Block Based Deconvolution Algorithm Using Spline Wavelet Packets

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…

Periodic Polynomial Splines

In this chapter, the spaces of periodic polynomial splines and the Spline Harmonic Analysis (SHA) in these spaces are briefly outlined. The stuff of this chapter is used for the design of periodic discrete-time splines and discrete-time-spline-based wavelets and wavelet packets. For a detailed description of the subject we refer to (Averbuch, Neittaanmaki and Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Springer, Berlin, 2014) [1]. Periodic polynomial splines provide an example of mixed discrete-continuous circular convolution.

Delineation of Malignant Skin Tumors by Hyperspectral Imaging

This chapter outlines a new non-invasive method for delineation of skin lesions such as lentigo maligna and lentigo maligna melanoma. The method is based on the analysis of hyperspectral (HS) images taken in vivo before surgical excision of the lesions. For this, characteristic features of the spectral signatures of diseased pixels and healthy pixels are extracted, which combine the intensities in a few selected wavebands with the coefficients of the wavelet frame transforms of the spectral curves. To reduce dimensionality and to reveal the internal structure of the datasets, the diffusion maps (DM) technique is applied. The averaged Nearest Neighbor and the Classification and Regression Tr…

Using affinity perturbations to detect web traffic anomalies

The initial training phase of machine learning algorithms is usually computationally expensive as it involves the processing of huge matrices. Evolving datasets are challenging from this point of view because changing behavior requires updating the training. We propose a method for updating the training profile efficiently and a sliding window algorithm for online processing of the data in smaller fractions. This assumes the data is modeled by a kernel method that includes spectral decomposition. We demonstrate the algorithm with a web server request log where an actual intrusion attack is known to happen. Updating the kernel dynamically using a sliding window technique, prevents the proble…

Polynomial Smoothing Splines

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.

Deconvolution by Regularized Matching Pursuit

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 …

Gear classification and fault detection using a diffusion map framework

This article proposes a system health monitoring approach that detects abnormal behavior of machines. Diffusion map is used to reduce the dimensionality of training data, which facilitates the classification of newly arriving measurements. The new measurements are handled with Nyström extension. The method is trained and tested with real gear monitoring data from several windmill parks. A machine health index is proposed, showing that data recordings can be classified as working or failing using dimensionality reduction and warning levels in the low dimensional space. The proposed approach can be used with any system that produces high-dimensional measurement data. peerReviewed

Periodic Discrete and Discrete-Time Splines

Periodic discrete splines with different periods and spans are introduced in Sect. 3.4 of Volume I (Averbuch, Neittaanmaki and Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Springer, Berlin, 2014) [2]. In this chapter, we regard periodic discrete splines as a base for the design of periodic discrete-time wavelets, wavelet packets and wavelet frames. Therefore, only the discrete splines whose spans are 2 are outlined. These discrete splines are linear combinations of the discrete B-splines. So also, the so-called discrete-time splines are discussed in the chapter that are linear combinations of the discrete-time B-splines. The discrete-time B-s…

An efficient VM-based software protection

This paper presents Truly-protect, a system, incorporating a virtual machine, that enables execution of encrypted programs. Our intention is to form a framework for a conditional access/digital rights management system.

Cubic Local Splines on Non-uniform Grid

In this chapter, two types of local cubic splines on non-uniform grids are described: 1. The simplest variation-diminishing splines and 2. The quasi-interpolating splines. The splines are computed by a simple fast computational algorithms that utilizes a relation between the splines and cubic interpolation polynomials. Those splines can serve as an efficient tool for real-time signal processing. As an input, they use either clean or noised arbitrarily-spaced samples. On the other hand, the capability to adapt the grid to the structure of an object and minimal requirements to the operating memory are great advantages for off-line processing of signals and multidimensional data arrays.

