A sparse regularization method for direction finding based on sparse estimation (SPARSE) is presented. It accounts for sensor position errors and general phase/gain mismatches in the generating model of the dictionary. Also, other model errors can be incorporated in this framework if their mutual independence can be assumed. Since the dictionary is derived from an ideal model, the regularization parameter of the sparse estimation problem has to impose a suitable tolerance level to handle the joint impacts of model errors and noise. This parameter is estimated as an upper bound of the MSE between the perturbed model, including errors such as phase and gain mismatches, and the ideal model. Moreover, it is directly connected to certain model errors, such that available prior knowledge of the system can be used to achieve selective regularization of specific errors. Hence, the regularization parameter can be adapted to the current situation in an automated fashion. Also, major parts of its derivation can be done offline and stored in a database.
The SPARSE method with proper regularization shows better robustness and higher resolution in the presence of sensor position errors and noise than the compared robust methods. However, its robustness to model errors is reduced when the source sig- nals are correlated, especially at low SNRs. A drawback of this regularization method is the need for estimating the source power, σ2
s, the noise power, σ2n, and the num-
ber of sources, K. Also, the regularization parameter depends on the confidence lev- els, αg, αφ, αX, and ˜αX, which are empirically determined. As a remedy, a combined
4.9 Conclusion 57
dictionary calibration and sparse estimation method, R-SPARSE, is presented. The calibrated dictionary takes the joint impacts of model errors and noise into account and provides stable and reliable sparse estimation results. Regularization alone can only adjust the tolerance level but the accuracy is limited by an erroneous dictionary. Therefore, dictionary calibration also improves the accuracy. Although regularization is still required for `1-minimization, the problem of choosing the regularization param-
eter (i.e. the confidence levels) is significantly alleviated. In particular, the sparse solution obtained by R-SPARSE is only marginally affected even by large variations of the confidence level and high success rates achieved for a wide range of SNRs. How- ever, the calibration performance depends on how good the covariance matrix and the number of sources can be estimated. Therefore, dictionary calibration is applicable when many snapshots and knowledge of the number of sources are available. It turns out that the joint impacts of correlation and model errors degrade the performance of both SPARSE and R-SPARSE. However, since R-SPARSE mitigates some model errors by calibration prior to sparse estimation, it is more robust to these effects. In a future research, a systematic approach to choosing to confidence levels for regular- ization can be considered. In addition, other types of model errors and off-grid sources can be taken into account. Also, uncertain dictionary parameters such a the speed of propagation or the signal frequency can be estimated. Related dictionary learning techniques are presented in the ensuing chapters.
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Chapter 5
Alternating Sparse Estimation and
Dictionary Learning
This chapter introduces a sparse estimation and dictionary learning framework based on alternating estimation (AE). While a sparse representation in the previous chapters is obtained based on a variant of `1-minimization, it is subsequently assumed that the
employed dictionary is highly redundant, such that the necessary RIP conditions are no longer fulfilled. Therefore, a dictionary pre-processing routine is applied to yield a modified dictionary with improved RIP conditions. Using the modified dictionary, a sparse representation of the signal is estimated using a greedy OMP-based algorithm. In Chapter 4, uncertainty in the dictionary is represented by a general error term, ∆D. The true dictionary is given by D + ∆D, where D is the dictionary obtained under ideal conditions. Different from this approach, this chapter considers uncertainty in terms of imperfectly known global and local dictionary parameters, θ, that represent particular global and local characteristics of the system. Hence, the dictionary can be written as a function of these parameters, i.e. D(θ). It is found in Chapter 4, that simply adjusting the tolerance level to account for uncertainty by regularization can improve robustness to model errors but it cannot improve the estimation performance. Therefore, in this chapter, AE-based estimation is used to iteratively estimate the dictionary parameters along with a sparse representation of the signal. While dictionary calibration in Chapter 4 is a static operation that is performed only once, the pre- processing routine in this chapter is applied in each AE iteration and the estimate of θ is successively improved. The generic sparse estimation and dictionary learning framework is specified for the application of fiber-optic sensing in combination with compressed sensing (CS) according to Chapter 3.2 and Chapter 3.3, respectively.
The material presented in this chapter is partly taken from [87]1, [88]2, [30]3 [131]4.
1C. Weiss and A. M. Zoubir, “Fiber Sensing Using UFWT-Lasers and Sparse Acquisition,” in Proc.
of the 21st European Signal Processing Conference (EUSIPCO), September, 2013.
2C. Weiss and A. M. Zoubir, “Fiber Sensing Using Wavelength-Swept Lasers: A Compressed
Sampling Approach,” in Proc. of the 3rd International Workshop on Compressed Sensing Theory and its Applications to Radar, Sonar and Remote Sensing (CoSeRa), June, 2015.
3C. Weiss and A. M. Zoubir, “A Compressed Sampling and Dictionary Learning Framework for
Wavelength-Division-Multiplexing-Based Distributed Fiber Sensing,” accepted for publication in Jour- nal of the Optical Society of America A, 2017 (assigned issue: vol. 34, no. 5).
4C. Weiss and A. M. Zoubir, “Dictionary Learning Strategies for Compressed Fiber Sensing Using
Chapter Outline
Section 5.1 gives an introduction and motivation and Section 5.2 provides an overview of state-of-the-art techniques and related work. In Section 5.3, the problem statement is introduced. Section 5.4 provides theoretical limits for sparse estimation and dictio- nary learning in terms of the Cram´er-Rao bound. In Section 5.5, a unified framework for compressed fiber sensing and dictionary learning is introduced. Statements about the computational complexity of the presented estimation algorithms are made in Sec- tion 5.6. Section 5.7 evaluates the applicability of the presented framework and the performance of the estimation algorithms in various scenarios. A discussion of the results and findings is provided in Section 5.8. Section 5.9 gives a conclusion for this chapter.