A multi-layered periodic structure is investigated for optimal shape design in diffraction gratings. A periodic dielectric material is used as the scattering profile for a planar incident wave. Designing optimal profiles for scattering is a type of inverse problem. The ability to fabricate such materials on the order of the wavelength of the incoming light is key for design strategies. We compute a finite element approximation on a variational setup of the forward problem. On the inverse and optimal design problem, we discuss the stability of the designs and develop computational strategies based on a level-set evolutionary approach.

Inverse problems in (exploration) seismology are known for their large to very large scale. For instance, certain sparsity-promoting inversion techniques involve vectors that easily exceed 230 unknowns while seismic imaging involves the construction and application of matrix-free discretized operators where single matrix-vector evaluations may require hours, days or even weeks on large compute clusters. For these reasons, software development in this field has remained the domain of highly technical codes programmed in low-level languages with little eye for easy development, code reuse and integration with (nonlinear) programs that solve inverse problems. Following ideas from the Symes’ Rice Vector Library and Bartlett’s C++ object-oriented interface, Thyra, and Reduction/Transformation operators (both part of the Trilinos software package), we developed a software-development environment based on overloading. This environment provides a pathway from in-core prototype development to out-of-core and MPI ’production’ code with a high level of code reuse. This code reuse is accomplished by integrating the out-of-core and MPI functionality into the dynamic object-oriented programming language Python. This integration is implemented through operator overloading and allows for the development of a coordinate-free solver framework that (i) promotes code reuse; (ii) analyses the statements in an abstract syntax tree and (iii) generates executable statements. In the current implementation, we developed an interface to generate executable statements for the out-of-core unix-pipe based (seismic) processing package RSF-Madagascar (rsf.sf.net). The modular design allows for interfaces to other seismic processing packages and to in-core Python packages such as numpy. So far, the implementation overloads linear operators and elementwise reduction/transformation operators. We are planning extensions towards nonlinear operators and integration with existing (parallel) solver frameworks such as Trilinos.

The use of fractal-based methods in imaging was first popularized with fractal image compression in the early 1990s. In this application, one seeks to approximate a given target image by the fixed point of a contractive operator called the fractal transform. Typically, one uses Local Iterated Function Systems with Grey-Level Maps (LIFSM), where the involved functions map a parent (domain) block in an image to a smaller child (range) block and the grey-level maps adjust the shading of the shrunken block. The fractal transform is defined by the collection of optimal parent-child pairings and parameters defining the grey-level maps. Iteration of the fractal transform on any initial image produces an approximation of the fixed point and, hence, an approximation of the target image. Since the parameters defining the LIFSM take less space to store than the target image does, image compression is achieved.This thesis extends the theoretical and practical frameworks of fractal imaging to one involving a particular type of multifunction that captures the idea that there are typically many near-optimal parent-child pairings. Using this extended machinery, we treat three application areas. After discussing established edge detection methods, we present a fractal-based approach to edge detection with results that compare favourably to the Sobel edge detector. Next, we discuss two methods of information hiding: first, we explore compositions of fractal transforms and cycles of images and apply these concepts to image-hiding; second, we propose and demonstrate an algorithm that allows us to securely embed with redundancy a binary string within an image. Finally, we discuss some theory of certain random fractal transforms with potential applications to texturing. / The Natural Sciences and Engineering Research Council and the University of Guelph helped to provide financial support for this research.

We are investigating a kernel-based autoconvolution problem, which has its origin in the physics of ultra short laser pulses. The task in this problem is to reconstruct a complex-valued function $x$ on a finite interval from measurements of its absolute value and a kernel-based autoconvolution of the form [[F(x)](s)=int k(s,t)x(s-t)x(t)de t.] This problem has not been studied in the literature. One reason might be that one has more information than in the classical autoconvolution case, where only the right hand side is available. Nevertheless we show that ill posedness phenomena may occur. We also propose an algorithm to solve the problem numerically and demonstrate its performance with artificial data. Since the algorithm fails to produce good results with real data and we suspect that the data for $|F(x)|$ are not dependable we also consider the whole problem with only $arg(F(x))$ given instead of $F(x)$.

<p>The emerging theory of compressed sensing has been nothing short of a revolution in signal processing, challenging some of the longest-held ideas in signal processing and leading to the development of exciting new ways to capture and reconstruct signals and images. Although the theoretical promises of compressed sensing are manifold, its implementation in many practical applications has lagged behind the associated theoretical development. Our goal is to elevate compressed sensing from an interesting theoretical discussion to a feasible alternative to conventional imaging, a significant challenge and an exciting topic for research in signal processing. When applied to imaging, compressed sensing can be thought of as a particular case of computational imaging, which unites the design of both the sensing and reconstruction of images under one design paradigm. Computational imaging tightly fuses modeling of scene content, imaging hardware design, and the subsequent reconstruction algorithms used to recover the images. </p><p>This thesis makes important contributions to each of these three areas through two primary research directions. The first direction primarily attacks the challenges associated with designing practical imaging systems that implement incoherent measurements. Our proposed snapshot imaging architecture using compressive coded aperture imaging devices can be practically implemented, and comes equipped with theoretical recovery guarantees. It is also straightforward to extend these ideas to a video setting where careful modeling of the scene can allow for joint spatio-temporal compressive sensing. The second direction develops a host of new computational tools for photon-limited inverse problems. These situations arise with increasing frequency in modern imaging applications as we seek to drive down image acquisition times, limit excitation powers, or deliver less radiation to a patient. By an accurate statistical characterization of the measurement process in optical systems, including the inherent Poisson noise associated with photon detection, our class of algorithms is able to deliver high-fidelity images with a fraction of the required scan time, as well as enable novel methods for tissue quantification from intraoperative microendoscopy data. In short, the contributions of this dissertation are diverse, further the state-of-the-art in computational imaging, elevate compressed sensing from an interesting theory to a practical imaging methodology, and allow for effective image recovery in light-starved applications.</p> / Dissertation

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Civil and Environmental Engineering." Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 101-103).

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 418-432). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 27-29). Also available on the Internet.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (p. [77]-79).