Broad Area Colloquium for Artificial Intelligence,
Geometry, Graphics, Robotics and Vision
The Art of Geodesics: Theory, Computational Framework, and
University of Minnesota
Monday, February 4, 2002, 4:15PM
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.
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
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
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
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
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