AMA Statistical Information Based Analysis of a Compressive Imaging System

Recent advances in optics and instrumentation have dramatically increased the amount of data, both spatial and spectral, that can be obtained about a target scene. The volume of the acquired data can and, in fact, often does far exceed the amount of intrinsic information present in the scene. In such cases, the large volume of data alone can impede the analysis and extraction of relevant information about the scene. One approach to overcoming this impedance mismatch between the volume of data and intrinsic information in the scene the data are supposed to convey is compressive sensing. Compressive sensing exploits the fact that most signals of interest, such as image scenes, possess natural correlations in their physical structure. These correlations, which can occur spatially as well as spectrally, can suggest a more natural sparse basis for compressing and representing the scene than standard pixels or voxels. A compressive sensing system attempts to acquire and encode the scene in this sparse basis, while preserving all relevant information in the scene. One criterion for assessing the content, acquisition, and processing of information in the image scene is Shannon information. This metric describes fundamental limits on encoding and reliably transmitting information about a source, such as an image scene. In this framework, successful encoding of the image requires an optimal choice of a sparse basis, while losses of information during transmission occur due to a finite system response and measurement noise. An information source can be represented by a certain class of image scenes, . e.g. those that have a common morphology. The ability to associate the recorded image with the correct member of the class that produced the image depends on the amount of Shannon information in the acquired data. In this manner, one can analyze the performance of a compressive imaging system for a specific class or ensemble of image scenes. We present such an information-based analysis of a compressive imaging system based on a new highly efficient and robust method that enables us to evaluate statistical entropies. Our method is based on the notion of density of states (DOS), which plays a major role in statistical mechanics by allowing one to express macroscopic thermal averages in terms of the number of configuration states of a system for a certain energy level. Instead of computing the number of microstates associated with a macroscopic energy of the system, however, we compute here the number of possible configurations (states) in the space of variables characteristic of an image scene and its observations that correspond to a certain probability value. This allows us to compute the statistical entropy of many correlated variables as an essentially one-dimensional (1D) probability integral, as we shall see presently. We assess the performance of a single pixel compressive sensing (CS) system based on the amount of information encoded and transmitted in parameters that characterize the information in the scene. Specifically, we shall study two applications of the CS approach, namely the problem of faint companion detection and the problem of satellite material disambiguation. Here, we compute the amount of statistical information the single-pixel data convey about the essential parameters of the scene, as a function of the choice of the projective measurement basis and the amount of measurement noise. The noise creates confusion when associating the recorded data with the correct member of the ensemble that produced the image. We show that multiple measurements enable one to mitigate this confusion noise.