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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.

Sampling: 60 Years After Shannon

M. Unser

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Plenary talk, Sixteenth International Conference on Digital Signal Processing (DSP'09), Σαντορίνη (Santorini), Ελλάδα (Greece), July 5-7, 2009.

The purpose of this talk is to present a modern, unifying perspective of sampling, while demonstrating that the research in this area is still alive and well. We concentrate on the traditional setup where the samples are taken on a uniform grid, but we explicitly take into account the non-ideal nature of the acquisition device and the fact that the measurements may be corrupted by noise. We argue in favor of a variational formulation where the optimal signal reconstruction is specified via a functional optimization problem. The cost to minimize is the sum of a discrete data term and a regularization functional that penalizes non-desirable solutions. We show that, when the regularization is quadratic, the optimal signal reconstruction (among all possible functions) is a generalized spline whose type is tied to the regularization operator. This leads to an optimal discretization and an efficient signal reconstruction in terms of generalized B-spline basis functions. A possible variation is to penalize the L1-norm of the derivative of the function (total variation), which can also be achieved within the spline framework via a suitable knot deletion process.
The theory of compressed sensing provides an alternative approach to sampling that is qualitatively similar to total-variation regularization. Here the idea to favor solutions that have a sparse representation in a wavelet basis. Practically, this is achieved by imposing a regularization constraint on the l1-norm of the wavelet coefficients. We show that the corresponding inverse problem can be solved efficiently via a multi-scale variant of the ISTA algorithm (iterative skrinkage-thresholding). 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.

Sampling and Approximation Theory

M. Unser

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Plenary talk, Summer School "New Trends and Directions in Harmonic Analysis, Approximation Theory, and Image Analysis," Inzell, Germany, September 17-21, 2007.

This tutorial will explain the modern, Hilbert-space approach for the discretization (sampling) and reconstruction (interpolation) of images (in two or higher dimensions). The emphasis will be on quality and optimality, which are important considerations for biomedical applications.

The main point in the modern formulation is that the signal model need not be bandlimited. In fact, it makes much better sense computationally to consider spline or wavelet-like representations that involve much shorter (e.g. compactly supported) basis functions that are shifted replicates of a single prototype (e.g., B-spline). We will show how Shannon's standard sampling paradigm can be adapted for dealing with such representations. In essence, this boils down to modifying the classical "anti-aliasing" prefilter so that it is optimally matched to the representation space (in practice, this can be accomplished by suitable digital post-filtering). Another important issue will be the assessment of interpolation quality and the identification of basis functions (and interpolators) that offer the best performance for a given computational budget.

Reference: M. Unser, "Sampling—50 Years After Shannon," Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, April 2000.

Sampling—50 Years After Shannon

M. Unser

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Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, April 2000.

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we re-interpret Shannon's sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of "shift-invariant" functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) pre-filters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., non-bandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multi-wavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.

Sampling and Interpolation for Biomedical Imaging

M. Unser

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Part I   Part II

2006 IEEE International Symposium on Biomedical Imaging, April 6-9, 2006, Arlington, Virginia, USA.

This tutorial will explain the modern, Hilbert-space approach for the discretization (sampling) and reconstruction (interpolation) of images (in two or higher dimensions). The emphasis will be on quality and optimality, which are important considerations for biomedical applications.
The main point in the modern formulation is that the signal model need not be bandlimited. In fact, it makes much better sense computationally to consider spline or wavelet-like representations that involve much shorter (e.g. compactly supported) basis functions that are shifted replicates of a single prototype (e.g., B-spline). We will show how Shannon's standard sampling paradigm can be adapted for dealing with such representations. In essence, this boils down to modifying the classical "anti-aliasing" prefilter so that it is optimally matched to the representation space (in practice, this can be accomplished by suitable digital post-filtering). We will also discuss efficient digital-filter-based solutions for high-quality image interpolation. Another important issue will be the assessment of interpolation quality and the identification of basis functions (and interpolators) that offer the best performance for a given computational budget. These concepts will be illustrated with various applications in biomedical imaging: tomographic reconstruction, 3D slicing and re-formatting, estimation of image differentials for feature extraction, and image registration (both rigid-body and elastic).

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