BACKGROUND: I was a mathematician ten years ago, but now I am a computer scientist. The goal of my research is to act as a bridge-builder between mathematics and computer graphics. That is, my mathematical background plays a good role in solving several problems induced in the field of computer graphics. In the graduate course, I studied knot theory, fractal theory, and approximation theory in the department of mathematics and also researched on computer graphics at the computer graphics laboratory in the department of computer science. I proposed a new method of visualization of polygonal knots and then developed a 3D rendering algorithm for 2D fractal images. Based on the differential geometry and approximation theory, I proposed a new collision detection algorithm of a continuous type in the Ph. D. thesis. The algorithm is to construct spherical voronoi diagrams and to transform the extreme vertex problem into the spherical point location problem, and to construct a distance function between objects. Hence, the time complexity of my algorithm is independent of the time step size of animation. At Electronics and Telecommunications Research Institute (ETRI), I participated in a development of human modeling and motion animation system as a post-doctoral position. At Dong-eui University, I have been studying geometric modeling. My current research interests are in the area of efficient representations of graphical objects: multi-resolution analysis and scattered data interpolation. Most of graphical objects in the Euclidean space may be represented as a polygonal mesh or a smooth surface. I have known that multi-resolution modeling is the most outstanding method to efficiently deal with polygonal models and that the smooth surface is generated by measuring the scattered data and interpolating them in practical applications.

MULTIRESOLUTION ANALYSIS: The term multiresolution has been intimately connected with the study of wavelets. Wavelets are useful to describe mathematical objects such as functions at different levels of resolution. Computer graphics as well as mathematics stands in need of hierarchical structures such as wavelets and subdivision. Hierarchical mesh representations (e.g., meshes with subdivision-connectivity, progressive meshes, etc.) offer the scalability of polygonal models, and various mesh applications (such as editing, deformation, compression, and parameterization) import these hierarchical approaches. I am interested in adapting multiresolution approach to the whole of modeling process with huge polygonal environments. The main area of my research consists of multiresolution mesh representation & manipulation, mesh morphing, mesh compression, and mesh parameterization. To do more active research, I am joining the computer graphics group of POSTECH (Pohang University of Science and Technology), Korea. Recently, we proposed a new mesh parameterization as a result of co-work.

SCATTERED DATA INTERPOLATION: Scattered data interpolation refers to the problem of fitting a smooth surface through a scattered, or non-uniform, distribution of data samples. This subject is of practical importance in many science and engineering fields, where data are often measured or generated at sparse and irregular positions. The goal of interpolation is to reconstruct an underlying function (e.g., surface) that may be evaluated at any desired set of positions. This serves to smoothly propagate the information associated with the scattered data onto all position in the domain. There are three principal sources of scattered data measured values of physical quantities, experimental results, and computational values. They are found in diverse and scientific and engineering applications. For example, in medical imaging, scattered data interpolation is essential to construct a closed surface from CT or MRI images of human organs. There are several methods to solve the scattered data interpolation problems. Specially, I am interested in two approaches: Multilevel B-spline interpolation and Radial Basis Function (RBF) interpolation. The multilevel B-spline interpolation method has several advantages such as a local-control property, fast computation, and so on. However, the method can be applied only to the functional data. On the other hand, the surface produced by radial basis function interpolation methods consists of a single patch. Hence, the surface has higher continuity than that produced by B-spline interpolation method. The main advantage of this method is that it dose not have any restriction of data form. So, this method has recently been applied to several medical applications. But, if the number of data is larger than about 2,000, then the time complexity is very slow. Recently, to resolve the disadvantages of two methods, I am developing a new hybrid method that fast interpolates 3D scattered data without any restriction. The method can resolve the disadvantage of two methods as well as preserve the advantages of multilevel B-spline interpolation methods. I think that I will get good results for the new data interpolation method. Moreover, this method can be applied to shape transformation, mesh simplification, and mesh compression.