This work introduces a system of algorithms to compute period matrices for general surfaces with arbitrary topologies. The algorithms are intrinsic to the geometry, and independent of surface representations. The computation is efficient, stable and practical for real applications. The algorithms are experimented on real surfaces including human faces and sculptures, and applied to surface identification problems. It is the first work that is both theoretically solid, and practically robust and accurate to handle real surfaces with arbitrary topologies.

This paper introduces a theoretic result that shows any surface in 3 dimensional Euclidean space can be determined by its conformal factor and mean curvature uniquely up to rigid motions. This theorem disproves the common belief that surfaces have three functional freedoms and immediately shows that one third of geometric data can be saved without loss of information.

The paper develops a practical algorithm to losslessly compress geometric surfaces based on Riemann surface structures. First we compute a global conformal parameterization of the surface. The surface can be segmented by holomorphic flows, where each segment can be conformally mapped to a rectangle on the parameter plane, which is guaranteed by circle-valued Morse theory. We construct a conformal geometry image for each segment, and record conformal factor and dihedral angle for each edge. In this way, we represent the surface using only two functions with canonical connectivity. We present the proofs of the theorems and the compression examples.

An information theoretical game is considered where both signal and noise are generalized Bernoulli sums with upper bounds on their mean values. It is shown that a pair of Poisson distributions is a Nash equilibrium pair.

We develop two different techniques to study volume mapping problem in Computer Graphics and Medical Imaging fields. The first one is to find a harmonic map from a 3 manifold to a 3D solid sphere and the second is a sphere carving algorithm which calculates the simplicial decomposition of volume adapted to surfaces. We derive the 3D harmonic energy equation and it can be easily extended to higher dimensions. We use a textrehedral mesh to represent the volume data. We demonstrate our method on various solid 3D models. We suggest that 3D harmonic mapping of volume can provide a canonical coordinate system for feature identification and registration for computer animation and medical imaging.

An image stored in image database systems is assumed to be associated with some content-based meta-data about that image, that is, information about objects in the image and absolute/relative spatial relationships among them. An image query for such an image database system can generally be handled in two ways: exact picture matching and approximate picture matching. We address the approximate picture matching problem of central interest in this paper, and present a stepwise approximation of intractable spatial constraints in an image query. Especially, this stepwise approximation may be pre-processed on an image query before an advanced picture matching algorithm is invoked. We then work out details of the stepwise approximation algorithm by analyzing, one by one, all possible 16 cases for results of the object matching step. Our analysis turns out that only 13 cases are valid, while the other 3 cases are identified impossible for finding an exact picture-matching between a query picture and a database picture. The worst-case running time complexity is given for each of them. In order to reduce the number of database pictures being matched by a user query, we also provide two suggestions to help enhance the effectiveness of image retrieval at the additional time cost.