Mathematical and statistical modeling of biological growth is an important problem in medical diagnostics. Here, we seek tools to analyze changes in anatomical parts using images collected over time. We introduce a structured model, called Growth by Random Iterated Diffeomorphisms (GRID), that treats a cumulative growth deformation as a composition of several elementary deformations. Each elementary deformation applies to a small region by capturing deformation local to that region and is characterized by a seed and a radial deformation pattern around that seed. These GRID variables--seed locations and radial deformation patterns---are estimated from observed images in two steps: 1) estimate a cumulative deformation over an observation interval; 2) estimate GRID variables using maximum-likelihood criterion from this estimated cumulative deformation. We demonstrate this framework using an MRI image data of a rat's brain growth. For future statistical analysis, we propose a time-varying Poisson process for the seed placements and a random drawing from a predetermined catalog of deformations for the radial deformation patterns.

Seeking probability models for images, we employ a spectral approach where the images are decomposed using bandpass filters and probability models are imposed on the filter outputs (also called spectral components). We employ a (two-parameter) family of probability densities, called Bessel K forms, for modeling the marginal densities of the spectral components, and demonstrate their fit to the observed histograms for video, infrared, and range images. Motivated by object-based models for image analysis, a relationship between the Bessel parameters and the imaged objects is established. Using L/sup 2/-metric on the set of Bessel K forms, we propose a pseudometric on the image space for quantifying image similarities/differences. Some applications, including clutter classification and pruning of hypotheses for target recognition, are presented.

We present a framework for modeling background clutter in natural images. Assuming that: (i) images are made up of 2D (projected) views of 3D (real) objects, and (ii) certain simplifying conditions hold, we present an analytical density for the derivatives of the natural images. This expression is shown to match well with the observed densities (histograms).

Spectral components, obtained via bandpass filtering of images, have become important tools in capturing image variability. We present a two-parameter family of probability densities, called the Bessel forms, to model the marginal densities of the spectral components. The two parameters, shape and scale parameters, are used to characterize each spectral component of the image. We derive an L/sup 2/-metric on the space of Bessel forms that leads to a metric on the space of natural images. The strength of these forms/metric is demonstrated via a study of natural clutter images.

We propose a framework for modeling clutter in natural images. Assuming that: 1) images are made up of 2D (projected) views of 3D (real) objects and 2) certain simplifying conditions hold, we derive an analytical density for natural images. This expression is shown to match well with the observed densities (histograms). In addition to deriving multidimensional densities, several extensions are also proposed.

Asymptotic approximations to the partition function of Gaussian random fields are derived. Textures are characterized via Gaussian random fields induced by stochastic difference equations determined by finitely supported, stationary, linear difference operators, adjusted to be nonstationary at the boundaries. It is shown that as the scale of the underlying shape increases, the log-normalizer converges to the integral of the log-spectrum of the operator inducing the random field. Fitting the covariance of the fields amounts to fitting the parameters of the spectrum of the differential operator-induced random field model. Matrix analysis techniques are proposed for handling textures with variable orientation. Examples of texture parameters estimated from training data via asymptotic maximum-likelihood are shown. Isotropic models involving powers of the Laplacian and directional models involving partial derivative mixtures are explored. Parameters are estimated for mitochondria and actin-myocin complexes in electron micrographs and clutter in forward-looking infrared images. Deformable template models are used to infer the shape of mitochondria in electron micrographs, with the asymptotic approximation allowing easy recomputation of the partition function as inference proceeds.

This article investigates the asymptotic performance of Bayesian target recognition algorithms using deformable-template representations. Rigid computer-aided design (CAD) models represent the underlying targets; low-dimensional matrix Lie-groups (rotation and translation) extend them to particular instances. Remote sensors observing the targets are modeled as projective transformations, converting three-dimensional scenes into random images. Bayesian target recognition corresponds to hypothesis selection in the presence of nuisance parameters; its performance is quantified as the Bayes' error. Analytical expressions for this error probability in small noise situations are derived, yielding asymptotic error rates for exponential error probability decay.

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 cases of both ground-based and airborne targets.

Dynamic programming algorithms are presented for automated generation of length minimizing geodesics and curves of extremal curvature on the neocortex of the macaque and the visible human. Probabilistic models of curve variation are constructed in terms of the variability in speed, curvature, and torsion in the Frenet representation.

This paper develops mathematical representations for neuro-anatomically significant substructures of the brain and their variability in a population. The focus of the paper is on the neuro-anatomical variation of the geometry and the "shape" of two-dimensional surfaces in the brain. As examples, we focus on the cortical and hippocampal surfaces in an ensemble of Macaque monkeys and human MRI brains. The "shapes" of the substructures are quantified via the construction of templates; the variations are represented by defining probabilistic deformations of the template. Methods for empirically estimating probability measures on these deformations are developed by representing the deformations as Gaussian random vector fields on the embedded sub-manifolds. The Gaussian random vector fields are constructed as quadratic mean limits using complete orthonormal bases on the sub-manifolds. The complete orthonormal bases are generated using modes of vibrations of the geometries of the brain sub-manifolds. The covariances are empirically estimated from an ensemble of brain data. Principal component analysis is presented for characterizing the "eigen-shape" of the hippocampus in an ensemble of MRI-MPRAGE whole brain images. Clustering based on eigen-shape is presented for two sub-populations of normal and schizophrenic.

Proposes a framework for simultaneous detection, tracking, and recognition of objects via data fused from multiple sensors. Complex dynamic scenes are represented via the concatenation of simple rigid templates. The variability of the infinity of pose is accommodated via the actions of matrix Lie groups extending the templates to individual instances. The variability of target number and target identity is accommodated via the representation of scenes as unions of templates of varying types, with the associated group transformations of varying dimension. We focus on recognition in the air-to-ground and ground-to-air scenarios. The remote sensing data is organized around both the coarse scale associated with detection as provided by tracking and range radars, along with the fine scale associated with pose and identity supported by high-resolution optical, forward looking infrared and delay-Doppler radar imagers. A Bayesian approach is adopted in which prior distributions on target scenarios are constructed via dynamical models of the targets of interest. These are combined with physics-based sensor models which define conditional likelihoods for the coarse/fine scale sensor data given the underlying scene. Inference via the Bayes posterior is organized around a random sampling algorithm based on jump-diffusion processes. New objects are detected and object identities are recognized through discrete jump moves through parameter space, the algorithm exploring scenes of varying complexity as it proceeds. Between jumps, the scale and rotation group transformations are generated via continuous diffusions in order to smoothly deform templates into individual instances of objects.