Wavelet Approach to Data Analysis, Manipulation, Compression, and Communication

Abstract : The main objective of our research program over the past three-year period is three-fold: firstly, mathematical theories and methods, as well as construction of basis functions, for multi-level approximation and analysis, with emphasis on scattered data interpolation and representation, were developed; secondly, based on minimum-energy criteria, new data processing tools, particularly variational algorithms and optimal wavelet thresholding methods, with applications to image restoration, were introduced; and finally, these developments were applied to data representation, manipulation, rendering, and communication, as well as to solutions of various specific problems in computer graphics. In particular, the standard approach to subdivision schemes by constructing templates of scalar values is extended to matrix-valued subdivisions to gain flexibility and smaller template sizes for the main purpose of achieving twice continuously differentiable surfaces with more desirable geometric shapes and arbitrary topologies. As a result, one-ring templates for interpolating surface subdivisions are introduced in this project, for a wide range of applications, including reversed engineering, surface interpolation of medical and geospatial data, and visualization of point clouds. The theories and methods developed in this project have also been applied to designing algorithms for various applications in computer graphics, including adaptive stroke-based sketching with editing features, non-photo realistic graphic drawings, rendering and animation, as well as wavelet-based digital image restoration.