Biorthogonal Wavelet Transforms

Wavelets in the polynomial and discrete spline spaces were introduced in Chaps. 8 and 10, respectively. In both cases, the wavelets’ design and implementation of the transforms were associated with perfect reconstruction (PR) filter banks. In this chapter, those associations are discussed in more detail. Biorthogonal wavelet bases generated by PR filter banks are investigated and a few examples of compactly supported biorthogonal wavelets are presented. Conditions for filters to restore and annihilate sampled polynomials are established (discrete vanishing moment property). In a sense, the material of this chapter is introductory to Chap. 12, where splines are used as a source for (non-spli…

Gear classification and fault detection using a diffusion map framework

Parameter Rating by Diffusion Gradient

Anomaly detection is a central task in high-dimensional data analysis. It can be performed by using dimensionality reduction methods to obtain a low-dimensional representation of the data, which reveals the geometry and the patterns that exist and govern it. Usually, anomaly detection methods classify high-dimensional vectors that represent data points as either normal or abnormal. Revealing the parameters (i.e., features) that cause detected abnormal behaviors is critical in many applications. However, this problem is not addressed by recent anomaly-detection methods and, specifically, by nonparametric methods, which are based on feature-free analysis of the data. In this chapter, we provi…

Wavelet Frames Generated by Spline Based p-Filter Banks

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-…

Discrete Periodic Spline Wavelets and Wavelet Packets

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.

Acoustic Detection of Moving Vehicles

This chapter outlines a robust algorithm to detect the arrival of a vehicle of arbitrary type when other noises are present. It is done via analysis of its acoustic signature against an existing database of recorded and processed acoustic signals. To achieve it with minimum number of false alarms, a construction of a training database of acoustic signatures of signals emitted by vehicles using the distribution of the energies among blocks of wavelet packet coefficients (waveband spectra, see Sect. 4.6) is combined with a procedure of random search for a near-optimal footprint (RSNOFP). The number of false alarms in the detection is minimized even under severe conditions such as: signals emi…

Introduction: Signals and Transforms

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].

Mixed Circular Convolutions and Zak Transforms

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

Analytic and directional wavelet packets in the space of periodic signals

The paper presents a versatile library of analytic and quasi-analytic complex-valued wavelet packets (WPs) which originate from discrete splines of arbitrary orders. The real parts of the quasi-analytic WPs are the regular spline-based orthonormal WPs designed in [2]. The imaginary parts are the so-called complementary orthonormal WPs, which, unlike the symmetric regular WPs, they are antisymmetric. Tensor products of 1D quasi-analytic WPs provide a diversity of 2D WPs oriented in multiple directions. For example, a set of the fourth-level WPs comprises 62 different directions. The designed computational scheme in the paper enables us to get fast and easy implementation of the WP transforms…

Non-periodic Discrete Splines

Discrete Splines with different spans were introduced in Sect. 3.3.1. This chapter focuses on a special case of discrete splines whose spans are powers of 2. These splines are discussed in more detail. The Zak transform provides an integral representation of such splines. Discrete exponential splines are introduced. Generators of the discrete-spline spaces are described whose properties are similar to properties of polynomial-spline spaces generators. Interpolating discrete splines provide efficient tools for upsampling 1D and 2D signals. An algorithm for explicit computation of discrete splines is described.

Biorthogonal Wavelet Transforms Originating from Discrete and Discrete-Time Splines

This chapter describes how to generate families of biorthogonal wavelet transforms in spaces of periodic signals using prediction p-filters originating from discrete-time and discrete splines. The transforms are generated by the lifting scheme (Sweldens (Wavelet applications in signal and image processing III, vol 2569, 1995, [7]), Sweldens (Appl Comput Harmon Anal 3:186–200, 1996, [8]), Sweldens (SIAM J Math Anal 29:511–546, 1997, [9]), see also Sect. 7.1 of this volume). The discrete-time wavelets related to those transforms are (anti)symmetric, well localized in time domain and have flat spectra. These families comprise wavelets with any number of local discrete vanishing moments (LDVMs)…

