Applied analysis: Mathematical imaging, image multifunctions, fractal-based methods in analysis

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Much of my research can be called "mathematical imaging," the use of mathematics to develop new methods of analyzing or processing images. My current attention has focussed on complex, high-dimensional, data sets arising from imaging, in particular, (1) "hyperspectral images" (HSI) obtained from remote sensing of the earth's surface and (2) diffusion magnetic resonance images (DMRI). A typical HSI can be a stack of well over 200 images of a region taken at different wavelengths. At each pixel representing a portion of this region, these 200+ reflectance values comprise the "spectral function" at that portion. From the spectral function, one can infer the materials on the surface, e.g., metals, water, foliage. A DMRI can also be viewed as a "stack" of many images representing the diffusion of water in different directions. At each voxel representing a small region of a patient, these many values give an idea of how water can diffuse in different directions. One important application of such images is "tractography", where the connectivity (or lack thereof) of neurons in the brain of a patient can be mapped. We are proposing a rather novel mathematical representation of these high-dimensional data sets which could lead to better algorithms for their processing, e.g., denoising, compression. I have also been interested in medical image compression - reducing the amount of computer memory needed to store a medical image. The question that remains unanswered is, "To what degree can a medical image be compressed before diagnostic information is lost?" Currently, most assessments of distortions produced by compression are subjective and performed by radiologists, making them extremely expensive and time-consuming. In collaboration with radiologists at McMaster University, we have been working on the problem of automating these assessments. 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 the co-author of the "structural similarity measure" (SSIM), recognized as one of the best measures of visual closeness to date. We have recently shown that SSIM performs much better in matching the subjective assessments of radiologists. It now remains to use SSIM effectively to determine new standards of medical image compression. I am also interested in the mathematical properties of SSIM. My research in mathematical imaging evolved from an earlier intensive research programme centered around "fractal analysis", in which one tries to express a mathematical 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.