J. Weickert and Ch. Schnoer. A theoretical framework for convex regularizers in pdebased computation of image motion. Technical Report Reine Informatik 13/2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and some global or local smoothness assumption on the flow field. In practice, flow fields are usually not smooth. The boundaries of moving objects will correspond to discontinuities in the motion field. Such motion discontinuities have been modeled implicitly by non quadratic robust estimators [2,17,14,28]. Other approaches tackled the problem of segmenting the motion field by treating the problems of motion estimation in disjoint sets and optimization of the motion boundaries separately [25,3,21,9] Some approaches are based on Markov Random Field formulations and the EM algorithm (cf. 12,1,29] ....

J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in PDE--based computation of image motion. Int. J. of Comp. Vis., 45(3):245--264, 2001.

....to be preserved. This opens interesting possibilities : We may for instance replace the simple function lagrangian term by more complex enhancement terms adapted to speci c regularization problems, even if it doesn t come from variational principles (as for instance those proposed in [39, 52, 61, 64, 68, 70]. On can also think to use this general equation (6) to solve other orthonormal constraints related problems (image matching, edge enhancement) From now on, we will study some particular cases of orthonormal vector sets, and the corresponding equations and applications. 4 Direction di usion ....

J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in pde-based computation of image motion. The International Journal of Computer Vision, 45(3):245-264, December 2001.

....(applications in medical images analysis) Here, a vector field models the pixels motion between the two images and a PDE is used to describe its evolution until it converges to the expected image transformation (Fig.1. 7) Interesting survey and references on this subject can be found in [5, 3, 6, 11, 13, 15, 17, 18, 49, 56, 67, 74, 94, 98, 109, 117, 124, 151, 172, 183, 192]. a) Direct superposing of two MRI images of the brain (b) Superposing after image registration [49] Figure 1.7: Image registration, treated as the evolution of a displacement field. Shape from Shading : This new and challenging problem consists in reconstructing a 3D representation of an ....

....tensors. In this chapter, we will often use the concept of diffusion tensor in order to designate real symmetric matrices, without considering the semi positive constraint, i.e tensors that may represent inverse diffusion. With the demonstration (3. 13) we extend the very recent work in [192], where the authors reached a particular case of diffusion tensors D, using a variational formulation : #(# )# # #(# )# # D = # (# ) # # ) # Here, the same function # R appeared for each eigenvalue of the obtained tensor D. With our ....

J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in pde-based computation of image motion. The International Journal of Computer Vision, 45(3):245--264, December 2001.

....without a loss of performance. The fact that confidences are driven by flow vector differences turns their influence in the anisotropic diffusion term of the first equation into a kind of flow driven regularizer (although still of a different kind than those described in the overview by Weickert [7]) The first terms in the second and third equations smooth the values of c. Just as with the correspondences themselves, one would expect the confidences to be smooth functions at most places. This concludes our discussion of the three basic eqs. 5) As a matter of fact, such a system is solved ....

J.Weickert and C.Schnorr. A theoretical framework for convex regularizers in pde-based computation of image motion. International Journal of Computer Vision, 45(3):245--264, 2001.

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J. Weickert and C. Schnrr. A theoretical framework for convex regularizers in PDE-based computations of image motion. International Journal of Computer Vision, 45(3):245-264, December 2001.

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J. Weickert and C. Schn orr. A theoretical framework for convex regularizers in PDE-based computation of image motion. International Journal of Computer Vision, 45(3):245--264, Dec. 2001.

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J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in pde-based computation of image motion. The International Journal of Computer Vision, 45(3):245--264, December 2001.

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J. Weickert and C. Schn orr. A theoretical framework for convex regularizers in pde-based computation of image motion. The International Journal of Computer Vision, 45(3):245--264, December 2001.

....motion information, which may arise for example due to partial occlusions or due to missing grey value structure. 1.1. Related work Discontinuity preserving motion estimation by variational models and related partial differential equations have a long tradition in computer vision research [2,17,18, 20,25,26]. These approaches are non generative in that they purely work in a data driven way. Moreover, they generally model the motion discontinuities implicitly in terms of appropriate (non quadratic) regularizers. There exist some variational approaches with explicit discontinuities for grey value ....

J. Weickert, C. Schno rr, A theoretical framework for convex regularizers in PDE-based computation of image motion, Int. J. Comput. Vis. 45 (3) (2001) 245 -- 264.

....(d) isotropic nonlinear regularization: e) anisotropic nonlinear regularization: with homogeneous Neumann boundary conditions. While this is very easy to verify for the cases (a) d) the proof for the case (e) is more involved. More details can be found in a recent paper [36] where these anisotropic nonlinear regularizers have been analyzed rst. We may regard the elliptic equations (18) 22) as fully implicit time discretizations of the parabolic di usion lters (1) 4) with initial value f and time step size . This connection has been used in [27, 24] to establish ....

