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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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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Sub-pixel distance maps and weighted distance transforms (1996)
    Springer
    Journal of mathematical imaging and vision 6 (1996), S. 223-233 
    Details
    ISSN: 1573-7683
    Keywords: distance transforms ; weighted distance ; curve evolution ; differential geometry ; continuous scale morphology
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new framework for computing the Euclidean distance and weighted distance from the boundary of a given digitized shape is presented. The distance is calculated with sub-pixel accuracy. The algorithm is based on a equal distance contour evolution process. The moving contour is embedded as a level set in a time varying function of higher dimension. This representation of the evolving contour makes possible the use of an accurate and stable numerical scheme, due to Osher and Sethian [22]. The relation between the classical shape from shading problem and the weighted distance transform is presented, as well as an algorithm that calculates the geodesic distance transform on surfaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00119840
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  • 3
    Electronic Resource
    Electronic Resource
    Shape from shading: Level set propagation and viscosity solutions (1995)
    Springer
    International journal of computer vision 16 (1995), S. 107-133 
    Details
    ISSN: 1573-1405
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We present a new implementation of an algorithm aimed at recovering a 3D shape from its 2D gray-level picture. In order to reconstruct the shape of the object, an almost arbitrarily initialized 3D function is propagated on a rectangular grid, so that a level set of this function tracks the height contours of the shape. The method imports techniques from differential geometry, fluid dynamics, and numerical analysis and provides an accurate shape from shading algorithm. The method solves some topological problems and gracefully handles cases of non-smooth surfaces that give rise to shocks in the propagating contours. Real and synthetic images of 3D profiles were submitted to the algorithm and the reconstructed surfaces are presented, demonstrating the effectiveness of the proposed method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01539551
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  • 4
    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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  • 5
    Electronic Resource
    Electronic Resource
    Analyzing and Synthesizing Images by Evolving Curves with the Osher-Sethian Method (1997)
    Springer
    International journal of computer vision 24 (1997), S. 37-55 
    Details
    ISSN: 1573-1405
    Keywords: shape from shading ; halftoning ; offsets ; distance maps ; minimal geodesics ; segmentation ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract Numerical analysis of conservation laws plays an important role in the implementation of curve evolution equations. This paper reviews the relevant concepts in numerical analysis and the relation between curve evolution, Hamilton-Jacobi partial differential equations, and differential conservation laws. This close relation enables us to introduce finite difference approximations, based on the theory of conservation laws, into curve evolution. It is shown how curve evolution serves as a powerful tool for image analysis, and how these mathematical relations enable us to construct efficient and accurate numerical schemes. Some examples demonstrate the importance of the CFL condition as a necessary condition for the stability of the numerical schemes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007970107971
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  • 6
    Electronic Resource
    Electronic Resource
    Global Minimum for Active Contour Models: A Minimal Path Approach (1997)
    Springer
    International journal of computer vision 24 (1997), S. 57-78 
    Details
    ISSN: 1573-1405
    Keywords: shape modeling ; deformable models—snakes ; path of minimal cost ; level sets ; segmentation ; feature extraction ; energy minimization ; partial differential equations ; curve evolution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract 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 topology-based saddle search routine. We show examples of our method applied to real aerial and medical images.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007922224810
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  • 7
    Book
    Book
    Numerical geometry of non-rigid shapes / (2008)
    New York, NY :Springer,
    Details
    Title: Numerical geometry of non-rigid shapes /
    Author: Bronstein, Alexander M.
    Contributer: Bronstein, Michael M. , Kimmel, Ron
    Publisher: New York, NY :Springer,
    Year of publication: 2008
    Pages: XXI, 340 S. : , Ill., graph. Darst.
    Series Statement: Monographs in computer science
    ISBN: 978-0-387-73300-5
    Type of Medium: Book
    Language: English
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  • 8
    Book
    Book
    Processing, Analyzing and Learning of Images, Shapes, and Forms : Part 2; 20 (2019)
    Amsterdam :Elsevier, North-Holland,
    Details
    Title: Processing, Analyzing and Learning of Images, Shapes, and Forms : Part 2; 20
    Author: Kimmel, Ron
    Contributer: Tai, Xue-Cheng
    Publisher: Amsterdam :Elsevier, North-Holland,
    Year of publication: 2019
    Pages: XIX, 684 Seiten
    Series Statement: Handbook of numerical analysis 20
    ISBN: 978-0-444-64140-3
    Type of Medium: Book
    Language: English
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  • 9
    Book
    Book
    Processing, Analyzing and Learning of Images, Shapes, and Forms : Part 1; 19 (2018)
    Amsterdam :Elsevier, North-Holland,
    Details
    Title: Processing, Analyzing and Learning of Images, Shapes, and Forms : Part 1; 19
    Author: Kimmel, Ron
    Contributer: Tai, Xue-Cheng
    Publisher: Amsterdam :Elsevier, North-Holland,
    Year of publication: 2018
    Pages: XI, 145 S.
    Series Statement: Handbook of numerical analysis 19
    ISBN: 978-0-444-64205-9
    Type of Medium: Book
    Language: English
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