M. Snyder, \On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction," PAMI 13, pp. 1105-1114, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....can be taken out of the in mum, and so cancelling the V (t; x) from both sides of the equation, dividing by and taking the limit as 0, we get the resulting Hamilton Jacobi as required. 4 Optical Flow Optical ow is the vector eld of apparent velocities of light patterns in an image; see [26, 65] and the references therein. There are a number of approaches to motion eld estimation which are based on trying to compute the intensity ow. One of the most popular approaches in this area is due to Horn and Schunck [27] who base their method on the assumption that the image intensities of ....

M. Snyder, \On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction," PAMI 13, pp. 1105-1114, 1991.

....ltering in image processing, we refer to [56] Homogeneous regularization arises as a special case of (15) when g(jrf j 2 ) 1 is considered. 2.1. 4 Anisotropic image driven regularization An early anisotropic modi cation of the Horn and Schunck functional is due to Nagel [35] see also [2, 17, 36, 38, 43, 44, 50]. The basic idea is to reduce smoothing across image boundaries, while encouraging smoothing along image boundaries. This is achieved by considering the regularizer V AI (rf; ru 1 ; ru 2 ) ru T 1 D(rf)ru 1 ru T 2 D(rf)ru 2 : 19) 6 D(rf) is a regularized projection matrix perpendicular ....

M.A. Snyder, On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 13, 1105-1114, 1991.

....ltering in image processing, we refer to [53] Homogeneous regularization arises as a special case of (15) when g(jrf j 2 ) 1 is considered. 2.1. 4 Anisotropic image driven regularization An early anisotropic modi cation of the Horn and Schunck functional is due to Nagel [34] see also [2, 17, 35, 37, 41, 42, 47]. The basic idea is to reduce smoothing across image boundaries, while encouraging smoothing along image boundaries. This is achieved by considering the regularizer V AI (rf; ru 1 ; ru 2 ) ru T 1 D(rf)ru 1 ru T 2 D(rf)ru 2 : 19) 6 D(rf) is a regularized projection matrix perpendicular ....

M.A. Snyder, On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 13, 1105-1114, 1991. 22

.... level information, we will focus our attention on the Nagel and Enkelmann method, rst, because of its simplicity, the underlying second order dioeerential operator is linear, and second, because this method has demonstrated its performance numerous times in the context of optical AEow estimations [2, 3, 7, 15, 34, 35, 36, 50, 52]. Let us mention here a link between the approach we propose in this article and the one proposed in [46] within the stereo framework. A simple application of the idea in [46] results in selecting a regularization term that takes into account the discontinuities within the disparity map. It ....

M.A. Snyder, On the mathematical foundations of smoothness constraints for the determination of optical AEow and for surface reconstruction, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 13, 11051114, 1991.

....introduced by Nagel and Enkelmann [31, 32] in the context of optic ow estimation. We use it here because of its simplicity (the underlying second order di erential operator is linear) and because this method has demonstrated its performance numerous times in the context of optical ow estimations [3, 4, 8, 14, 31, 32, 45, 47]. 2.4. Minimizing the Energy In order to minimize our energy functional, we solve its associated Euler Lagrange equation C div (D (rI l ) r ) I l (x) I r (x) a Ir y (x) b Ir x (x) p a 2 b 2 = 0; 21) where x : x; y) t , I r (x) I r (x ....

M. A. Snyder, On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction, IEEE Trans. Pattern Anal. Mach. Intell. 13, 1991, 1105-1114.

....and Anandan [11, 12] Blanc F eraud et al. 14] Heitz and Bouthemy [26] and M emin and P erez [36] The image driven anisotropic Nagel Enkelmann approach has been subject to many subsequent studies. Examples include later work by Nagel [40, 41] as well as research by Schn orr [47, 48] and Snyder [51]. A multigrid realization of this method has been described by Enkelmann [19] and a related pyramid framework is studied by Anandan [5] An isotropic image driven optic ow regularization is investigated by Alvarez et al. 1] With respect to embeddings into a linear scale space framework our ....

M.A. Snyder, On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 13, 1105-1114, 1991.

....the points given by (x nd; i n)T ) CHAPTER 2. BACKGROUND 17 the smoothness term captures our knowledge of the motion field since it varies slowly and expresses the motion of a part of the image which is strongly correlated with the neighboring parts. The smoothness term can be deterministic[13] or probabilistic[25] 4] The motion field reflects the movement of objects within the scene. Therefore the motion field within a part of the image can be a collection of closely correlated motion estimates. Of course this is not true across motion discontinuities and object boundaries. However, ....

....a part of the image can be a collection of closely correlated motion estimates. Of course this is not true across motion discontinuities and object boundaries. However, the work of Dubois and Depommier[7] Horn and Schunck[27] for example have shown that it is a reasonable assumption. Snyder [13] has investigated the mathematical foundations of the smoothness constraint term. He starts with three basic assumptions regarding this term: 1. invariance under a change of the image coordinate system. 2. positive definiteness so that a minimum has to exist. 3. it can be expressed as a linear ....

Snyder M., On the Mathematical Foundation of Smoothness Constraints for the Determination of Optical Flow and for Surface Reconstruction, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 11, November 1991, pp. 1105-1114.

