Likelihood-informed dimension reduction for nonlinear inverse problems

The intrinsic dimensionality of an inverse problem is affected by prior information, the accuracy and number of observations, and the smoothing properties of the forward operator. From a Bayesian perspective, changes from the prior to the posterior may, in many problems, be confined to a relatively low-dimensional subspace of the parameter space. We present a dimension reduction approach that defines and identifies such a subspace, called the "likelihood-informed subspace" (LIS), by characterizing the relative influences of the prior and the likelihood over the support of the posterior distribution. This identification enables new and more efficient computational methods for Bayesian inference with nonlinear forward models and Gaussian priors. In particular, we approximate the posterior distribution as the product of a lower-dimensional posterior defined on the LIS and the prior distribution marginalized onto the complementary subspace. Markov chain Monte Carlo sampling can then proceed in lower dimensions, with significant gains in computational efficiency. We also introduce a Rao-Blackwellization strategy that de-randomizes Monte Carlo estimates of posterior expectations for additional variance reduction. We demonstrate the efficiency of our methods using two numerical examples: inference of permeability in a groundwater system governed by an elliptic PDE, and an atmospheric remote sensing problem based on Global Ozone Monitoring System (GOMOS) observations

Mathematics behind a Class of Image Restoration Algorithms

The restoration techniques are usually oriented toward modeling the type of degradation in order to infer the inverse process for recovering the given image. This approach usually involves the option for a criterion to numerically evaluate the quality of the resulted image and consequently the restoration process can be expressed in terms of an optimization problem. Most of the approaches are essentially based on additional hypothesis concerning the statistical properties of images. However, in real life applications, there is no enough information to support a certain particular image model, and consequently model-free developments have to be used instead. In our approaches the problem of image denoising/restoration is viewed as an information transmission/processing system, where the signal representing a certain clean image is transmitted through a noisy channel and only a noise-corrupted version is available. The aim is to recover the available signal as much as possible by using different noise removal techniques that is to build an accurate approximation of the initial image. Unfortunately, a series of image qualities, as for instance clarity, brightness, contrast, are affected by the noise removal techniques and consequently there is a need to partially restore them on the basis of information extracted exclusively from data. Following a brief description of the image restoration framework provided in the introductory part, a PCA-based methodology is presented in the second section of the paper. The basics of a new informational-based development for image restoration purposes and scatter matrix-based methods are given in the next two sections. The final section contains concluding remarks and suggestions for further work

Automated Target Acquisition, Recognition and Tracking (ATTRACT)

The primary objective of phase 1 of this research project is to conduct multidisciplinary research that will contribute to fundamental scientific knowledge in several of the USAF critical technology areas. Specifically, neural networks, signal processing techniques, and electro-optic capabilities are utilized to solve problems associated with automated target acquisition, recognition, and tracking. To accomplish the stated objective, several tasks have been identified and were executed

Edge and Line Feature Extraction Based on Covariance Models

age segmentation based on contour extraction usually involves three stages of image operations: feature extraction, edge detection and edge linking. This paper is devoted to the first stage: a method to design feature extractors used to detect edges from noisy and/or blurred images. The method relies on a model that describes the existence of image discontinuities (e.g. edges) in terms of covariance functions. The feature extractor transforms the input image into a “log-likelihood ratio” image. Such an image is a good starting point of the edge detection stage since it represents a balanced trade-off between signal-to-noise ratio and the ability to resolve detailed structures. For 1-D signals, the performance of the edge detector based on this feature extractor is quantitatively assessed by the so called “average risk measure”. The results are compared with the performances of 1-D edge detectors known from literature. Generalizations to 2-D operators are given. Applications on real world images are presented showing the capability of the covariance model to build edge and line feature extractors. Finally it is shown that the covariance model can be coupled to a MRF-model of edge configurations so as to arrive at a maximum a posteriori estimate of the edges or lines in the image