On Fresnelets, interference fringes, and digital holography

In this thesis, we describe new approaches and methods for reconstructing complex-valued wave fields from digital holograms. We focus on Fresnel holograms recorded in an off-axis geometry, for which operational real-time acquisition setups readily exist. The three main research directions presented are the following. First, we derive the necessary tools to port methods and concepts of wavelet-based approaches to the field of digital holography. This is motivated by the flexibility, the robustness, and the unifying view that such multiresolution procedures have brought to many applications in image processing. In particular, we put emphasis on space-frequency processing and sparse signal representations. Second, we propose to decouple the demodulation from the propagation problem, which are both inherent to digital Fresnel holography. To this end, we derive a method for retrieving the amplitude and phase of the object wave through a local analysis of the hologram's interference fringes. Third, since digital holography reconstruction algorithms involve a number of parametric models, we propose automatic adjustment methods of the corresponding parameters. We start by investigating the Fresnel transform, which plays a central role in both the modeling of the acquisition procedure and the reconstruction of complex wave fields. The study of the properties that are central to wavelet and multiresolution analysis leads us to derive Fresnelets, a new family of wavelet-like bases. Fresnelets permit the analysis of holograms with a good localization in space and frequency, in a way similar to wavelets for images. Since the relevant information in a Fresnel off-axis hologram may be separated both in space and frequency, we propose an approach for selectively retrieving the information in the Fresnelet domain. We show that in certain situations, this approach is superior to others that exclusively rely on the separation in space or frequency. We then derive a least-squares method for the estimation of the object wave's amplitude and phase. The approach, which is reminiscent of phase-shifting techniques, is sufficiently general to be applied in a wide variety of situations, including those dictated by the use of microscopy objectives. Since it is difficult to determine the reconstruction distance manually, we propose an automatic procedure. We take advantage of our separate treatment of the phase retrieval and propagation problems to come up with an algorithm that maximizes a sharpness metric related to the sparsity of the signal's expansion in distance-dependent Fresnelet bases. Based on a simulation study, we suggest a number of guidelines for deciding which algorithm to apply to a given problem. We compare existing and the newly proposed solutions in a wide variety of situations. Our final conclusion is that the proposed methods result in flexible algorithms that are competitive with preexisting ones and superior to them in many cases. Overall, they may be applied in a wide range of experimental situations at a low computational cost

Phase extraction of non-stationary signals produced in dynamic interferometry involving speckle waves

It is now widely acknowledged, among communities of researchers and engineers of very different horizons, that speckle interferometry (SI) offers powerful techniques to characterize mechanical rough surfaces with a submicronic accuracy in static or quasi-static regime, when small displacements are involved (typically several microns or tens of microns). The issue of dynamic regimes with possibly large deformations (typically several hundreds of microns) is still topical and prevents an even more widespread use of speckle techniques. This is essentially due to the lack of efficient processing schemes able to cope with non-stationary AM-FM interferometric signals. In addition, decorrelation-induced phase errors represent an hindrance to accurate measurement when such large displacements and classical fringe analysis techniques are considered. This work is an attempt to address those issues and to endeavor to make the most of speckle interferometry signals. Our answers to those problems are located on two different levels. First of all, we adopt the temporal analysis approach, i.e. the analysis of the temporal signal of each pixel of the sensor area used to record the interferograms. A return to basics of phase extraction is operated to properly identify the conditions under which the computed phase is meaningful and thus give some insight on the physical phenomenon under analysis. Due to their intrinsic non-stationary nature, a preprocessing tool is missing to put the SI temporal signals in a shape which ensures an accurate phase computation, whichever technique is chosen. This is where the Empirical Mode Decomposition (EMD) intervenes. This technique, somehow equivalent to an adaptive filtering technique, has been studied and tailored to fit with our expectations. The EMD has shown a great ability to remove efficiently the random fluctuating background intensity and to evaluate the modulation intensity. The Hilbert tranform (HT) is the natural quadrature operator. Its use to build an analytical signal from the so-detrended SI signal, for subsequent phase computation, has been studied and assessed. Other phase extraction techniques have been considered as well for comparison purposes. Finally, our answer to the decorrelation-induced phase error relies on the well-known result that the higher the pixel modulation intensity, the lower the random phase error. We took benefit from this result – not only linked to basic SNR considerations, but more specifically to the intrinsic phase structure of speckle fields – with a novel approach. The regions within the pixel signal history classified as unreliable because under-modulated, are purely and simply discarded. An interpolation step with the Delaunay triangulation is carried out with the so-obtained non-uniformly sampled phase maps to recover a smooth phase which relies on the most reliable available data. Our schemes have been tested and discussed with simulated and experimental SI signals. We eventually have developed a versatile, accurate and efficient phase extraction procedure, perfectly able to tackle the challenge of dynamic behaviors characterization, even for displacements and/or deformations beyond the classical limit of the correlation dimensions

Contributions à la segmentation d'image : phase locale et modèles statistiques

Ce document presente une synthèse de mes travaux apres these, principalement sur la problematique de la segmentation d’images

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

NASA thesaurus. Volume 1: Hierarchical Listing

There are over 17,000 postable terms and nearly 4,000 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary and Volume 3 - Definitions

NASA Thesaurus. Volume 1: Hierarchical listing

There are 16,713 postable terms and 3,716 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary