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Sparsity and Inverse Problems: Think Analog, and Act Digital

M. Unser

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IEEE International Conference on on Acoustics, Speech, and Signal Processing (ICASSP), March 20-25, 2016, Shanghai, China.

Sparsity and compressed sensing are very popular topics in signal processing. More and more researchers are relying on l1-type minimization scheme for solving a variety of ill-posed problems in imaging. The paradigm is well established with a solid mathematical foundation, although the arguments that have been put forth in the past are deterministic and finite-dimensional for the most part.

In this presentation, we shall promote a continuous-domain formulation of the problem ("think analog") that is more closely tied to the physics of imaging and that also lends it itself better to mathematical analysis. For instance, we shall demonstrate that splines (which are inherently sparse) are global optimizers of linear inverse problems with total-variation (TV) regularization constraints.

Alternatively, one can adopt an infinite-dimensional statistical point of view by modeling signals as sparse stochastic processes. The guiding principle it then to discretize the inverse problem by projecting both the statistical and physical measurement models onto a linear reconstruction space. This leads to the specification of a general class of maximum a posteriori (MAP) signal estimators complemented with a practical iterative reconstruction scheme ("act digital"). While the framework is compatible with the traditional methods of Tikhonov and TV, it opens the door to a much broader class of potential functions that are inherently sparse, while it also suggests alternative Bayesian recovery procedures. We shall illustrate the approach with the reconstruction of images in a variety of modalities including MRI, phase-contrast tomography, cryo-electron tomography, and deconvolution microscopy. In recent years, significant progress has been achieved in the resolution of ill-posed linear inverse problems by imposing l1/TV regularization constraints on the solution. Such sparsity-promoting schemes are supported by the theory of compressed sensing, which is finite dimensional for the most part.

Sparsity and the optimality of splines for inverse problems: Deterministic vs. statistical justifications

M. Unser

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Invited talk: Mathematics and Image Analysis (MIA'16), 18-20 January, 2016, Institut Henri Poincaré, Paris, France.

In recent years, significant progress has been achieved in the resolution of ill-posed linear inverse problems by imposing l1/TV regularization constraints on the solution. Such sparsity-promoting schemes are supported by the theory of compressed sensing, which is finite dimensional for the most part.

In this talk, we take an infinite-dimensional point of view by considering signals that are defined in the continuous domain. We claim that non-uniform splines whose type is matched to the regularization operator are optimal candidate solutions. We show that such functions are global minimizers of a broad family of convex variational problems where the measurements are linear and the regularization is a generalized form of total variation associated with some operator L. We then discuss the link with sparse stochastic processes that are solutions of the same type of differential equations.The pleasing outcome is that the statistical formulation yields maximum a posteriori (MAP) signal estimators that involve the same type of sparsity-promoting regularization, albeit in a discretized form. The latter corresponds to the log-likelihood of the projection of the stochastic model onto a finite-dimensional reconstruction space.

Sparse stochastic processes: A statistical framework for compressed sensing and biomedical image reconstruction

M. Unser

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Plenary. IEEE International Symposium on Biomedical Imaging (ISBI), 16-19 April, 2015, New York, USA.

Sparsity is a powerful paradigm for introducing prior constraints on signals in order to address ill-posed image reconstruction problems.

In this talk, we first present a continuous-domain statistical framework that supports the paradigm. We consider stochastic processes that are solutions of non-Gaussian stochastic differential equations driven by white Lévy noise. We show that this yields intrinsically sparse signals in the sense that they admit a concise representation in a matched wavelet basis.

We apply our formalism to the discretization of ill-conditioned linear inverse problems where both the statistical and physical measurement models are projected onto a linear reconstruction space. This leads to the specification of a general class of maximum a posteriori (MAP) signal estimators complemented with a practical iterative reconstruction scheme. While our family of estimators includes the traditional methods of Tikhonov and total-variation (TV) regularization as particular cases, it opens the door to a much broader class of potential functions that are inherently sparse and typically nonconvex. We apply our framework to the reconstruction of images in a variety of modalities including MRI, phase-contrast tomography, cryo-electron tomography, and deconvolution microscopy.

