Applications of compressed sensing in computational physics

Conventional sampling theory is dictated by Shannon's celebrated sampling theorem: For a signal to be reconstructed from samples, it must be sampled with at least twice the maximum frequency found in the signal. This principle is key in all modern signal acquisition, from consumer electronics to medical imaging devices. Recently, a new theory of signal acquisition has emerged in the form of Compressed Sensing, which allows for complete conservation of the information in a signal using far fewer samples than Shannon's theorem dictates. This is achieved by noting that signals with information are usually structured, allowing them to be represented with very few coefficients in the proper basis, a property called sparsity. In this thesis, we survey the existing theory of compressed sensing, with details on performance guarantees in terms of the Restricted Isometry Property. We then survey the state-of-the-art applications of the theory, including improved MRI using Total Variation sparsity and restoration of seismic data using curvelet and wave atom sparsity. We apply Compressed Sensing to the problem of finding statistical properties of a signal based CS methods, by attempting to measure the Hurst exponent of rough surfaces by partial measurements. We suggest an improvement on previous results in seismic data restoration, by applying a learned dictionary of signal patches for restoration

Uses of Complex Wavelets in Deep Convolutional Neural Networks

Image understanding has long been a goal for computer vision. It has proved to be an exceptionally difficult task due to the large amounts of variability that are inherent to objects in a scene. Recent advances in supervised learning methods, particularly convolutional neural networks (CNNs), have pushed forth the frontier of what we have been able to train computers to do. Despite their successes, the mechanics of how these networks are able to recognize objects are little understood, and the networks themselves are often very difficult and time-consuming to train. It is very important that we improve our current approaches in every way possible. A CNN is built from connecting many learned convolutional layers in series. These convolutional layers are fairly crude in terms of signal processing - they are arbitrary taps of a finite impulse response filter, learned through stochastic gradient descent from random initial conditions. We believe that if we reformulate the problem, we may achieve many insights and benefits in training CNNs. Noting that modern CNNs are mostly viewed from and analyzed in the spatial domain, this thesis aims to view the convolutional layers in the frequency domain (viewing things in the frequency domain has proved useful in the past for denoising, filter design, compression and many other tasks). In particular, we use complex wavelets (rather than the Fourier transform or the discrete wavelet transform) as basis functions to reformulate image understanding with deep networks. In this thesis, we explore the most popular and well-developed form of using complex wavelets in deep learning, the ScatterNet from Stephane Mallat. We explore its current limitations by building a DeScatterNet and found that while it has many nice properties, it may not be sensitive to the most appropriate shapes for understanding natural images. We then develop a locally invariant convolutional layer, a combination of a complex wavelet transform, a modulus operation, and a learned mixing. To do this, we derive backpropagation equations and allow gradients to flow back through the (previously fixed) ScatterNet front end. Connecting several such locally invariant layers allows us to build learnable ScatterNet, a more flexible and general form of the ScatterNet (while still maintaining its desired properties). We show that the learnable ScatterNet can provide significant improvements over the regular ScatterNet when being used as a front end for a learning system. Additionally, we show that the locally invariant convolutional layer can directly replace convolutional layers in a deep CNN (and not just at the front-end). The locally invariant convolutional layers naturally downsample the input (because of the complex modulus) while increasing the channel dimension (because of the multiple wavelet orientations used). This is an operation that often happens in a CNN by a combination of a pooling and convolutional layer. It was at these locations in a CNN where the learnable ScatterNet performed best, implying it may be useful as learnable pooling layer. Finally, we develop a system to learn complex weights that act directly on the wavelet coefficients of signals, in place of a convolutional layer. We call this layer the wavelet gain layer and show it can be used alongside convolutional layers. The network designer may then choose to learn in the pixel or wavelet domains. This layer shows a lot of promise and affords more control over what regions of the frequency space we want our layer to learn from. Our experiments show that it can improve on learning in the pixel domain for early layers of a CNN

