I. Cohen, Nonlinear variational method for optical ow computation, in Proc. Eighth Scandinavian Conference on Image Analysis, vol. 1, Troms, Norway, May 1993, pp. 523-530.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [31, 23] and it has rst been used for image smoothing with simultaneous edge enhancement [26] Later on, close connections to regularization methods have been discovered [29] and related nonlinear methods have also entered computer vision elds such as motion analysis in image sequences [8] or interactive segmentation [4, 20] In this paper we shall learn about the basic ideas behind these methods, but also about their theoretical foundation and their adequate numerical realization. It is important to note that the requirements for a good numerical scheme in image processing or ....

....y; z) where (x; y) denotes the location and z 2 [0; Z] is the time. We are looking for the optic ow eld 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 ow problem; see e.g. [14, 22, 8, 37]. In [36] 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 0 = df(x(z) y(z) z) dz = f x u f y v f z : 19) 2. As second assumption we impose a ....

I. Cohen, Nonlinear variational method for optical ow computation, in Proc. Eighth Scandinavian Conference on Image Analysis, vol. 1, Troms, Norway, May 1993, pp. 523-530.

.... that the optic ow eld should vary smoothly in space [26] Such a term may be modi ed in an image driven way in order to suppress smoothing at or across image boundaries [1, 35] As an alternative, ow driven modi cations have been proposed which reduce smoothing across ow discontinuities [8, 12, 14, 30, 42, 45, 57]. Most smoothness terms require only spatial smoothness. Spatiotemporal smoothness terms have been considered to a much smaller extent [7, 34, 37, 59] Since smoothness terms ll in information from regions where reliable ow estimates exist to regions where no estimates are possible, they create ....

....a system of coupled di usion reactions equations for the two ow components. The fastly emerging use 2 of PDE based image restoration methods [23, 40] such as nonlinear di usion ltering and total variation denoising, has motivated many researchers to apply similar ideas to estimate optic ow [1, 4, 12, 14, 25, 30, 39, 42, 45, 57]. A systematic framework that links the di usion and optic ow paradigms, however, has not been studied so far. Furthermore, from the framework of di usion ltering it is also well known that anisotropic lters with a di usion tensor have more degrees of freedom than isotropic ones with ....

[Article contains additional citation context not shown here]

I. Cohen, Nonlinear variational method for optical ow computation, Proc. Eighth Scandinavian Conf. on Image Analysis (SCIA '93, Troms, May 25-28,

.... and Schunck [19] who proposed to calculate an approximate solution of (2) that minimizes the functional J HS ( w) 1 2 Z (jru(x)j 2 jrv(x)j 2 ) dx : 3) Recently there has been a trend to use more sophisticated constraints to preserve edges and corners in the motion eld (see e.g. [23, 8, 27, 30, 2, 31]) This can be achieved by considering e.g. penalizing functionals like J NE ( w) Z trace (r w(x) t D 2 (rI) x) r w(x) dx ; 4) with D 2 (rI) x) 1 jrI(x)j 2 2 2 ( I x2 (x) I x1 (x) I x2 (x) I x1 (x) t 2 E ) Here and in the sequel of ....

....denotes the Euclidean norm and E denotes the unitary matrix. The motivation for using such penalizing functionals comes from anisotropic di usion ltering. For some background on this topic we refer to [29] Another frequently used edge preserving technique is via BV penalizing functionals like [8] J BV ( w) Z (jru(x)j jrv(x)j) dx ; 5) where both R jruj dx and R jrvj dx are understood as the bounded variations semi norms of u and v. For a de nition of the space of functions of bounded variation and the semi norm we refer to [13] Following the standard way of solving ....

I. Cohen, Nonlinear variational method for optical ow computation. In Proc. Eighth Scandinavian Conf. on Image Analysis (SCIA '93, Troms, May 25-28,

.... and Schunck [19] who proposed to calculate an approximate solution of (2) that minimizes the functional J HS ( w) 1 2 Z (jru(x)j 2 jrv(x)j 2 ) dx : 3) Recently there has been a trend to use more sophisticated constraints to preserve edges and corners in the motion eld (see e.g. [23, 8, 27, 30, 2, 31]) This can be achieved by considering e.g. penalizing functionals like J NE ( w) Z trace (r w(x) t D 2 (rI) x) r w(x) dx ; 4) with D 2 (rI) x) 1 jrI(x)j 2 2 2 ( I x 2 (x) I x 1 (x) I x 2 (x) I x 1 (x) t 2 E ) Here and in the sequel ....

