Bradley J Lucier
My research started in numerical methods for partial differential equations, motivated by scalar hyperbolic conservation laws, which can have rough (even discontinuous) solutions. This study led in 1985 to moving-grid numerical methods with rigorous second-order error bounds, the only such methods known. Later I worked with Ron DeVore, then at the University of South Carolina, on applying some ideas from nonlinear approximation theory (specifically, the recently-developed theory of approximation by free-knot splines) to prove new regularity results for conservation laws. These results state that if the nonlinear flux of the conservation law is smooth and convex, and the initial data has smoothness order K for any K>0, then even after discontinuities develop, the solution maintains that smoothness. One must measure smoothness in the space Lq for q=1/(K+1) for this result to hold, however, and these spaces are non-convex.
Around the same time Ron was working with Bjorn Jawerth and Vasil Popov to develop the theory of nonlinear compression of wavelet decompositions of functions; this theory in one dimension was closely related to the results on approximation by free-knot splines. I wrote a series of papers with Ron and Bjorn applying these ideas to image and surface compression, and data compression in general; this included the first paper to apply nonlinear approximation theory to image processing, specifically to analyze the effects of quantization of wavelet coefficients in image compression. This paper introduced a number of other techniques, including integer-to-integer wavelet transforms, nonlinear wavelet transforms of binary images, etc. The paper shows how different strategies of quantizating wavelet coefficients are equivalent to minimizing the error in different Lp spaces.
Later work continued on other areas of applying wavelets to image processing, including noise removal, image reconstruction, etc. This work was in done in collaboration with, among others, Ron, Antonin Chambolle, and Nam-Yong Lee.
I've collaborated loosely, but over a considerable time, with Maria Kallergi and other researchers at the University of South Florida Moffit Cancer Research Center. For one study, I designed a wavelet compression method specifically for mammography, by learning from radiologists which image features are important for diagnosis and tailoring the quantization strategy so that those features are given importance as data is removed. The previous mathematical study was helpful here, in that it interpreted the compression method as, first, determining the relative importance of features at different scales, and second, determining the quantization strategy so that all important feature scales degrade at the same rate. The study concluded that, after compressing the digitized mammograms at an average rate of over 50-1 (individual compression rates ranged from 14-to-1 to over 2000-to-1, depending on the complexity of the individual mammographic image), radiologists interpreted the compressed mammograms more accurately than they did the originals. This study was published in Radiology.
I have a number of other interests of a more amateur nature, mainly in computing, including random number generation, the Scheme programming language, algorithms for computing with large integers, computation of elementary functions, high-performance computing, etc. Scientific computation has motivated nearly all aspects of my research program.