Finally, we investigate the possibility of specifying signal estimators that are optimal in the MSE sense. There, we consider the simpler denoising problem and present a direct solution for first-order processes based on message passing that serves as our goldstandard. We also point out some of the pittfalls of the MAP paradigm (in the non-Gaussian setting) and indicate future directions of research.

Sparse stochastic processes: A statistical framework for compressed sensing and biomedical image reconstruction

M. Unser

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4 hours tutorial, Inverse Problems and Imaging Conference, Institut Henri Poincaré, Paris, April 7-11, 2014.

We introduce an extended family of continuous-domain sparse processes that are specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. We present the functional tools for their characterization. We show that their transform-domain probability distributions are infinitely divisible, which induces two distinct types of behavior‐Gaussian vs. sparse‐at the exclusion of any other. This is the key to proving that the non-Gaussian members of the family admit a sparse representation in a matched wavelet basis.

Next, we apply our continuous-domain characterization of the signal to the discretization of ill-conditioned linear inverse problems where both the statistical and physical measurement models are projected onto a linear reconstruction space. This leads the derivation of a general class of maximum a posteriori (MAP) signal estimators. While the formulation is compatible with the standard methods of Tikhonov and l1-type regularizations, which both appear as particular cases, it open the door to a much broader class of sparsity-promoting regularization schemes that are typically nonconvex. We illustrate the concept with the derivation of algorithms for the reconstruction of biomedical images (deconvolution microscopy, MRI, X-ray tomography) from noisy and/or incomplete data. The proposed framework also suggests alternative Bayesian recovery procedures that minimize t he estimation error.

Reference

Sparse stochastic processes: A statistical framework for modern signal processing

M. Unser

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Plenary talk, Int. Conf. Syst. Sig. Im. Proc. (IWSSIP), Bucharest, July 7-9, 2013.

We introduce an extended family of sparse processes that are specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. We present the mathematical tools for their characterization. The two leading threads of the exposition are

  • the statistical property of infinite divisibility, which induces two distinct types of behavior-Gaussian vs. sparse-at the exclusion of any other;
  • the structural link between linear stochastic processes and splines.
This allows us to prove that these processes admit a parsimonious representation in some matched wavelet-like basis. We show that these models have predictive power for image compression and that they are applicable to the derivation of statistical algorithms for solving ill-posed inverse problems, including compressed sensing.

Towards a theory of sparse stochastic processes, or when Paul Lévy joins forces with Nobert Wiener

M. Unser

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Mathematics and Image Analysis 2012 (MIA'12), Paris, January 16-18, 2012

The current formulations of compressed sensing and sparse signal recovery are based on solid variational principles, but they are fundamentally deterministic. By drawing on the analogy with the classical theory of signal processing, it is likely that further progress may be achieved by adopting a statistical (or estimation theoretic) point of view. Here, we shall argue that Paul Lévy (1886- 1971), who was one of the very early proponents of Haar wavelets, was in advance over his time, once more. He is the originator of the Lévy-Khinchine formula, which happens to be the perfect (non-Gaussian) ingredient to support a continuous-domain theory of sparse stochastic processes.

Specifically, we shall present an extended class of signal models that are ruled by stochastic differential equations (SDEs) driven by white Léevy noise. Léevy noise is a highly singular mathematical entity that can be interpreted as the weak derivative of a Lévy process. A special case is Gaussian white noise which is the weak derivative of the Wiener process (a.k.a. Brownian motion). When the excitation (or innovation) is Gaussian, the proposed model is equivalent to the traditional one. Of special interest is the property that the signals generated by non-Gaussian linear SDEs tend to be sparse by construction; they also admit a concise representation in some adapted wavelet basis. Moreover, these processes can be (approximately) decoupled by applying a discrete version of the whitening operator (e.g., a finite-difference operator). The corresponding log-likelihood functions, which are nonquadratic, can be specified analytically. In particular, this allows us to uncover a Lęevy processes that results in a maximum a posteriori (MAP) estimator that is equivalent to total variation. We make the connection with current methods for the recovery of sparse signals and present some examples of MAP reconstruction of MR images with sparse priors.

