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39
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 1073 (43 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.
Global Minimum for Active Contour Models: A Minimal Path Approach
, 1997
"... A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model’s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the ..."
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Cited by 196 (65 self)
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A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model’s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the internal regularization term in the external potential term. Our method is based on finding a path of minimal length in a Riemannian metric. We then make use of a new efficient numerical method to find this shortest path. It is shown that the proposed energy, though based only on a potential integrated along the curve, imposes a regularization effect like snakes. We explore the relation between the maximum curvature along the resulting contour and the potential generated from the image. The method is capable to close contours, given only one point on the objects’ boundary by using a topologybased saddle search routine. We show examples of our method applied to real aerial and medical images.
Greedy optimal homotopy and homology generators
 Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms
, 2005
"... Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops t ..."
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Cited by 88 (12 self)
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Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdi`ere and Lazarus.
Fast extraction of minimal paths in 3D images and applications to virtual endoscopy
, 2001
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Minimal Surfaces Based Object Segmentation
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1997
"... A geometric approach for 3D object segmentation and representation is presented. The segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3D image. The model is based on curvature motion and the computation of surfaces with minimal areas, better known as m ..."
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Cited by 70 (13 self)
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A geometric approach for 3D object segmentation and representation is presented. The segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3D image. The model is based on curvature motion and the computation of surfaces with minimal areas, better known as minimal surfaces. The space where the surfaces are computed is induced from the 3D image (volumetric data) in which the objects are to be detected. The model links between classical deformable surfaces obtained via energy minimization, and intrinsic ones derived from curvature based flows. The new approach is stable, robust, and automatically handles changes in the surface topology during the deformation. Index Terms3D segmentation, minimal surfaces, deformable models, mean curvature motion, medical images.  F  1I NTRODUCTION ONE of the basic problems in image analysis is object detection. It can be associated with the problem of boundary detection, when boundaries are defined as curves or surfaces separating homogeneous regions. "Snakes," or active contours, were proposed by Kass et al. in [16] to solve this problem, and were later extended to 3D surfaces. The classical snakes and 3D deformable surfaces approach are based on deforming an initial contour or surface towards the boundary of the object to be detected. The deformation is obtained by minimizing a functional designed so that its (local) minima is at the boundary of the object [3], [33]. The energy usually involves two terms, one that controls the smoothness of the surface and the other that attracts it to the object's boundary. The topology of the final surface is, in general, as that of the initial one, unless special procedures are used to detect possible spli...
Optimal Algorithm for Shape from Shading and Path Planning
, 2001
"... An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A cons ..."
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Cited by 68 (3 self)
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An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A consistent numerical scheme based on Sethian’s fast marching method is used to compute the reconstructed surface. The surface is a viscosity solution of an Eikonal equation for the vertical light source case. For the oblique light source case, the reconstructed surface is the viscosity solution to a different partial differential equation. A modification of the fast marching method yields a numerically consistent, computationally optimal, and practically fast algorithm for the classical shape from shading problem. Next, the fast marching method coupled with a back tracking via gradient descent along the reconstructed surface is shown to solve the path planning problem in robot navigation.
Image segmentation using deformable models
 Handbook of Medical Imaging. Vol.2 Medical Image Processing and Analysis
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ShapeBased 3D Surface Correspondence Using Geodesics and Local Geometry
, 2000
"... This paper describes a new method for determining correspondence between points on pairs of surfaces based on shape using a combination of geodesic distance and surface curvature. An initial sparse set of corresponding points are generated using a shapebased matching procedure. Geodesic interpolatio ..."
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Cited by 60 (4 self)
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This paper describes a new method for determining correspondence between points on pairs of surfaces based on shape using a combination of geodesic distance and surface curvature. An initial sparse set of corresponding points are generated using a shapebased matching procedure. Geodesic interpolation is employed in order to capture the complex surface. In addition, surface correspondence and triangulation are computed simultaneously in a hierarchical way. Results applied to human cerebral cortical surfaces are shown to evaluate the approach. 1 Introduction Determining the correspondence of 3D points between pairs of surfaces has many important applications such as for comparing shape between deformable objects, nonrigid registration, developing probabilistic models and atlases, etc. While shape provides the basis for such a correspondence, this problem remains a di#cult one due to ambiguity when the surfaces are complex and variable, such as with the human cerebral cortex. In additi...
Multiple Contour Finding and Perceptual Grouping as a set of Energy Minimizing Paths
 Journal of Mathematical Imaging and Vision
, 2001
"... We address the problem of finding a set of contour curves in an image. ..."
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Cited by 41 (30 self)
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We address the problem of finding a set of contour curves in an image.