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 Contributions

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.

Period: 2010-ongoing

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

Major Publications

[1] 

C.S. Seelamantula, N. Pavillon, C. Depeursinge, M. Unser, "Local Demodulation of Holograms Using the Riesz Transform with Application to Microscopy," Journal of the Optical Society of America A, vol. 29, no. 10, pp. 2118-2129, October 2012.

[2] 

U. Kamilov, E. Bostan, M. Unser, "Wavelet Shrinkage with Consistent Cycle Spinning Generalizes Total Variation Denoising," IEEE Signal Processing Letters, vol. 19, no. 4, pp. 187-190, April 2012.

[3] 

J.P. Ward, H. Kirshner, M. Unser, "Is Uniqueness Lost for Under-Sampled Continuous-Time Auto-Regressive Processes?," IEEE Signal Processing Letters, vol. 19, no. 4, pp. 183-186, April 2012.

[4] 

P.D. Tafti, M. Unser, "On Regularized Reconstruction of Vector Fields," IEEE Transactions on Image Processing, vol. 20, no. 11, pp. 3163-3178, November 2011.

[5] 

U.S. Kamilov, P. Pad, A. Amini, M. Unser, "MMSE Estimation of Sparse Lévy Processes," IEEE Transactions on Signal Processing, vol. 61, no. 1, pp. 137-147, January 1, 2013.

[6] 

M. Unser, N. Chenouard, "A Unifying Parametric Framework for 2D Steerable Wavelet Transforms," SIAM Journal on Imaging Sciences, vol. 6, no. 1, pp. 102-135, 2013.

[7] 

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, M. Unser, "A Practical Inverse-Problem Approach to Digital Holographic Reconstruction," Optics Express, vol. 21, no. 3, pp. 3417-3433, February 11, 2013.

[8] 

A. Kazerouni, U.S. Kamilov, E. Bostan, M. Unser, "Bayesian Denoising: From MAP to MMSE Using Consistent Cycle Spinning," IEEE Signal Processing Letters, vol. 20, no. 3, pp. 249-252, March 2013.

[9] 

J.P. Ward, K.N. Chaudhury, M. Unser, "Decay Properties of Riesz Transforms and Steerable Wavelets," SIAM Journal on Imaging Sciences, vol. 6, no. 2, pp. 984-998, 2013.

[10] 

A. Amini, U.S. Kamilov, E. Bostan, M. Unser, "Bayesian Estimation for Continuous-Time Sparse Stochastic Processes," IEEE Transactions on Signal Processing, vol. 61, no. 4, pp. 907-920, February 15, 2013.

[11] 

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, "Fast Iterative Reconstruction of Differential Phase Contrast X-Ray Tomograms," Optics Express, vol. 21, no. 5, pp. 5511-5528, March 11, 2013.

[12] 

S. Lefkimmiatis, J.P. Ward, M. Unser, "Hessian Schatten-Norm Regularization for Linear Inverse Problems," IEEE Transactions on Image Processing, vol. 22, no. 5, pp. 1873-1888, May 2013.

[13] 

E. Bostan, U.S. Kamilov, M. Nilchian, M. Unser, "Sparse Stochastic Processes and Discretization of Linear Inverse Problems," IEEE Transactions on Image Processing, vol. 22, no. 7, pp. 2699-2710, July 2013.

[14] 

A. Amini, P. Thévenaz, J.P. Ward, M. Unser, "On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes," IEEE Transactions on Information Theory, vol. 59. no. 8, pp. 5063-5074, August 2013.

[15] 

N. Chenouard, M. Unser, "3D Steerable Wavelets in Practice," IEEE Transactions on Image Processing, vol. 21, no. 11, pp. 4522-4533, November 2012.

[16] 

A. Amini, U.S. Kamilov, M. Unser, "The Analog Formulation of Sparsity Implies Infinite Divisibility and Rules Out Bernoulli-Gaussian Priors," Proceedings of the 2012 IEEE Information Theory Workshop (ITW'12), Lausanne VD, Swiss Confederation, September 3-7, 2012, pp. 687-691.

[17] 

E. Bostan, U. Kamilov, M. Unser, "Reconstruction of Biomedical Images and Sparse Stochastic Modeling," Proceedings of the Ninth IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI'12), Barcelona, Kingdom of Spain, May 2-5, 2012, pp. 880-883.

[18] 

U. Kamilov, A. Amini, M. Unser, "MMSE Denoising of Sparse Lévy Processes via Message Passing," Proceedings of the Thirty-Seventh IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'12), 京都市 (Kyoto), Japan, March 25-30, 2012, pp. 3637-3640.

[19] 

A. Amini, U. Kamilov, M. Unser, "Bayesian Denoising of Generalized Poisson Processes with Finite Rate of Innovation," Proceedings of the Thirty-Seventh IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'12), 京都市 (Kyoto), Japan, March 25-30, 2012, pp. 3629-3632.

[20] 

U. Kamilov, E. Bostan, M. Unser, "Generalized Total Variation Denoising via Augmented Lagrangian Cycle Spinning with Haar Wavelets," Proceedings of the Thirty-Seventh IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'12), 京都市 (Kyoto), Japan, March 25-30, 2012, pp. 909-912.

Related Topics

• Medical Imaging
• Advanced Image Processing in Biology
• Splines
• Mathematical Imaging
• Wavelets