RUI: Development of Frame Extensions and their Application

点击翻译按钮获取中文摘要

0103762LiA program of research related to the application of pseudoframes for subspaces (PFFS) and relevant interpolation techniques is proposed. Theme 1 of the program studies the contribution of PFFSs to artifact reduction in data/image compression. Key observations lie in the fact that there are symmetric and compactly supported biorthogonal wavelets, but not within an MRA (of compactly supported wavelets). It shows the "lack of room" within the MRA subspace structure. PFFS is a tool to explore the insight story of such a phenomenon. Theme 2 focuses on a new construction approach of interpolatory multiresolution analysis (I-MRA). The key observation is that an interpolatory sequence can be defined by sampled reproducing kernel function of a subspace V of an MRA. It is expected that such an interpolatory sequence can easily form a basis or frame of V, giving rise to an interpolatory "interface" to the MRA, and a construction methodology of I-MRA. Theme 3 studies applications of PFFSs in the Sinc-type methods for interpolations and quadratures. The motivation is to explore the advantage of fast decaying properties of PFFSs in the Sinc-type applications. In theme 4, the P.I. expects to apply similar interpolation techniques as described in theme 2 to study multi-band samplings using frames and/or PFFSs as fundamental tools. The technique of PFFSs with their flexibility is expected to play a role.Each research topic is highly motivated by its applications in data/image compression, sampling and interpolation for efficient numerical implementations. PFFS is a tool to study a subspace V while "standing" outside of it. In doing so, one is given the freedom and additional "room" to perform tasks in V by using advanced tools not available in V. In signal/image processing, such advanced properties of PFFS are extremely useful because it is known that biorthogonal wavelets of finite length with symmetry are desirable for reducing ``blocking'' and ``ringing'' effects, and yet such biorthogonal wavelets do not reside in its dual wavelet subspace. We believe that PFFS shall not only provide a class of biorthogonal wavelet examples, the additional information (known from the construction) contributed by PFFS will provide practical insight about "blocking" and "ringing" artifacts, and thereby delivers new solutions and enhanced signal/image compression rate and communication speed. Other applications of PFFSs in irregular sampling (for multi-band functions) as well as in the Sinc-type methods (for interpolation and quadratures) are all based on applying flexible properties of PFFSs. The expectations include enhanced practical efficiency and improved rate of convergence. The project on interpolatory multiresolution analysis has practical applications in data acquisition/sampling while simultaneously performing the wavelet transform. One of the key ingredients of the study is to build an interpolatory "interface" to a given MRA and still keep the underlying desirable wavelet properties.