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Sampling 
Stochastic Models for Sparse and PiecewiseSmooth Signals 

M. Unser 

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

We introduce an extended family of continuousdomain stochastic models for sparse, piecewisesmooth 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 nonstandard aspect is that the models are driven by nonGaussian noise (impulsive Poisson or alphastable) 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 nonGaussian (sparse) counterpart of fractional Brownian motion; they are nonstationary and have the same ωtype spectral signature. We prove that our generalized processes have a sparse representation in a waveletlike basis subject to some mild matching condition. Finally, we discuss implications for sampling and sparse signal recovery. 



Sampling: 60 Years After Shannon 

M. Unser 

Plenary talk, Sixteenth International Conference on Digital Signal Processing (DSP'09), Σαντορίνη (Santorini), Ελλάδα (Greece), July 57, 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 nonideal 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 nondesirable 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 Bspline basis functions. A possible variation is
to penalize the L1norm of the derivative of the function (total variation), which can also be
achieved within the spline framework via a suitable knot deletion process.


Sampling and Approximation Theory 

M. Unser 

Plenary talk, Summer School "New Trends and Directions in Harmonic Analysis, Approximation Theory, and Image Analysis," Inzell, Germany, September 1721, 2007. 

This tutorial will explain the modern, Hilbertspace 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 waveletlike representations that involve much shorter (e.g. compactly supported) basis functions that are shifted replicates of a single prototype (e.g., Bspline). 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 "antialiasing" prefilter so that it is optimally matched to the representation space (in practice, this can be accomplished by suitable digital postfiltering). 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. 569587, April 2000. 



Sampling—50 Years After Shannon 

M. Unser 

Proceedings of the IEEE, vol. 88, no. 4, pp. 569587, 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, Hilbertspace formulation, we reinterpret 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 "shiftinvariant" 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 (antialiasing) prefilters 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., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. 

Sampling and Interpolation for Biomedical Imaging 

M. Unser 
Part I Part II 

2006 IEEE International Symposium on Biomedical Imaging, April 69, 2006, Arlington, Virginia, USA. 

This tutorial will explain the modern, Hilbertspace 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.



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