Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2014 | Buch

Introduction: Signals and Transforms Kapitel lesen Erstes Kapitel lesen

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

Volume I: Periodic Splines

verfasst von: Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev

Verlag: Springer Netherlands

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

This volume provides universal methodologies accompanied by Matlab software to manipulate numerous signal and image processing applications. It is done with discrete and polynomial periodic splines. Various contributions of splines to signal and image processing from a unified perspective are presented. This presentation is based on Zak transform and on Spline Harmonic Analysis (SHA) methodology. SHA combines approximation capabilities of splines with the computational efficiency of the Fast Fourier transform. SHA reduces the design of different spline types such as splines, spline wavelets (SW), wavelet frames (SWF) and wavelet packets (SWP) and their manipulations by simple operations. Digital filters, produced by wavelets design process, give birth to subdivision schemes. Subdivision schemes enable to perform fast explicit computation of splines' values at dyadic and triadic rational points. This is used for signals and images up sampling. In addition to the design of a diverse library of splines, SW, SWP and SWF, this book describes their applications to practical problems. The applications include up sampling, image denoising, recovery from blurred images, hydro-acoustic target detection, to name a few. The SWF are utilized for image restoration that was degraded by noise, blurring and loss of significant number of pixels. The book is accompanied by Matlab based software that demonstrates and implements all the presented algorithms. The book combines extensive theoretical exposure with detailed description of algorithms, applications and software.