Biorthogonal Multiwavelets Originated from Hermite Splines

This chapter presents multiwavelet transforms that manipulate discrete-time signals. The transforms are implemented in two phases: 1. Pre (post)-processing, which transforms a scalar signal into a vector signal (and back). 2. Wavelet transforms of the vector signal. Both phases are performed in a lifting way. The cubic interpolating Hermite splines are used as a predicting aggregate in the vector wavelet transform. Pre(post)-processing algorithms which do not degrade the approximation accuracy of the vector wavelet transforms are presented. A scheme of vector wavelet transforms and three pre(post)-processing algorithms are described. As a result, we get fast biorthogonal algorithms to trans…

Biorthogonal Wavelet Transforms Originating from Splines

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…

Application of Periodic Frames to Image Restoration

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.

Introduction: Periodic Filters and Filter Banks

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.

Spline-Based Wavelet Transforms

The Lifting Scheme introduced in (Sweldens, Appl. Comput. Harmon. Anal. 3(2), 186–200 (1996) and Sweldens, SIAM J. Math. Anal. 29(2), 511–546 (1997).) [3, 4] is a method that constructs bi-orthogonal wavelet transforms of signals and provides their efficient implementation. The main feature of the lifting scheme is that all the constructions are derived directly in the spatial domain and therefore can be custom designed to more general and irregular settings such as non-uniformly spaced data samples and bounded intervals. In this chapter, we outline the lifting scheme and describe how to use the local quasi-interpolating splines, introduced in Chap. 6, for the construction of wavelet transf…

Hydro-Acoustic Target Detection

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.

Multiwavelet Frames Originated From Hermite Splines

The chapter presents a method for the construction of multiwavelet frame transform for manipulation of discrete-time signals. The frames are generated by three-channel perfect reconstruction oversampled multifilter banks. The design of the multifilter bank starts from a pair of interpolating multifilters, which originate from the cubic Hermite splines. The remaining multifilters are designed by factoring polyphase matrices. Input to the oversampled analysis multifilter bank is a vector-signal, which is derived from an initial scalar signal by one out of three pre-processing algorithms. The post-processing algorithms convert the vector output from the synthesis multifilter banks into a scala…

Two-Dimensional Orthogonal Wavelets and Wavelet Packets

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.

Periodic Spline Wavelets and Wavelet Packets

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.

Image inpainting using directional wavelet packets originating from polynomial splines

The paper presents a new algorithm for the image inpainting problem. The algorithm is using a recently designed versatile library of quasi-analytic complex-valued wavelet packets (qWPs) which originate from polynomial splines of arbitrary orders. Tensor products of 1D qWPs provide a diversity of 2D qWPs oriented in multiple directions. For example, a set of the fourth-level qWPs comprises 62 different directions. The properties of the presented qWPs such as refined frequency resolution, directionality of waveforms with unlimited number of orientations, (anti-)symmetry of waveforms and windowed oscillating structure of waveforms with a variety of frequencies, make them efficient in image pro…

Discrete-Time Periodic Wavelet Packets

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.

Mixed Convolutions and Zak Transforms

In this chapter we introduce the mixed continuous–discrete and discrete–discrete convolutions. Important special cases of such convolutions are the polynomial and discrete splines, respectively. The Zak transforms, which are introduced in the chapter, provide integral representation of signals, which, in the following chapters, serves as a tool for the design of splines and spline-wavelets and operations over them. The exponential splines, which are the Zak transforms of polynomial and discrete B-splines are introduced. Explicit formulas for the characteristic functions of splines’ spaces are derived.

Fast Computation by Subdivision of Multidimensional Splines and Their Applications

We present theory and algorithms for fast explicit computations of uni- and multi-dimensional periodic splines of arbitrary order at triadic rational points and of splines of even order at diadic rational points. The algorithms use the forward and the inverse Fast Fourier transform (FFT). The implementation is as fast as FFT computation. The algorithms are based on binary and ternary subdivision of splines. Interpolating and smoothing splines are used for a sample rate convertor such as resolution upsampling of discrete-time signals and digital images and restoration of decimated images that were contaminated by noise. The performance of the rate conversion based spline is compared with the…

Periodic Orthogonal Wavelets and Wavelet Packets

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.

