Many geophysical inverse problems are ill-posed and have to be regularized. The most often used solution methods for solving ill-posed problems are based on the use of quadratic regularization that results in smooth solutions. Solutions of this type are not to be suitable when the model parameter is piecewise continuous blocky and edges are desired in the regularized solution. To avoid the smoothing of edges, which are very important attributes of an image, an edge-preserving regularization (non-quadratic regularization) term has to be employed. Total Variation (TV) regularization is one of the most effective regularization techniques for allowing sharp edges and the existence of discontinuities in the solutions. The edge-preserving regularization based on the TV method for small-scale geophysical inverse problems to the problem of estimating the acoustic velocity perturbation from a multi-source-receiver geophysical experiment is studied. The acoustic velocity perturbation is assumed to be piecewise continuous and blocky. The problem is based on linearization acoustic modeling using the framework of the single-scattering Born approximation from a known constant background medium. To solve this non-linear and ill-posed problem, an iterative scheme based on the conjugate gradient method is employed. The TV regularization method provides us with the opportunity to recover more useful information of velocity profiles from the measured seismic data. Though it requires more effort in implementing the TV term to control the smoothing and regularization parameter, the algorithm possesses the strong ability of marking the discontinuities and ensures their preservation from over-smoothing. / Geophysics

Let X, Y be locally compact Hausdorff spaces. A linear operator T from C0(X,E) to C0(Y,F) is called disjointness preserving if coz(Tf)¡äcoz(Tg) = whenever coz(f)¡äcoz(g) = ∅. We discuss some cases on these compact disjointness preserving operators T and prove that if £f0 is a nonzero point of £m(T), then £f0 is an eigenvalue of T and we find a projection ∏: C0(X,E) ¡÷C0(X,E), such that for Y1 = ∏C0(X;E) and Y2 = (1-∏)C0(X;E), the operator T|Y1 -£f0 is a nilpotent and £f0-T|Y2 is invertible.

Images may be degraded by some random error which is called noise. Noise may occur during image capture, transmission or processing and its elimination is achieved by smoothing filters. Linear smoothing filters blur the edges and edges are important characteristics of images and must be preserved. Various edge preserving smoothing filters are proposed in the literature. In this thesis, most common smoothing and edge preserving smoothing filters are discussed and a new method is proposed by modifying Ambrosio Tortorelli approximation of Mumford Shah Model. New method takes into edge direction consistency account and produces sharper results at comparable scales.

In this thesis, we study the question when a local automorphism of continuous functions, or in general, of an operator algebra, is an automorphism. We also study the question how to write an n-disjointness preserving operator as a finite sum of orthomorphisms locally.

This item was digitized as part of the Million Books Project led by Carnegie Mellon University and supported by grants from the National Science Foundation (NSF). Cornell University coordinated the participation of land-grant and agricultural libraries in providing historical agricultural information for the digitization project; the University of Arizona Libraries, the College of Agriculture and Life Sciences, and the Office of Arid Lands Studies collaborated in the selection and provision of material for the digitization project.

This item was digitized as part of the Million Books Project led by Carnegie Mellon University and supported by grants from the National Science Foundation (NSF). Cornell University coordinated the participation of land-grant and agricultural libraries in providing historical agricultural information for the digitization project; the University of Arizona Libraries, the College of Agriculture and Life Sciences, and the Office of Arid Lands Studies collaborated in the selection and provision of material for the digitization project.

This item was digitized as part of the Million Books Project led by Carnegie Mellon University and supported by grants from the National Science Foundation (NSF). Cornell University coordinated the participation of land-grant and agricultural libraries in providing historical agricultural information for the digitization project; the University of Arizona Libraries, the College of Agriculture and Life Sciences, and the Office of Arid Lands Studies collaborated in the selection and provision of material for the digitization project.

This item was digitized as part of the Million Books Project led by Carnegie Mellon University and supported by grants from the National Science Foundation (NSF). Cornell University coordinated the participation of land-grant and agricultural libraries in providing historical agricultural information for the digitization project; the University of Arizona Libraries, the College of Agriculture and Life Sciences, and the Office of Arid Lands Studies collaborated in the selection and provision of material for the digitization project.

This item was digitized as part of the Million Books Project led by Carnegie Mellon University and supported by grants from the National Science Foundation (NSF). Cornell University coordinated the participation of land-grant and agricultural libraries in providing historical agricultural information for the digitization project; the University of Arizona Libraries, the College of Agriculture and Life Sciences, and the Office of Arid Lands Studies collaborated in the selection and provision of material for the digitization project.

This item was digitized as part of the Million Books Project led by Carnegie Mellon University and supported by grants from the National Science Foundation (NSF). Cornell University coordinated the participation of land-grant and agricultural libraries in providing historical agricultural information for the digitization project; the University of Arizona Libraries, the College of Agriculture and Life Sciences, and the Office of Arid Lands Studies collaborated in the selection and provision of material for the digitization project.