Real and Complex Analysis

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Time-Frequency Analysis.

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

Zeros of orthogonal polynomials

Identification and rational L 2 approximation: a gradient algorithm

This article explores an optimization procedure for microwave filters and multiplexers. The procedure is initialized by a classical filter synthesis based on a segmented electromagnetic synthesis that provides the basic dimensions of the structure. The optimization loop, which combines a global electromagnetic analysis and a coupling identification, improves the structure response compared to an empirical optimization.

Direct electromagnetic optimization of microwave filters

We consider the inverse problems of locating pointwise or small size conductivity defaults in a plane domain, from overdetermined boundary measurements of solutions to the Laplace equation. We express these issues in terms of best rational or meromorphic approximation problems on the boundary, with poles constrained to belong to the domain. This approach furnishes efficient and original resolution schemes.

Recovery of pointwise sources or small inclusions in 2D domains and rational approximation

We exhibit new links between approximation theory in the complex domain and a family of inverse problems for the 2D Laplacian related to non-destructive testing.

How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian

[1] Scanning magnetic microscopes are being increasingly utilized in paleomagnetic studies of geological samples. These instruments typically map a single component of the sample's magnetic field at close proximity with submillimeter horizontal spatial resolution. However, in most applications, an image of the magnetization distribution within the sample is desired rather than its external magnetic field. This requires carefully solving an ill-posed inverse problem to obtain solutions that are nearly free of artifacts and consistent with both natural and laboratory magnetization processes. We present a new, fast inversion technique based on classic methods developed for the Fourier domain that retrieves planar unidirectional magnetization distributions from magnetic field maps. Whereas our approach considers the subtle peculiarities of scanning magnetic microscopy which otherwise can complicate this technique, much of the formalism and algorithms described in this work can also be directly applied to province-scale magnetic field data from aeromagnetic surveys and may be used as an initial step in the modeling of magnetic sources with complex three-dimensional geometries. We discuss sources of inaccuracy observed in practical implementations of the technique and present strategies to improve the quality of inversions. Numerous examples of inversion of both synthetic and experimental data demonstrate the performance of the technique under different conditions. In particular, we retrieve magnetization distributions of a Hawaiian basalt and compare it to inversions calculated in a previous work. We conclude by showing a reconstructed magnetization for the eucrite meteorite ALHA81001 that displays in high resolution the spatial distribution of high-coercivity grains within the sample.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/jgrb.50229

Laurent Baratchart

Papers

Identification and rational L 2 approximation: a gradient algorithm

Direct electromagnetic optimization of microwave filters

Recovery of pointwise sources or small inclusions in 2D domains and rational approximation

How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian

Fast inversion of magnetic field maps of unidirectional planar geological magnetization