Calculation of Splines Values by Subdivision

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.

Non-periodic Discrete-Spline Wavelets

This chapter describes wavelet analysis in the spaces of discrete splines whose spans are powers of 2. This wavelet analysis is similar to wavelet analysis in the polynomial-spline spaces. The transforms are based on relations between exponential discrete splines from different resolution scales. Generators of discrete-spline wavelet spaces are described. The discrete-spline wavelet transforms generate wavelet transforms in signal space. Practically, wavelet transforms of signals are implemented by multirate filtering of signals by two-channel filter banks with the downsampling factor 2 (critically sampled filter banks). The filtering implementation is accelerated by switching to the polyph…

Non-periodic Polynomial Splines

In this chapter, we outline the essentials of the splines theory. By themselves, they are of interest for signal processing research. We use the Zak transform to derive an integral representation of polynomial splines on uniform grids. The integral representation facilitated design of different generators of spline spaces and their duals. It provides explicit expressions for interpolating and smoothing splines of any order. In forthcoming chapters, the integral representation of splines will be used for the constructions of efficient subdivision schemes and so also for the design spline-based wavelets and wavelet frames.

Quasi-interpolating and Smoothing Local Splines

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…

Detection of Incipient Bearing Fault in a Slowly Rotating Machine Using Spline Wavelet Packets

This chapter describes a successful application of spline-based wavelet packet transforms (WPTs) described in Chap. 4 to a complicated problem of detection of incipient defects in rolling element bearings by the analysis of recorded vibration signals. The methodology presented in this chapter is applied to the analysis of vibration data recorded from large bearings working in real unfavorable operation conditions in presence of strong noise and vibrations from multiple internal and external sources. It relies on properties of discrete spline-based wavelet packets such as orthogonality, near-rectangular spectra, transient oscillating shapes of testing waveforms and fast implementation of tra…

Polar Classification of Nominal Data

Many modern systems record various types of parameter values. Numerical values are relatively convenient for data analysis tools because there are many methods to measure distances and similarities between them. The application of dimensionality reduction techniques for data sets with such values is also a well known practice. Nominal (i.e., categorical) values, on the other hand, encompass some problems for current methods. Most of all, there is no meaningful distance between possible nominal values, which are either equal or unequal to each other. Since many dimensionality reduction methods rely on preserving some form of similarity or distance measure, their application to such data sets…

Spline Algorithms for Deconvolution and Inversion of Heat Equation

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.

Snapshot Spectral Imaging

This chapter describes an application of the spline-based wavelet frames to the spectral imaging. It presents a method that enables to convert a regular digital camera into a snapshot spectral imager by equipping the camera with a dispersive diffuser and with a compressed sensing-based algorithm for digital processing. The method relies on the assumption that typical images can be sparsely represented in the frame domain. The solution is found from the constrained \(l_{1}\) minimization of a functional by Bregman iterations. Results of optical experiments are reported. The chapter is based on the paper (Golub et al., Appl. Opt. 55, 432–443, (2016), [11]).

Data Compression Using Wavelet and Local Cosine Transforms

The chapter describes an algorithm that compresses two-dimensional data arrays, which are piece-wise smooth in one direction and have oscillating events in the other direction. Seismic, hyper-spectral and fingerprints data, for example, have such a mixed structure. The transform part of the compression process is an algorithm that combines wavelet and local cosine transform (LCT). The quantization and the entropy coding parts of the compression are taken from the SPIHT codec. To efficiently apply the SPIHT codec to a mixed coefficients array, reordering of the LCT coefficients takes place. On the data arrays, which have the mixed structure, this algorithm outperforms other algorithms that a…

Polynomial Spline-Wavelets

This chapter presents wavelets in the spaces of polynomial splines. The wavelets’ design is based on the Zak transform, which provides an integral representation of spline-wavelets. The exponential wavelets which participate in the integral representation are counterparts of the exponential splines that were introduced in Chap. 4. Fast algorithms for the wavelet transforms of splines are presented. Generators of spline-wavelet spaces are described, such as the B-wavelets and their duals and the Battle-Lemarie wavelets whose shifts form orthonormal bases of the spline-wavelet spaces.

