Search results for "Spline"
showing 10 items of 170 documents
Wavelet Frames Generated by Spline Based p-Filter Banks
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 oversampled perfect reconstruction periodic filter banks. The filter banks are derived from discrete-time and discrete periodic splines. Each filter bank comprises one linear phase low-pass filter (in most cases interpolating) and one high-pass filter, whose magnitude response mirrors that of a low-pass filter. In addition, these filter banks comprise a number of band-pass filters. In this chapter, frames generated by four-channel filter banks are briefly outlined (see Chap. 17 in [2] for details) and tight frames generated by six-…
Zur numerischen Lösung gewöhnlicher Differential-gleichungen mit Splines in einem Sonderfall
1980
In an earlier paper [1] a general procedure has been presented to obtain polynomial spline approximations for the solution of the initial value problem for ordinary differential equations. In this paper the general procedure is described by an equivalent one step method. Furthermore two convergence theorems are proved for a special case which is not included in the general convergence or divergence theory given in [1].
More wavelet-like orthonormal bases for the lowest Landau level: Some considerations
1994
In a previous work, Antoine and I (1994) have discussed a general procedure which 'projects' arbitrary orthonormal bases of L2(R) into orthonormal bases of the lowest Landau level. In this paper, we apply this procedure to a certain number of examples, with particular attention to the spline bases. We also discuss Haar, Littlewood-Paley and Journe bases.
Block-Based Inversion of the Heat Equations
2014
This chapter presents robust methods, which refine the algorithms, in Sect. 7.2, for inversion of the heat equations. The idea behind the algorithms is to solve the inversion problem separately in different frequency bands. This is achieved by using spline wavelet packets. The solutions that minimize some parameterized quadratic functionals, are derived as linear combinations of the wavelet packets. Choice of parameters, which is performed automatically, determines the trade-off between the solution regularity and the initial data approximation. The Spline Harmonic Analysis (SHA) technique provides a unified computational scheme for the fast implementation of the algorithm and an explicit r…
Biorthogonal Wavelet Transforms Originating from Splines
2015
This chapter describes how to design families of biorthogonal wavelet transforms of signals and respective biorthogonal Wavelet bases in the signal space using spline-based prediction filters. Although the designed Wavelets originate from splines, they are not splines themselves. The design and implementation of the biorthogonal Wavelet transforms is done using the Lifting scheme. Most of the filters participating in the expansion of signals over the presented bases have infinite impulse responses and are implemented by recursive filtering whose computational cost is competitive with the FIR filtering cost. Properties of the designed Wavelets, such as symmetry, flat spectra, good time domai…
Quasi-interpolating and Smoothing Local Splines
2015
In this chapter, local quasi-interpolating and smoothing splines are described. Although approximation properties of local spline are similar to properties of the global interpolating and smoothing splines, their design does not require the IIR filtering of the whole data array. The computation of a local spline value at some point utilizes only a few adjacent grid samples. Therefore, local splines can be used for real-time processing of signals and for the design of FIR filter banks generating wavelets and wavelet frames (Chaps. 12 and 14). In the chapter, local splines of different orders are designed and their approximation properties are established which are compared with the propertie…
Periodic Orthogonal Wavelets and Wavelet Packets
2018
In this chapter, we discuss how to derive versatile families of periodic discrete-time orthogonal wavelets and wavelet packets from discrete and discrete-time splines outlined in Chap. 3. These wavelets and wavelet packets, although not having compact supports, are well localized in the time domain. They can have any number of discrete vanishing moments. Their DFT spectra tend to have a rectangular shape when the spline order grows and provide a collection of refined splits of the Nyquist frequency band. The wavelet and wavelet packet transforms are implemented in a fast way using the FFT.
Two-Dimensional Orthogonal Wavelets and Wavelet Packets
2018
This chapter extends the design of spline-based orthogonal discrete-time wavelets and wavelet packets to two-dimensional case. The corresponding transforms are implemented by using the 2D FFT.
Discrete-Time Periodic Wavelet Packets
2014
Direct and inverse wavelet and wavelet packet transforms of a spline are implemented by filtering the spline’s coordinates by two-channel critically sampled p-filter banks. In this chapter, those p-filter banks are utilized for processing discrete-time signals. The p-filter banks generate discrete-time wavelets and wavelet packets in the spaces of 1D and 2D periodic signals.
Block Based Deconvolution Algorithm Using Spline Wavelet Packets
2010
This paper presents robust algorithms to deconvolve discrete noised signals and images. The idea behind the algorithms is to solve the convolution equation separately in different frequency bands. This is achieved by using spline wavelet packets. The solutions are derived as linear combinations of the wavelet packets that minimize some parameterized quadratic functionals. Parameters choice, which is performed automatically, determines the trade-off between the solution regularity and the initial data approximation. This technique, which id called Spline Harmonic Analysis, provides a unified computational scheme for the design of orthonormal spline wavelet packets, fast implementation of the…