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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
    Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours (1999)
    Springer
    Acta applicandae mathematicae 59 (1999), S. 45-77 
    Details
    ISSN: 1572-9036
    Keywords: affine invariant detection ; denoising ; segmentation ; affine scale-space ; affine gradient ; active contours ; gradient flows ; geodesics ; invariant metrics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1006295328209
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  • 3
    Electronic Resource
    Electronic Resource
    Affine-Invariant Distances, Envelopes and Symmetry Sets (1998)
    Springer
    Geometriae dedicata 71 (1998), S. 237-261 
    Details
    ISSN: 1572-9168
    Keywords: symmetry set ; affine invariance ; skew symmetry ; singularities ; dual map ; affine invariant distance ; conics.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005099011913
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  • 4
    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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  • 5
    Electronic Resource
    Electronic Resource
    The ubiquitous ellipse (1995)
    Springer
    Acta applicandae mathematicae 38 (1995), S. 149-161 
    Details
    ISSN: 1572-9036
    Keywords: 35Q80 ; 41A15 ; 52B99 ; 53A15 ; affine invariant ; multi-scale smoothing ; geometric heat flows ; polygons ; B-splines ; ellipses
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We discuss three different affine invariant evolution processes for smoothing planar curves. The first one is derived from ageometric heat-type flow, both the initial and the smoothed curves being differentiable. The second smoothing process is obtained from a discretization of this affine heat equation. In this case, the curves are represented by planarpolygons. The third process is based onB-spline approximations. For this process, the initial curve is a planar polygon, and the smoothed curves are differentiable and even analytic. We show that, in the limit, all three affine invariant smoothing processes collapse any initial curve into anelliptic point.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00992844
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  • 6
    Electronic Resource
    Electronic Resource
    Affine invariant scale-space (1993)
    Springer
    International journal of computer vision 11 (1993), S. 25-44 
    Details
    ISSN: 1573-1405
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract A newaffine invariant scale-space for planar curves is presented in this work. The scale-space is obtained from the solution of a novel nonlinear curve evolution equation which admits affine invariant solutions. This flow was proved to be the affine analogue of the well knownEuclidean shortening flow. The evolution also satisfies properties such ascausality, which makes it useful in defining a scale-space. Using an efficient numerical algorithm for curve evolution, this continuous affine flow is implemented, and examples are presented. The affine-invariant progressive smoothing property of the evolution equation is demonstrated as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01420591
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  • 7
    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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  • 8
    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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  • 9
    Electronic Resource
    Electronic Resource
    A Subdivision Scheme for Continuous-Scale B-Splines and Affine-Invariant Progressive Smoothing (1997)
    Springer
    Journal of mathematical imaging and vision 7 (1997), S. 23-40 
    Details
    ISSN: 1573-7683
    Keywords: B-spline representations ; subdivision schemes ; continuous scale ; affine invariant ; progressive smoothing ; computer implementation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Multiscale representations and progressive smoothing constitutean important topic in different fields as computer vision, CAGD,and image processing. In this work, a multiscale representationof planar shapes is first described. The approach is based oncomputing classical B-splines of increasing orders, andtherefore is automatically affine invariant. The resultingrepresentation satisfies basic scale-space properties at least ina qualitative form, and is simple to implement. The representation obtained in this way is discrete in scale,since classical B-splines are functions in $$C^{k - 2}$$ , where k isan integer bigger or equal than two. We present a subdivisionscheme for the computation of B-splines of finite support atcontinuous scales. With this scheme, B-splines representationsin $$C^r$$ are obtained for any real r in [0, ∞), andthe multiscale representation is extended to continuous scale. The proposed progressive smoothing receives a discrete set ofpoints as initial shape, while the smoothed curves arerepresented by continuous (analytical) functions, allowing astraightforward computation of geometric characteristics of theshape.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008261923192
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  • 10
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    Measuring robustness of brain networks in autism spectrum disorder with Ricci curvature (2020)
    Springer Nature
    Details
    Publication Date: 2020-07-02
    Electronic ISSN: 2045-2322
    Topics: Natural Sciences in General
    Published by Springer Nature
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