Distributed approximating functionals, multi-resolution wavelet analysis and medical image processing.

摘要

This work mainly focused on the fundamental, mathematical properties of the Distributed Approximating Functionals and DAF approach to wavelet transforms in the field of signal processing. Different DAFs are considered, belonging to two classes: the well-tempered or smoothing DAFs and the interpolating DAFs. These DAFs have been proved to indeed yield Unity Approximations. Therefore, they are rigorously generalized delta sequences, as claimed previously based on computational research. Two different ways at extending the DAF theory to the multi-dimensional cases have been given. The non-Cartesian (non-separable) Distributed Approximating Functionals are presented as an example. Following this, the second part of this work deals with the relationship between DAF theory and the multi-resolution analysis and wavelet transforms. A brief review of the wavelet transform has been included. A wavelet generator based on Holschneider's ideas is demonstrated and utilized to generate wavelets using the DAFs as the scaling function. Numerical experiments have been carried out to show the power of this technique for image processing, especially for edge detection. Also in this part of the dissertation, research on low-pass filter design was carried out utilizing the HDAFs. Infinite smoothness and controllable accuracy are the advantages of this approach to filter design, which yields as many vanishing moments for the generated wavelets as one desires. In the last part of this dissertation, the theory is implemented in the field of signal processing, particularly for various types of medical image processing. An integrated system based on continuous wavelet transforms, DAF theory, Visual Group Normalization, and Soft Logic Masking techniques is presented, the system is named "DAF Sparkle." As to use for the multi-dimensional applications, the non-Cartesian DAF (NCDAF) has been used as the "surround function" in an application to mammogram enhancement, based on the "Retinex mechanism." The details, of the NCDAF are given and the results of numerical experiments demonstrate the usefulness of DAF theory.