Bayesian inference for inverse problems

Traditionally, the MaxEnt workshops start by a tutorial day. This paper summarizes my talk during 2001'th workshop at John Hopkins University. The main idea in this talk is to show how the Bayesian inference can naturally give us all the necessary tools we need to solve real inverse problems: starting by simple inversion where we assume to know exactly the forward model and all the input model parameters up to more realistic advanced problems of myopic or blind inversion where we may be uncertain about the forward model and we may have noisy data. Starting by an introduction to inverse problems through a few examples and explaining their ill posedness nature, I briefly presented the main classical deterministic methods such as data matching and classical regularization methods to show their limitations. I then presented the main classical probabilistic methods based on likelihood, information theory and maximum entropy and the Bayesian inference framework for such problems. I show that the Bayesian framework, not only generalizes all these methods, but also gives us natural tools, for example, for inferring the uncertainty of the computed solutions, for the estimation of the hyperparameters or for handling myopic or blind inversion problems. Finally, through a deconvolution problem example, I presented a few state of the art methods based on Bayesian inference particularly designed for some of the mass spectrometry data processing problems.Comment: Presented at MaxEnt01. To appear in Bayesian Inference and Maximum Entropy Methods, B. Fry (Ed.), AIP Proceedings. 20pages, 13 Postscript figure

Near real-time stereo vision system

The apparatus for a near real-time stereo vision system for use with a robotic vehicle is described. The system is comprised of two cameras mounted on three-axis rotation platforms, image-processing boards, a CPU, and specialized stereo vision algorithms. Bandpass-filtered image pyramids are computed, stereo matching is performed by least-squares correlation, and confidence ranges are estimated by means of Bayes' theorem. In particular, Laplacian image pyramids are built and disparity maps are produced from the 60 x 64 level of the pyramids at rates of up to 2 seconds per image pair. The first autonomous cross-country robotic traverses (of up to 100 meters) have been achieved using the stereo vision system of the present invention with all computing done onboard the vehicle. The overall approach disclosed herein provides a unifying paradigm for practical domain-independent stereo ranging

Likelihood-Ratio-Based Biometric Verification

The paper presents results on optimal similarity measures for biometric verification based on fixed-length feature vectors. First, we show that the verification of a single user is equivalent to the detection problem, which implies that, for single-user verification, the likelihood ratio is optimal. Second, we show that, under some general conditions, decisions based on posterior probabilities and likelihood ratios are equivalent and result in the same receiver operating curve. However, in a multi-user situation, these two methods lead to different average error rates. As a third result, we prove theoretically that, for multi-user verification, the use of the likelihood ratio is optimal in terms of average error rates. The superiority of this method is illustrated by experiments in fingerprint verification. It is shown that error rates below 10/sup -3/ can be achieved when using multiple fingerprints for template construction

Bayesian data assimilation in shape registration

In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions\ud for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterisation in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum p0 and the reparameterisation vector field v, informed by regularity results about the forward model. Having done this, we illustrate how Maximum Likelihood Estimators (MLEs) can be used to find regions of high posterior density, but also how we can apply recently developed MCMC methods on function spaces to characterise the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem