Multiscale, statistical anomaly detection analysis and algorithms for linearized inverse scattering problems

In this paper we explore the utility of multiscale and statistical techniques for detecting and characterizing the structure of localized anomalies in a medium based upon observations of scattered energy obtained at the boundaries of the region of interest. Wavelet transform techniques are used to provide an efficient and physically meaningful method for modeling the non-anomalous structure of the medium under investigation. We employ decision-theoretic methods both to analyze a variety of difficulties associated with the anomaly detection problem and as the basis for an algorithm to perform anomaly detection and estimation. These methods allow for a quantitative evaluation of the manner in which the performance of the algorithms is impacted by the amplitudes, spatial sizes, and positions of anomalous areas in the overall region of interest. Given the insight provided by this work, we formulate and analyze an algorithm for determining the number, location, and magnitudes associated with a set of anomaly structures. This approach is based upon the use of a Generalized, M-ary Likelihood Ratio Test to successively subdivide the region as a means of localizing anomalous areas in both space and scale. Examples of our multiscale inversion algorithm are presented using the Born approximation of an electrical conductivity problem formulated so as to illustrate many of the features associated with similar detection problems arising in fields such as geophysical prospecting, ultrasonic imaging, and medical imaging.