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41
Area and Length Minimizing Flows for Shape Segmentation
, 1997
"... A number of active contour models have been proposed which unify the curve evolution framework with classical energy minimization techniques for segmentation, such as snakes. The essential idea is to evolve a curve (in 2D) or a surface (in 3D) under constraints from image forces so that it clings to ..."
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Cited by 74 (10 self)
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A number of active contour models have been proposed which unify the curve evolution framework with classical energy minimization techniques for segmentation, such as snakes. The essential idea is to evolve a curve (in 2D) or a surface (in 3D) under constraints from image forces so that it clings to features of interest in an intensity image. Recently the evolution equation has been derived from first principles as the gradient flow that minimizes a modified length functional, tailored to features such as edges. However, because the flow may be slow to converge in practice, a constant (hyperbolic) term is added to keep the curve/surface moving in the desired direction. In this paper, we derive a modification of this term based on the gradient flow derived from a weighted area functional, with image dependent weighting factor. When combined with the earlier modified length gradient flow we obtain a pde which offers a number of advantages, as illustrated by several examples of shape segm...
Image Segmentation by ReactionDiffusion Bubbles
 Proc. ICCV
, 1995
"... FigureGround segmentation is a fundamental problem in computer vision. The main difficulty is the integration of lowlevel, pixelbased local image features to obtain global objectbased descriptions. Active contours in the form of snakes, balloons, and levelset modeling techniques have been propo ..."
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Cited by 52 (2 self)
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FigureGround segmentation is a fundamental problem in computer vision. The main difficulty is the integration of lowlevel, pixelbased local image features to obtain global objectbased descriptions. Active contours in the form of snakes, balloons, and levelset modeling techniques have been proposed that satisfactorily address this question for certain applications. However, these methods require manual initialization, do not always perform well near sharp protrusions or indentations, or often cross gaps. We propose an approach inspired by these methods and a shockbased representation of shape in terms of parts, protrusions, and bends. Since initially it is not clear where the objects or their parts are, parts are hypothesized in the form of fourth order shocks randomly initialized in homogeneous areas of images. These shocks then form evolving contours, or bubbles, which grow, shrink, merge, split and disappear to capture the objects in the image. In the homogeneous areas of the i...
ScaleSpace Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor
 Laboratory of Technomathematics, University of Kaiserslautern, P.O
, 1994
"... In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in recent years. The goal of the present paper is to provide a mathematical foundation for continuous nonlinear diffusion filtering as a scalespace transformation which is f ..."
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Cited by 21 (3 self)
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In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in recent years. The goal of the present paper is to provide a mathematical foundation for continuous nonlinear diffusion filtering as a scalespace transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By studying the Lyapunov functionals, it is shown that nonlinear diffusion reduces L p norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it satisfies an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m dimensional vectorvalued images. The results are juxtaposed to linear and morphological scalespaces. . Keywords....
Shape from shading: Level set propagation and viscosity solutions, Int
 J. Comput. Vision
, 1995
"... ron @ tx.teclmion.ac.il ..."
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 18 (8 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
Non Uniform Multiresolution Method for Optical Flow and Phase Portrait Models: Environmental Applications
 International Journal of Computer Vision
, 1999
"... . In this paper we dene a complete framework for processing large image sequences for a global monitoring of short range oceanographic and atmospheric processes. This framework is based on the use of a non quadratic regularization technique in optical ow computation that preserves ow discontinuities ..."
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Cited by 13 (2 self)
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. In this paper we dene a complete framework for processing large image sequences for a global monitoring of short range oceanographic and atmospheric processes. This framework is based on the use of a non quadratic regularization technique in optical ow computation that preserves ow discontinuities. We also show that using an appropriate tessellation of the image according to an estimate of the motion eld can improve optical ow accuracy and yields more reliable ows. This method denes a non uniform multiresolution approach for coarse to ne grid generation. It allows to locally increase the resolution of the grid according to the studied problem. Each added node renes the grid in a region of interest and increases the numerical accuracy of the solution in this region. We make use of such a method for solving the optical ow equation with a non quadratic regularization scheme allowing the computation of optical ow eld while preserving its discontinuities. The second part of th...
