Personal Research Statement describing scientific interests and research plans

My general research interests include image processing, numerical analysis, algebra, combinatorics, differential equations, neural networks, pattern recognition, and statistics. I have taken a multitude of courses at the graduate level in mathematics, applied mathematics, computer science and statistics. I am particularly interested in mathematical problems that are driven by applications. Because of my studies in computer science, I have become very interested in applied areas. After I took courses in computer vision and pattern recognition, I worked on a project involving planar curve classification and developed algorithms using elastic transformations for curve classification. Those transformations have become very important in many areas such as recognition of hand written characters and cardiac curves, where rigid transformations and scaling do not work well. My current research focuses on the development of the image transforms on arrays of hexagonal lattices. While traditional image processing algorithms and digital image transforms are carried out on pixels of square lattices, hexagonal lattices have proven to be advantageous in several application domains. In comparison to rectangular grids, hexagonal grids provide for higher packing density of discs and well defined adjacency. Physical pixels such as printer dots and electron beams usually have circular shapes and thus operate more effectively on hexagonal grids. If the data are circularly bandlimited, then the minimum sampling density for a hexagonal lattice is 13.4% less than for a rectangular lattice [1]. Furthermore, if a planar domain is circular and sampled on a square lattice, then the discrete Fourier transform (DFT) on that domain mat not be invertible. We need to extend the circular domain to a square domain when sampled on a square lattice, or to a regular hexagonal domain when sampled on a hexagonal lattice. The area of the required hexagonal domain is less than that of the square domain. Hence hexagonal grids are used by a variety of researchers active in areas such as geoscience, computer graphics, and ecology. For instance, they are being used in the soil moisture and ocean salinity space mission [2,3]. The Keck ten-meter telescope utilized a main mirror composed of 36 hexagonal segments [4]. The problems I am working on were communicated to my advisors by a research group at the company named Pyxis Innovation Inc. This research group is developing a multiresolutional and uniform sampling scheme of the Earth. The Pyxis structure is a kind of data structure used in digital Earth. It consists of elegant hierarchical hexagons which reference the sphere in multiresolutions, and a natural indexing scheme. The components of such a digital Earth include 1) compact, multi-scale reference to any location on the Earth, 2) on demand mapping of dynamic sensor data, and 3) web-based and distributed spacial information and applications. As some experts (working on Remote Sensing, Earth and Planetary Atmosphere, and Global Biological Diversity) pointed out, this scheme eliminates most of the problems in projection conversion, visualization and minimizing efforts on the integration of global DEM, raster and vector data. The indices of the hierarchical hexagons of the Pyxis structure and their (vector) addition are very important for quick data retrieval and translation. We developed efficient algorithms which have linear computational complexity to add any two labels