Deformable template representations of observed imagery, model the variability of target pose via the actions of the matrix Lie groups on rigid templates. In this paper, we study the construction of minimum mean squared error estimators on the special orthogonal group, SO(n), for pose estimation. Due to the nonflat geometry of SO(n), the standard Bayesian formulation, of optimal estimators and their characteristics, requires modifications. By utilizing Hilbert-Schmidt metric defined on GL(n), a larger group containing SO(n), a mean squared criterion is defined on SO(n). The Hilbert-Schmidt estimate (HSE) is defined to be a minimum mean squared error estimator, restricted to SO(n). The expected error associated with the HSE is shown to be a lower bound, called the Hilbert-Schmidt bound (HSB), on the error incurred by any other estimator. Analysis and algorithms are presented for evaluating the HSE and the HSB in case of both ground-based and airborne targets.

Abstract — The emergent role of information theory in image formation is surveyed. Unlike the subject of information-theoretic communication theory, information-theoretic imaging is far from a mature subject. The possible role of information theory in problems of image formation is to provide a rigorous framework for defining the imaging problem, for defining measures of optimality used to form estimates of images, for addressing issues associated with the development of algorithms based on these optimality criteria, and for quantifying the quality of the approximations. The definition of the imaging problem consists of an appropriate model for the data and an appropriate model for the reproduction space, which is the space within which image estimates take values. Each problem statement has an associated optimality criterion that measures the overall quality of an estimate. The optimality criteria include maximizing the likelihood function and minimizing mean squared error for stochastic problems, and minimizing squared error and discrimination for deterministic problems. The development of algorithms is closely tied to the definition of the imaging problem and the associated optimality criterion. Algorithms with a strong information-theoretic motivation are obtained by the method of expectation maximization. Related alternating minimization algorithms are discussed. In quantifying the quality of approximations, global and local measures are discussed. Global measures include the (mean) squared error and discrimination between an estimate and the truth, and probability of error for recognition or hypothesis testing problems. Local measures include Fisher information. Index Terms—Image analysis, image formation, image processing, image reconstruction, image restoration, imaging, inverse problems, maximum-likelihood estimation, pattern recognition. I.

The variabilities in orientations and positions of rigid objects can be modeled by applying rotation and translation groups on their surface manifolds. Following the deformable template theory the rigid templates, given by two-dimensional surface descriptions, are rotated and translated to conform to individual objects in a particular scene. The fundamental group generating rigid motion is special Euclidean group SE(n), the semi-direct product of the special orthogonal group SO(n) and the translation group IR n . Under this model the scene representations take values in Cartesian products of the curved Lie group, SE(n). Given the observations of a scene obtained from a set of standard remote sensors, we generate the conditional mean estimates of transformation groups modeling that scene. Techniques, based on ergodic jumping stochastic gradient flows, are developed which accommodate the curved geometry of these groups. Algorithms and simulation results are presented in the context o...

We formulate a Bayesian approach to the joint tracking and recognition of airborne targets via reflected commercial television and FM radio signals measured by an array of sensors. Such passive systems may remain covert, whereas traditional active systems must reveal their presence and location by their transmissions. Since the number of aircraft in the scene is not known a priori, and targets may enter and leave the scene at unknown times, the parameter space is a union of subspaces of varying dimension as well as varying target classes. Targets tracks are parameterized via both positions and orientations, with the orientations naturally represented as elements of the special orthogonal group SO(3). A prior on target tracks is constructed from Newtonian equations of motion. This prior results in a coupling between the position and orientation estimates, yielding a coupling between the tracking and recognition problems. A likelihood function is formulated which incorporates the sensor...

