RUI: Development of Frame Extensions and their Application

提出了子空间伪帧（PFFS）及其插值技术应用的lia研究计划。该计划的主题1研究了pffs对减少数据/图像压缩中的伪影的贡献。关键的观察结果在于存在对称和紧支持双正交小波，但不在（紧支持小波的）MRA内。它显示了MRA子空间结构中的“空间不足”。PFFS是一种探索这种现象的洞察力故事的工具。主题2重点介绍了一种新的插值多分辨率分析（I-MRA）构建方法。关键的观察是插值序列可以通过采样再现MRA的子空间V的核函数来定义。期望这样的插值序列可以很容易地形成V的基或框架，从而形成对MRA的插值“接口”，并形成I-MRA的构建方法。主题3研究了pffs在c型插值和正交方法中的应用。其动机是探索pffs在c型应用中的快速衰减特性的优势。在主题4中，pi期望应用主题2中描述的类似插值技术来研究使用帧和/或pffs作为基本工具的多频带采样。pffs技术以其灵活性有望发挥作用。每个研究课题都受到其在数据/图像压缩、采样和插值方面的应用的高度激励，以实现高效的数值实现。PFFS是一种在子空间V之外研究子空间V的工具。在这样做的过程中，人们被赋予了自由和额外的“空间”，通过使用V中不可用的先进工具来执行V中的任务。在信号/图像处理中，PFFS的这种先进特性非常有用，因为众所周知，具有对称性的有限长度双正交小波对于减少“阻塞”和“环”效应是理想的，然而这种双正交小波并不存在于其对偶小波子空间中。我们相信PFFS不仅可以提供一类双正交小波示例，PFFS提供的附加信息（从构建中已知）将为“阻塞”和“响”伪影提供实用的见解，从而提供新的解决方案，提高信号/图像压缩率和通信速度。pffs在不规则采样（用于多频带函数）以及sinctype方法（用于插值和正交）中的其他应用都是基于应用pffs的柔性特性。期望包括提高实际效率和提高收敛速度。插值多分辨率分析项目在数据采集/采样中有实际应用，同时执行小波变换。该研究的关键组成部分之一是建立一个插值“接口”到给定的MRA，仍然保持潜在的理想的小波特性。

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.