Probabilistic modeling of wavelet coefficients for processing of image and video signals

Statistical estimation and detection techniques are widely used in signal processing including wavelet-based image and video processing. The probability density function (PDF) of the wavelet coefficients of image and video signals plays a key role in the development of techniques for such a processing. Due to the fixed number of parameters, the conventional PDFs for the estimators and detectors usually ignore higher-order moments. Consequently, estimators and detectors designed using such PDFs do not provide a satisfactory performance. This thesis is concerned with first developing a probabilistic model that is capable of incorporating an appropriate number of parameters that depend on higher-order moments of the wavelet coefficients. This model is then used as the prior to propose certain estimation and detection techniques for denoising and watermarking of image and video signals. Towards developing the probabilistic model, the Gauss-Hermite series expansion is chosen, since the wavelet coefficients have non-compact support and their empirical density function shows a resemblance to the standard Gaussian function. A modification is introduced in the series expansion so that only a finite number of terms can be used for modeling the wavelet coefficients with rendering the resulting PDF to become negative. The parameters of the resulting PDF, called the modified Gauss-Hermite (NIGH) PDF, are evaluated in terms of the higher-order sample-moments. It is shown that the MGH PDF fits the empirical density function better than the existing PDFs that use a limited number of parameters do. The proposed MGH PDF is used as the prior of image and video signals in designing maximum a posteriori and minimum mean squared error-based estimators for denoising of image and video signals and log-likelihood ratio-based detector for watermarking of image signals. The performance of the estimation and detection techniques are then evaluated in terms of the commonly used metrics. It is shown through extensive experimentations that the estimation and detection techniques developed utilizing the proposed MGH PDF perform substantially better than those that utilize the conventional PDFs. These results confirm that the superior fit of the MGH PDF to the empirical density function resulting from the flexibility of the MGH PDF in choosing the number of parameters, which are functions of higher-order moments of data, leads to the better performance. Thus, the proposed MGH PDF should play a significant role in wavelet-based image and video signal processin

Quantitative analysis of algorithms for compressed signal recovery

Compressed Sensing (CS) is an emerging paradigm in which signals are recovered from undersampled nonadaptive linear measurements taken at a rate proportional to the signal's true information content as opposed to its ambient dimension. The resulting problem consists in finding a sparse solution to an underdetermined system of linear equations. It has now been established, both theoretically and empirically, that certain optimization algorithms are able to solve such problems. Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2007), which is the focus of this thesis, is an established CS recovery algorithm which is known to be effective in practice, both in terms of recovery performance and computational efficiency. However, theoretical analysis of IHT to date suffers from two drawbacks: state-of-the-art worst-case recovery conditions have not yet been quantified in terms of the sparsity/undersampling trade-off, and also there is a need for average-case analysis in order to understand the behaviour of the algorithm in practice. In this thesis, we present a new recovery analysis of IHT, which considers the fixed points of the algorithm. In the context of arbitrary matrices, we derive a condition guaranteeing convergence of IHT to a fixed point, and a condition guaranteeing that all fixed points are 'close' to the underlying signal. If both conditions are satisfied, signal recovery is therefore guaranteed. Next, we analyse these conditions in the case of Gaussian measurement matrices, exploiting the realistic average-case assumption that the underlying signal and measurement matrix are independent. We obtain asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. By generalizing the notion of xed points, we extend our analysis to the variable stepsize Normalised IHT (NIHT) (Blumensath and Davies, 2010). For both stepsize schemes, comparison with previous results within this framework shows a substantial quantitative improvement. We also extend our analysis to a related algorithm which exploits the assumption that the underlying signal exhibits tree-structured sparsity in a wavelet basis (Baraniuk et al., 2010). We obtain recovery conditions for Gaussian matrices in a simplified proportional-dimensional asymptotic, deriving bounds on the oversampling rate relative to the sparsity for which recovery is guaranteed. Our results, which are the first in the phase transition framework for tree-based CS, show a further significant improvement over results for the standard sparsity model. We also propose a dynamic programming algorithm which is guaranteed to compute an exact tree projection in low-order polynomial time