....denotes the Euclidean norm and E denotes the unitary matrix. The motivation for using such penalizing functionals comes from anisotropic di usion ltering. For some background on this topic we refer to [29] Another frequently used edge preserving technique is via BV penalizing functionals like [8] J BV ( w) Z (jru(x)j jrv(x)j) dx ; 5) 2 where both R jruj dx and R jrvj dx are understood as the bounded variations seminorms of u and v. For a de nition of the space of functions of bounded variation and the semi norm we refer to [13] Following the standard way of solving a ....

I. Cohen, Nonlinear variational method for optical ow computation. In Proc. Eighth Scandinavian Conf. on Image Analysis (SCIA '93, Troms, May 25-28, 1993), Vol. 1, pages 523-530, 1993.

.... that the optic ow eld should vary smoothly in space [25] Such a term may be modi ed in an image driven way in order to suppress smoothing at or across image boundaries [1, 34] As an alternative, ow driven modi cations have been proposed which reduce smoothing across ow discontinuities [8, 12, 14, 29, 40, 43, 54]. Most smoothness terms require only spatial smoothness. Spatiotemporal smoothness terms have been considered to a much smaller extent [7, 33, 36, 56] Since smoothness terms ll in information from regions where reliable ow estimates exist to regions where no estimates are possible, they create ....

....a system of coupled di usion reactions equations for the two ow components. The fastly emerging use of PDE based image restoration methods [22, 39] such as nonlinear di usion ltering and total variation denoising, has motivated many researchers to apply similar ideas to estimate optic ow [1, 4, 12, 14, 24, 29, 38, 40, 43, 54]. A systematic framework that links the di usion and optic ow paradigms, however, has not been studied so far. Furthermore, from the framework of di usion ltering it is also well known that anisotropic lters with a di usion tensor have more degrees of freedom than isotropic ones with ....

[Article contains additional citation context not shown here]

I. Cohen, Nonlinear variational method for optical ow computation, Proc. Eighth Scandinavian Conf. on Image Analysis (SCIA '93, Troms, May 25-28, 1993), Vol. 1, 523-530, 1993.

....optical ow eld varies smoothly in space. However, since many natural image sequences are better described in terms of piecewise smooth ow elds separated by discontinuities, much research has been done to modify the Horn and Schunck approach in order to permit such discontinuous ow elds; see [6, 11, 12, 14, 17, 18, 24, 26, 34, 36, 42, 43, 46, 49, 55] and the references therein. An important improvement in this direction has been achieved by Nagel and Enkelmann [42] in 1986 (see also [39] They consider the following minimization problem: ENE (h) Z R 2 (I 1 (x u(x; y) y v(x; y) I 2 (x; y) 2 dx (4) C Z R 2 trace rh T D ....

....smoothing is ow driven while ours is imagedriven. Another PDE technique that is similar in vein to the work of Proesmans et al. is a stereo method by Shah [50] Other ow driven regularizations with discontinuitypreserving properties include the work of Aubert et al. 6] Cohen [17], Deriche et al. 18] Hinterberger [27] Kumar et al. 34] Schn orr [49] Weickert [55] and Weickert and Schn orr [57] Related stochastic regularization approaches have been studied by Black and Anandan [11, 12] Blanc F eraud et al. 14] Heitz and Bouthemy [26] and M emin and P erez [36] ....

I. Cohen, Nonlinear variational method for optical ow computation, Proc. Eighth Scandinavian Conf. on Image Analysis (SCIA '93, Troms, May 25-28, 1993), Vol. 1, 523-530, 1993.

....that the optical ow eld is smooth. However, since many natural image sequences are better described in terms of piecewise smooth ow elds separated by discontinuities, much research has been done to modify the Horn and Schunck approach in order to permit such discontinuous ow elds; see e.g. [8, 10, 11, 21, 24, 25, 27, 32] and the references therein. An important improvement in this direction has been achieved by Nagel and Enkelmann [24] in 1986. They consider the following minimization problem: ENE (h) Z R 2 (I 1 (x u(x; y) y v(x; y) I 2 (x; y) 2 dx (3) C Z R 2 trace (rh) T D (rI 1 ) rh) ....

I. Cohen, Nonlinear variational method for optical ow computation, Proc. Eighth Scandinavian Conf. on Image Analysis (SCIA '93, Troms, May 25-28, 1993), Vol. 1, 523-530, 1993.

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