Jianer Chen

One of the theoretically intriguing problems in computer-aided geometric modeling comes from the stitching of the tensor product Bezier patches. When they share an extraordinary vertex, it is not possible to obtain continuity C1 or G1 along the edges emanating from that extraordinary vertex. Unfortunately, this stitching problem cannot be solved by using higher degree or rational polynomials. In this paper, we present a modified de Casteljau subdivision algorithm that can provide a solution to this problem. Our modified de Casteljau subdivision, when combined with topological modeling, provides a framework for interactive real-time modeling of piecewise smooth manifold meshes with arbitrary topology. The main advantage of the modified subdivision is that the continuity C1 on a given boundary edge does not depend on the positions of the control points on other boundary edges. The modified subdivision allows us to obtain the desired C1 continuity along the edges emanating from the extraordinary vertices along with the desired G1 continuity in the extraordinary vertices.

One of the key problems in dealing with color in rendering, shading, compositing, or image manipulation is that we do not have algebraic structures that support operations over colors. In this paper, we present an all-encompassing framework that can support a set of algebraic structures with associativity, commutativity, and inverse properties. To provide these three properties, we build our algebraic structures on an extension of projective space by allowing for negative and complex numbers. These properties are important for (1) manipulating colors as periodic functions, (2) solving inverse problems dealing with colors, and (3) being consistent with the wave representation of the color. Allowance of negative and complex numbers is not a problem for practical applications, since we can always convert the results into desired range for display purposes as we do in High Dynamic Range imaging. This set of algebraic structures can be considered as a generalization of the Minkowski norm Lp in projective space. These structures also provide a new version of the generalized Holder average with associativity property. Our structures provide inverses of any operation by allowing for negative and complex numbers. These structures provide all properties of the generalized Holder average by providing a continuous bridge between the classical weighted average, harmonic mean, maximum, and minimum operations using a single parameter p.