BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

ERC FUN-SP: A functional framework for sparse, non-gaussian signal processing and bioimaging

Medical Imaging
Bioimaging
Splines
Mathematical Imaging
Wavelets

Principal Investigators: Michael Unser, Pouya Tafti, Arash Amini, John-Paul Ward, Pedram Pad, Ulugbek S. Kamilov, Emrah Bostan, Masih Nilchian, Julien Fageot

Summary

In recent years, the research focus in signal processing has shifted away from the classical linear paradigm, which is intimately linked with the theory of stationary Gaussian processes. Instead of considering Fourier transforms and performing quadratic optimization, researchers are presently favoring wavelet-like representations and have adopted "sparsity" as design paradigm. The goal of this project is to develop a unifying operator-based framework for signal processing that provides the "sparse" counterpart of the classical Gaussian theory and supports this recent shift of paradigm.

Main Contribution

We have introduced a generalized innovation model which is equivalent to defining stochastic processes as solutions of stochastic differential equations driven by white Lévy noise. Based on functional arguments, we have shown that the non-Gaussian processes in this family are intrinsically sparse. We have uncovered a conceptual parallel between generalized stochastic processes and splines, which has helped us transposing mathematical methods from one field to the other. This has led us to the construction of new operator-like wavelet transforms that have the ability to decouple sparse processes. We have also proposed a novel operator-based framework for the design of steerable wavelets, both in 2-D and 3-D, and have deployed these tools for the analysis of biomedical images.
We have applied our stochastic models to the design of new Bayesian signal-reconstruction algorithms and have established links with the kind of sparsity-promoting schemes that are currently in favor in the field. We have used the framework to develop new 3-D reconstruction algorithms for fluorescence deconvolution microscopy, digital holography microscopy, and X-ray phase-contrast microscopy, in close collaboration with imaging scientists.

Available algorithms
Steerable wavelets in 2D. Circular Wavelets . A complete parametric framework and set of matlab tools for computing steerable wavelet frames in 2-D. Specific designs include Simoncelli's pyramid, Marr and monogenic wavelets, Prolate spheroidal wavelets, and curvelets.

Collaboration:

Period: 2010-ongoing

Funding: ERC-2010-AdG 267439-FUN-SP

Major Publications