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73
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 1400 (52 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
Bayesian Interpolation
 Neural Computation
, 1991
"... Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison is demonstrated by studying the inference problem of interpolating noisy data. T ..."
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Cited by 523 (18 self)
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Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison is demonstrated by studying the inference problem of interpolating noisy data. The concepts and methods described are quite general and can be applied to many other problems. Regularising constants are set by examining their posterior probability distribution. Alternative regularisers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. `Occam's razor' is automatically embodied by this framework. The way in which Bayes infers the values of regularising constants and noise levels has an elegant interpretation in terms of the effective number of parameters determined by the data set. This framework is due to Gull and Skilling. 1 Data modelling and Occam's razor In science, a central task is to develop and compare models to a...
Regularization networks and support vector machines
 Advances in Computational Mathematics
, 2000
"... Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization a ..."
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Cited by 269 (33 self)
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Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik’s theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case.
Bayesian color constancy
 Journal of the Optical Society of America A
, 1997
"... The problem of color constancy may be solved if we can recover the physical properties of illuminants and surfaces from photosensor responses. We consider this problem within the framework of Bayesian decision theory. First, we model the relation among illuminants, surfaces, and photosensor response ..."
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Cited by 138 (18 self)
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The problem of color constancy may be solved if we can recover the physical properties of illuminants and surfaces from photosensor responses. We consider this problem within the framework of Bayesian decision theory. First, we model the relation among illuminants, surfaces, and photosensor responses. Second, we construct prior distributions that describe the probability that particular illuminants and surfaces exist in the world. Given a set of photosensor responses, we can then use Bayes’s rule to compute the posterior distribution for the illuminants and the surfaces in the scene. There are two widely used methods for obtaining a single best estimate from a posterior distribution. These are maximum a posteriori (MAP) and minimum meansquarederror (MMSE) estimation. We argue that neither is appropriate for perception problems. We describe a new estimator, which we call the maximum local mass (MLM) estimate, that integrates local probability density. The new method uses an optimality criterion that is appropriate for perception tasks: It finds the most probable approximately correct answer. For the case of low observation noise, we provide an efficient approximation. We develop the MLM estimator for the colorconstancy problem in which flat matte surfaces are uniformly illuminated. In simulations we show that the MLM method performs better than the MAP estimator and better than a number of standard colorconstancy algorithms. We note conditions under which even the optimal estimator produces poor estimates: when the spectral properties of the surfaces in the scene are biased. © 1997 Optical Society of America [S07403232(97)016074] 1.
Hierarchic Voronoi Skeletons
, 1995
"... Robust and timeefficient skeletonization of a (planar) shape, which is connectivity preserving and based on Euclidean metrics, can be achieved by first regularizing the Voronoi diagram (VD) of a shape's boundary points, i.e., by removal of noisesensitive parts of the tessellation and then by estab ..."
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Cited by 123 (3 self)
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Robust and timeefficient skeletonization of a (planar) shape, which is connectivity preserving and based on Euclidean metrics, can be achieved by first regularizing the Voronoi diagram (VD) of a shape's boundary points, i.e., by removal of noisesensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents. Each component of the VD is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise. The second processing step, a hierarchic clustering of skeleton branches, leads to a multiresolution representation of the skeleton, termed skeleton pyramid.
ObjectCentered Surface Reconstruction: Combining MultiImage Stereo and Shading
 International Journal of Computer Vision
, 1995
"... Our goal is to reconstruct both the shape and reflectance properties of surfaces from multiple images. We argue that an objectcentered representation is most appropriate for this purpose because it naturally accommodates multiple sources of data, multiple images (including motion sequences of a rig ..."
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Cited by 120 (19 self)
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Our goal is to reconstruct both the shape and reflectance properties of surfaces from multiple images. We argue that an objectcentered representation is most appropriate for this purpose because it naturally accommodates multiple sources of data, multiple images (including motion sequences of a rigid object), and selfocclusions. We then present a specific objectcentered reconstruction method and its implementation. The method begins with an initial estimate of surface shape provided, for example, by triangulating the result of conventional stereo. The surface shape and reflectance properties are then iteratively adjusted to minimize an objective function that combines information from multiple input images. The objective function is a weighted sum of stereo, shading, and smoothness components, where the weight varies over the surface. For example, the stereo component is weighted more strongly where the surface projects onto highly textured areas in the images, and less strongly othe...
Single Lens Stereo with a Plenoptic Camera
, 1992
"... Ordinary cameras gather light across the area of their lens aperture, and the light striking a given subregion of the aperture is structured somewhat differently than the light striking an adjacent subregion. By analyzing this optical structure, one can infer the depths of objects in the scene, i.e. ..."
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Cited by 116 (0 self)
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Ordinary cameras gather light across the area of their lens aperture, and the light striking a given subregion of the aperture is structured somewhat differently than the light striking an adjacent subregion. By analyzing this optical structure, one can infer the depths of objects in the scene, i.e., one can achieve "single lens stereo." We describe a novel camera for performing this analysis. It incorporates a single main lens along with a lenticular array placed at the sensor plane. The resulting "plenoptic camera" provides information about how the scene would look when viewed from a continuum of possible viewpoints bounded by the main lens aperture. Deriving depth information is simpler than in a binocular stereo system because the correspondence problem is minimized. The camera extracts information about both horizontal and vertical parallax, which improves the reliability of the depth estimates.
A theory of cortical responses
, 2005
"... This article concerns the nature of evoked brain responses and the principles underlying their generation. We start with the premise that the sensory brain has evolved to represent or infer the causes of changes in its sensory inputs. The problem of inference is well formulated in statistical terms. ..."
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Cited by 101 (21 self)
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This article concerns the nature of evoked brain responses and the principles underlying their generation. We start with the premise that the sensory brain has evolved to represent or infer the causes of changes in its sensory inputs. The problem of inference is well formulated in statistical terms. The statistical fundaments of inference may therefore afford important constraints on neuronal implementation. By formulating the original ideas of Helmholtz on perception, in terms of modernday statistical theories, one arrives at a model of perceptual inference and learning that can explain a remarkable range of neurobiological facts. It turns out that the problems of inferring the causes of sensory input (perceptual inference) and learning the relationship between input and cause (perceptual learning) can be resolved using exactly the same principle. Specifically, both inference and learning rest on minimizing the brain’s free energy, as defined in statistical physics. Furthermore, inference and learning can proceed in a biologically plausible fashion. Cortical responses can be seen as the brain’s attempt to minimize the free energy induced by a stimulus and thereby encode the most likely cause of that stimulus. Similarly, learning emerges from changes in synaptic efficacy that minimize the free energy, averaged over all stimuli encountered. The underlying scheme rests on empirical Bayes and hierarchical models
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 82 (5 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