On Signal Recovery: Independent Component Analysis for Multiple Measurement Vectors and Source Separation
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In many applications such as natural sciences, medicine or imaging, some hidden features are to be extracted using some measurements. This arises inverse problem where the cause required to be obtained using the measurement. Two well-known inverse problems are multiple measurement vector (MMV) and blind source separation (BSS). In both topics, modeling the signals and the structure among the signals have significant effect on the recovery performance. Classical algorithms for the MMV problem assume either independent columns for the solution matrix or certain models of correlation among the columns. The correlation structure in the previous MMV formulation does not capture the signals well for some applications like remote photoplethysmography (rPPG) signal extraction where the signals are independent and linearly mixed in a certain manner. In practice, the mixtures of these signals are observed through different channels. Based on this structure, first, a uniqueness condition is derived and it is proved that this condition can be much less restrictive than the general MMV uniqueness condition. Then, two algorithms based on sparse Bayesian learning (SBL) are proposed. The corresponding features of their cost function are analyzed. Moreover two algorithms are proposed based on orthogonal matching pursuit (OMP) and basis pursuit (BP). Then, the exact recovery conditions for those algorithms are also provided. In the case that low number of measurements are available, using sparse recovery methods is necessary in rPPG estimation. However when a large number of samples are available, one can employ BSS approach to remove noise and motion artifacts and extract the rPPG signal as well. For this purpose, a BSS algorithm is introduced where the time-varying conditional variances of the signals are modeled. A Cramér Rao lower bound (CRLB) for the mixing matrix corresponding to the model is derived. The CRLB derived is more general than previous CRLBs as it considers time-varying conditional variance. This presented research contribution introduces a new class of MMV problem which has an impact on the signal recovery performance for the applications like rPPG estimation. The algorithms developed in this research may be used in other areas as well due to similarity between the structure complied with the applications such as image separation and direction of arrival estimation DOA of multiple sources.