Sparse modelling with the ℓ p norm of 0 ≤ p ≤ 1 requires the availability of the proximity operator of the ℓ p norm. The proximity operators of the ℓ 0 and ℓ 1 norms are the well‐known hard‐ and soft‐thresholding estimators, respectively. In this study, the authors give a complete study on the properties of the proximity operator of the ℓ p norm. Based on these properties, explicit formulas of the proximity operators of the ℓ 1/2 norm and ℓ 2/3 norm are derived with simple proofs; for other values of p , an iterative Newton's method is developed to compute the proximity operator of the ℓ p norm by fully exploring the available proximity operators of the ℓ 0 , ℓ 1/2 , ℓ 2/3 , and ℓ 1 norms. As applications, the proximity operator of the ℓ p norm with 0 ≤ p ≤ 1 is applied to the ℓ p ‐regularisation for compressive sensing and image restoration.

Since all packets in linear network coding are subject to linear combinations, in all existing network coding schemes, a full rank of received packets is required to start decoding. This requirement unfortunately results in long delays and low throughputs. In this work we propose two classes of rank deficient decoders that work for rank deficient received packets. Within either class, different decoding strategies have been proposed for tradeoffs between delay/throughput and data accuracy. The decoders of the first class take advantage of the sparsity inherent in data and produce the data vectors with the smallest Hamming weight. Since these decoders have high complexities, we propose a class of decoders with polynomial complexities based on linear programming. Both classes of decoders can recover data from fewer received packets and hence achieve higher throughputs and shorter delays than the full rank decoder.

In this paper, we propose an image inpainting optimization model whose objective function is a smoothed l(1) norm of the weighted nondecimated discrete cosine transform (DCT) coefficients of the underlying image. By identifying the objective function of the proposed model as a sum of a differentiable term and a nondifferentiable term, we present a basic algorithm inspired by Beck and Teboulle's recent work on the model. Based on this basic algorithm, we propose an automatic way to determine the weights involved in the model and update them in each iteration. The DCT as an orthogonal transform is used in various applications. We view the rows of a DCT matrix as the filters associated with a multiresolution analysis. Nondecimated wavelet transforms with these filters are explored in order to analyze the images to be inpainted. Our numerical experiments verify that under the proposed framework, the filters from a DCT matrix demonstrate promise for the task of image inpainting.

While random linear network coding is a powerful tool for disseminating information in communication networks, it is highly susceptible to errors caused by various sources. Due to error propagation, errors greatly deteriorate the throughput of network coding and seriously undermine both reliability and security of data. Hence, error control for network coding is vital. Recently, constant-dimension codes (CDCs), especially Kötter-Kschischang (KK) codes, have been proposed for error control in random linear network coding. KK codes can also be constructed from Gabidulin codes, an important class of rank metric codes. Rank metric decoders have been recently proposed for both Gabidulin and KK codes, but they have high computational complexities. Furthermore, it is not clear whether such decoders are feasible and suitable for hardware implementations. In this paper, we reduce the complexities of rank metric decoders and propose novel decoder architectures for both codes. The synthesis results of our decoder architectures for Gabidulin and KK codes with limited error-correcting capabilities over small fields show that our architectures not only are affordable, but also achieve high throughput.

This paper studies a problem of image restoration that observed images are contaminated by Gaussian and impulse noise. Existing methods for this problem in the literature are based on minimizing an objective functional having the l(1) fidelity term and the Mumford-Shah regularizer. We present an algorithm on this problem by minimizing a new objective functional. The proposed functional has a content-dependent fidelity term which assimilates the strength of fidelity terms measured by the l(1) and l(2) norms. The regularizer in the functional is formed by the l(1) norm of tight framelet coefficients of the underlying image. The selected tight framelet filters are able to extract geometric features of images. We then propose an iterative framelet-based approximation/sparsity deblurring algorithm (IFASDA) for the proposed functional. Parameters in IFASDA are adaptively varying at each iteration and are determined automatically. In this sense, IFASDA is a parameter-free algorithm. This advantage makes the algorithm more attractive and practical. The effectiveness of IFASDA is experimentally illustrated on problems of image deblurring with Gaussian and impulse noise. Improvements in both PSNR and visual quality of IFASDA over a typical existing method are demonstrated. In addition, Fast_IFASDA, an accelerated algorithm of IFASDA, is also developed.