....[41] This procedure is motivated by the observation that object boundaries often coincide with intensity edges. For the same reason, an oriented smoothness constraint has been proposed [68] 67] that applies smoothing only along the direction of a locally constant intensity. Investigations in [78] show that the oriented smoothness constraint is the only plausible one 5 According to Hadamard s definition, a problem is called well posed if it has a unique solution that continuously depends on the data. TO APPEAR IN IEEE SIGNAL PROCESSING MAGAZINE (JULY 1999) 19 among all separable ....

M. Snyder, "On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction," IEEE Trans. Pattern Anal. Machine Intell., vol. 13, pp. 1105--1114, Nov. 1991.

....for instance Sobey and Srinivasan [ 19 ] merge spatial filtering into their basic algorithm which is a differential technique. Some algorithmic components within the pipeline are not amenable to parallelization on medium grain hardware. Relaxation methods have been explored extensively [ 20 ] as a way of propagating results stemming from work by Horn and Schunk [ 21 ] Repeated synchronization and transfer of data within each pixel neighbourhood is necessary making the algorithms particularly appropriate for SIMD processors [ 22 ] or the more recent embodiment of SIMD ....

M. A. Snyder. On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(11):1105--1114, November 1991.

....of the normal flow, many researchers have proposed several algorithms based on offering additional smoothness constraints. These algorithms can be classified into three main groups; regularization based techniques, multigradient equations based technique, and multipoint techniques [3] 2] [26], and [32] 1.1.1 Regularization based techniques (or global optimization) In this approach, the (iterative) solutions can be obtained by minimizing a functional where the additional smoothness constraint is defined over the whole image. The result is dense optical flow fields (i.e. a motion ....

M.A. Snyder. On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction. IEEE transactions on pattern analysis and machine intelligence, 13(11):1105-- 1114, November 1991.

.... in which not the image intensity function itself but functions of it are assumed to be preserved, have been introduced by [Buxton and Buxton, 1984; Waxman et al. 1988] A comparative study of minimization approaches using a smoothness term applied to gradient based techniques can be found in [Snyder, 1989]. A second class of techniques to derive locally available image information is based on correlation. Assuming the conservation of the local intensity distribution, a small window in the first image is searched for a corresponding window in the next image, where the search criterion is maximal ....

M. Snyder. On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction. In Proc. IEEE Workshop on Visual Motion, pages 107--115, 1989.

....Using intensity cues to disable or reduce smoothing across intensity boundaries can only enhance the resulting disparity or motion estimates. The first attempts at incorporating dependence on the observations into the prior were made by Nagel and Enkelmann [6] and more recently by Snyder [7], both in the context of motion estimation. Both Nagel and Snyder use in their definition of a prior term a linear transformation having one eigenvector parallel and the second one perpendicular to the image gradient. Yet, the associated eigenvalues, although different, are constant and as a ....

....goes one step further by yielding diffusion equations which become isotropic in the limit as the image gradient vanishes, and stay otherwise anisotropic with the degree of anisotropy depending on the image gradient magnitude. In that sense, it blends the simple formulation of Nagel [6] and Snyder [7] with the space varying kernel approach of Nitzberg and Shiota [4] 3. PROPOSED APPROACH In the case of disparity or motion estimation, we assume that image intensity boundaries provide strong cues as to where disparity motion boundaries are located. Similarly to others [6] 7] we propose to ....

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M. Snyder, "On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction," IEEE Trans. Pattern Anal. Machine Intell., vol. 13, pp. 1105--1114, Nov. 1991.

....[42] This procedure is motivated by the observation that object boundaries often coincide with intensity edges. For the same reason, an oriented smoothness constraint has been proposed [67, 66] that applies smoothing only along the direction of a locally constant intensity. Investigations in [77] show that the oriented smoothness constraint is the only plausible one among all separable constraints of the same order. As another extension, boundaries present in motion fields have been preserved by nonstationary autoregressive modelling in [27] Explicit estimation of a line process ....

M. Snyder, "On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction," IEEE Trans. Pattern Anal. Machine Intell., vol. 13, pp. 1105--1114, Nov. 1991.

....equation based on a combination of the perspective projection and notions from differential geometry. First order derivatives of image flows were studied by Subbarao [133] A study of smoothness constraints for determination of optical flow and for surface reconstruction was made by Snyder [127]. Willick and Yang [147] examined motion constraint equations for the computation of optical flow. Three constraint equations are presented and studied: the original motion constraint equation by Horn and Schunck, Schunck s equation, and Nagel s equation. Experimental results indicate that the ....

Snyder, M.A., "On the Mathematical Foundations of Smoothness Constraints for the Determination of Optical Flow and for Surface Reconstruction", IEEE Transactions on Pattern Analysis and Machine Vision, vol. 13, no.11, 1991, pp. 1105-1113.

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M. A. Snyder. \On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction". IEEE Transactions on Pattern Analysis and Machine Intelligence, 13:1105-1114, 1991.

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M. A. Snyder. \On the mathematical foundations of smoothness constraints for the determination of optical ow and for surface reconstruction". IEEE Transactions on Pattern Analysis and Machine Intelligence, 13:1105-1114, 1991.

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Snyder, M.A. (1991). On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction. IEEE Trans. PAMI 13 (11), 1105--1114.

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