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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 98 (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.
Shapeadapted smoothing in estimation of 3D shape cues from affine distortions of local 2D brightness structure
, 2001
"... This article describes a method for reducing the shape distortions due to scalespace smoothing that arise in the computation of 3D shape cues using operators (derivatives) de ned from scalespace representation. More precisely, we are concerned with a general class of methods for deriving 3D shap ..."
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Cited by 64 (3 self)
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This article describes a method for reducing the shape distortions due to scalespace smoothing that arise in the computation of 3D shape cues using operators (derivatives) de ned from scalespace representation. More precisely, we are concerned with a general class of methods for deriving 3D shape cues from 2D image data based on the estimation of locally linearized deformations of brightness patterns. This class
Velocity Likelihoods in Biological and Machine Vision
 In Probabilistic Models of the Brain: Perception and Neural Function
, 2001
"... Recent approaches to estimating twodimensional image motion and to modeling the perception of image motion have achieved success with Bayesian formulations. With a Bayesian approach, the goal is to compute the posterior probability distribution of velocity, which is proportional to a likelihood ..."
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Cited by 38 (4 self)
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Recent approaches to estimating twodimensional image motion and to modeling the perception of image motion have achieved success with Bayesian formulations. With a Bayesian approach, the goal is to compute the posterior probability distribution of velocity, which is proportional to a likelihood function and a prior. The likelihood function describes the probability of observing the image data given the image velocity; surprisingly, there is still disagreement about the right likelihood function to use. Here we derive a likelihood function by starting from a generative model. We assume that the scene translates, conserving image brightness, while the image is equal to the projected scene plus noise. We discuss the connection between the resulting likelihood function and existing models of motion analysis. We show that the likelihood can be calculated by a population of units whose response properties are similar to \motion energy" units. This suggests that a population o...
Computing Optic Flow by ScaleSpace Integration of Normal Flow
 in `Proc. ScaleSpace'01
, 2001
"... In this paper we will present a least committed multiscale method for computation of optic flow fields. We extract optic flow fields from normal flow, by fitting the normal components of a local polynomial model of the optic flow to the normal flow. This fitting is based on an analytically solvable ..."
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Cited by 4 (0 self)
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In this paper we will present a least committed multiscale method for computation of optic flow fields. We extract optic flow fields from normal flow, by fitting the normal components of a local polynomial model of the optic flow to the normal flow. This fitting is based on an analytically solvable optimization problem, in which an integration scalespace over the normal flow field regularizes the solution. An automatic local scale selection mechanism is used in order to adapt to the local structure of the flow field. The performance profile of the method is compared with that of existing optic flow techniques and we show that the proposed method performs at least as well as the leading algorithms on the benchmark image sequences proposed by Barron et al. [3]. We also do a performance comparison on a synthetic fire particle sequence and apply our method to a real sequence of smoke circulation in a pigsty. Both consist of highly complex nonrigid motion.
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 4 (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.
Evaluating OpticalFlow Algorithms on a Parallel Machine
 Image and Vision Comp
"... Algorithmic development of opticalow routines is hampered by slow turnaround times (to iterate over testing, evaluation, and adjustment of the algorithm). To ease the problem, parallel implementation on a convenient generalpurpose parallel machine is possible. A generic parallel pipeline struct ..."
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Algorithmic development of opticalow routines is hampered by slow turnaround times (to iterate over testing, evaluation, and adjustment of the algorithm). To ease the problem, parallel implementation on a convenient generalpurpose parallel machine is possible. A generic parallel pipeline structure, suitable for distributedmemory machines, has enabled parallelisation to be quickly achieved. Gradient, correlation, and phasebased methods of opticalow detection have been constructed to demonstrate the approach.
unknown title
, 2001
"... Shapeadapted smoothing in estimation of 3D shape cues from aÆne distortions of local 2D brightness structure ..."
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Shapeadapted smoothing in estimation of 3D shape cues from aÆne distortions of local 2D brightness structure
14 Bayesian MultiScale Dierential Optical Flow
"... 14.2 Dierential formulation..................... 398 14.3 Uncertainty Model........................ 400 14.4 Coarsetone estimation.................... 404 14.5 Implementation issues...................... 411 ..."
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14.2 Dierential formulation..................... 398 14.3 Uncertainty Model........................ 400 14.4 Coarsetone estimation.................... 404 14.5 Implementation issues...................... 411
1The Computation of Optical Flow
"... Twodimensional image motion is the projection of the threedimensional motion of objects, relative to a visual sensor, onto its image plane. Sequences of timeordered images allow the estimation of projected twodimensional image motion as either instantaneous image velocities or discrete image dis ..."
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Twodimensional image motion is the projection of the threedimensional motion of objects, relative to a visual sensor, onto its image plane. Sequences of timeordered images allow the estimation of projected twodimensional image motion as either instantaneous image velocities or discrete image displacements. These are usually called the optical
ow eld or the image velocity eld. Provided that optical
ow is a reliable approximation to twodimensional image motion, it may then be used to recover the threedimensional motion of the visual sensor (to within a scale factor) and the threedimensional surface structure (shape or relative depth) through assumptions concerning the structure of the optical
ow eld, the threedimensional environment and the motion of the sensor. Optical
ow may also be used to perform motion detection, object segmentation, timetocollision and focus of expansion calculations, motion compensated encoding and stereo disparity measurement. We investigate the computation of optical
ow in this survey: widely known methods for estimating optical
ow are classied and examined by scrutinizing the hypotheses and