Interpolatory frames in signal space

IEEE Trans. Signal Process.

Zheludev

Cohen

2007

We present a method for construction of multiwavelet frames for manipulation of discrete signals. The frames are generated by three-channel perfect reconstruction oversampled multifilter banks. The design of the multifilter bank starts from a pair of interpolatory multifilters. We derive these interpolatory multifilters from the cubic Hermite splines. We use original preprocessing algorithms, which transform scalar signals into vector arrays that serve as inputs to the oversampled analysis multifilter banks. These pre-processing algorithms do not degrade the approximation accuracy of the transforms of the vectors by multifilter banks. The postprocessing algorithms convert the vector output of the synthesis multifilter banks into scalar signal. The discrete framelets, generated by the designed filter banks, are symmetric and have short support. The analysis framelets have four vanishing moments, whereas the synthesis framelets converge to Hermite splines supported on the interval [-1,1].

Section: Discussionmentioning

confidence: 99%

“…Oversampled filter banks provide a natural source to devise wavelet-type frames in the signal space [6,10,4]. In the multiwavelet case, the filter banks consist of matrix filters, the so-called multifilters.…”

Section: Introductionmentioning

confidence: 99%

Multiwavelet Frames in Signal Space Originated From Hermite Splines

IEEE Trans. Signal Process.

Zheludev

Cohen

2007

“…We introduced in previous works 6,21 new classes of tight and bi-framelets as well as multiframelets 22 that combine properties that are needed in signal processing, such as symmetry, at spectra, sufficient number of vanishing moments, interpolation and moderate redundancy, set aside an efficient implementation. Some of them are described in Sec.…”

Section: Discussionmentioning

confidence: 99%

Symmetric Interpolatory Framelets and Their Erasure Recovery Properties

Amrani

Int. J. Wavelets Multiresolut Inf. Process.

Cohen

et al. 2007

A new class of wavelet-type frames in signal space that uses (anti)symmetric waveforms is presented. The construction employs interpolatory filters with rational transfer functions. These filters have linear phase. They are amenable either to fast cascading or parallel recursive implementation. Robust error recovery algorithms are developed by utilizing the redundancy inherent in frame expansions. Experimental results recover images when (as much as) 60% of the expansion coefficients are either lost or corrupted. The proposed approach inflates the size of the image through framelet expansion and multilevel decomposition thus providing redundant representation of the image. Finally, the frame-based error recovery algorithm is compared with a classical coding approach.

“…If the number of channels in p-filter banksH and H is S = 2, then the number of elements in the frames is N , which equals the dimension of the space Π [N ]. Thus, in that case, the sets ψ s The signalsψ s [1] and ψ s [1] , s = 0, . .…”

Section: One-level Frame Transformmentioning

confidence: 99%

Wavelet Frames Generated by Spline Based p-Filter Banks

Spline and Spline Wavelet Methods With Applications to Signal and Image Processing

Neittaanmäki

Zheludev

2014

This chapter presents a design scheme to generate tight and so-called semi-tight frames in the space of discrete-time periodic signals. The frames originate from three-and four-channel perfect reconstruction periodic filter banks. The filter banks are derived from interpolating and quasi-interpolating polynomial splines and from discrete splines. Each filter bank comprises one linear phase low-pass filter (in most cases interpolating) and one high-pass filter, whose magnitude's response mirrors that of a low-pass filter. In addition, these filter banks comprise one or two band-pass filters. In the semi-tight frames case, all the filters have linear phase and (anti)symmetric impulse response, while in the tight frame case, some of band-pass filters are slightly asymmetric. The design scheme enables to design framelets with any number of LDVMs.The computational complexity of the framelet transforms practically does not depend on the number of LDVMs and on the size of the impulse response of filters. Recently frames or redundant expansions of signals have attracted considerable interest from researchers working in signal processing although one particular class of frames, the Gabor systems, has been applied and investigated since 1946 [11]. As the requirement of one-to-one correspondence between the signal and its transform coefficients is dropped, there is more freedom to design and implement frame transforms. Frame expansions of signals demonstrate resilience to quantization noise and to coefficients losses [12][13][14]. Thus, frames may serve as a tool for error correction of signals that are transmitted through lossy/noisy channels. Recently, overcomplete representation of signals was applied to image reconstruction [5,6]. Combination of wavelet frames with the Bregman iterations techniques [2,19] provided a new impact to the image processing applications such as deconvolution, inpainting, denoising, to name a few [3,4,9,17,18]. It was mentioned in Sect. 2.2 that oversampled PR filter banks generate a specific kind of frames in the periodic signal space. This chapter presents families of frames whose generating three-and four-channel p-filter banks originate from polynomial and discrete splines. These frames have properties such as symmetry, interpolation, and flat spectra, which are attractive for signal processing.