Multibiometric security in wireless communication systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 05/08/2010.This thesis has aimed to explore an application of Multibiometrics to secured wireless communications. The medium of study for this purpose included Wi-Fi, 3G, and WiMAX, over which simulations and experimental studies were carried out to assess the performance. In specific, restriction of access to authorized users only is provided by a technique referred to hereafter as multibiometric cryptosystem. In brief, the system is built upon a complete challenge/response methodology in order to obtain a high level of security on the basis of user identification by fingerprint and further confirmation by verification of the user through text-dependent speaker recognition. First is the enrolment phase by which the database of watermarked fingerprints with memorable texts along with the voice features, based on the same texts, is created by sending them to the server through wireless channel. Later is the verification stage at which claimed users, ones who claim are genuine, are verified against the database, and it consists of five steps. Initially faced by the identification level, one is asked to first present one’s fingerprint and a memorable word, former is watermarked into latter, in order for system to authenticate the fingerprint and verify the validity of it by retrieving the challenge for accepted user. The following three steps then involve speaker recognition including the user responding to the challenge by text-dependent voice, server authenticating the response, and finally server accepting/rejecting the user. In order to implement fingerprint watermarking, i.e. incorporating the memorable word as a watermark message into the fingerprint image, an algorithm of five steps has been developed. The first three novel steps having to do with the fingerprint image enhancement (CLAHE with 'Clip Limit', standard deviation analysis and sliding neighborhood) have been followed with further two steps for embedding, and extracting the watermark into the enhanced fingerprint image utilising Discrete Wavelet Transform (DWT). In the speaker recognition stage, the limitations of this technique in wireless communication have been addressed by sending voice feature (cepstral coefficients) instead of raw sample. This scheme is to reap the advantages of reducing the transmission time and dependency of the data on communication channel, together with no loss of packet. Finally, the obtained results have verified the claims

3D Object Recognition Based On Constrained 2D Views

The aim of the present work was to build a novel 3D object recognition system capable of classifying man-made and natural objects based on single 2D views. The approach to this problem has been one motivated by recent theories on biological vision and multiresolution analysis. The project's objectives were the implementation of a system that is able to deal with simple 3D scenes and constitutes an engineering solution to the problem of 3D object recognition, allowing the proposed recognition system to operate in a practically acceptable time frame. The developed system takes further the work on automatic classification of marine phytoplank- (ons, carried out at the Centre for Intelligent Systems, University of Plymouth. The thesis discusses the main theoretical issues that prompted the fundamental system design options. The principles and the implementation of the coarse data channels used in the system are described. A new multiresolution representation of 2D views is presented, which provides the classifier module of the system with coarse-coded descriptions of the scale-space distribution of potentially interesting features. A multiresolution analysis-based mechanism is proposed, which directs the system's attention towards potentially salient features. Unsupervised similarity-based feature grouping is introduced, which is used in coarse data channels to yield feature signatures that are not spatially coherent and provide the classifier module with salient descriptions of object views. A simple texture descriptor is described, which is based on properties of a special wavelet transform. The system has been tested on computer-generated and natural image data sets, in conditions where the inter-object similarity was monitored and quantitatively assessed by human subjects, or the analysed objects were very similar and their discrimination constituted a difficult task even for human experts. The validity of the above described approaches has been proven. The studies conducted with various statistical and artificial neural network-based classifiers have shown that the system is able to perform well in all of the above mentioned situations. These investigations also made possible to take further and generalise a number of important conclusions drawn during previous work carried out in the field of 2D shape (plankton) recognition, regarding the behaviour of multiple coarse data channels-based pattern recognition systems and various classifier architectures. The system possesses the ability of dealing with difficult field-collected images of objects and the techniques employed by its component modules make possible its extension to the domain of complex multiple-object 3D scene recognition. The system is expected to find immediate applicability in the field of marine biota classification

