Research statement - UCLA Department of Mathematics

filter banks or wavelet decomposition are widely used techniques. The drawback here, however, is
that one computes coefficients of far more components than actually present, and that the shape of the
components is often overly restricted and predefined. Adaptive band decomposition techniques have
gained importance, such as the empirical mode decomposition (EMD) algorithm.
We are currently working on a variational model for adaptive mode decomposition, where from the
input signal we estimate a number of modes which are mostly band-separated (but not strictly), and
whose carrier frequency and bandwidth are determined on-line, adaptively. Indeed, we perform
Wiener filtering on the demodulated input signal for each individual mode, and the resulting optimization
problems are a mixture of adaptive Gabor-wavelet filtering and spectral Gaussian mixture
models.
Future Research: Vision and Objectives
My research is driven by the quest for better models and tools for image understanding. I want to
continue working on a broad spectrum of image processing and computer vision problems, that all
have their useful application in science and society. Image segmentation, decomposition, denoising,
restoration, illumination normalization—these tasks are all routinely performed in medical image
scanners as well as consumer electronics, but research has not come to an end yet. As the imaging
modalities become more and more complex, so do the questions to be answered by images. And the
computational complexity of the inverse problems involved doesn’t stop growing.
More challenging forms of images, such as omni-directional images or maps on more complicated
surfaces such as the human cortex are becoming more important, and most of the established image
processing tools do not directly translate to these new domains. Also, generalizing even further, data
points on graphs can be considered images, and many machine learning tasks have a structure very
similar to image processing problems, such as clustering or classification.
Answering upcoming medical imaging problems still requires a lot of original research in applied
mathematics, a lot of mathematical modeling intuition as well as the synthesis of many powerful
existing concepts, like convex optimization tools, compressed sensing, Beltrami regularization or nonlocal
operators, coupled with new insights from optimization, numerical optimization and scientific
computing. I am strongly willing to further enlarge my toolbox in applied mathematics, and dedicate
myself to research for new image understanding solutions.
Dec. 2012, Dominique Zosso
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