The Matlab software can be downloaded from http://extras.springer.com

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction: Signals and Transforms
Abstract
In this chapter we outline some well known facts about periodic signals and transforms, which are needed throughout the book. For details we refer to the classical textbook Oppenheim and Schafer [2].
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 2. Introduction: Periodic Filters and Filter Banks
Abstract
In this chapter filtering of periodic signals is outlined. Periodic filters and periodic filter banks are defined. Perfect reconstruction filter banks are characterized via their polyphase matrices.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 3. Mixed Circular Convolutions and Zak Transforms
Abstract
In this chapter the notion of mixed circular convolution is introduced. The polynomial and discrete periodic splines defined on uniform grids are special cases of such convolutions. The so-called Zak transforms provide tools to handle mixed circular convolutions
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 4. Periodic Polynomial Splines
Abstract
In this chapter the spaces of periodic polynomial splines, which are introduced in Sect. 3.​2.​2, are discussed in more details. It is shown that the periodic exponential splines generate a specific form of harmonic analysis in these spaces. A family of generators of the spaces of periodic splines is presented.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 5. Polynomial Smoothing Splines
Abstract
Interpolating splines is a perfect tool for approximation of a continuous-time signal \(f(t)\) in the case when samples \(x[k]=f(k),\;k\in \mathbb {Z}\) are available. However, frequently, the samples are corrupted by random noise. In such case, the so-called smoothing splines provide better approximation. In this chapter we describe periodic smoothing splines in one and two dimensions. The SHA technique provides explicit expression of such splines and enables us to derive optimal values of the regularization parameters.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 6. Calculation of Splines Values by Subdivision
Abstract
Assume, the samples of a spline \(S(t)\in {}^{p}\fancyscript{S}\) on the grid \(\mathbf{g} =\{k\}_{k\in \mathbb {Z}}\) are available: \(S(k)=y[k]\). Subdivision schemes are proposed to calculate the spline’s values at dyadic and triadic rational points \(S(k/2^m)\) and \(S(k/3^m)\). The SHA technique provides fast and explicit implementation of the subdivision for one- and two-dimensional periodic splines.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 7. Spline Algorithms for Deconvolution and Inversion of Heat Equation
Abstract
In this chapter, we present algorithms based on Tikhonov regularization for solving two related problems: deconvolution and inversion of heat equation. The algorithms, which utilize the SHA technique, provide explicit solutions to the problems in one and two dimensions.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 8. Periodic Spline Wavelets and Wavelet Packets
Abstract
This chapter presents wavelets and wavelet packets in the spaces of periodic splines of arbitrary order, which, in essence, are the multiple generators for these spaces. The SHA technique provides explicit representation of the wavelets and wavelet packets and fast implementation of the transforms in one and several dimensions.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 9. Discrete-Time Periodic Wavelet Packets
Abstract
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.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 10. Deconvolution by Regularized Matching Pursuit
Abstract
In this chapter, an efficient method that restores signals from strongly noised blurred discrete data is presented. The method can be characterized as a Regularized Matching Pursuit (RMP), where dictionaries consist of spline wavelet packets. It combines ideas from spline theory, wavelet analysis and greedy algorithms. The main distinction from the conventional matching pursuit is that different dictionaries are used to test the data and to approximate the solution. In addition, oblique projections of data onto dictionary elements are used instead of orthogonal projections, which are used in the conventional Matching Pursuit (MP). The slopes of the projections and the stopping rule for the algorithm are determined automatically.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 11. Block-Based Inversion of the Heat Equations
Abstract
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 representation of the solutions. The presented algorithms provide stable solutions that accurately approximate the initial temperature distribution.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 12. Hydro-Acoustic Target Detection
Abstract
This chapter presents an example of utilization of the discrete–time wavelet packets, which are described in Sect. 9.​1, to classification of acoustic signals and detection of a target. The methodology based on wavelet packets is applied to a problem of detection of a boat of a certain type when other background noises are present. The solution is obtained via analysis of boat’s hydro-acoustic signature against an existing database of recorded and processed hydro-acoustic signals. The signals are characterized by the distribution of their energies among blocks of wavelet packet coefficients.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 13. Periodic Discrete Splines
Abstract
Periodic discrete splines with different periods and spans were introduced in Sect.  3.​4. In this chapter, we discuss families of periodic discrete splines, whose periods and spans are powers of 2. As in the polynomial splines case, the Zak transform is extensively employed. It results in the Discrete Spline Harmonic Analysis (DSHA). Utilization of the Fast Fourier transform (FFT) enables us to implement all the computations in a fast explicit way.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 14. Discrete Periodic Spline Wavelets and Wavelet Packets
Abstract
Similarly to periodic polynomial splines, existence of the set of embedded discrete periodic splines spaces \(\varPi [N]= \fancyscript{S}_{[0]}\supset {}^{2r} \fancyscript{S}_{[1]}\supset \cdots \supset {}^{2r} \fancyscript{S}_{[m]}\cdots \), combined with the DSHA provides flexible tools for design and implementation of wavelet and wavelet packet transforms. As in the polynomial case, all the calculations consist of fast direct and inverse Fourier transforms (FFT and IFFT, respectively) and simple arithmetic operations. Raising the splines order does not increase the computation complexity.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 15. Biorthogonal Wavelet Transforms
Abstract
In this chapter, design and implementation of biorthogonal wavelet transforms of periodic signals is described. For this purpose, perfect reconstruction (PR) p-filter banks are used. In particular, design of compactly supported wavelets, such as popular 5/3 and 9/7 wavelets is outlined. Adaptation of the concepts of polynomials restoration and of vanishing moments to the discrete periodic setting is discussed.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 16. Biorthogonal Wavelet Transforms Originating from Splines
Abstract
This section describes how to generate families of biorthogonal wavelet transforms in spaces of periodic signals using prediction p-filters originating from polynomial and discrete splines. The wavelets related to those transforms are (anti)symmetric, well localized in time domain and have flat spectra. The families contain low-pass p-filters, which locally restore sampled polynomials of any degree, while the respective high-pass p-filters locally eliminate polynomials of the same degrees.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 17. Wavelet Frames Generated by Spline Based p-Filter Banks
Abstract
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.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Chapter 18. Application of Periodic Frames to Image Restoration
Abstract
In this chapter, we present examples of image restoration using periodic frames. Images to be restored were degraded by blurring, aggravated by random noise and random loss of significant number of pixels. The images are transformed by periodic frames designed in Sects. 17.​2 and 17.​4, which are extended to the 2D setting in a standard tensor product way. In the presented experiments, performances of different tight and semi-tight frames are compared between each other in identical conditions.
Amir Z. Averbuch, Pekka Neittaanmaki, Valery A. Zheludev
Backmatter
Metadaten
Titel
Spline and Spline Wavelet Methods with Applications to Signal and Image Processing
verfasst von
Amir Z. Averbuch
Pekka Neittaanmaki
Valery A. Zheludev
Copyright-Jahr
2014
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-8926-4
Print ISBN
978-94-017-8925-7
DOI
https://doi.org/10.1007/978-94-017-8926-4

Neuer Inhalt

    Bildnachweise