Mathematical methods in imaging - superresolution, coregistration, fusion - with applications; fractal-based methods of analysis

点击翻译按钮获取中文摘要

Much of my research could be called mathematical imaging, the use of mathematics to develop new methods of image processing (e.g. denoising, deblurring) or image analysis.  It has evolved from an earlier research programme in which I explored the use of fractal geometry to approximate images and signals.  For example, we are interested in the problem of ``super-resolution'' -- taking images with one or several resolutions and combining their information to produce an image with a higher resolution.  No matter how quickly technology is moving, specialists would always like to see things at even higher resolution.  I am currently working with two such types of specialists:  a medical physicist at a local cancer centre and engineers at my university who are interested in the properties of porous media such as rock, concrete, wood and cartilage, and how fluid flows through these media.  But these two sets of researchers are also faced with another problem, that of comparing images that are obtained from different imaging devices, for example, magnetic resonance spectroscopy (MRI), positron emission tomography (PET), computerized tomography (CT).  Each of these devices captures different properties of what is being imaged. The challenge is to combine the unique information offered from each  mage to get a better total picture of what is going on in the sample/subject.  This is  called ``data fusion.'' The problem is even more difficult in medical imaging since it is not at all the case that a patient will be in exactly the same position at different times and in different machines! This is the problem of ``image registration:''  warping the images - internal organs included - appropriately so that they can overlap each other.  We have been working on the important problem of registering local ``regions of interest'' as well as possible, which is of particular interest to diagosticians. We have also returned to fractal imaging with fresh eyes to find that images are in general quite ``self-similar'' -- given a piece ``A'' of an image, there are generally many other pieces of the image that can be made to look like ``A''.  We are also looking into ways of characterizing images based upon their degrees of self-similarity.