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This dissertation is concerned with surface representations which record surface properties as a function of surface orientation. The Extended Gaussian Image (EGI) of an object records the variation of surface area with surface orientation. When the object is polyhedral, the EGI takes the form of a set of vectors, one for each face, parallel to the outer surface normal of the face. The length of a vector is the area of the corresponding face. The EGI uniquely represents convex objects and is easily derived from conventional models of an object. An iterative algorithm is described which converts an EGI into an object model in terms of coordinates of vertices, edges, and faces. The algorithm converges to a solution by constrained optimization. There are two aspects to describing shape for polyhedral objects: first, the way in which faces intersect each other, termed the adjacency structure, and, second, the location of the faces in space. The latter may change without altering the former, but not vice versa. The algorithm for shape recovery determines both elements of shape. The continuous support function is described in terms of the area function for curves, permitting a qualitative comparison of the smoothness of the two functions. The next Section describes a method of curve segmentation based on extrema of the support function. Because the support function varies with translation, its behaviour under translation is studied, leading to proposals for several candidate centres of support. The study of these ideas suggests some interesting problems in computational geometry. The EGI has been applied to determine object attitude, the rotation in 3-space bringing a sample object into correspondence with a prototype. The methods developed for the inversion problem can be applied to attitude determination. Experiments show attitude determination using the new method to be more robust than area matching methods. The method given here can be applied at lower resolution of orientation, so that it is possible to sample the space of attitudes more densely, leading to increased accuracy in attitude determination. The discussion finally is broadened to include non-convex objects, where surface orientation is not unique. The generalizations of the EGI do not support shape reconstruction for arbitrary non-convex objects. However, surfaces of revolution do allow a natural generalization of the EGI. The topological structure of regions of constant sign of curvature is invariant under Euclidean motion, and may be useful for recognition tasks. / Science, Faculty of / Computer Science, Department of / Graduate

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Thesis (Ph. D.)--Lehigh University, 2006. / Includes vita. Includes bibliographical references (leaves 254-261).

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La Spectroscopie de Masse d'Ions Secondaires (SIMS) permet d'obtenir des images de distributions d'atomes à la surface d'un échantillon. La réponse impulsionnelle (RI) de l'instrument est inconnue. La déconvolution en aveugle a pour but d'enlever le flou associé. Ce problème mal conditionné est résolu en contraignant sa solution (régularisation). Le degré optimum de régularisation dépend d'un paramètre à déterminer. Il est trouvé, ainsi que ceux de la RI, par la méthode de validation croisée généralisée. Une étape de calibrage restreint l'espace de recherche des paramètres de la RI et les calculs sont accélérés en exploitant le modèle gaussien. L'image est déconvoluée en résolvant un grand système linéaire par la méthode du gradient conjugué. Un préconditionnement exploitant la séparabilité de la RI (isotrope ou anisotrope) en accélère la convergence. On montre comment utiliser plusieurs images d'un échantillon pour avoir une résolution plus fine (super-résolution). / Secondary Ion Mass Spectrometry (SIMS) creates images of atomic distributions on a sample's surface. The point spread function (PSF) is unknown. Blind deconvolution is used to remove the associated blur. This ill-conditionned problem is solved by constraining its solution (regularization). The optimum degree of regularization depends on a parameter to be determined. This parameter is found, as well as those of the PSF, by the generalized cross validation method. A calibration phase reduces the search space for the PSF parameters. The gaussian model used for the PSF is exploited to accelerate the computations. The image is deconvolved by solving a large linear system with the conjugate gradient method. A preconditionner making use of the PSF separability (isotropic or anisotropic) speeds up convergence.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2014 / Matrix models, and their associated integrals, are encoded with a rich structure, especially when studied in the large N limit. In our project we study the dynamics of a Gaussian ensemble of m complex matrices or 2m hermitian matrices for d = 0 and d = 1 systems. We rst investigate the two hermitian matrix model parameterized in \matrix valued polar coordinates", and study the integral and the quantum mechanics of this system. In the Hamiltonian picture, the full Laplacian is derived, and in the process, the radial part of the Jacobian is identi ed. Loop variables which depend only on the eigenvalues of the radial matrix turn out to form a closed subsector of the theory. Using collective eld theory methods and a density description, this Jacobian is independently veri ed. For potentials that depend only on the eigenvalues of the radial matrix, the system is shown to be equivalent to a system of non-interacting (2+1)-dimensional \radial fermions" in a harmonic potential. The matrix integral of the single complex matrix system, (d = 0 system), is studied in the large N semi-classical approximation. The solutions of the stationary condition are investigated on the complex plane, and the eigenvalue density function is obtained for both the single and symmetrically extended intervals of the complex plane. The single complex matrix model is then generalized to a Gaussian ensemble of m complex matrices or 2m hermitian matrices. Similarly, for this generalized ensemble of matrices, we study both the integral of the system and the Hamiltonian of the system. A closed sector of the system is again identi ed consisting of loop variables that only depend on the eigenvalues of a matrix that has a natural interpretation as that of a radial matrix. This closed subsector possess an enhanced U(N)m+1 symmetry. Using the Schwinger-Dyson equations which close on this radial sector we derive the Jacobian of the change of variables to this radial sector. The integral of the system of m complex matrices is evaluated in the large N semi-classical approximation in a density description, where we observe the emergence of a new logarithmic term when m 2. The solutions of the stationary condition of the system are investigated on the complex plane, and the eigenvalue density functions for m 2 are obtained in the large N limit. The \fermionic description" of the Gaussian ensemble of m complex matrices in radially invariant potentials is developed resulting in a sum of non-interacting Hamiltonians in (2m + 1)-dimensions with an induced singular term, that acts on radially anti-symmetric wavefunctions. In the last chapter of our work, the Hamiltonian of the system of m complex matrices is formulated in the collective eld theory formalism. In this density description we will study the large N background and obtain the eigenvalue density function.