Generalized Sparse Covariance-based Estimation

In this work, we extend the sparse iterative covariance-based estimator (SPICE), by generalizing the formulation to allow for different norm constraints on the signal and noise parameters in the covariance model. For a given norm, the resulting extended SPICE method enjoys the same benefits as the regular SPICE method, including being hyper-parameter free, although the choice of norms are shown to govern the sparsity in the resulting solution. Furthermore, we show that solving the extended SPICE method is equivalent to solving a penalized regression problem, which provides an alternative interpretation of the proposed method and a deeper insight on the differences in sparsity between the extended and the original SPICE formulation. We examine the performance of the method for different choices of norms, and compare the results to the original SPICE method, showing the benefits of using the extended formulation. We also provide two ways of solving the extended SPICE method; one grid-based method, for which an efficient implementation is given, and a gridless method for the sinusoidal case, which results in a semi-definite programming problem

High resolution sparse estimation of exponentially decaying two-dimensional signals

In this work, we consider the problem of high-resolution estimation of the parameters detailing a two-dimensional (2-D) signal consisting of an unknown number of exponentially decaying sinusoidal components. Interpreting the estimation problem as a block (or group) sparse representation problem allows the decoupling of the 2-D data structure into a sum of outer-products of 1-D damped sinusoidal signals with unknown damping and frequency. The resulting non-zero blocks will represent each of the 1-D damped sinusoids, which may then be used as non-parametric estimates of the corresponding 1-D signals; this implies that the sought 2-D modes may be estimated using a sequence of 1-D optimization problems. The resulting sparse representation problem is solved using an iterative ADMM-based algorithm, after which the damping and frequency parameter can be estimated by a sequence of simple 1-D optimization problems

Computationally Efficient Estimation of Multi-Dimensional Spectral Lines

In this work, we propose a computationally efficient algorithm for estimating multi-dimensional spectral lines. The method treats the data tensor's dimensions separately, yielding the corresponding frequency estimates for each dimension. Then, in a second step, the estimates are ordered over dimensions, thus forming the resulting multidimensional parameter estimates. For high dimensional data, the proposed method offers statistically efficient estimates for moderate to high signal to noise ratios, at a computational cost substantially lower than typical non-parametric Fourier-transform based periodogram solutions, as well as to state-of-the-art parametric estimators

Joint DOA and Multi-Pitch Estimation Using Block Sparsity

In this paper, we propose a novel method to estimate the fundamental frequencies and directions-of-arrival (DOA) of multi-pitch signals impinging on a sensor array. Formulating the estimation as a group sparse convex optimization problem, we use the alternating direction of multipliers method (ADMM) to estimate both temporal and spatial correlation of the array signal. By first jointly estimating both fundamental frequencies and time-of-arrivals (TOAs) for each sensor and sound source, we then form a non-linear least squares estimate to obtain the DOAs. Numerical simulations indcate the preferable performance of the proposed estimator as compared to current state-of-the-art methods

Multi-Pitch Estimation Exploiting Block Sparsity

We study the problem of estimating the fundamental frequencies of a signal containing multiple harmonically related sinusoidal components using a novel block sparse signal representation. An efficient algorithm for solving the resulting optimization problem is devised exploiting a novel variable step-size alternating direction method of multipliers (ADMM). The resulting algorithm has guaranteed convergence and shows notable robustness to the f 0 vs f0/2f0/2 ambiguity problem. The superiority of the proposed method, as compared to earlier presented estimation techniques, is demonstrated using both simulated and measured audio signals, clearly indicating the preferable performance of the proposed technique

An Adaptive Penalty Approach to Multi-Pitch Estimation

This work treats multi-pitch estimation, and in particular the common misclassification issue wherein the pitch at half of the true fundamental frequency, here referred to as a sub-octave, is chosen instead of the true pitch. Extending on current methods which use an extension of the Group LASSO for pitch estimation, this work introduces an adaptive total variation penalty, which both enforce group- and block sparsity, and deal with errors due to sub-octaves. The method is shown to outperform current state-of-the-art sparse methods, where the model orders are unknown, while also requiring fewer tuning parameters than these. The method is also shown to outperform several conventional pitch estimation methods, even when these are virtued with oracle model orders

Sparse Chroma Estimation for Harmonic Audio

This work treats the estimation of the chromagram for harmonic audio signals using a block sparse reconstruction framework. Chroma has been used for decades as a key tool in audio analysis, and is typically formed using a Fourier-based framework that maps the fundamental frequency of a musical tone to its corresponding chroma. Such an approach often leads to problems with tone ambiguity, which we avoid by taking into account the harmonic structure and perceptional attributes in music. The performance of the proposed method is evaluated using real audio files, clearly showing preferable performance as compared to other commonly used methods

Sparse Modeling Heuristics for Parameter Estimation - Applications in Statistical Signal Processing

This thesis examines sparse statistical modeling on a range of applications in audio modeling, audio localizations, DNA sequencing, and spectroscopy. In the examined cases, the resulting estimation problems are computationally cumbersome, both as one often suffers from a lack of model order knowledge for this form of problems, but also due to the high dimensionality of the parameter spaces, which typically also yield optimization problems with numerous local minima. In this thesis, these problems are treated using sparse modeling heuristics, with the resulting criteria being solved using convex relaxations, inspired from disciplined convex programming ideas, to maintain tractability. The contributions to audio modeling and estimation focus on the estimation of the fundamental frequency of harmonically related sinusoidal signals, which is commonly used model for, e.g., voiced speech or tonal audio. We examine both the problems of estimating multiple audio sources assuming the expected harmonic structure, as well as the problem of robustness to the often occurring inharmonic structure, such that the higher order sinusoidal components deviate in an unknown way from the expected multiples of the fundamental frequency. This is a problem commonly occurring for, for instance, string instruments, which, if not properly accounted for, will degrade the performance of most pitch estimators noticeably. We also consider the problem of localizing audio sources in an unknown and possibly reverberant acoustic environment, allowing for simultaneous localization of far-field and near-field signals. The DNA sequencing contribution, presented in the more general setting of arbitrary categorical sequences, is inspired by the problem of identifying segments in the genome, which are characterized by the highly periodic behavior of the sequence. In each of the contributions, an appropriate computationally efficient algorithm is proposed. Specifically for the sparse models, alternating directions method of multipliers and cyclic coordinate descent implementations are suggested, since the proposed convex criteria are in practice easier to solve than the standard interior point solvers would suggest. The suggested methods are in all cases compared with previously proposed algorithms and/or measured data, as appropriate

Sparse Estimation Of Spectroscopic Signals

This work considers the semi-parametric estimation of sparse spec- troscopic signals, aiming to form a detailed spectral representation of both the frequency content and the spectral line widths of the oc- curring signals. Extending on the recent FOCUSS-based SLIM al- gorithm, we propose an alternative prior for a Bayesian formulation of this sparse reconstruction method, exploiting a proposed suitable prior for the noise variance. Examining three common models for spectroscopic signals, the introduced technique allows for reliable estimation of the characteristics of these models. Numerical sim- ulations illustrate the improved performance of the proposed tech- nique