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59
Fast approximate energy minimization via graph cuts
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when v ..."
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Cited by 2127 (61 self)
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In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an αβswap: for a pair of labels α, β, this move exchanges the labels between an arbitrary set of pixels labeled α and another arbitrary set labeled β. Our first algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an αexpansion: for a label α, this move assigns an arbitrary set of pixels the label α. Our second
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...
Efficient GraphBased Energy Minimization Methods In Computer Vision
, 1999
"... ms (we show that exact minimization in NPhard in these cases). These algorithms produce a local minimum in interesting large move spaces. Furthermore, one of them nds a solution within a known factor from the optimum. The algorithms are iterative and compute several graph cuts at each iteration. Th ..."
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Cited by 115 (6 self)
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ms (we show that exact minimization in NPhard in these cases). These algorithms produce a local minimum in interesting large move spaces. Furthermore, one of them nds a solution within a known factor from the optimum. The algorithms are iterative and compute several graph cuts at each iteration. The running time at each iteration is eectively linear due to the special graph structure. In practice it takes just a few iterations to converge. Moreover most of the progress happens during the rst iteration. For a certain piecewise constant prior we adapt the algorithms developed for the piecewise smooth prior. One of them nds a solution within a factor of two from the optimum. In addition we develop a third algorithm which nds a local minimum in yet another move space. We demonstrate the eectiveness of our approach on image restoration, stereo, and motion. For the data with ground truth, our methods signicantly outperform standard methods. Biographical Sketch Olga
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.
BIOLOGICAL IMAGE MOTION PROCESSING: A REVIEW
, 1985
"... Motion as a fundamental visual dimension Functional aspects of image motion processing (I) Encoding of the third dimension (2) Time to collision (TTC) (3) Image segmentation (4) Motion as a proprioceptive sense (5) Motion as a stimulus to drive eye movements (6) Motion as required for pattern vision ..."
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Cited by 79 (0 self)
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Motion as a fundamental visual dimension Functional aspects of image motion processing (I) Encoding of the third dimension (2) Time to collision (TTC) (3) Image segmentation (4) Motion as a proprioceptive sense (5) Motion as a stimulus to drive eye movements (6) Motion as required for pattern vision (7) Image motion processing as useful for perceiving real moving objects Multiplicity of functional roles Motion blindness? Parallel and serial processing within an early motion system: a skeletal model Random dot stimuli D ImAX Dm, Intermediate values of velocity Experiments using sinusoidal gratings Common spacetime framework to account for random dot and grating data Motion hyperacuity Metrical encoding of velocity Fourier domain description of moving images Chromatic input to the motion system? Computational theories of motion processing Early models Recent models Beyond the simple pair? Single cell analysis of image motion Definitional issues Movement, nonmovement and premovement units Motion processing at the extrastriate level Orientation tuning in the motion system An oblique effect for motion? The aperture problem Temporal integration of velocity signals Higherorder computations on the optical flow field Derivatives of velocity Interocular comparison of motion signals: motion in depth
Slow and Smooth: a Bayesian theory for the combination of of local motion signals in human vision
, 1998
"... In order to estimate the motion of an object, the visual system needs to combine multiple local measurements, each of which carries some degree of ambiguity. We present a model of motion perception whereby measurements from dierent image regions are combined according to a Bayesian estimator: the ..."
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Cited by 69 (3 self)
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In order to estimate the motion of an object, the visual system needs to combine multiple local measurements, each of which carries some degree of ambiguity. We present a model of motion perception whereby measurements from dierent image regions are combined according to a Bayesian estimator: the estimated motion maximizes the posterior probability assuming a prior favoring slow and smooth velocities. In reviewing a large number of previously published phenomena we nd that the Bayesian estimator predicts a wide range of psychophysical results. This suggests that the seemingly complex set of illusions arise from a single computational strategy that is optimal under reasonable assumptions. 1 Introduction Estimating motion in scenes containing multiple, complex motions remains a dicult problem for computer vision systems, yet is performed eortlessly by human observers. Motion analysis in such scenes imposes conicting demands on the design of a vision system (Braddick, 1993)....
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.
Recursive Filters for Optical Flow
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1995
"... : Working toward ecient (realtime) implementations of optical ow methods, we have applied simple recursive lters to achieve temporal smoothing and dierentiation of image intensity, and to compute 2d ow from component velocity constraints using spatiotemporal leastsquares minimization. Accuracy in ..."
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Cited by 45 (1 self)
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: Working toward ecient (realtime) implementations of optical ow methods, we have applied simple recursive lters to achieve temporal smoothing and dierentiation of image intensity, and to compute 2d ow from component velocity constraints using spatiotemporal leastsquares minimization. Accuracy in simulation is similar to that obtained in the study by Barron et al. [3], while requiring much less storage, less computation, and shorter delays. 1 Introduction Many methods exist for computing optic ow, but few currently run at frame rates on reasonably priced, conventional hardware. The goal of this paper is to outline simplications to a successful gradientbased approach that reduce computational expense with little degradation in accuracy. Our specic concerns include temporal smoothing and dierentiation of image intensity, and temporal integration of component velocity constraints to solve for 2d velocity. More generally, we are working toward ecient implementations of dierent...
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 39 (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...