This paper presents diffeomorphic transformations of three-dimensional (3-D) anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery. These transformations are generated in a hierarchical manner, accommodating both global and local anatomical detail. The initial low-dimensional registration is accomplished by constraining the transformation to be in a low-dimensional basis. The basis is defined by the Green's function of the elasticity operator placed at predefined locations in the anatomy and the eigenfunctions of the elasticity operator. The high-dimensional large deformations are vector fields generated via the mismatch between the template and target-image volumes constrained to be the solution of a Navier--Stokes fluid model. As part of this procedure, the Jacobian of the transformation is tracked, insuring the generation of diffeomorphisms. It is shown that transformations constrained by quadratic regularization methods such as the Laplacian, biharmonic, and linear elasticity models, do not ensure that the transformation maintains topology and, therefore, must only be used for coarse global registration.

Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least-squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore investigators usually resort to numerical simulations to examine properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson sta...

Most expectation-maximization (EM) type algorithms for penalized maximum-likelihood image reconstruction converge slowly, particularly when one incorporates additive background effects such as scatter, random coincidences, dark current, or cosmic radiation. In addition, regularizing smoothness penalties (or priors) introduce parameter coupling, rendering intractable the M-steps of most EM-type algorithms. This paper presents space-alternating generalized EM (SAGE) algorithms for image reconstruction, which update the parameters sequentially using a sequence of small "hidden" data spaces, rather than simultaneously using one large complete-data space. The sequential update decouples the M-step, so the maximization can typically be performed analytically. We introduce new hidden-data spaces that are less informative than the conventional completedata space for Poisson data and that yield significant improvements in convergence rate. This acceleration is due to statistical considerations, not numerical overrelaxation methods, so monotonic increases in the objective function are guaranteed. We provide a general global convergence proof for SAGE methods with nonnegativity constraints.

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.

Many important problems in engineering and science are well-modeled by Poisson processes. In many applications it is of great interest to accurately estimate the intensities underlying observed Poisson data. In particular, this work is motivated by photon-limited imaging problems. This paper studies a new Bayesian approach to Poisson intensity estimation based on the Haar wavelet transform. It is shown that the Haar transform provides a very natural and powerful framework for this problem. Using this framework, a novel multiscale Bayesian prior to model intensity functions is devised. The new prior leads to a simple, Bayesian intensity estimation procedure. Furthermore, we characterize the correlation behavior of the new prior and show that it has 1/f spectral characteristics. The new framework is applied to photon-limited image estimation and its potential to improve nuclear medicine imaging is examined.

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e. for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shiftvariant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shiftvariant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

This paper analyzes the performance of Bayesian object recognition algorithms in the context of deformable templates. Rigid CAD surface models represent the underlying targets; low-dimensional matrix Lie groups (rotation and translation) extend them to the particular instance of pose and position. For a target ff, I ff represents its templates and sI ff is the target template at the pose/location denoted by the parameter s. The remote sensors observing the objects are modeled by the projective transformation T , that is, T sI ff is the signature of target ff at pose s when viewed by the sensor T . The observations I D are modeled as a random fields with mean T sI ff . In a Bayesian approach, object recognition and pose estimation are basically optimizations for a given cost function related to the posterior. Recognition performance is analyzed through probability of error: given a target ff 0 at pose s 0 what is the probability of it being recognized as ff 1 . Asymptotic ex...

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 apply the complexity--regularization principle to statistical ill-posed inverse problems in imaging. The class of problems studied includes restoration of images corrupted by Gaussian or Poisson noise and nonlinear transforms. We formulate a natural distortion measure in image space and develop nonasymptotic bounds on estimation performance in terms of an index of resolvability that characterizes the compressibility of the true image. These bounds extend previous results that were obtained in the literature under simpler observational models. We present a connection between complexity-regularized estimation and rate-distortion theory, which suggests a method for constructing optimal codebooks. However, the design of computationally tractable complexity--regularized image estimators is quite challenging; we present some of the issues involved and illustrate them with a Poisson-imaging application. Keywords: nonparametric estimation, compression, minimum description length principle, ...

Multiresolution transforms, including a wavelet transform, are applied to image visualization, image restoration, filtering and compression, and object detection. Variance stabilization is used, when appropriate, to cater for common astronomical image noise models. We discuss validation of such methods in the case of astronomical image processing. A range of examples illustrate the effectiveness of this approach in handling point source and extended astronomical objects. 1 Introduction We describe a general framework for image analysis based on the following methodological components: multiresolution; noise separation; object support image; and Minkowski operators. Image analysis applications, most of which will be touched on below, include: restoration; noise filtering; compression; artifact filtering; object detection; and object description. General characteristics of astronomical images include: the presence of noise, to be addressed in the next paragraph; point sources (stars or ...