Project Description
In this study the student will develop a new efficient method based upon wavelets decomposition for the numerical solution of ill-posed Fredholm integral equations of the first kind. A comparative analysis of the performance of a few wavelet collocation methods will be carried out. A special regularization method combined with wavelet collocation methods are applied to handle the ill-posed Fredholm problems. Through numerical examples, the performance of the present method will be investigated. The results of the numerical tests will show better accuracy of the proposed method based on the Linear Legendre multiwavelets for a variety of benchmark problems.
Tasks and Responsibilites
The student will work on the following tasks: -Review current literature on the use of wavelets in solving integral type problems –such as the work done by Kessler in scattering calculations. - Determine a suitable wavelet system, likely orthogonal, that provides the desired efficiency for solving the EC convolution integral equation. -Develop software for the two-dimensional case using a regularized least squares method for image reconstruction. -Test the efficiency and accuracy of wavelet-based image reconstruction using virtual data that is deliberately contaminated with noise. - test the overall wavelet scheme by attempting reconstruction from actual lab data on standard gelatin phantoms. -will focus on doing simulations to demonstrate that wavelets can indeed be used for the inversion of two and three -dimensional version of the convolution integral and aims at achieving the goals of the project. -will write the paper summarizing all the results.