Abstract:

Image processing is a gigantic field, whose applications range from the purely recreational (Television / DVD, etc.), to military (satellite imaging), medical (ultra sound, computed tomography, MRI), and the scientific realm (telescopes, high resolution microscopes).

Traditionally, most image processing has been done by Electrical Engineers, and Computer Scientists. The addition of statistical models, and more recently, variational and PDE-based approaches have introduced more mathematicians into the field (and introduced image processing to more mathematicians!)

In this talk, I will give an introduction to the use of nonlinear PDEs, and in particular total variation (TV) methods for restoration of noisy and blurred images. The main advantage of these methods over conventional linear frequency domain methods is the preservation of discontinuities (edges) in the images. Since edges define the locations of objects, edge preservation is crucial for automatic detection, tracking, and object classification. We would not want to confuse a baby milk factory with an enemy tank, healthy tissue with a brain tumor, etc...

The latter part of the talk will be concerned with my work on extensions of the gray scale TV method: (i) to vector valued (color) images, (ii) to enhance multi-scale capturing, and detail retention, using an adaptive scheme, and (iii) to reduce the "staircasing effect" visible in TV restored images.

There will be plenty of images to go with the equations!!!