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The area known as signal processing covers a large range of pure and
applied mathematical techniques and disciplines. This has had a wide
spectrum of applications
in science and technology.
One can for instance point to the determination of the structure of DNA (around 1953) by scientist such as Crick, Watson, Wilkins and Franklyn. One can make a long list of other important applications , such as the problems of medical imaging, gephysical prospecting, remote sensing, reconstruction and denoising of images, etc. where tools such as Fourier analysis or expansions in some other basis of functions such as orthogonal polynomials or wavelets play an important role.
The work of Crick and Watson and mainly that of Crick's mentor Max Perutz constitute the basis for the routine determination of the three dimensional structure of proteins and other biological units of great interest. M. Perutz solved the structure of hemoglobin in the fifties. With the exception of R. Franklyn who died very young, all these researchers were honored with different Nobel Prizes in Medicine and Biology for carrying out work with a strong mathematical content.
The strong relation between the theory of orthogonal polynomials and
prediction theory for stochastic processes, an area developed more or less
simultaneously by N. Wiener in the USA and A. Kolmogorov in the Soviet Union
in the 40's is well known and constitutes the core of several important articles,
such as those of T. Kkailath in the 70'. The theory was extended to the case
of a signal with a continuous time parameter in a seminal article by M. G. Krein.
The title of this paper is "On a generalization of investigations of Stieltjes",
Doklady ,1952. T. Stieltjes is one of the founders of the theory of orthogonal
polynomials and his work was done mainly towards the end of the 19th century.
The choice of this title for Krein's paper speaks clearly of the close connection
between the two subjects.
The work we have referred to deals almost exclusively with one dimensional signals that have some extra restrictive properties such as being Gaussian or Markovian. These signal are usually scalar valued.
The purpose of this workshop is to detail the present state of the art, to describe the mathematical, scientific and technological issues behind these developments and to point the way in the direction of extending this work to the case of signals that depend on a multi dimensional parameter and that take values that are not just scalars but might be vector or matrix valued.
Particular attention will be devoted to the study of special functions with matrix values, an area with a large development in the last three years, and its connection with "hyperspectral imaging" one of the main areas of interest in the case of matrix valued signals: using passive sensors that register at a given time several two dimensional images of different parts in the electromagnetic spectrum one can form "data cubes" with very useful information. This is being used in areas that include geology, hydrology and urban planning as well as cartography, among others. Among possible future uses of this type of mathematical analysis one has the emerging area of "tensor tomography" where one aims at mapping the deformation tensor in different regions of the human heart and other organs.
We will start on Moday Octuber 15, at 9:30 and finish on Wednesday October 17 at 16:30.
