Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Measurements, Models, Systems and Design

531 s.

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

Methods for Photoacoustic Image Reconstruction Exploiting Properties of Curvelet Frame

Curvelet frame is of special significance for photoacoustic tomography (PAT) due to its sparsifying and microlocalisation properties. In this PhD project, we explore the methods for image reconstruction in PAT with flat sensor geometry using Curvelet properties. This thesis makes five distinct contributions: (i) We investigate formulation of the forward, adjoint and inverse operators for PAT in Fourier domain. We derive a one-to-one map between wavefront directions in image and data spaces in PAT. Combining the Fourier operators with the wavefront map allows us to create the appropriate PAT operators for solving limited-view problems due to limited angular sensor sensitivity. (ii) We devise a concept of wedge restricted Curvelet transform, a modification of standard Curvelet transform, which allows us to formulate a tight frame of wedge restricted Curvelets on the range of the PAT forward operator for PAT data representation. We consider details specific to PAT data such as symmetries, time oversampling and their consequences. We further adapt the wedge restricted Curvelet to decompose the wavefronts into visible and invisible parts in the data domain as well as in the image domain. (iii) We formulate a two step approach based on the recovery of the complete volume of the photoacoustic data from the sub-sampled data followed by the acoustic inversion, and a one step approach where the photoacoustic image is directly recovered from the subsampled data. The wedge restricted Curvelet is used as the sparse representation of the photoacoustic data in the two step approach. (iv) We discuss a joint variational approach that incorporates Curvelet sparsity in photoacoustic image domain and spatio-temporal regularization via optical flow constraint to achieve improved results for dynamic PAT reconstruction. (v) We consider the limited-view problem due to limited angular sensitivity of the sensor (see (i) for the formulation of the corresponding fast operators in Fourier domain). We propose complementary information learning approach based on splitting the problem into visible and invisible singularities. We perform a sparse reconstruction of the visible Curvelet coefficients using compressed sensing techniques and propose a tailored deep neural network architecture to recover the invisible coefficients

Efficient compression of motion compensated residuals

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Learning Equivariant Representations

State-of-the-art deep learning systems often require large amounts of data and computation. For this reason, leveraging known or unknown structure of the data is paramount. Convolutional neural networks (CNNs) are successful examples of this principle, their defining characteristic being the shift-equivariance. By sliding a filter over the input, when the input shifts, the response shifts by the same amount, exploiting the structure of natural images where semantic content is independent of absolute pixel positions. This property is essential to the success of CNNs in audio, image and video recognition tasks. In this thesis, we extend equivariance to other kinds of transformations, such as rotation and scaling. We propose equivariant models for different transformations defined by groups of symmetries. The main contributions are (i) polar transformer networks, achieving equivariance to the group of similarities on the plane, (ii) equivariant multi-view networks, achieving equivariance to the group of symmetries of the icosahedron, (iii) spherical CNNs, achieving equivariance to the continuous 3D rotation group, (iv) cross-domain image embeddings, achieving equivariance to 3D rotations for 2D inputs, and (v) spin-weighted spherical CNNs, generalizing the spherical CNNs and achieving equivariance to 3D rotations for spherical vector fields. Applications include image classification, 3D shape classification and retrieval, panoramic image classification and segmentation, shape alignment and pose estimation. What these models have in common is that they leverage symmetries in the data to reduce sample and model complexity and improve generalization performance. The advantages are more significant on (but not limited to) challenging tasks where data is limited or input perturbations such as arbitrary rotations are present

Discrete Wavelet Transforms

The discrete wavelet transform (DWT) algorithms have a firm position in processing of signals in several areas of research and industry. As DWT provides both octave-scale frequency and spatial timing of the analyzed signal, it is constantly used to solve and treat more and more advanced problems. The present book: Discrete Wavelet Transforms: Algorithms and Applications reviews the recent progress in discrete wavelet transform algorithms and applications. The book covers a wide range of methods (e.g. lifting, shift invariance, multi-scale analysis) for constructing DWTs. The book chapters are organized into four major parts. Part I describes the progress in hardware implementations of the DWT algorithms. Applications include multitone modulation for ADSL and equalization techniques, a scalable architecture for FPGA-implementation, lifting based algorithm for VLSI implementation, comparison between DWT and FFT based OFDM and modified SPIHT codec. Part II addresses image processing algorithms such as multiresolution approach for edge detection, low bit rate image compression, low complexity implementation of CQF wavelets and compression of multi-component images. Part III focuses watermaking DWT algorithms. Finally, Part IV describes shift invariant DWTs, DC lossless property, DWT based analysis and estimation of colored noise and an application of the wavelet Galerkin method. The chapters of the present book consist of both tutorial and highly advanced material. Therefore, the book is intended to be a reference text for graduate students and researchers to obtain state-of-the-art knowledge on specific applications

Image Analysis via Applied Harmonic Analysis : Perceptual Image Quality Assessment, Visual Servoing, and Feature Detection

Certain systems of analyzing functions developed in the field of applied harmonic analysis are specifically designed to yield efficient representations of structures which are characteristic of common classes of two-dimensional signals, like images. In particular, functions in these systems are typically sensitive to features that define the geometry of a signal, like edges and curves in the case of images. These properties make them ideal candidates for a wide variety of tasks in image processing and image analysis. This thesis discusses three recently developed approaches to utilizing systems of wavelets, shearlets, and alpha-molecules in specific image analysis tasks. First, a perceptual image similarity measure is introduced that is solely based on the coefficients obtained from six discrete Haar wavelet filters but yields state of the art correlations with human opinion scores on large benchmark databases. The second application concerns visual servoing, which is a technique for controlling the motion of a robot by using feedback from a visual sensor. In particular, it will be investigated how the coefficients yielded by discrete wavelet and shearlet transforms can be used as the visual features that control the motion of a robot with six degrees of freedom. Finally, a novel framework for the detection and characterization of features such as edges, ridges, and blobs in two-dimensional images is presented and evaluated in extensive numerical experiments. Here, versatile and robust feature detectors are obtained by exploiting the special symmetry properties of directionally sensitive analyzing functions in systems created within the recently introduced alpha-molecule framework