Mathematical Sciences: Computational Mathematics Issues in Nonlinear Diffusion Modules in Image Processing

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Chan 9626755 The investigator and his colleagues study computational mathematics issues in nonlinear diffusion models in image processing. These problems are computationally intensive and accurate and efficient methods are needed. Traditionally, the standard methods involve computation in the frequency domain, facilitated by efficient FFT and wavelet algorithms. Recently, there has been a new movement towards a partial differential equation (PDE) based approach, which is motivated by a more systematic approach to restoring images with sharp edges, as well as for image segmentation. The image is diffused (denoised) according to a nonlinear anisotropic diffusion PDE, designed to diffuse less near edges. Moreover, the PDEs are designed to possess certain desirable geometrical properties such as affine invariance and causality. From a computational standpoint, the PDE formulations call for new computational techniques that are different from the traditional frequency domain and algebraic approaches. As yet, the nonlinear diffusion models are considered somewhat expensive compared to traditional methods, and it is one of the goals of this project to make them more efficient while retaining their desirable geometric properties. Additionally, attempts are made to improve and extend these methods to color and other vector-valued images. Specific computational algorithms being studied include primal-dual minimization methods, preconditioning techniques, and wavelet algorithms. Image processing has many important applications in both the physical and the medical sciences. In the current revolution in communication and the advent of the information highway, more and more images are being transmitted and better mathematical algorithms are needed to compress and remove noise and other distortions occurring in the transmission. Applications in the medical sciences and biotechnology field range from computer topography to processing of microscopic images of molecular structures. In the environmental area, satellite imaging has been used to map natural resources as well as environmental pollution. In the area of manufacturing, imaging systems are used to detect defects automatically. In all of these applications, a key process is that of image restoration, namely, cleaning up an image polluted by noise and blurring. This is the main subject of this project. These problems are very computationally intensive due to the large number of pixels and the possibility of sequences of images (e.g. videos), solving them requires clever mathematical algorithms as well as high performance computers.