....is also the case for the vector valued setting, so we refrain from showing experimental results, since they can hardly be distinguished from those for di usion ltering. 2. 3 Application: Variational Image Sequence Analysis Let us now apply the preceding concepts to the analysis of image sequences [36]. One of the main goals of image sequence analysis is the recovery of the socalled optic ow eld. Optic ow describes the apparent motion of structures in the image plane. It can be used in a large variety of applications ranging from the recovery of motion parameters in robotics to the design ....

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J. Weickert and C. Schn orr, A theoretical framework for convex regularizers in PDE-based computation of image motion, International Journal of Computer Vision, 45 (2001), pp. 245-264. 23

....prior. 1 Related Work Discontinuity preserving motion estimation by variational methods and related partial di erential equations have a long tradition in computer vision. In some approaches the motion discontinuities are modeled implicitly in terms of appropriate (non quadratic) regularizers [14, 2, 12, 11, 17]. Other approaches pursue separate steps of variational motion estimation on disjoint sets with a shape optimization procedure [16, 4, 15, 8] For the case of grey value segmentation, there exist some region based variational approaches with explicit discontinuities (cf. 13] and extensions to ....

J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in PDE{based computation of image motion. Int. J. of Comp. Vis., 45(3):245-264, 2001.

....This can improve segmentation of a known object in cases of missing or misleading motion information. Related Work Discontinuity preserving motion estimation by variational models and related partial di erential equations have a long tradition in computer vision [18] 23] 2] 16] 15] [25]. These approaches are non generative in that they purely work in a data driven way. Moreover, they generally model the motion discontinuities implicitly in terms of appropriate (non quadratic) regularizers. There exist some variational approaches with explicit discontinuities for grey value ....

J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in PDE{based computation of image motion. Int. J. of Comp. Vision, 45(3):245-264, 2001.

....where (x, y) ## denotes the location and z # [0, Z] is the time. We are looking for the optic flow field # u(x,y,z) v(x,y,z) # which describes the correspondence of image structures at di#erent times. Variational methods constitute one possibility to solve the optic flow problem; see e.g. [8, 14, 22, 37]. In [38] a method is considered which is based on the following two assumptions: 1. Image structures do not change their grey value over time. Therefore, along their path (x(z) y(z) one obtains (19) 0 = df(x(z) y(z) z) dz = f x u f y v f z . 2. As second assumption we impose a ....

.... # 3 u 2 # 3 v 2 # # dx dy dz. This functional can be regarded as a special representative of a much larger class of optic flow functionals for which one can establish general well posedness results in H 1(# (0, T ) H 1(# (0, T ) For more details the reader is referred to [37]. The steepest descent equations for (21) with a di#erentiable regularizer # are u t = # 3 # # # # # 3 u 2 # 3 v 2 # # 3 u # 1 # f x (f x u f y v f z ) 22) v t = # 3 # # # # # 3 u 2 # 3 v 2 # # 3 v # 1 # f y (f x u f y v f z ) 23) This is ....

Weickert J. and Schnorr C., A theoretical framework for convex regularizers in PDE-based computation of image motion, Tech. Rep. 13/2000, Computer Science Series, University of Mannheim, Germany, June 2000, to appear in International Journal of Computer Vision.

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J. Weickert and Ch. Schnoer. A theoretical framework for convex regularizers in pdebased computation of image motion. Technical Report Reine Informatik 13/2000.

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J. Weickert and C. Schn orr. A theoretical framework for convex regularizers in pde-based computation of image motion. Int. J. Comput. Vision, 45(3):245--264, 2001.

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Joachim Weickert and Christoph Schn orr, "A theoretical framework for convex regularizers in pde-based computation of image motion," Int. J. Comput. Vision, vol. 45, no. 3, pp. 245--264, 2001.

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J. Weickert and C. Schnrr. A theoretical framework for convex regularizers in pde-based computation of image motion. The International Journal of Computer Vision, 45(3):245264, December 2001.

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J. Weickert and C. Schn orr. A theoretical framework for convex regularizers in pdebased computation of image motion. The International Journal of Computer Vision, 45(3):245--264, December 2001.

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J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in PDE--based computation of image motion. Int. J. of Comp. Vis., 45(3):245--264, 2001.

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J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in PDE--based computation of image motion. IJCV, 45(3):245--264, 2001.

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J. Weickert and C. Schn orr. A theoretical framework for convex regularizers in PDE--based computation of image motion. Int. J. of Comp. Vis., 45(3):245--264, 2001.

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J. Weickert and C. Schnorr. A theoretical framework for convex regularizers in PDE--based computation of image motion. IJCV, 45(3):245--264, 2001.

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J. Weickert and C. Schn orr. A theoretical framework for convex regularizers in PDE--based computation of image motion. Int. J. of Comp. Vis., 45(3):245--264, 2001.

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