Introduction: Digital Filters and Filter Banks

A basic operation of spectr is filterin. In this introductory chapter, processing of one- and two-dimensional signals by digital filters and filter banks is outlined. A polyphase implementation of multirate filtering is described. The application of filtering with IIR filters, whose transfer functions are rational, is described. Bases and frames in the signals’ space that are generated by perfect reconstruction filter banks are discussed. The Butterworth filters, which are used in further constructions, are introduced.

Wavelet Frames Generated by Perfect Reconstruction Filter Banks

It was indicated in Sect. 2.2 that oversampled perfect reconstruction (PR) filter banks generate specific types of frames in the signal space. In this chapter, a family of tight and semi-tight frames is presented. The three- and four-channel filter banks that generate the framed originate from polynomial and discrete splines. Those frames have properties, which are attractive for signal processing, such as symmetry, interpolation, flat spectra. These properties are combined with fine time-domain localization and efficient implementation. This family includes framelets, that have any number of discrete vanishing moments. Non-compactness of their supports is compensated by exponential decay o…

Local Splines on Non-uniform Grid

In this Chapter and in the next Chap. 7, we deal with continuous rather than discrete and discrete-time splines. In these and only these chapters, we abandon the assumption that the grid, on which the splines are constructed, is uniform and consider splines on arbitrary grids. Two types of local cubic and quadratic splines on non-uniform grids are described: 1. The simplest variation-diminishing splines and 2. The quasi-interpolating splines. The splines are computed by simple fast computational algorithms that utilize relations between the splines and interpolation polynomials. In addition, these relations provide sharp estimations of splines’ approximation accuracy. These splines can serv…

Introduction: Periodic Signals and Filters

In this chapter we briefly outline some well-known facts about Discrete-time periodic signals, their transforms and periodic digital filters and filter banks. For details we refer to the classical textbook A. V. Oppenheim and R. W. Schafer (Discrete-Time Signal Processing, Prentice Hall, New York, 2010, [3]) and Volume I of our book (Averbuch, Neittaanmaki and Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Periodic Splines, vol. 1 (Springer, Berlin, 2014)) [1] Throughout the volume, unless other indicated, \(N=2^{j}, \;j\in \mathbb {N}\).

Periodic Discrete Splines

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.

Splines Computation by Subdivision

In this chapter, fast stable algorithms are presented, which compute splines’ values at dyadic and triadic rational points starting from their samples at integer grid points. The algorithms are implemented by the causal-anticausal recursive filtering of initial data samples, which is followed by iterated application of FIR filters. Extension of the algorithms to the multidimensional case is straightforward. A natural application of the presented subdivision algorithms is for upsampling of signals and images. A few upsampling examples are provided.

Acoustic detection and classification of river boats

We present a robust algorithm to detect the arrival of a boat of a certain type when other background noises are present. It is done via the analysis of its acoustic signature against an existing database of recorded and processed acoustic signals. We characterize the signals by the distribution of their energies among blocks of wavelet packet coefficients. To derive the acoustic signature of the boat of interest, we use the Best Discriminant Basis method. The decision is made by combining the answers from the Linear Discriminant Analysis (LDA) classifier and from the Classification and Regression Trees (CART) that is also accompanied with an additional unit, called Aisles, that reduces fal…

Local cubic splines on non-uniform grids and real-time computation of wavelet transform

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of s…

Block-Based Inversion of the Heat Equations

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…