CAREER: Subdivision Schemes and Wavelets: New Tools, New Settings

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The proposed research will focus on three partly relateddirections in multiresolution methods: (I) subdivision schemesand wavelets in the geometric setting, (II) the advantage of redundantsystems and (III) ideal data representation in higher dimensions.Subdivision and multiresolution methods in the geometric settingis a recently launched research area.Prompted by the extremely high interests in applications such as terrainmodeling and 3-D scanning technology, this research direction has attractedthe attention of both computer scientists and applied mathematicians.Existing subdivision schemes for surface generation do not promisewell-defined curvature everywhere in a resulted surface. In variousapplications it is desirable to work with curvature smooth surfaces.In the regular setting the proposer recently constructed a family ofC^2 subdivision schemes with a small interpolating stencil.Underlying this construction is a variety of tools from optimization,wavelet theory and multivairate interpolation theory. This project willfurther refine these tools in order to resolve the original open problemand such that these tools can be applied to other settings as well.For various applications in image reconstruction/processing, itwas found that highly redundant wavelet-like image representationssignificantly outperform the standard non-redundant orthogonalwavelet transforms used ubiquitously in the image codingcommunity. On the one hand, there is currently no complete theoryfor explaining the fundamental advantages of redundancy. On theother hand, researchers have just started to realize thefundamental limits of wavelet transforms in higher dimensions andhave proposed a wide variety of novel data representation schemesthat address these fundamental limits. Research activityunder this project seeks to develop mathematical theory andcomputational tools in the general area of adaptive methods ofrepresenting and analyzing images and other multidimensional data.The educational plan proposes a new way of teachingcomputational science. The projectwill develop teaching materials for our numerical computing coursesthat better suit the needs, interests and mathematical abilityof computer science students. The goal ofthis project is to develop a set ofelectronic lecture notes for our future senior numerical computingcourse. These lecture notes will add two new dimensions to any ofthe existing textbooks: (i) An interactive computationalenvironment (based on QPE's such as Matlab or Octave)will be integrated into the notes so that a student can easily reproduceany computational result presented and derive any newcomputational experiments. (ii) Applicationsareas such as computer graphics, computer vision, data-mining andcomputational fluid dynamics will beused in various places ofthis notes to illustrate the role of numerical methods in industry.