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Toward a theory of shape from specular flow
 In IEEE International Conference on Computer Vision
, 2007
"... Abstract The image of a curved, specular (mirrorlike) ..."
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Cited by 33 (6 self)
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Abstract The image of a curved, specular (mirrorlike)
Manifold models for signals and images
 COMPUTER VISION AND IMAGE UNDERSTANDING
, 2009
"... This article proposes a new class of models for natural signals and images. The set of patches extracted from the data to analyze is constrained to be close to a low dimensional manifold. This manifold structure is detailed for various ensembles suitable for natural signals, images and textures mode ..."
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Cited by 29 (3 self)
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This article proposes a new class of models for natural signals and images. The set of patches extracted from the data to analyze is constrained to be close to a low dimensional manifold. This manifold structure is detailed for various ensembles suitable for natural signals, images and textures modeling. These manifolds provide a lowdimensional parameterization of the local geometry of these datasets. These manifold models can be used to regularize inverse problems in signal and image processing. The restored signal is represented as a smooth curve or surface traced on the manifold that matches the forward measurements. A manifold pursuit algorithm computes iteratively a solution of the manifold regularization problem. Numerical simulations on inpainting and compressive sensing inversion show that manifolds models bring an improvement for the recovery of data with geometrical features. Key words: signal processing, image modeling, texture, manifold. PACS: code, code Capturing the complex geometry of signals and images is at the core of recent advances in sound and natural image processing. Edges and texture patterns create complex nonlocal interactions. This paper studies these geometries for several sounds, images and textures models. The set of local patches in the dataset is modeled using smooth manifolds. These local features trace a continuous curve (resp. surface) on the manifold, which is a prior that can be used to solve inverse problems.
Hyperbolic planforms in relation to visual edges and textures perception. Plos Computational Biology, 2009. Accepted for publication 11/04/2009
"... We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g., optical imaging, an ..."
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Cited by 25 (13 self)
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We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g., optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific secondorder tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the nonEuclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or Hplanforms that are reminiscent of Euclidean planar waves and of the planforms that were used in previous work to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the builtin or acquired invariance of the neural organization to the action of the corresponding subgroups.
Curvaturedriven PDE methods for matrixvalued images
, 2004
"... Abstract. Matrixvalued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DTMRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edgelike structures ..."
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Cited by 20 (9 self)
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Abstract. Matrixvalued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DTMRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edgelike structures in tensor fields, we first generalise Di Zenzo’s concept of a structure tensor for vectorvalued images to tensorvalued data. This structure tensor allows us to extend scalarvalued mean curvature motion and selfsnakes to the tensor setting. We present both twodimensional and threedimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on threedimensional DTMRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalarvalued counterparts. Keywords: DTMRI, denoising, segmentation, edge detection, structure tensor, mean curvature motion, selfsnakes, active contours
Geometric Rectification of Cameracaptured Document Images
, 2007
"... Compared to typical scanners, handheld cameras offer convenient, flexible, portable, and noncontact image capture, which enables many new applications and breathes new life into existing ones. However, cameracaptured documents may suffer from distortions caused by nonplanar document shape and per ..."
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Cited by 18 (1 self)
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Compared to typical scanners, handheld cameras offer convenient, flexible, portable, and noncontact image capture, which enables many new applications and breathes new life into existing ones. However, cameracaptured documents may suffer from distortions caused by nonplanar document shape and perspective projection, which lead to failure of current OCR technologies. We present a geometric rectification framework for restoring the frontalflat view of a document from a single cameracaptured image. Our approach estimates 3D document shape from texture flow information obtained directly from the image without requiring additional 3D/metric data or prior camera calibration. Our framework provides a unified solution for both planar and curved documents and can be applied in many, especially mobile, camerabased document analysis applications. Experiments show that our method produces results that are significantly more OCR compatible than the original images. Index Terms Camerabased OCR, image rectification, shape estimation, texture flow analysis.
Surface Geometric Constraints for Stereo in Belief Propagation
, 2006
"... Belief propagation has been shown to be a powerful inference mechanism for stereo correspondence. However the classical formulation of belief propagation implicitly imposes the frontal parallel plane assumption in the compatibility matrix for exploiting contextual information, since the priors perfe ..."
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Cited by 15 (1 self)
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Belief propagation has been shown to be a powerful inference mechanism for stereo correspondence. However the classical formulation of belief propagation implicitly imposes the frontal parallel plane assumption in the compatibility matrix for exploiting contextual information, since the priors perfer no depth (disparity) change in surrounding neighborhoods. This results in systematic errors for slanted or curved surfaces. To eliminate these errors we propose to use contextual information geometrically, and show how to encode surface differential geometric properties in the compatibility matrix for stereo correspondence. This enforces consistency for both depth and surface normal, extending the traditional formulation beyond consistency for (constant) depth. With such geometric contextual information, the belief propagation algorithm shows dramatic improvement on generic nonfrontal parallel scenes. Several such examples are provided.