Multibiometric security in wireless communication systems

This thesis has aimed to explore an application of Multibiometrics to secured wireless communications. The medium of study for this purpose included Wi-Fi, 3G, and WiMAX, over which simulations and experimental studies were carried out to assess the performance. In specific, restriction of access to authorized users only is provided by a technique referred to hereafter as multibiometric cryptosystem. In brief, the system is built upon a complete challenge/response methodology in order to obtain a high level of security on the basis of user identification by fingerprint and further confirmation by verification of the user through text-dependent speaker recognition. First is the enrolment phase by which the database of watermarked fingerprints with memorable texts along with the voice features, based on the same texts, is created by sending them to the server through wireless channel. Later is the verification stage at which claimed users, ones who claim are genuine, are verified against the database, and it consists of five steps. Initially faced by the identification level, one is asked to first present one’s fingerprint and a memorable word, former is watermarked into latter, in order for system to authenticate the fingerprint and verify the validity of it by retrieving the challenge for accepted user. The following three steps then involve speaker recognition including the user responding to the challenge by text-dependent voice, server authenticating the response, and finally server accepting/rejecting the user. In order to implement fingerprint watermarking, i.e. incorporating the memorable word as a watermark message into the fingerprint image, an algorithm of five steps has been developed. The first three novel steps having to do with the fingerprint image enhancement (CLAHE with 'Clip Limit', standard deviation analysis and sliding neighborhood) have been followed with further two steps for embedding, and extracting the watermark into the enhanced fingerprint image utilising Discrete Wavelet Transform (DWT). In the speaker recognition stage, the limitations of this technique in wireless communication have been addressed by sending voice feature (cepstral coefficients) instead of raw sample. This scheme is to reap the advantages of reducing the transmission time and dependency of the data on communication channel, together with no loss of packet. Finally, the obtained results have verified the claims.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Scalable video compression with optimized visual performance and random accessibility

This thesis is concerned with maximizing the coding efficiency, random accessibility and visual performance of scalable compressed video. The unifying theme behind this work is the use of finely embedded localized coding structures, which govern the extent to which these goals may be jointly achieved. The first part focuses on scalable volumetric image compression. We investigate 3D transform and coding techniques which exploit inter-slice statistical redundancies without compromising slice accessibility. Our study shows that the motion-compensated temporal discrete wavelet transform (MC-TDWT) practically achieves an upper bound to the compression efficiency of slice transforms. From a video coding perspective, we find that most of the coding gain is attributed to offsetting the learning penalty in adaptive arithmetic coding through 3D code-block extension, rather than inter-frame context modelling. The second aspect of this thesis examines random accessibility. Accessibility refers to the ease with which a region of interest is accessed (subband samples needed for reconstruction are retrieved) from a compressed video bitstream, subject to spatiotemporal code-block constraints. We investigate the fundamental implications of motion compensation for random access efficiency and the compression performance of scalable interactive video. We demonstrate that inclusion of motion compensation operators within the lifting steps of a temporal subband transform incurs a random access penalty which depends on the characteristics of the motion field. The final aspect of this thesis aims to minimize the perceptual impact of visible distortion in scalable reconstructed video. We present a visual optimization strategy based on distortion scaling which raises the distortion-length slope of perceptually significant samples. This alters the codestream embedding order during post-compression rate-distortion optimization, thus allowing visually sensitive sites to be encoded with higher fidelity at a given bit-rate. For visual sensitivity analysis, we propose a contrast perception model that incorporates an adaptive masking slope. This versatile feature provides a context which models perceptual significance. It enables scene structures that otherwise suffer significant degradation to be preserved at lower bit-rates. The novelty in our approach derives from a set of "perceptual mappings" which account for quantization noise shaping effects induced by motion-compensated temporal synthesis. The proposed technique reduces wavelet compression artefacts and improves the perceptual quality of video