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  • 1
    Electronic Resource
    Electronic Resource
    Minimal surfaces: a geometric three dimensional segmentation approach (1997)
    Springer
    Numerische Mathematik 77 (1997), S. 423-451 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):53A10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050294
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  • 2
    Electronic Resource
    Electronic Resource
    A geometric model for active contours in image processing (1993)
    Springer
    Numerische Mathematik 66 (1993), S. 1-31 
    Details
    ISSN: 0945-3245
    Keywords: 49F22 ; 53A10 ; 82A60 ; 76T05 ; 49A50 ; 80A15 ; 40F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We propose a new model for active contours based on a geometric partial differential equation. Our model is intrinsec, stable (satisfies the maximum principle) and permits a rigorous mathematical analysis. It enables us to extract smooth shapes (we cannot retrieve angles) and it can be adapted to find several contours simultaneously. Moreover, as a consequence of the stability, we can design robust algorithms which can be engineed with no parameters in applications. Numerical experiments are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385685
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  • 3
    Electronic Resource
    Electronic Resource
    Vector Median Filters, Inf-Sup Operations, and Coupled PDE's: Theoretical Connections (2000)
    Springer
    Journal of mathematical imaging and vision 12 (2000), S. 109-119 
    Details
    ISSN: 1573-7683
    Keywords: vector median filtering ; inf-sup operations ; asymptotic behavior ; anisotropic diffusion ; curvature motion ; coupled PDE's
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we formally connect between vector median filters, inf-sup morphological operations, and geometric partial differential equations. Considering a lexicographic order, which permits to define an order between vectors in RN, we first show that the vector median filter of a vector-valued image is equivalent to a collection of infimum-supremum morphological operations. We then proceed and study the asymptotic behavior of this filter. We also provide an interpretation of the infinitesimal iteration of this vectorial median filter in terms of systems of coupled geometric partial differential equations. The main component of the vector evolves according to curvature motion, while, intuitively, the others regularly deform their level-sets toward those of this main component. These results extend to the vector case classical connections between scalar median filters, mathematical morphology, and mean curvature motion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008310305351
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  • 4
    Electronic Resource
    Electronic Resource
    Topographic Maps and Local Contrast Changes in Natural Images (1999)
    Springer
    International journal of computer vision 33 (1999), S. 5-27 
    Details
    ISSN: 1573-1405
    Keywords: topographic map ; mathematical morphology ; level set ; junctions ; contrast changes ; digitization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We call “natural” image any photograph of an outdoor or indoor scene taken by a standard camera. We discuss the physical generation process of natural images as a combination of occlusions, transparencies and contrast changes. This description fits to the phenomenological description of Gaetano Kanizsa according to which visual perception tends to remain stable with respect to these basic operations. We define a contrast invariant presentation of the digital image, the topographic map, where the subjacent occlusion-transparency structure is put into evidence by the interplay of level lines. We prove that each topographic map represents a class of images invariant with respect to local contrast changes. Several visualization strategies of the topographic map are proposed and implemented and mathematical arguments are developed to establish stability properties of the topographic map under digitization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008144113494
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  • 5
    Electronic Resource
    Electronic Resource
    Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case (2000)
    Springer
    International journal of computer vision 36 (2000), S. 149-161 
    Details
    ISSN: 1573-1405
    Keywords: directions ; isotropic and anisotropic diffusion ; multiscale representations ; harmonic maps ; general non-flat manifolds ; liquid crystals ; gradients ; optical flow ; color images
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L norm in general and an L1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non-flat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008152115986
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  • 6
    Electronic Resource
    Electronic Resource
    Geodesic Active Contours (1997)
    Springer
    International journal of computer vision 22 (1997), S. 61-79 
    Details
    ISSN: 1573-1405
    Keywords: dynamic contours ; variational problems ; differential geometry ; Riemannian geometry ; geodesics ; curve evolution ; topology free boundary detection
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007979827043
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  • 7
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    Uniqueness of the Cheeger set of a convex body (2009)
    Elsevier
    Details
    Publication Date: 2009-01-01
    Print ISSN: 0362-546X
    Electronic ISSN: 1873-5215
    Topics: Mathematics
    Published by Elsevier
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  • 8
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    Anisotropic curvature-driven flow of convex sets (2006)
    Elsevier
    Details
    Publication Date: 2006-10-01
    Print ISSN: 0362-546X
    Electronic ISSN: 1873-5215
    Topics: Mathematics
    Published by Elsevier
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  • 9
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    Minimal surfaces: a geometric three dimensional segmentation approach (1997)
    Springer
    Details
    Publication Date: 1997-10-01
    Print ISSN: 0029-599X
    Electronic ISSN: 0945-3245
    Topics: Mathematics
    Published by Springer
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  • 10
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    A geometric model for active contours in image processing (1993)
    Springer
    Details
    Publication Date: 1993-12-01
    Print ISSN: 0029-599X
    Electronic ISSN: 0945-3245
    Topics: Mathematics
    Published by Springer
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