Wavelets, sparsity and biomedical image reconstruction

M. Unser

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Imaging Seminar, University of Bern, Inselspital November 13, 2012.

Our purpose in this talk is to advocate the use of wavelets for advanced biomedical imaging. We start with a short tutorial on wavelet bases, emphasizing the fact that they provide a sparse representation of images. We then discuss a simple, but remarkably effective, image-denoising procedure that essentially amounts to discarding small wavelet coefficients (soft-thresholding). The crucial observation is that this type of ‚Äúsparsity-promoting‚ÄĚ algorithm is the solution of a l1-norm minimization problem. The underlying principle of wavelet regularization is a powerful concept that has been used advantageously for compressed sensing and for reconstructing images from limited and/or noisy measurements. We illustrate the point by presenting wavelet-based algorithms for 3D deconvolution microscopy, and MRI reconstruction (with multiple coils and/or non-Cartesian k-space sampling). These methods were developed at the EPFL in collaboration with imaging scientists and are, for the most part, providing state-of-the-art performance.

Stochastic Models for Sparse and Piecewise-Smooth Signals

M. Unser

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Sparse Representations and Efficient Sensing of Data, Schloss Dagstuhl, Jan. 30 - Feb. 4, 2011

We introduce an extended family of continuous-domain stochastic models for sparse, piecewise-smooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; this is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The non-standard aspect is that the models are driven by non-Gaussian noise (impulsive Poisson or alpha-stable) and that the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete distributional characterization of these processes. We also introduce signals that are the non-Gaussian (sparse) counterpart of fractional Brownian motion; they are non-stationary and have the same ω-type spectral signature. We prove that our generalized processes have a sparse representation in a wavelet-like basis subject to some mild matching condition. Finally, we discuss implications for sampling and sparse signal recovery.

Recent Advances in Biomedical Imaging and Signal Analysis

M. Unser

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Proceedings of the Eighteenth European Signal Processing Conference (EUSIPCO'10), Ålborg, Denmark, August 23-27, 2010, EURASIP Fellow inaugural lecture.

Wavelets have the remarkable property of providing sparse representations of a wide variety of "natural" images. They have been applied successfully to biomedical image analysis and processing since the early 1990s.

In the first part of this talk, we explain how one can exploit the sparsifying property of wavelets to design more effective algorithms for image denoising and reconstruction, both in terms of quality and computational performance. This is achieved within a variational framework by imposing some ℓ1-type regularization in the wavelet domain, which favors sparse solutions. We discuss some corresponding iterative skrinkage-thresholding algorithms (ISTA) for sparse signal recovery and introduce a multi-level variant for greater computational efficiency. We illustrate the method with two concrete imaging examples: the deconvolution of 3-D fluorescence micrographs, and the reconstruction of magnetic resonance images from arbitrary (non-uniform) k-space trajectories.

In the second part, we show how to design new wavelet bases that are better matched to the directional characteristics of images. We introduce a general operator-based framework for the construction of steerable wavelets in any number of dimensions. This approach gives access to a broad class of steerable wavelets that are self-reversible and linearly parameterized by a matrix of shaping coefficients; it extends upon Simoncelli's steerable pyramid by providing much greater wavelet diversity. The basic version of the transform (higher-order Riesz wavelets) extracts the partial derivatives of order N of the signal (e.g., gradient or Hessian). We also introduce a signal-adapted design, which yields a PCA-like tight wavelet frame. We illustrate the capabilities of these new steerable wavelets for image analysis and processing (denoising).

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