Centroidal Voronoi Tessellation Algorithms for Image Compression, Segmentation, and Multichannel Restoration

Centroidal Voronoi tessellations (CVT's) are special Voronoi tessellations for which the generators of the tessellation are also the centers of mass (or means) of the Voronoi cells or clusters. CVT's have been found to be useful in many disparate and diverse settings. In this paper, CVT-based algorithms are developed for image compression, image segmenation, and multichannel image restoration applications. In the image processing context and in its simplest form, the CVT-based methodology reduces to the well-known k-means clustering technique. However, by viewing the latter within the CVT context, very useful generalizations and improvements can be easily made. Several such generalizations are exploited in this paper including the incorporation of cluster dependent weights, the incorporation of averaging techniques to treat noisy images, extensions to treat multichannel data, and combinations of the aforementioned. In each case, examples are provided to illustrate the efficiency, flexibility, and effectiveness of CVT-based image processing methodologies. Qiang Du is a Professor of Mathematics at the Pennsylvania State University. He received his Ph.D. from the Carnegie Mellon University in 1988. Since then, he has held academic positions at several institutions such as the University of Chicago and the Hong Kong University of Science and Technology. He has published over 100 papers on numerical algorithms and their various applications. His recent research works include studies of bio-membranes, complex fluids, quantized vortices, micro-structure evolution, image and data analysis, mesh generation and optimization, and approximations of partial differential equations. Max Guzburger is the Frances Eppes Professor of Computational Science and Mathematics at Florida State University. He received his Ph.D. degree from New York University in 1969 and has held positions at the University of Tennessee, Carnegie Mellon University, Virginia Tech, and Iowa State University. He is the author of five books and over 225 papers. His research interest include computational methods for partial differential equations, control of complex systems, superconductivity, data mining, computational geometry, image processing, uncertainty quantification, and numerical analysis. Lili Ju is an Assistant Professor of Mathematics at the University of South Carolina, Columbia. He received a B.S. degree in Mathematics from Wuhan University in China in 1995, a M.S. degree in Computational Mathematics from the Chinese Academy of Sciences in 1998, and a Ph.D. in Applied Mathematics from Iowa State University in 2002. From 2002 to 2004, he was an Industrial Postdoctoral Researcher at the Institute of Mathematics and Its Applications at the University of Minnesota. His research interests include numerical analysis, scientific computation, parallel computing, and medical image processing. Xiaoqiang Wang is a graduate student in mathematics at the Pennsylvania State University, working under the supervision of Qiang Du. Starting in September 2005, he will be an Industrial Postdoctoral Researcher at the Institute of Mathematics and its Applications at the University of Minnesota. His research interests are in the fields of applied mathematics and scientific computation. His work involves numerical simulation and analysis, algorithms for image processing and data mining, parallel algorithms, and high-performance computing.