Three Dimensional Generalization of Buildings based on ScaleSpaces',Report, Chair for Photogrammetry and Remote
 Sensing, Technische Universität München.Online: [http://serv.photo.verm.tumuenchen.de/b3d/s98html.html] (1 March 2005) Ordnance Survey 2004, 'OS MasterMap Integrated Transport Network Layer', Online: [http://www.ordnancesurvey.co.uk/oswebsite/products/os
, 1998
"... ..."
Stochastic Differential Equations and Geometric Flows
 IEEE TRANSACTIONS ON IMAGE PROCESSING
"... In recent years curve evolution, applied to a single contour or to the level sets of an image via partial differential equations, has emerged as an important tool in image processing and computer vision. Curve evolution techniques have been utilized in problems such as image smoothing, segmentation, ..."
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Cited by 10 (1 self)
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In recent years curve evolution, applied to a single contour or to the level sets of an image via partial differential equations, has emerged as an important tool in image processing and computer vision. Curve evolution techniques have been utilized in problems such as image smoothing, segmentation, and shape analysis. We give a local stochastic interpretation of the basic curve smoothing equation, the so called geometric heat equation, and show that this evolution amounts to a tangential diffusion movement of the particles along the contour. Moreover, assuming that a priori information about the shapes of objects in an image is known, we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise. We also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features, in contrast to the original geometric heat flow which tends to circularize any closed curve.
ShockBased ReactionDiffusion Bubbles for Image Segmentation
, 1994
"... FigureGround segmentation is a fundamental problem in computer vision. The main difficulty is the integration of lowlevel, pixelbased local image features to obtain global objectbased descriptions. Active contours in the form of snakes, balloons, and levelset modeling techniques have been propos ..."
Abstract

Cited by 8 (2 self)
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FigureGround segmentation is a fundamental problem in computer vision. The main difficulty is the integration of lowlevel, pixelbased local image features to obtain global objectbased descriptions. Active contours in the form of snakes, balloons, and levelset modeling techniques have been proposed that satisfactorily address this question for certain applications. However, these methods require manual initialization, do not always perform well near sharp protrusions or indentations, or often cross gaps. We propose an approach inspired by these methods and a shockbased representation of shape in terms of parts, protrusions, and bends. In this representation parts are related to fourth order shocks. Since initially it is not clear where the objects or their parts are, parts are hypothesized in the form of fourth order shocks randomly initialized in homogeneous areas of images. These shocks then form evolving contours, or bubbles, which grow, shrink, merge, split and disappear to cap...
Toward a Shock Grammar for Recognition
 IEEE Conf. on Computer Vision and Pattern Recognition
, 1995
"... The recognition of objects from their projected twodimensional shapes is a challenging problem owing to the spectrum of possible variations reflected in the image domain, e.g., those caused by movement of parts, changes in viewing geometry, occlusion, etc. This motivates a need for quantitative as ..."
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Cited by 8 (0 self)
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The recognition of objects from their projected twodimensional shapes is a challenging problem owing to the spectrum of possible variations reflected in the image domain, e.g., those caused by movement of parts, changes in viewing geometry, occlusion, etc. This motivates a need for quantitative as well as qualitative descriptions of shape in terms of structural relations between components; the latter remain largely invariant under the above changes. In this paper we confront the theoretical and practical difficulties of computing such a representation, based on the detection of shocks or singularities that arise as a shape is deformed, as organized in two stages. First, we develop subpixel local detectors for the detection of shocks and a classification of them into four types. Second, we show that shock patterns are not arbitrary, but obey the rules of a grammar which limits the possible shock combinations. In addition, shock patterns satisfy specific topological and geometric constraints. We develop this shock grammar and exploit the topological and geometric constraints to enforce global consistency: shock hypotheses that violate the grammar or are topologically or geometrically invalid are pruned, and survivors are organized into higher level structures. The result is a computational method for the detection, classification, and grouping of shocks. This leads to a description of shape as a hierarchical graph of shock groups. The graph is computed in the reactiondiffusion space, where diffusion plays a role of regularization to determine the significance of each shockgroup. The representation is stable with rotations, scale changes, occlusion, movement of parts, noise and other variations, even at very low resolutions. We illustrate the suitability of this repres...