Mathematics
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Item Novel integro-differential schemes for multiscale image representation(2009) Athavale, Prashant Vinayak; Tadmor, Eitan; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Multiscale representation of a given image is the problem of constructing a family of images, where each image in this family represents a scaled version of the given image. This finds its motivation from biological vision studies. Using the hierarchical multiscale image representation proposed by Tadmor et. al. [32], an image is decomposed into sums of simpler `slices', which extract more refined information from the previous scales. This approach motivates us to propose a novel integro-differential equation (IDE), for a multiscale image representation. We examine various properties of this IDE. The advantage of formulating the IDE this way is that, although this IDE is motivated by variational approach, we no longer need to be associated with any minimization problem and can modify the IDE, suitable to our image processing needs. For example, we may need to find different scales in the image, while retaining or enhancing prominent edges, which may define boundaries of objects. We propose some edge preserving modifications to our IDE. One of the important problems in image processing is deblurring a blurred image. Images get blurred due to various reasons, such as unfocused camera lens, relative motion between the camera and the object pictured, etc. The blurring can be modeled with a continuous, linear operator. Recovering a clean image from a blurry image, is an ill-posed problem, which is solved using Tikhonov-like regularization. We propose a different IDE to solve the deblurring problem. We propose hierarchical multiscale scheme based on (BV; L1) decomposition, proposed by Chan, Esedoglu, Nikolova and Alliney [12, 25, 3]. We finally propose another hierarchical multiscale representation based on a novel weighted (BV;L1) decomposition.Item Algorithms for Solving Linear and Polynomial Systems of Equations over Finite Fields with Applications to Cryptanalysis(2007-06-07) Bard, Gregory Van; Washington, Lawrence C; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation contains algorithms for solving linear and polynomial systems of equations over GF(2). The objective is to provide fast and exact tools for algebraic cryptanalysis and other applications. Accordingly, it is divided into two parts. The first part deals with polynomial systems. Chapter 2 contains a successful cryptanalysis of Keeloq, the block cipher used in nearly all luxury automobiles. The attack is more than 16,000 times faster than brute force, but queries 0.62 × 2^32 plaintexts. The polynomial systems of equations arising from that cryptanalysis were solved via SAT-solvers. Therefore, Chapter 3 introduces a new method of solving polynomial systems of equations by converting them into CNF-SAT problems and using a SAT-solver. Finally, Chapter 4 contains a discussion on how SAT-solvers work internally. The second part deals with linear systems over GF(2), and other small fields (and rings). These occur in cryptanalysis when using the XL algorithm, which converts polynomial systems into larger linear systems. We introduce a new complexity model and data structures for GF(2)-matrix operations. This is discussed in Appendix B but applies to all of Part II. Chapter 5 contains an analysis of "the Method of Four Russians" for multiplication and a variant for matrix inversion, which is log n faster than Gaussian Elimination, and can be combined with Strassen-like algorithms. Chapter 6 contains an algorithm for accelerating matrix multiplication over small finite fields. It is feasible but the memory cost is so high that it is mostly of theoretical interest. Appendix A contains some discussion of GF(2)-linear algebra and how it differs from linear algebra in R and C. Appendix C discusses algorithms faster than Strassen's algorithm, and contains proofs that matrix multiplication, matrix squaring, triangular matrix inversion, LUP-factorization, general matrix in- version and the taking of determinants, are equicomplex. These proofs are already known, but are here gathered into one place in the same notation.Item Network Tomography(2006-08-24) Gavilanez, Franklin; Berenstein, Carlos A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)While conventional tomography is associated to the Radon transform in Euclidean spaces, electrical impedance tomography, or EIT, is associated to the Radon transform in the hyperbolic plane. In this dissertation, we discuss some recent work on network tomography that can be associated to a problem similar to EIT on graphs and indicate how in some sense it may be also associated to the Radon transform on trees. We develop a strategy to determine the weight w for the case of general weighted graphs. We begin by considering relatively simple regions of interest in a graph and suitable choices for the data of the w-Neumann boundary value problem to produce a linear system of equations for the values of w.Item Entropy-Based Moment Closures in Semiconductor Models(2006-04-27) Hauck, Cory D.; Levermore, Charles D.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We investigate aspects of entropy-based moment closures which are used to simplify kinetic models of particle systems. Closures of this type use variational principles to formally generate balance laws for velocity moments of a kinetic density. These balance laws form a symmetric hyperbolic system of partial differential equations that satisfies an analog of Boltzmann's famous H-Theorem. However, in spite of this elegant structure, practical implementation of entropy-based closures requires that several analytical and computational issues be settled. Our presentation is devoted to the development of electron transport models in semiconductor devices. In this context, balance laws for velocity moments are generally referred to as hydrodynamic models. Such models provide a reasonable alternative to kinetic and Monte Carlo approaches, which are usually expensive, and the well-known drift-diffusion model, which is much simpler but a has a limited range of validity. We first analyze the minimization problem that defines the entropy closure. It is known that there are physically relevant cases for which this problem is ill-posed. Using a dual formulation, we find so-called complementary slackness conditions which give a geometric interpretation of ill-posed cases in terms of the Lagrange multipliers of the minimization problem. Under reasonable assumptions, we show that these cases are rare in a very precise sense. We also develop pertubations of well-posed entropy-based closures, thereby making them useful for modeling systems with heat flux and anisotropic stress. Heat flux has long been known to be an important component of electron transport in semiconductors. However, we also observe that anisotropy in the stress tensor also plays an important role in regions of high electric field. This conclusion is made based on our simulations of two different devices. Finally, we devise a new split scheme for hydrodynamic models. The splitting is based on the balance of forces in the hydrodynamic model that recovers the drift-diffusion equation in the asymptotic limit of small mean-free-path. This scheme removes numerically stiffness and excessive dissipation typically associated with standard shock-capturing schemes in the drift-diffusion limit. In addition, it significantly reduces numerical current oscillations near material junctions.Item Registration Methods for Quantitative Imaging(2005-08-03) Rohde, Gustavo Kunde; Berenstein, Carlos A; Healy, Dennis M; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)At the core of most image registration problems is determining a spatial transformation that relates the physical coordinates of two or more images. Registration methods have become ubiquitous in many quantitative imaging applications. They represent an essential step for many biomedical and bioengineering applications. For example, image registration is a necessary step for removing motion and distortion related artifacts in serial images, for studying the variation of biological tissue properties, such as shape and composition, across different populations, and many other applications. Here fully automatic intensity based methods for image registration are reviewed within a global energy minimization framework. A linear, shift-invariant, stochastic model for the image formation process is used to describe several important aspects of typical implementations of image registration methods. In particular, we show that due to the stochastic nature of the image formation process, most methods for automatic image registration produce answers biased towards `blurred' images. In addition we show how image approximation and interpolation procedures necessary to compute the registered images can have undesirable effects on subsequent quantitative image analysis methods. We describe the exact sources of such artifacts and propose methods through which these can be mitigated. The newly proposed methodology is tested using both simulated and real image data. Case studies using three-dimensional diffusion weighted magnetic resonance images, diffusion tensor images, and two-dimensional optical images are presented. Though the specific examples shown relate exclusively to the fields of biomedical imaging and biomedical engineering, the methods described are general and should be applicable to a wide variety of imaging problems.