"Mathematical imaging, image multifunctions, diagnostically lossless image compression, fractal-based methods of analysis and approximation"

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Much of my research can be called "mathematical imaging,'' the use of mathematics to develop new methods of image processing or image analysis. One such image processing method, which still receives a great deal of attention, is ''denoising'', the removal of troublesome noise from images (e.g., instrument noise from an MRI). I have also returned to the research area of image compression -- reducing the amount of computer memory needed to store a digital image. (The standard "JPEG" method used to compress images in digital cameras is based on a well-known mathematical principle.) The efficient storage, transmission, retrieval and display of images - in particular medical images - in large-scale databases has become a major challenge. The question that remains unanswered is "To what degree can a medical image be compressed before diagnostic information is lost?'' Currently, most assessments of such distortions are done by radiologists, making them subjective, extremely expensive and time-consuming. In collaboration with a radiologist at McMaster University and a software designer at Agfa HealthCare, we are working on the problem of automating this assessment, i.e., predicting when diagnostically critical distortions will occur. This leads to another problem in image processing - assessing the "visual quality" of images. There is a standard, mathematically-based method of computing the "distance" between two images. However, two images that are close in this distance may not be close visually. One of my collaborators at UW is a co-author of the "structural similarity measure," (SSIM) recognized as one of the best measures of visual closeness to date. We have been working on the use of SSIM to denoise images and now plan to use it in the medical image compression problem outlined above. I am also interested in the mathematical properties of SSIM. My research in mathematical imaging evolved from an earlier research programme centered around "fractal analysis.'' In fractal analysis, one tries to express an "object'' as a union of smaller, possibly distorted copies of itself. I continue to pursue this area of research which has interesting applications, particularly in imaging.