075]
32 Definition of optimal bases In this section, the proposed projection bases for motion and image spaces are introduced as solutions of an ad hoc constrained minimization problem. [sent-148, score-0.556]
33 1), and it is then particularized to the case of motion estimation (Subsection 3. [sent-150, score-0.193]
34 Continuous formulation of the projection basis Let F be a Hilbert functional space, with given inner product ? [sent-154, score-0.213]
35 , ψn the optimal waveforms, that are solutions of the following constrained minimization problem: ⎪⎨ ⎪⎧ ⎪ (B? [sent-167, score-0.067]
36 n (15) where B is a linear operator on F and δj,k is the delta Kronwehcekreer symbol. [sent-173, score-0.045]
37 , ψn that arise from (15) coincide with the n first solutions of the following progressive unbounded minimization problem: ⎪⎧⎨⎪ ⎩⎪ ⎪. [sent-189, score-0.067]
38 = a rg ψ m ∈ Fi n B Q B (ψ ),|ψ |2 = 1 ,ψ ⊥ ψj1,∀<(16k) In practice, the proposed waveforms are computed as eigenfunctions of a symmetric square matrix which is calculated from the discretized versions of the operators L and cBu. [sent-191, score-0.432]
39 Given the definition of Ω, the bases are depending on the choice of the functional Q tahned operator B de, wpehnicdhin hga oven t toh e be c hdoeificnee odf fi tnh eor fduenr cttoi ensure tahned required properties hoafv image ea dndef mineodtio inn ofiredlders. [sent-197, score-0.363]
40 t As to the scalar waveforms that we use for image sub- space, we consider solutions of the minimization problem (16), with: Q(ψ) ? [sent-198, score-0.584]
41 Ω|∇ψ(r)|2dr for ψ : Ω → R being a scalar vector field and B(ψ) applying :N Ωeum →an Rn boundary caolnardi vteiocntos on image d Ba(tψa. [sent-200, score-0.257]
42 E apx-amples of the resulting waveforms are displayed in Fig. [sent-201, score-0.547]
43 2, reconstructions obtained with bases of 50, 100 and 500 elements are displayed. [sent-204, score-0.171]
44 number of elements according to the size of studied structures on images. [sent-208, score-0.072]
45 As to vector waveforms related to the motion subspace, we will still consider solutions of the minimization problem (16), with: Q(φ) ? [sent-209, score-0.64]
46 As to the operator B :(φ Ω), →diff Rerent situations may be defined. [sent-212, score-0.045]
47 3, in which vector fields are represented by streamlines. [sent-216, score-0.062]
48 The projection bases proposed in this subsection have some similarity with the wavelets described in [12, 13], which have been used for optical flow estimation [6, 7]. [sent-217, score-0.601]
49 Specifically, a family of bi-orthogonal diadic projection functions is derived in [12, 13] as the curl of standard biorthogonal wavelets, on a square domain. [sent-218, score-0.132]
50 Several mathematical properties of those wavelets are described: the curl of these bi-orthogonal wavelets are still bi-orthogonal, and they satisfy the Dirichlet conditions on the boundaries ofthe Figure2. [sent-219, score-0.433]
51 Topt bot m:Asateli mage nditsreconstrucion with 50, 100, and 500 elements respectively. [sent-220, score-0.077]
52 However, unlike the projection basis proposed here, the wavelets proposed in [12, 13] are not suitable to represent vector fields that are defined on an irregular domain Ω, since the boundary conditions are not met on the boundary ∂Ω in the case where Ω is not square. [sent-222, score-0.497]
53 is sufficiently small in the initial condition, it is not guaranteed that it will stay small when a simulation is run with the reduced model. [sent-231, score-0.17]
54 As well known, errors in the boundary condition cause the simulation to become unstable after a certain number of time steps. [sent-232, score-0.094]
55 by [12, 13], which is overcome by the projection basis proposed in this article. [sent-235, score-0.179]
56 In addition, in [12, 13] only vector waveforms are proposed, therefore their framework cannot be used to represent the pseudo-image component of the state vector. [sent-236, score-0.483]
57 In contrast, the framework presented here can be used both for vector and scalar components of the state vector. [sent-237, score-0.224]
58 Results on motion estimation The reduced model obtained with the scalar basis for images and the divergence-free basis for motion fields has been used for estimating motion on a sequence of six satellite images acquired by NOAA/AVHRR sensors on May 14th and 15th 2005. [sent-239, score-1.175]
59 The number of elements for the scalar basis is 240 and the one of the vector basis is 24. [sent-242, score-0.404]
60 As no ground-truth is available for satellite data, a twinexperiment is first defined in order to quantify the method. [sent-246, score-0.171]
61 A simulation is performed with the motion field displayed in Fig. [sent-247, score-0.416]
62 For a clearer rendering of the motion, arrows that represent the velocity field are superposed to the usual color representation. [sent-250, score-0.103]
63 Statistics of that motion field are given in Table 1: minimal, maximal and average values Figure4. [sent-251, score-0.205]
64 Four elements ψnof the scalar basis, for n = 30, n = 60, n = 120, and n = 240. [sent-254, score-0.171]
65 Statistics on the norm of the initial motion field. [sent-261, score-0.152]