In this paper, we propose an approach to manage network resources for a direct sequence code division multiple access (DS-CDMA) visual sensor network where nodes monitor scenes with varying levels of motion. It uses cross-layer optimization across the physical layer, the link layer, and the application layer. Our technique simultaneously assigns a source coding rate, a channel coding rate, and a power level to all nodes in the network based on one of two criteria that maximize the quality of video of the entire network as a whole, subject to a constraint on the total chip rate. One criterion results in the minimal average end-to-end distortion amongst all nodes, while the other criterion minimizes the maximum distortion of the network. Our experimental results demonstrate the effectiveness of the cross-layer optimization.

In this paper we design novel generalized likelihood ratio test (GLRT)-type packet-data detectors for general multiaccess/multiuser digital communication systems and we develop analytical performance evaluation tools for finite data packet sizes. For the known channel case, we derive a coherent GLRT packet-data detector while for the unknown channel case we derive both a coherent pilot assisted GLRT packet-data detector and a differential phase-shift-keying (DPSK) GLRT packet-data detector. Efficient suboptimum implementations of the above schemes that exhibit complexity linear in the packet size are also considered. Simulation studies evaluate the performance of the proposed schemes in the context of packet-data code-division multiple access (CDMA) communications.

Motivated by the necessity of having a good clock synchronization amongst the nodes of wireless ad-hoc sensor networks, the joint maximum likelihood (JML) estimator for clock phase offset and skew under exponential noise model for reference broadcast synchronization (RBS) protocol is formulated and found via a direct algorithm. The Gibbs sampler is also proposed for joint clock phase offset and skew estimation and shown to provide superior performance relative to JML- estimator. Lower and upper bounds for the mean-square errors (MSE) of JML-estimator and Gibbs Sampler are introduced in terms of the MSE of the uniform minimum variance unbiased (UMVU) estimator and the conventional best linear unbiased estimator (BLUE), respectively.

When data symbols modulate a signature waveform to move across a channel in the presence of disturbance, the signature that maximizes the signal-to-interference-plus-noise ratio (SINR) at the output of the maximum-SINR filter is the smallest-eigenvalue eigenvector of the disturbance autocovariance matrix. In digital communication systems the signature alphabet is finite and digital signature optimization is NP-hard. In this paper, we present a formal search procedure of cost, upon eigenvector decomposition, log-linear in the signature code length that returns the maximum-SINR binary signature vector near arcs of least SINR decrease from the real maximum SINR solution in the Euclidean vector space. The quality of the proposed adaptive binary designs is measured against the theoretical upper bound of the complex/real eigenvector maximizer.

Recently, a few efficient timing synchronization protocols for wireless sensor networks (WSNs) have been proposed with the goal of maximizing the accuracy and minimizing the power utilization. This paper proposes novel clock skew estimators assuming different delay environments to achieve energy-efficient network-wide synchronization for WSNs. The proposed clock skew correction mechanism significantly increases the re-synchronization period, which is a critical factor in reducing the overall power consumption. The proposed synchronization scheme can be applied to the conventional protocols without additional overheads. Moreover, this paper derives the Cramer-Rao lower bounds and the maximum likelihood estimators under different delay models and assumptions. These analytical metrics serves as good benchmarks for the thus far reported experimental results

We consider robust signal processing techniques for inference-centric distributed sensor networks operating in the presence of possible sensor and/or communication failures. Motivated by the multiple description (MD) principle, we develop robust distributed quantization schemes for a decentralized detection system. Specifically, focusing on a two-sensor system, our design criterion mirrors that of MD principle: if one of the two transmissions fails, we can guarantee an acceptable performance, while enhanced performance can be achieved if both transmissions are successful. Different from the conventional MD problem is the distributed nature of the problem as well as the use of error probability as the performance measure. Two different optimization criteria are used in the distributed quantizer design, the first a constrained optimization problem, and the second using an erasure channel model. We demonstrate that these two formulations are intrinsically related to each other. Further, using a person-by-person optimization approach, we propose an iterative algorithm to find the optimal local quantization thresholds. A design example is provided to illustrate the validity of the iterative algorithm and the improved robustness compared to the classical distributed detection approach that disregards the possible transmission losses.

