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Copyright:
1996, 1999, 2006
J.A. Sethian

Applications to Computer Graphics
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topic outlined in red. 		Overview of and references for papers on theory Overview of and references for papers on link to hyperbolic equations Overview of and references for level_set methods Overview of and references for on stationary formulation Overview of and references for Narrow Band formulation Overview of and references for papers on Fast Marching Methods Work on unstructured mesh versions of level set and fast marching methods Coupling interface methods to complex physics Adaptive mesh refinement Applications to semiconductor modeling Applications to geometry Applications to medical imaging Applications to constructing geodesics on surfaces Applications to seismology and travel times Applications to combustion Applications to fluid mechanics Applications to materials sciences Applications to robotics Applications to computer graphics Applications to CAD/CAM Applications to mesh generation

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References:

  1. An Optimal Time Algorithm for Shape from Shading : Kimmel, R., and Sethian, J.A., LBNL-41660, Lawrence Berkeley National Laboratory, Berkeley, California, April 1998.
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  2. Fast Voronoi Diagrams and Offsets on Triangulated Surfaces : Kimmel, R., and Sethian, J.A., Proceedings, AFA Conference on Curves and Surfaces, Saint-Malo, France, July, 1999
    Abstract

    We apply the Fast Marching Method on triangulated domains to efficiently compute Voronoi diagrams and offset curves on triangulated manifolds. The computational complexity of the proposed algorithm is optimal, O(M log M), where M is the number of vertices. The algorithm also applies to weighted domains in which a different cost is assigned to each surface point.

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  3. Optimal Algorithm for Shape from Shading : Kimmel, R., and Sethian, J.A., Proc. of 4-th Asian Conf. on Computer Vision. Taipei, Taiwan, January 2000.
    Abstract

    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 image surface is a weighted distance. A first order 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 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.

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  4. Subjective surfaces: A method for completing missing boundaries : Sarti, A., Malladi, R., and Sethian, J.A., Proc. Nat. Acad. Sciences, 97, pp. 6258-6263, 2000.
    Abstract

    We present a model and algorithm for segmentation of images with missing boundaries. In many situations, the human visual system fills in missing gaps in edges and boundaries, building and completing information that is not present. This presents a considerable challenge in computer vision, since most algorithms attempt to exploit existing data. Completion models, which postulate how to construct missing data, are popular, but are often trained and specific to particular images. In this paper, we take the following perspective: we consider a reference point within an image as given, and then develop an algorithm which tries to build missing information on the basis of the given point of view and the available information as boundary data to the algorithm. We test the algorithm on some standard images, including the classical triangle of Kanizsa and low signal/noise ratio medical images.

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  5. Subjective Surfaces: a Geometric Model for Boundary Completion : Sarti, A., Malladi, R., and Sethian, J.A., IJCV, 46, 3, pp.201-221, 2002,
    Abstract

    We present a geometric model and a computational method for segmentation of images with missing boundaries. In many situations, the human visual system fills in missing gaps in edges and boundaries, building and completing information that is not present. These situations have been widely studied by Gestalt psycologists both in the case of modal and amodal completion. Boundary completion presents a considerable challenge in computer vision, since most algorithms attempt to exploit existing data. A large body of work concerns completion models, which postulate how to construct missing data; these models are often trained and specific to particular images. In this paper, we take the following, alternative perspective: we consider a reference point within an image as given, and then develop an algorithm which tries to build missing information on the basis of the given point of view and the available information as boundary data to the algorithm. Starting from the point of view, a surface is constructed. Then it is evolved with the mean curvature flow in the metric induced by the image until a piecewise constant We test the computational model to modal completion, amodal completion, texture, photo and medical images. We extend the geometric model and the algorithm to 3D in order to extract shapes from low signal/noise ratio medical volumes. Results in 3D echocardiograpghy and 3D fetal echography are presented.

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