Our work has focused on deformable template representations of geometric variability in automatic target recognition (ATR). Within this framework we have proposed the generation of conditional mean estimates of pose of ground-based targets remotely sensed via forward-looking infrared radar (FLIR) systems. Using the rotation group parameterization of the orientation space and a Bayesian estimation framework, conditional mean estimators are defined on the rotation group with minimum mean squared error (MMSE) performance bounds calculated following [1]. This paper focuses on the accommodation of thermodynamic variation. Our new approach relaxes assumptions of the target's underlying thermodynamic state, expanding thermodynamic state as a scalar field. Estimation within the deformable template setting poses geometric and thermodynamic variation as a joint inference. MMSE pose estimators for geometric variation are derived, demonstrating the "cost" of accommodating thermodynamic variability...

this paper was to provide a summary of the stateof -the-art theory on Bayesian model selection and the application of MCMC algorithms. It has been shown how applications of considerable complexity can be handled successfully within this framework. Several methods for dealing with the use of default, improper priors in the Bayesian model selection 506 Andrieu, Doucet et al. framework has been shown. Special care has been taken to pinpoint the subtleties of jumping from one parameter space to another, and in general, to show the construction of MCMC samplers in such scenarios. The focus in the paper was on the reversible jump MCMC algorithm as this is the most widely used of all existing methods; it is easy to use, flexible and has nice properties. Many references have been cited, with the emphasis being given to articles with signal processing applications. A Notation

We propose a multi-view stereo reconstruction algorithm which recovers urban scenes as a combination of meshes and geometric primitives. It provides a compact model while preserving details: irregular elements such as statues and ornaments are described by meshes whereas regular structures such as columns and walls are described by primitives (planes, spheres, cylinders, cones and tori). A Jump-Diffusion process is designed to sample these two types of elements simultaneously. The quality of a reconstruction is measured by a multi-object energy model which takes into account both photo-consistency and semantic considerations (i.e. geometry and shape layout). The sampler is embedded into an iterative refinement procedure which provides an increasingly accurate hybrid representation. Experimental results on complex urban structures and large scenes are presented and compared to multi-view based meshing algorithms.

We present an original multi-view stereo reconstruction algorithm which allows the 3D-modeling of urban scenes as a combination of meshes and geometric primitives. The method provides a compact model while preserving details: irregular elements such as statues and ornaments are described by meshes whereas regular structures such as columns and walls are described by primitives (planes, spheres, cylinders, cones and tori). We adopt a two-step strategy consisting first in segmenting the initial meshbased surface using a multi-label Markov Random Field based model and second, in sampling primitive and mesh components simultaneously on the obtained partition by a Jump-Diffusion process. The quality of a reconstruction is measured by a multi-object energy model which takes into account both photo-consistency and semantic considerations (i.e. geometry and shape layout). The segmentation and sampling steps are embedded into an iterative refinement procedure which provides an increasingly accurate hybrid representation. Experimental results on complex urban structures and large scenes are presented and compared to the state-of-the-art multi-view stereo meshing algorithms.

Many FLIR simulation efforts have been undertaken with the goals of training and testing automatic target recognition algorithms and predicting performance. In our pattern theoretic Grenander/Bayesian approach to the ATR problem, simulation provides the heart of the ATR algorithm itself. An hypothesized scene, simulated from the emissive characteristics of the hypothesized scene elements, is compared with the collected data by a likelihood function based on sensor statistics. This likelihood is combined with a prior distribution defined over the set of possible scenes to form a posterior distribution. Jump-diffusion processes which sample from the posterior provide the engine of the inference algorithm. New objects are detected and object types are recognized through discrete jump moves. Between jumps, the location and orientation of objects are estimated via continuous diffusions. Both the diffusion and jump operations involve the simulation of scenes produced by hypothesized configur...

INTRODUCTION Many automatic target recognition (ATR) algorithms are intimately tied to the particular sensor they are designed for, and are not readily adapted to other kinds of sensors. Grenander's pattern theory 1--3 seeks a conceptual separation between the underlying representation of a scene, the sensors used to observe that scene, and the algorithm used to perform inference using the underlying representation and the sensor model. In this paradigm, a hypothesized scene, simulated from the characteristics of the hypothesized scene elements, is compared to the collected data by a likelihood function based on sensor statistics. The likelihood is combined with prior knowledge to form a Bayesian posterior distribution. One can explore di#erent algorithms which exploit the same underlying representation and sensor model to determine which algorithm is the most e#cient. Similarly, by employing a common representation, a particular algorithm designed for one sen