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394
Geodesic Active Contours
, 1997
"... A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both in ..."
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Cited by 1427 (47 self)
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A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes ” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.
Robust Anisotropic Diffusion
, 1998
"... Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edgestopping" function in the ani ..."
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Cited by 363 (18 self)
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Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edgestopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edgestopping" function based on Tukey's biweight robust estimator, that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the ...
Gradient flows and geometric active contour models
 in Proc. of the 5th International Conference on Computer Vision
, 1995
"... In this paper, we analyze the geometric active contour models discussed in [6, 181 from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel snake paradigm in which the feature of interes ..."
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Cited by 239 (18 self)
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In this paper, we analyze the geometric active contour models discussed in [6, 181 from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus the snake is attracted very naturally and eficiently to the desired feature. Moreover, we consider some 30 active surface models based on these ideas. 1
A geometrical framework for low level vision
 IEEE Trans. on Image Processing
, 1998
"... Abstract—We introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented. We consider intensity images as surfaces in the space. The image is, thereby, a twodimensional (2D) surface in threedimensional (3D) space for graylevel images, an ..."
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Cited by 223 (35 self)
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Abstract—We introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented. We consider intensity images as surfaces in the space. The image is, thereby, a twodimensional (2D) surface in threedimensional (3D) space for graylevel images, and 2D surfaces in five dimensions for color images. The new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast, and results in new and efficient schemes. Extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms. Index Terms — Color image processing, image enhancement, image smoothing, nonlinear image diffusion, scalespace. I.
Scalespace theory: A basic tool for analysing structures at different scales
 Journal of Applied Statistics
, 1994
"... An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims at describing the structure of unknown realworld signals, then a multiscale representation of data is of crucial importance. This chapter gives a tutorial review of ..."
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Cited by 167 (11 self)
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An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims at describing the structure of unknown realworld signals, then a multiscale representation of data is of crucial importance. This chapter gives a tutorial review of a special type of multiscale representation, linear scalespace representation, which has been developed by thecomputer vision community in order to handle image structures at di erent scales in a consistent manner. The basic idea is to embed the original signal into a oneparameter family of gradually smoothed signals, in which the ne scale details are successively suppressed. Under rather general conditions on the type of computations that are to performed at the rst stages of visual processing, in what can be termed the visual front end, it can be shown that the Gaussian kernel and its derivatives are singled out as the only possible smoothing kernels. The conditions that
Implicit, Nonparametric Shape Reconstruction from Unorganized Points Using A Variational Level Set Method
 Computer Vision and Image Understanding
, 1998
"... In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, ..."
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Cited by 156 (22 self)
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In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, and/or curves, and/or surface patches. We do not assume any knowledge of the ordering, connectivity or topology of the data sets or of the true shape. Only the location of each point or general Hausdorff distance to the data set is known. The key idea in our approach is to find an implicit nonparametric representation of the curve or surface on a fixed rectangular grid. With this representation of surfaces we can easily (a) find the closest point and distance from any point to the surface (useful in illumination and many other applications), (b) find the intersection curve of two surfaces which is guaranteed to lie on both surfaces in our representation, and (c) perform any Boolean operatio...
Fast surface reconstruction using the level set method
 In VLSM ’01: Proceedings of the IEEE Workshop on Variational and Level Set Methods
, 2001
"... In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data ..."
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Cited by 149 (12 self)
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In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surfacelike model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.
CoherenceEnhancing Diffusion Filtering
, 1999
"... The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operato ..."
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Cited by 137 (3 self)
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The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operator (secondmoment matrix, structure tensor). An mdimensional formulation of this method is analysed with respect to its wellposedness and scalespace properties. An efficient scheme is presented which uses a stabilization by a semiimplicit additive operator splitting (AOS), and the scalespace behaviour of this method is illustrated by applying it to both 2D and 3D images.
Conformal Curvature Flows: From Phase Transitions to Active Vision
, 1995
"... In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be consid ..."
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Cited by 137 (34 self)
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In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edgeseeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the AllenCahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3D active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a levelset approach. Key words: Active vision, antiphase boundary, visual tracking, edge detection, segmentation, gradient flows, Riemannian metrics, viscosity solutions, geometric heat equ...
An Axiomatic Approach to Image Interpolation
, 1998
"... We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The ..."
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Cited by 128 (8 self)
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We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range.