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Splines, Noise, Fractals and Optimal Signal Reconstruction 

M. Unser 

Plenary talk, Seventh International Workshop on Sampling Theory and Applications (SampTA'07), Thessaloniki, Greece, June 15, 2007. 

We consider the generalized sampling problem with nonideal acquisition device. The task is to “optimally” reconstruct a continuouslyvarying input signal from its discrete, noisy measurements in some integershiftinvariant space. We propose three formulations of the problem—variational/Tikhonov, minimax, and minimum mean square error estimation—and derive the corresponding solutions for a given reconstruction space. We prove that these solutions are also globallyoptimal, provided that the reconstruction space is matched to the regularization operator (deterministic signal) or, alternatively, to the whitening operator of the process (stochastic modeling). Moreover, the three formulations lead to the same generalized smoothing spline reconstruction algorithm, but only if the reconstruction space is chosen optimally. We then show that fractional splines and fractal processes (fBm) are solutions of the same type of differential equations, except that the context is different: deterministic versus stochastic. We use this link to provide a solid stochastic justification of splinebased reconstruction algorithms. Finally, we propose a novel formulation of vectorsplines based on similar principles, and demonstrate their application to flow field reconstruction from nonuniform, incomplete ultrasound Doppler data. This is joint work with Yonina Eldar, Thierry Blu, and Muthuvel Arigovindan. 



The Colored Revolution of Bioimaging 

C. Vonesch, F. Aguet, J.L. Vonesch, M. Unser 

IEEE Signal Processing Magazine, vol. 23, no. 3, pp. 2031, May 2006. 

With the recent development of fluorescent probes and new highresolution microscopes, biological imaging has entered a new era and is presently having a profound impact on the way research is being conducted in the life sciences. Biologists have come to depend more and more on imaging. They can now visualize subcellular components and processes in vivo, both structurally and functionally. Observations can be made in two or three dimensions, at different wavelengths (spectroscopy), possibly with timelapse imaging to investigate cellular dynamics. The observation of many biological processes relies on the ability to identify and locate specific proteins within their cellular environment. Cells are mostly transparent in their natural state and the immense number of molecules that constitute them are optically indistinguishable from one another. This makes the identification of a particular protein a very complex task—akin to finding a needle in a haystack. 

Wavelet Demystified 

M. Unser 

Invited presentation, Technical University of Eindhoven, The Netherlands, May 31, 2006. 

This 2hour tutorial focuses on wavelet bases : it covers the concept of multiresolution analysis, the construction of wavelets,
filterbank algorithms, as well as an indepth discussion of fundamental wavelet properties. The presentation is progressive starting with the example
of the Haar transform and essentially self contained. 



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. 

Wavelet Games 

M. Unser, M. Unser 

The Wavelet Digest, vol. 11, Issue 4, April 1, 2003. 

Summary: A dialogue between father and son about wavelets, Legos, and other existential questions. 

Fractional Splines and Wavelets 

M. Unser, T. Blu 

SIAM Review, vol. 42, no. 1, pp. 4367, March 2000. 

We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees α > 1. These splines, which involve linear combinations of the onesided power functions x_{+}^{α} = max(0, x)^{α}, belong to L^{1} and are αHölder continuous for α > 0. We construct the corresponding Bsplines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains. We show that these functions satisfy most of the properties of the traditional Bsplines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the Bsplines which are not compactly supported for nonintegral α's. Their most astonishing feature (in reference to the StrangFix theory) is that they have a fractional order of approximation α + 1 while they reproduce the polynomials of degree [α]. For α > 1/2, they satisfy all the requirements for a multiresolution analysis of L^{2} (Riesz bounds, two scale relation) and may therefore be used to build new families of wavelet bases with a continuouslyvarying order parameter. Our construction also yields symmetrized fractional Bsplines which provide the connection with Duchon's general theory of radial (m,s)splines (including thinplate splines). In particular, we show that the symmetric version of our splines can be obtained as solution of a variational problem involving the norm of a fractional derivative. 


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