This paper analyzes the effects of the level of cooperation (LOC) on the performance of a practical cooperative coding scheme for frequency division half-duplex relay links in general wireless ad-hoc networks. The end-to-end bit error probability (BEP) and the corresponding optimum LOG minimizing the end-to-end BEP are investigated for specific realizations using a family of rate compatible punctured convolutional (RCPC) codes. This paper shows that the BEP of the cooperative coding scheme depends on the LOC and there exists an optimum LOC for every specific realization. Further, it is shown that more than a certain level of cooperation is required to achieve possible cooperative gains in practical signal-to-noise ratio (SNR) regions, and increasing LOC does not effect much on the BEP performance for sufficiently large LOCs

We consider in this paper local sensor quantizer design for large-scale bandwidth and/or energy constrained wireless sensor networks (WSNs) operating in fading channels. In particular, under the Neyman-Pears on framework, we address the design of binary local sensor quantizers for a binary hypothesis problem in the asymptotic regime where the number of sensors is large. Motivated by the sensor censoring idea for reduced communication rate, each sensor either transmits `1' to a fusion center or remains silent. By adopting energy detector as the fusion rule, we develop a procedure to obtain local sensor threshold that maximizes the Kullback-Leibler distance of the distributions of the fusion statistic under the two hypotheses. The proposed quantizer design is well suited for the emerging large scale resource-constrained WSNs applications. Numerical results based on Gaussian and exponential observations are presented to demonstrate the design procedure

Recently, a few protocols for synchronizing the nodes of wireless sensor networks (WSNs) to a common time frame have been proposed with the goal of maximizing the accuracy and minimizing the power utilization. Thus far, the performance of the existing protocols for time synchronization is assessed only through computer simulations and experiments without using any rigorous analytical metrics. The goal of this paper is to fill up this gap by deriving the Cramer-Rao lower bound (CRLB) for the clock offset in one of the most popular synchronization algorithms, namely timing-sync protocol for sensor networks (TPSN), assuming commonly used exponential and Gaussian delay models, respectively. Furthermore, this paper proposes novel and practical clock skew estimators requiring no prior information of the fixed portion of delays, which makes the TPSN algorithm very suitable to synchronization in light of its power efficiency constraint.

When data symbols modulate a signature waveform/pattern to move across a channel in the presence of disturbance, as in common spread-spectrum (SS) or code-division multiple-access (CDMA) or multiuser ultra-wideband (UWB) communication systems, the signature/spreading code that maximizes the signal-to- interference-plus-noise ratio (SINR) at the output of the maximum SINR filter is the smallest-eigenvalue eigenvector of the disturbance autocovariance matrix. In digital communication systems, however, the signature alphabet is finite and digital signature optimization is NP-hard. In this paper, we present a formal search procedure of cost linear in the signature length that returns the maximum-SINR binary signature near lines of least SINR decrease in the Euclidean vector space. The quality of the proposed adaptive binary design is compared against the theoretical upper bound of the complex/real eigenvector maximizer and other known static and adaptive binary signature designs.

In this paper we propose novel generalized likelihood ratio test (GLRT) packet-data detectors for general multiaccess/multiuser digital communication systems and we develop analytical performance evaluation tools for finite data packet sizes. We evaluate the performance of the proposed GLRT schemes in the context of packet-data CDMA communications.

The architecture and circuit design are the two most effective means of reducing power in CMOS VLSI. Mathematical manipulations, based on applying ideas from multirate signal processing have been applied to create high performance, low power architectures. To illustrate this approach, two case studies are presented-one concerns the design of a fast Fourier transform (FFT) device, while the other one is concerned with the design of analog-to-digital converters.

We study why prefiltering is necessary for discrete multiwavelet transforms. We propose a method to generate prefilters given a set of multiwavelet filterbanks, where they are called good prefilters.

