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Reliable Estimation of Dense Optical Flow Fields with Large Displacements
, 2001
"... In this paper we show that a classic optical ow technique by Nagel and Enkelmann (1986) can be regarded as an early anisotropic diusion method with a diusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and ..."
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Cited by 122 (14 self)
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In this paper we show that a classic optical ow technique by Nagel and Enkelmann (1986) can be regarded as an early anisotropic diusion method with a diusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and the smoothness term in dierent images, (ii) use a linear scalespace focusing strategy from coarse to ne scales for avoiding convergence to physically irrelevant local minima, and (iii) create an energy functional that is invariant under linear brightness changes. Applying a gradient descent method to the resulting energy functional leads to a system of diusion{reaction equations. We prove that this system has a unique solution under realistic assumptions on the initial data, and we present an ecient linear implicit numerical scheme in detail. Our method creates ow elds with 100 % density over the entire image domain, it is robust under a large range of parameter variations, and it c...
A Theoretical Framework for Convex Regularizers in PDEBased Computation of Image Motion
, 2000
"... Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness consta ..."
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Cited by 99 (25 self)
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Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for datadriven and flowdriven, isotropic and anisotropic, as well as spatial and spatiotemporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are wellposed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flowdriven regularizers is identified, and a design criterion is proposed for constructing anisotropic flowdriven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.
Diffusion and Regularization of Vector and MatrixValued Images
, 2002
"... The goal of this paper is to present a unified description of diffusion and regularization techniques for vectorvalued as well as matrixvalued data fields. In the vectorvalued setting, we first review a number of existing methods and classify them into linear and nonlinear as well as isotropic an ..."
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Cited by 55 (16 self)
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The goal of this paper is to present a unified description of diffusion and regularization techniques for vectorvalued as well as matrixvalued data fields. In the vectorvalued setting, we first review a number of existing methods and classify them into linear and nonlinear as well as isotropic and anisotropic methods. For these approaches we present corresponding regularization methods. This taxonomy is applied to the design of regularization methods for variational motion analysis in image sequences. Our vectorvalued framework is then extended to the smoothing of positive semidefinite matrix fields. In this context a novel class of anisotropic di usion and regularization methods is derived and it is shown that suitable algorithmic realizations preserve the positive semidefiniteness of the matrix field without any additional constraints. As an application, we present an anisotropic nonlinear structure tensor and illustrate its advantages over the linear structure tensor.
Accurate optical flow computation under nonuniform brightness varations
 Computer Vision and Image Understanding
, 2005
"... In this paper, we present a very accurate algorithm for computing optical
ow with nonuniform brightness variations. The proposed algorithm is based on a generalized dynamic image model (GDIM) in conjunction with a regularization framework to cope with the problem of nonuniform brightness variati ..."
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Cited by 7 (0 self)
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In this paper, we present a very accurate algorithm for computing optical
ow with nonuniform brightness variations. The proposed algorithm is based on a generalized dynamic image model (GDIM) in conjunction with a regularization framework to cope with the problem of nonuniform brightness variations. To alleviate
ow constraint errors due to image aliasing and noise, we employ a reweighted leastsquares method to suppress unreliable
ow constraints, thus leading to robust estimation of optical
ow. In addition, a dynamic smoothness adjustment scheme is proposed to eciently suppress the smoothness constraint in the vicinity of the motion and brightness variation discontinuities, thereby preserving motion boundaries. We also employ a constraint renement scheme, which aims at reducing the approximation errors in the rstorder dierential
ow equation, to rene the optical
ow estimation especially for large image motions. To eciently minimize the resulting energy function for optical ow computation, we utilize an incomplete Cholesky preconditioned conjugate gradient algorithm to solve the large linear system. Experimental results on some synthetic and real image sequences show that the proposed algorithm compares favorably to most existing techniques
A Theoretical Framework for Convex Regularizers in PDEBased Computation of Image Motion
, 2000
"... Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness consta ..."
Abstract

Cited by 5 (1 self)
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Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for datadriven and flowdriven, isotropic and anisotropic, as well as spatial and spatiotemporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are wellposed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flowdriven regularizers is identified, and a design criterion is proposed for constructing anisotropic flowdriven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.
Internal Tutorial: Computing 2D and 3D Optical Flow. J.L.Barron and N.A.Thacker.
"... Optical ow is an approximation of the local image motion based upon local derivatives in a given sequence of images. That is, in 2D it species how much each image pixel moves between adjacent images while in 3D in species how much each volume voxel moves between adjacent volumes. The 2D image sequen ..."
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Optical ow is an approximation of the local image motion based upon local derivatives in a given sequence of images. That is, in 2D it species how much each image pixel moves between adjacent images while in 3D in species how much each volume voxel moves between adjacent volumes. The 2D image sequences used here are formed under perspective projection via the relative motion of a camera and scene objects. The 3D volume sequences