Signal Reconstruction From Lower Dimensional Measurements

This is the era of information-explosion, billions of data are produced, collected and then stored in our daily life. The manners of collecting the data sets are various but always following the criteria-the less data while the more information. Thus the most favorite way is to directly measure the information, which, commonly, resides in a lower dimensional space than its carrier, namely, the data (signals or states). This method is thus called information measuring, and conceptually can be concluded in a framework with the following three steps:(1) modeling, to condense the information relevant to signals to a small subspace;(2) measuring, to preserve the information in lower dimensional measurement space; and (3) restoring, to re-construct signals from the lower dimensional measurements. From this vein, the main contributions of this thesis, saying observer and model based Bayesian com-pressive sensing can be well unified in the framework of information measuring:the main concerned problems of both applications can be decomposed into the above three aspects. In the first part, the problem is resided in the domain of control systems where the objective of observer design is located in the observability to determine whether the system states are recoverable and observation of the system states from the lower dimensional measurements (commonly but not restrictively). Specifically, a class of switched systems is considered with high switching frequency, or even with Zeno phenomenon, where the transitions of the discrete state are too high to be captured. However, the averaged value obtained through filtering the transitions can be easily sensed as the partial knowledge. Consequently, only with this partial knowledge, the observability for the switched systems is discussed respectively from differential geometric approach and algebraic approach, and the corresponding ob-servers are designed as well. In the second part, the topic is switched to compressive sensing which is objected to sampling the sparse signals directly in a compressed manner, where the central fundamentals are resided in signal modeling according to available priors, constructing sensing matrix satisfying the so-called restricted isometry property and restoring the original sparse signals using sparse regularized linear inversion algorithms. Respectively, considering the properties of CS related to modeling, measuring and restoring, we propose to (1) exploit the chaotic se- quences to construct the sensing matrix (or measuring operator) which is called chaotic sensing matrix,(2) further consider the sparsity model and then rebuild the signal model to consider structures underlying the sparsity patterns, and (3) propose three non-parametric algorithms for structured sparse signals through the hierarchical Bayesian method. And the experimental results prove that the chaotic sensing matrix is with the similar property to sub-Gaussian random matrix and the additional consideration on structures underlying sparsity patterns largely improves the performances of reconstruction and robustness.