Date of Award


Degree Type

Honors College Thesis


Computing; Mathematics

First Advisor

James V. Lambers, Ph.D.

Advisor Department



In this paper, we propose an automatic numerical method for solving a nonlinear partialdifferential- equation (PDE) based image-processing model. The Perona-Malik diffusion equation (PME) accounts for both forward and backward diffusion regimes so as to perform simultaneous denoising and deblurring depending on the value of the gradient. One of the limitations of this equation is that a large value of the gradient for backward diffusion can lead to singularity formation or staircasing. Guidotti-Kim-Lambers (GKL) came up with a bound for backward diffusion to prevent staircasing, where the backward diffusion is only limited to a specific range beyond which backward diffusion is stopped and forward diffusion begins. Our model combines the PME model and GKL model for automatic sharpening of blurred text-images using Nelder-Mead optimization, a derivative free optimization method that uses n+1 test points arranged as a simplex for n-dimensional optimization. We solve our model by discretizing the PDE in space using finite difference approximation scheme. Then, we enhance the image in each iteration using Backward Euler time-stepping and Minimum Residual Method (MINRES) in MATLAB. Likewise, we propose a gradientbased sharpness metric for our text-images, which also serves as an objective function for our Nelder-Mead optimizer. Our result shows that our proposed model is accurate in enhancing text images and predicting the unknown value of the blurring kernel for automatic sharpening. Numerical results show that the proposed objective sharpness measure coincide with the subjective sharpness of the enhanced image.