Unveiling The Tree: A Convex Framework for Sparse Problems

This paper presents a general framework for generating greedy algorithms for solving convex constraint satisfaction problems for sparse solutions by mapping the satisfaction problem into one of graph traversal on a rooted tree of unknown topology. For every pre-walk of the tree an initial set of generally dense feasible solutions is processed in such a way that the sparsity of each solution increases with each generation unveiled. The specific computation performed at any particular child node is shown to correspond to an embedding of a polytope into the polytope received from that nodes parent. Several issues related to pre-walk order selection, computational complexity and tractability, and the use of heuristic and/or side information is discussed. An example of a single-path, depth-first algorithm on a tree with randomized vertex reduction and a run-time path selection algorithm is presented in the context of sparse lowpass filter design

On the eigenfilter design method and its applications: a tutorial

The eigenfilter method for digital filter design involves the computation of filter coefficients as the eigenvector of an appropriate Hermitian matrix. Because of its low complexity as compared to other methods as well as its ability to incorporate various time and frequency-domain constraints easily, the eigenfilter method has been found to be very useful. In this paper, we present a review of the eigenfilter design method for a wide variety of filters, including linear-phase finite impulse response (FIR) filters, nonlinear-phase FIR filters, all-pass infinite impulse response (IIR) filters, arbitrary response IIR filters, and multidimensional filters. Also, we focus on applications of the eigenfilter method in multistage filter design, spectral/spacial beamforming, and in the design of channel-shortening equalizers for communications applications

LMMSE Estimation and Interpolation of Continuous-Time Signals from Discrete-Time Samples Using Factor Graphs

The factor graph approach to discrete-time linear Gaussian state space models is well developed. The paper extends this approach to continuous-time linear systems/filters that are driven by white Gaussian noise. By Gaussian message passing, we then obtain MAP/MMSE/LMMSE estimates of the input signal, or of the state, or of the output signal from noisy observations of the output signal. These estimates may be obtained with arbitrary temporal resolution. The proposed input signal estimation does not seem to have appeared in the prior Kalman filtering literature

Implementation of the Trigonometric LMS Algorithm using Original Cordic Rotation

The LMS algorithm is one of the most successful adaptive filtering algorithms. It uses the instantaneous value of the square of the error signal as an estimate of the mean-square error (MSE). The LMS algorithm changes (adapts) the filter tap weights so that the error signal is minimized in the mean square sense. In Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS), two new versions of LMS algorithms, same formulations are performed as in the LMS algorithm with the exception that filter tap weights are now expressed using trigonometric and hyperbolic formulations, in cases for TLMS and HLMS respectively. Hence appears the CORDIC algorithm as it can efficiently perform trigonometric, hyperbolic, linear and logarithmic functions. While hardware-efficient algorithms often exist, the dominance of the software systems has kept those algorithms out of the spotlight. Among these hardware- efficient algorithms, CORDIC is an iterative solution for trigonometric and other transcendental functions. Former researches worked on CORDIC algorithm to observe the convergence behavior of Trigonometric LMS (TLMS) algorithm and obtained a satisfactory result in the context of convergence performance of TLMS algorithm. But revious researches directly used the CORDIC block output in their simulation ignoring the internal step-by-step rotations of the CORDIC processor. This gives rise to a need for verification of the convergence performance of the TLMS algorithm to investigate if it actually performs satisfactorily if implemented with step-by-step CORDIC rotation. This research work has done this job. It focuses on the internal operations of the CORDIC hardware, implements the Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS) algorithms using actual CORDIC rotations. The obtained simulation results are highly satisfactory and also it shows that convergence behavior of HLMS is much better than TLMS.Comment: 12 pages, 5 figures, 1 table. Published in IJCNC; http://airccse.org/journal/cnc/0710ijcnc08.pdf, http://airccse.org/journal/ijc2010.htm

Image interpolation using Shearlet based iterative refinement

This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three main steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through iterative thresholding, and (c) extracting high frequency information from the approximation to refine the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multiscale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.8 dB is observed over a dataset of 200 images

Adaptive multichannel control of time-varying broadband noise and vibrations

This paper presents results obtained from a number of applications in which a recent adaptive algorithm for broadband multichannel active noise control is used. The core of the algorithm uses the inverse of the minimum-phase part of the secondary path for improvement of the speed of convergence. A further improvement of the speed of convergence is obtained by using double control filters for elimination of adaptation loop delay. Regularization was found to be necessary for robust operation. The regularization technique which is used preserves the structure to eliminate the adaptation loop delay. Depending on the application at hand, a number of extensions are used for this algorithm. For an application with rapidly changing disturbance spectra, the core algorithm was extended with an iterative affine projection scheme, leading to improved convergence rates as compared to the standard nomalized lms update rules. In another application, in which the influence of the parametric uncertainties was critical, the core algorithm was extended with low authority control loops operating at high sample rates. In addition, results of other applications are given, such as control of acoustic energy density and control of time-varying periodic and non-periodic vibrations

A recursive scheme for computing autocorrelation functions of decimated complex wavelet subbands

This paper deals with the problem of the exact computation of the autocorrelation function of a real or complex discrete wavelet subband of a signal, when the autocorrelation function (or Power Spectral Density, PSD) of the signal in the time domain (or spatial domain) is either known or estimated using a separate technique. The solution to this problem allows us to couple time domain noise estimation techniques to wavelet domain denoising algorithms, which is crucial for the development of blind wavelet-based denoising techniques. Specifically, we investigate the Dual-Tree complex wavelet transform (DT-CWT), which has a good directional selectivity in 2-D and 3-D, is approximately shift-invariant, and yields better denoising results than a discrete wavelet transform (DWT). The proposed scheme gives an analytical relationship between the PSD of the input signal/image and the PSD of each individual real/complex wavelet subband which is very useful for future developments. We also show that a more general technique, that relies on Monte-Carlo simulations, requires a large number of input samples for a reliable estimate, while the proposed technique does not suffer from this problem

Exploiting Image Local And Nonlocal Consistency For Mixed Gaussian-Impulse Noise Removal

Most existing image denoising algorithms can only deal with a single type of noise, which violates the fact that the noisy observed images in practice are often suffered from more than one type of noise during the process of acquisition and transmission. In this paper, we propose a new variational algorithm for mixed Gaussian-impulse noise removal by exploiting image local consistency and nonlocal consistency simultaneously. Specifically, the local consistency is measured by a hyper-Laplace prior, enforcing the local smoothness of images, while the nonlocal consistency is measured by three-dimensional sparsity of similar blocks, enforcing the nonlocal self-similarity of natural images. Moreover, a Split-Bregman based technique is developed to solve the above optimization problem efficiently. Extensive experiments for mixed Gaussian plus impulse noise show that significant performance improvements over the current state-of-the-art schemes have been achieved, which substantiates the effectiveness of the proposed algorithm.Comment: 6 pages, 4 figures, 3 tables, to be published at IEEE Int. Conf. on Multimedia & Expo (ICME) 201