Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in`2(Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a necessary and sufficient condition for perfect reconstruction using FIR filters after an FIR analysis. Complete parameterizations of oversampled filter banks satisfying these conditions are given. Further, we study the condition under which the frame dual to the frame associated with an FIR filter bank is also FIR and give a parameterization of a class of filter banks satisfying this property. Then, we focus on nonsubsampled filter banks. Nonsubsampled filter banks implement transforms similar to continuous-time transforms and allow for very flexible design. We investigate relations of these filter banks to continuous-time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two-channel filter banks that yield highly regular wavelets with a given number of vanishing moments.

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The purpose of this study is to enable high spatial resolution voxel-wise quantitative analysis of myocardial perfusion in dynamic contrast-enhanced cardiovascular MR, in particular by finding the most favorable quantification algorithm in this context. Four deconvolution algorithms--Fermi function modeling, deconvolution using B-spline basis, deconvolution using exponential basis, and autoregressive moving average modeling--were tested to calculate voxel-wise perfusion estimates. The algorithms were developed on synthetic data and validated against a true gold-standard using a hardware perfusion phantom. The accuracy of each method was assessed for different levels of spatial averaging and perfusion rate. Finally, voxel-wise analysis was used to generate high resolution perfusion maps on real data acquired from five patients with suspected coronary artery disease and two healthy volunteers. On both synthetic and perfusion phantom data, the B-spline method had the highest error in estimation of myocardial blood flow. The autoregressive moving average modeling and exponential methods gave accurate estimates of myocardial blood flow. The Fermi model was the most robust method to noise. Both simulations and maps in the patients and hardware phantom showed that voxel-wise quantification of myocardium perfusion is feasible and can be used to detect abnormal regions.