The goal of this thesis is to gain a deep understanding of inverse problems arising from 2D image and 3D surface reconstruction, and to design effective techniques for solving them. Both computational and theoretical issues are studied and efficient numerical algorithms are proposed. The first part of this thesis is concerned with the recovery of 2D images, e.g., de-noising and de-blurring. We first consider implicit methods that involve solving linear systems at each iteration. An adaptive Huber regularization functional is used to select the most reasonable model and a global convergence result for lagged diffusivity is proved. Two mechanisms---multilevel continuation and multigrid preconditioning---are proposed to improve efficiency for large-scale problems. Next, explicit methods involving the construction of an artificial time-dependent differential equation model followed by forward Euler discretization are analyzed. A rapid, adaptive scheme is then proposed, and additional hybrid algorithms are designed to improve the quality of such processes. We also devise methods for more challenging cases, such as recapturing texture from a noisy input and de-blurring an image in the presence of significant noise. It is well-known that extending image processing methods to 3D triangular surface meshes is far from trivial or automatic. In the second part of this thesis we discuss techniques for faithfully reconstructing such surface models with different features. Some models contain a lot of small yet visually meaningful details, and typically require very fine meshes to represent them well; others consist of large flat regions, long sharp edges (creases) and distinct corners, and the meshes required for their representation can often be much coarser. All of these models may be sampled very irregularly. For models of the first class, we methodically develop a fast multiscale anisotropic Laplacian (MSAL) smoothing algorithm. To reconstruct a piecewise smooth CAD-like model in the second class, we design an efficient hybrid algorithm based on specific vertex classification, which combines K-means clustering and geometric a priori information. Hence, we have a set of algorithms that efficiently handle smoothing and regularization of meshes large and small in a variety of situations.

It is difficult for people to recognize currencies from different countries. Our aim is to help people solve this problem. However, currency recognition systems that are based on image analysis entirely are not sufficient. Our system is based on image processing and makes the process automatic and robust. We use SEK and Chinese RMB as examples to illustrate the technique. Color and shape information are used in our algorithm.

Inexpensive high quality fundus camera systems enable imaging of retina for vision related health management and diagnosis at large scale. A computer based analysis system can help establish the general baseline of normal conditions vs. anomalous ones, so that different classes of retinal conditions can be classified. Advanced applications, ranging from disease screening algorithms, aging vs. disease trend modeling and prediction, and content-based retrieval systems can be developed. In this dissertation, I propose an analytical framework for the modeling of retina blood vessels to capture their statistical properties, so that based on these properties one can develop blood vessel mapping algorithms with self-optimized parameters. Then, other image objects can be registered based on vascular topology modeling techniques. On the basis of these low level analytical models and algorithms, the third major element of this dissertation is a high level population statistics application, in which texture classification of macular patterns is correlated with vessel structures, which can also be used for retinal image retrieval. The analytical models have been implemented and tested based on various image sources. Some of the algorithms have been used for clinical tests. The major contributions of this dissertation are summarized as follows: (1) A concise, accurate feature representation of retinal blood vessel on retinal images by proposing two feature descriptors Sp and Ep derived from radial contrast transform. (2) A new statistical model of lognormal distribution, which captures the underlying physical property of the levels of generations of the vascular network on retinal images. (3) Fast and accurate detection algorithms for retinal objects, which include retinal blood vessel, macular-fovea area and optic disc, and (4) A novel population statistics based modeling technique for correlation analysis of blood vessels and other image objects that only exhibit subtle texture changes.

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Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 187-201). Available also in a digital version from Dissertation Abstracts.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 78-80).

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (p. 95-97). Also available in print.

Thesis (Ph. D.)--University of Chicago, Dept of Physics, 2001. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 131-143).