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Recent years have seen significant advances in the application of sophisticated mathematical theories to the problems arising in image processing. As yet, even very low level visual processing by computers remains a challenging problem. Both planar and three-dimensional images are of importance, and these arise in a wide variety of applications, including medical imagery---ultrasound, nuclear magnetic resonance, X-ray computed topography, etc.--- military and industrial imaging, and film restoration and animation. Basic issues include image smoothing and denoising, image enhancement, morphology, image compression, segmentation (determining boundaries of objects--- including problems of camera distortion and partial occlusion). Several mathematical approaches have emerged, including methods based on nonlinear partial differential equations, stochastic and statistical methods, and signal processing techniques, including wavelets and other transform theories. Partial differential equations are used to describe the evolution of shapes under curvature-controlled diffusions, providing a multi-scale representation that is based upon curvature flows of fundamental importance in differential geometry. These methods have proven successful in noise reduction while maintaining edge retention. Applications to segmentation are based on a variational formulation of the method of "snakes" or moving contours, in which an initial contour converges to the object boundary via a gradient descent flow based on a conformally Riemannian metric. Wavelets have applications to practical image compression methods, and texture characterization. Statistical methods such as the EM algorithm have been successfully applied to a variety of low level vision problems. A primary goal of this workshop is to educate and interest mathematicians in the mathematical and scientific problems that arise in basic image processing.

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2000-2001 Program: Mathematics in Multimedia

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