We first prove that the vector Karhunen-Loeve (VKL) transform for any finite many vector-valued signal, x/sub 1/,/spl middot//spl middot//spl middot/,x/sub L/, exists. The VKL transform is equivalent to the scalar KL transform for the scalar-valued signal X=(x/sub 1//sup T/,/spl middot//spl middot//spl middot/,x/sub L//sup T/)/sup T/. Based on VKL transforms, we provide a necessary and sufficient condition for the existence of the optimal vector transforms (unitary). With the condition, one can see that the optimal unitary vector transforms do not exist in most cases, and therefore needs to use suboptimal unitary vector transforms. We then prove that the optimal nonunitary vector transform for x/sub 1/,/spl middot//spl middot//spl middot/,x/sub L/ exists when all eigenvalues of the correlation matrix of the signal X are nonzero. We formulate the optimal vector transforms via the VKL transforms.>

This paper presents a modification of the alias-free generalized discrete time-frequency distribution (AF-GDTFD) which reduces the computations necessary to implement the algorithm from O(N/sup 3/) to O(N/sup 2/log N). With this new implementation, we examine the problem of designing/modifying kernels defined in the ambiguity plane for use with the AF-GDTFD. We present three different methods to allow the use of these kernels and demonstrate them using the Butterworth kernel.>

The theory of lapped transforms is extended to include asymmetrical overlaps between adjacent intervals. Thus, this result will permit lapped transforms to be used between two intervals of greatly differing size.>

Presents a systematic method to construct two dimensional spatial-varying filter banks that preserve the perfect reconstruction property. These filter banks are constructed from arbitrary two dimensional (separable or nonseparable) local perfect reconstruction FIR filter banks on different rectangular regions, where overlaps between neighboring regions are used to eliminate the blocking effects. A numerical example is given to illustrate the theory.>

In this paper, the interrelation between the Zak transform and the Generalized Discrete Time-Frequency Distribution (GDTFD) is examined. Starting with the discrete Zak transform, its definition is broadened to include arbitrary windows creating the Windowed Zak (WZ) transform. The WZ is then combined with the spectrogram. It is demonstrated that the spectrogram based upon the WZ, called the Zak-Spectrogram, is a generalization of the standard spectrogram. Next, building upon the idea of the weighted spectrogram, the weighted Zak-Spectrogram is used to produce a new class of GDTFD, called the Decimated GDTFD (DGDTFD). The Decimated GDTFD is similar to the GDTFD except it trades bandwidth for computational speed.>

The major goal of this research was to investigate speech coding techniques in an attempt to achieve high quality speech transmittable at 4800 bits per second. The approach taken to achieve this goal was to code the frequency domain representation of speech. Speech was represented by a sparse set of frequency components. Four frequency selections schemes were implemented, and the resulting frequency coefficients (magnitude and phase) were coded in an efficient manner for transmission. Specific techniques involved in the speech coder included: (1) a recurrent neural architecture to make a periodic/noiselike decision, (2) the use of variable length windows for analysis and synthesis, and (3) a representation of noiselike speech using frequency banded energy information. The quality of the reconstructed speech was tested using listening tests which compared the different frequency selection schemes, along with original and sampled speech. The system did not achieve 'toll quality' speech; however, the resulting speech was highly intelligible. Specific quality degradation was noted at window transitions.

A new iterative maximum picking neural net (MAXnet) is presented. This formulation determines the value and the location either for a unique maximum or for multiple maxima. This new net converges, for many commonly occurring distributions, in O(log M) iterations using only simple computing elements.

The multilayer perceptron, when trained as a classifier using backpropagation, is shown to approximate the Bayes optimal discriminant function. The result is demonstrated for both the two-class problem and multiple classes. It is shown that the outputs of the multilayer perceptron approximate the a posteriori probability functions of the classes being trained. The proof applies to any number of layers and any type of unit activation function, linear or nonlinear.

A new Hankel transform algorithm designed for uniformly sampled data is presented. Although data of this type occur frequently, previous algorithms require interpolations and/or numerical evaluations of Bessel functions. These difficulties can be avoided by using a Tchebycheff transform followed by a Fourier transform. The basic structure and performance of any Hankel transform algorithm derived from this two-step process depends on the combined results from the numerical methods used to compute the Tchebycheff and Fourier transforms. The algorithm presented here is the most elementary of several algorithms derived from this procedure. Examples are presented and errors associated with the results are discussed.

Some fundamental results in the area of computability theory are presented. These include the fact that a finite state machine can find the residue of an arbitrarily long sequence using any modulus and any radix. This leads to the consideration of using modular arithmetic on arbitrarily long sequences with a finite state machine. A finite state machine can perform modular addition, subtraction, multiplication, and, if defined, division of a pair of arbitrarily large numbers, using any modulus and any radix.