66 The simulation provides a sequence of six pseudo-images, taken at same dates than the real acquisitions. [sent-263, score-0.052]
67 These pseudo-images are then assimilated with the reduced model in order to estimate the underlying motion. [sent-264, score-0.167]
68 Value at date 0, named estimation, is displayed in Fig. [sent-265, score-0.189]
69 The error between motion estimations and ground-truth is quantified in Table 2. [sent-269, score-0.189]
70 1 58 0 a L2 regularisation of motion [11, 18] or on a second-order regularisation of the divergence [3, 19]. [sent-284, score-0.258]
71 In order to bet- ter visualize the differences between methods, we defined five characteristics points on the first observation, which are displayed as red crosses on Fig. [sent-285, score-0.159]
72 These characteristic points are then advected by the ground-truth motion field (displayed in red), the one obtained with our method (displayed in green), and the result of Sun et al. [sent-287, score-0.317]
73 The characteristic points obtained at the end of the whole advection process are visualized on the last observation on Fig. [sent-290, score-0.112]
74 The color of the ellipse surrounding each set of points gives an additional information on the quality of the result: a green ellipse codes that our method gives the best result, while a blue one means that Sun’s algorithm provides a better result. [sent-292, score-0.068]
75 The sequence of real satellite observations partly displayed on Fig. [sent-293, score-0.365]
76 4 has then been processed for estimating its motion with the reduced model. [sent-294, score-0.27]
77 An additional experiment has been conducted with a scalar basis of size 120. [sent-297, score-0.229]
78 This demonstrates that if small scales of image data are not taken into account in the basis, motion can not be correctly retrieved. [sent-298, score-0.152]
79 On the other hand, experiment with a scalar basis of size 480 demonstrates that further increasing the subspace dimension does not allow to improve results and has for consequence to increase the size of the reduced model and com- putational cost. [sent-299, score-0.385]
80 The right number of elements should be chosen accordingly to the size of the spatial structures that impact motion. [sent-300, score-0.072]
81 10 displays the result obtained when satellite images are assimilated with the full model described by System (1). [sent-302, score-0.255]
82 As the full model considers a local description of motion, under the divergencefree constraint, global characteristics such as the two main vortices are underestimated, due to smoothing created by the data assimilation process. [sent-303, score-0.542]
83 On the other hand, having determined the size of motion basis in accordance to the minimal size of structures to be retrieved allows a better modeling and a good retrieval of these vortices. [sent-304, score-0.282]
84 8 have been advected by Sun motion field (displayed on top of Fig. [sent-306, score-0.284]
85 Positions on the last observation are displayed on Fig. [sent-310, score-0.159]
86 Figure 12 displays a second satellite sequence of Sea Surface Temperature images acquired on July 27th and 28th 2007. [sent-312, score-0.202]
87 Figure 13 provides results of motion estimation: Sun et al. [sent-313, score-0.152]
88 Again, it can be observed that structures are better assessed and retrieved by the reduced model. [sent-315, score-0.186]
89 Conclusion and Perspectives The main contribution of the paper is the new set of orthogonal projection functions introduced for reduced models. [sent-317, score-0.247]
90 The new waveforms have been obtained by maximizing smoothness while imposing desirable properties, such as zero divergence and suitable boundary conditions. [sent-318, score-0.43]
91 The proposed waveforms enable us to write down a reduced model, which has been used for image assimilation. [sent-319, score-0.506]
92 As we see from the experimental results, structures are better assessed with our approach, while computational cost becomes low, even for large size basins. [sent-320, score-0.068]
93 A major perspective of this work concerns the improvement of the properties imposed to the image and motion bases. [sent-321, score-0.188]
94 As to scalar waveforms, for the image subspace, constraints to allow finer scales at positions of fine structures will be investigated from a long term satellite images data base. [sent-322, score-0.33]
95 As to vector waveforms, for the motion subspace, we will investigate how to use a data base of analysis, obtained by data assimilation in an oceanographic model, to derive properties to be imposed to the basis elements. [sent-323, score-0.828]
96 Wavelets to reconstruct turbulence multifractals from experimental image sequences. [sent-377, score-0.089]
97 [9] [10] [11] [12] [13] mal motion estimation. [sent-392, score-0.152]
98 Learning reduced models for motion estimation on long temporal image sequences. [sent-397, score-0.311]
99 Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. [sent-426, score-0.596]
100 A semiLagrangian discontinuous Galerkin method for scalar advection by incompressible flows. [sent-451, score-0.208]
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