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Broad Area Colloquium for Artificial Intelligence,
Geometry, Graphics, Robotics and Vision

The Art of Geodesics: Theory, Computational Framework, and Applications

Guillermo Sapiro
University of Minnesota

Monday, February 4, 2002, 4:15PM
Gates B01


Computing distance functions and geodesics in high-dimensional surfaces has applications in numerous areas in mathematical physics, image processing, medical imaging, computer vision, robotics, computer graphics, computational geometry, optimal control, knowledge discovery, and brain research. Geodesics are used for example for path planning in robotics, brain flattening and brain warping in computational neuroscience, crests, valleys, and silhouettes computations in computer graphics and brain research, mesh generation, segmentation in medical imaging, and many applications in mathematical physics. Last but not least, distances and geodesics in high dimensions are fundamental for problems in data mining, dimensionality reduction, and recognition. In addition, generalized geodesics, following the theory of harmonic maps, also found applications in numerous fields, including but not limited to brain warping, color image processing, 3D object recognition, information visualization, inverse problems like those arising from EEG/MEG, and computer graphics.

In this talk we will discuss computational techniques for finding geodesics and generalized geodesics in any dimension. We will address the problem mainly for implicit hyper-surfaces and hyper-surfaces defined from unorganized points. We will discuss computationally optimal and efficient techniques to compute these geodesics, presenting both the underlying theory and numerous examples. We will also briefly comment how to our surprise, some of the mathematical ideas used to derive these techniques are connected with mathematical techniques to study problems in super-conductivity and nanoscales. We conclude the talk describing our current efforts in applying these computational frameworks.

About the Speaker

Guillermo Sapiro received his B.Sc. (summa cum laude), M.Sc., and Ph.D. from the Department of Electrical Engineering at the Technion, Israel Institute of Technology, in 1989, 1991, and 1993 respectively. After post-doctoral research at MIT, Dr. Sapiro became Member of Technical Staff at the research facilities of HP Labs in Palo Alto, California. He is currently with the Department of Electrical and Computer Engineering at the University of Minnesota.

G. Sapiro works on differential geometry and geometric partial differential equations, both in theory and applications in computer vision, computer graphics, medical imaging, brain imaging, scientific computation, and image analysis. He recently co-edited a special issue of IEEE Image Processing in this topic and a second one in the Journal of Visual Communication and Image Representation.

G. Sapiro was awarded the Gutwirth Scholarship for Special Excellence in Graduate Studies in 1991, the Ollendorff Fellowship for Excellence in Vision and Image Understanding Work in 1992, the Rothschild Fellowship for Post-Doctoral Studies in 1993, the Office of Naval Research Young Investigator Award in 1998, the Presidential Early Career Awards for Scientist and Engineers (PECASE) in 1988, and the National Science Foundation Career Award in 1999.


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