Texture Synthesis with Grouplets
, 2009
"... This paper proposes a new method to synthesize and inpaint geometric textures. The texture model is composed of a geometric layer that drives the computation of a new grouplet transform. The geometry is an orientation flow that follows the patterns of the texture to analyze or synthesize. The groupl ..."
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Cited by 10 (0 self)
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This paper proposes a new method to synthesize and inpaint geometric textures. The texture model is composed of a geometric layer that drives the computation of a new grouplet transform. The geometry is an orientation flow that follows the patterns of the texture to analyze or synthesize. The grouplet transform extends the original construction of Mallat [1] and is adapted to the modeling of natural textures. Each grouplet atoms is an elongated stroke located along the geometric flow. These atoms exhibit a wide range of lengths and widths, which is important to match the variety of structures present in natural images. Statistical modeling and sparsity optimization over these grouplet coefficients enable the synthesis of texture patterns along the flow. This article explores texture inpainting and texture synthesis, which both require the joint optimization of the geometric flow and the grouplet coefficients.
Hue fields and color curvatures: A perceptual organization approach to color image denoising
 IEEE conference on CVPR
, 2003
"... The denoising of color images is an increasingly studied problem whose stateoftheart solutions employ a variety of diffusion schemes. Specifying the correct diffusion is difficult, however, in part because of the subtleties of color interactions. We address this difficulty by proposing a perceptu ..."
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Cited by 9 (3 self)
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The denoising of color images is an increasingly studied problem whose stateoftheart solutions employ a variety of diffusion schemes. Specifying the correct diffusion is difficult, however, in part because of the subtleties of color interactions. We address this difficulty by proposing a perceptual organization approach to color denoising based on the principle of good continuation. We exploit the periodic chromatic (hue) component of the color in its representation as a frame field. We derive two hue curvatures and use them to construct a local model for the behavior of the color, which in turn specifies consistency constraints between nearby color measurements. These constraints are then used to replace noisy pixels by examining their spatial context. Such a contextual analysis (combined with standard methods to handle the scalar channels, saturation and lightness), results in a robust noise removal process that preserves discontinuities, singularities, and fine chromatic structures, including those that diffusion processes are prone to distort. We demonstrate our approach on a variety of synthetic and natural images. 1.
Contextual Inference in ContourBased Stereo Correspondence
, 2006
"... Standard approaches to stereo correspondence have difficulty when scene structure does not lie in or near the frontal parallel plane, in part because an orientation disparity as well as a positional disparity is introduced. We propose a correspondence algorithm based on differential geometry, that t ..."
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Cited by 9 (3 self)
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Standard approaches to stereo correspondence have difficulty when scene structure does not lie in or near the frontal parallel plane, in part because an orientation disparity as well as a positional disparity is introduced. We propose a correspondence algorithm based on differential geometry, that takes explicit advantage of both disparities. The algorithm relates the 2D differential structure (position, tangent, and curvature) of curves in the left and right images to the Frenet approximation of the (3D) space curve. A compatibility function is defined via transport of the Frenet frames, and they are matched by relaxing this compatibility function on overlapping neighborhoods along the curve. The remaining false matches are concurrently eliminated by a model of "near" and "far" neurons derived from neurobiology. Examples on scenes with complex 3D structures are provided.
Shape from Specular Flow
"... Abstract—An image of a specular (mirrorlike) object is nothing but a distorted reflection of its environment. When the environment is unknown, reconstructing shape from such an image can be very difficult. This reconstruction task can be made tractable when, instead of a single image, one observes ..."
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Cited by 8 (1 self)
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Abstract—An image of a specular (mirrorlike) object is nothing but a distorted reflection of its environment. When the environment is unknown, reconstructing shape from such an image can be very difficult. This reconstruction task can be made tractable when, instead of a single image, one observes relative motion between the specular object and its environment, and therefore, a motion field—or specular flow—in the image plane. In this paper, we study the shape from specular flow problem and show that observable specular flow is directly related to surface shape through a nonlinear partial differential equation. This equation has the key property of depending only on the relative motion of the environment while being independent of its content. We take first steps toward understanding and exploiting this PDE, and we examine its qualitative properties in relation to shape geometry. We analyze several cases in which the surface shape can be recovered in closed form, and we show that, under certain conditions, specular shape can be reconstructed when both the relative motion and the content of the environment are unknown. We discuss numerical issues related to the proposed reconstruction algorithms, and we validate our findings using both real and synthetic data. Index Terms—Specular objects, specular flow, shape reconstruction, environment motion field, Gaussian curvature, parabolic points, specular curvature. Ç 1