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The embryonic mouse brain undergoes drastic changes in establishing basic anatomical compartments and laying out major axonal connections of the developing brain. Correlating anatomical changes with gene-expression patterns is an essential step toward understanding the mechanisms regulating brain development. Traditionally, this is done in a cross-sectional manner, but the dynamic nature of development calls for probing gene-neuroanatomy interactions in a combined spatiotemporal domain. Here, we present a four-dimensional (4D) spatiotemporal continuum of the embryonic mouse brain from E10.5 to E15.5 reconstructed from diffusion magnetic resonance microscopy (dMRM) data. This study achieved unprecedented high-definition dMRM at 30- to 35-μm isotropic resolution, and together with computational neuroanatomy techniques, we revealed both morphological and microscopic changes in the developing brain. We transformed selected gene-expression data to this continuum and correlated them with the dMRM-based neuroanatomical changes in embryonic brains. Within the continuum, we identified distinct developmental modes comprising regional clusters that shared developmental trajectories and similar gene-expression profiles. Our results demonstrate how this 4D continuum can be used to examine spatiotemporal gene- neuroanatomical interactions by connecting upstream genetic events with anatomical changes that emerge later in development. This approach would be useful for large-scale analysis of the cooperative roles of key genes in shaping the developing brain. [ABSTRACT FROM AUTHOR]

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. Index Terms - Pose estimation, ATR, Hilbert-Schmidt bounds, Bayesian approach, performance analysis, orthogonal groups.

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. Index Terms - Brain mapping, global shape models, medical imaging, pattern theory.

Random sampling algorithm based on jump-diffusion dynamics enables simultaneous detection, tracking and automatic target recognition in air-to-ground and ground-to-air scenarios. A Bayesian approach is adopted for inference of scenes. New objects are detected, and discrete jump moves through parameter space are studied for identification of objects. The algorithm explores scenes of different complexities. Continuous diffusions generate scale and rotation group transformations between jumps. The method combines dynamics-based tracking within one estimation framework.

A novel approach for accommodating local shape variation associated with mapping a two-dimensional or three-dimensional template image into alignment with an image similar in topology has been developed. Vector-field transformation is used to accommodate local shape variability and to constrain the transformation. Unlike quadratic penalty methods, the stress associated with restraining motion relaxes over time to allow large-magnitude deformations.

A maximum a posteriori (MAP) algorithm is presented for the estimation of spin-density and spin-spin decay distributions from frequency and phase-encoded magnetic resonance imaging data. Linear spatial localization gradients are assumed: the y-encode gradient applied during the phase preparation time of duration [Tau] before measurement collection, and the x-encode gradient applied during the full data collection time t [greater than or equal to] 0. The MRI signal model developed in [22] is used in which a signal resulting from M phase encodes (rows) and N frequency encode dimensions (columns) is modeled as a super-position of MN sinc-modulated exponentially decaying sinusoids with unknown spin-density and spin-spin decay parameters. The nonlinear least-squares MAP estimate of the spin density and spin-spin decay distributions solves for the 2MN spin-density and decay parameters minimizing the squared-error between the measured data and the sinc-modulated exponentially decay signal model using an iterative expectation-maximization algorithm. A covariance diagonalizing transformation is derived which decouples the joint estimation of MN sinusoids into M separate N sinusoid optimizations, yielding an order of magnitude speed up in convergence. The MAP solutions are demonstrated to deliver a decrease in standard deviation of image parameter estimates on brain phantom data of greater than a factor of two over Fourier-based estimators of the spin density and spin-spin decay distributions. A parallel processor implementation is demonstrated which maps the N sinusoid coupled minimization to separate individual simple minimizations, one for each processor.

An iterative algorithm associated with maximum likelihood (ML) Toeplitz covariance estimation was used to calculate ML estimates of complex sinusoids when subjected to stationary unknown Gaussian noise. This algorithm iterated between mean estimations for a given current covariance and vice versa subsequent to estimating the initial Toeplitz covariance and mean. These covariant estimates were compared with Cramer-Rao bounds and conventional estimators.

The theory of large deviations for Gibbs random fields is used to show that the asymptotic number of bits per symbol for Ziv-Lempel codes in two dimensions is given by the maximal entropy of all Gibbs fields with the same interaction. The error probability is shown to converge exponentially fast to zero. In addition, the stronger version of the Shannon-McMillan theorem proved by Ornstein and Weiss is formulated and proved in terms of the exponential decay of the probability of the nontypical sequences. Index Terms--Errorless coding, image compression.

The entropies and combinatories of trees that branch according to fixed but finite numbers of rules are studied. Context-free grammars are used to categorize the ways in which nodes branch to yield daughter nodes, thus providing an organized setting to examine the entropies for random branching processes whose realizations are trees and whose probabilities are determined by probabilities associated to the substitution rules of the grammar. Normalized entropy rates H are derived for the critical branching rate (p = 1) and supercritical branching rate (p > 1) processes. An equipartition theorem is proven for the supercritical processes proving that L-generation trees normalized by their number of nodes have log probability converging to the entropy rate H with L, almost everywhere in the nonextinction set. A strong departure from classical theorems for Markov sources occurs for super-critical branching processes p > 1 as the typical sets have super-geometric growth rates. Defining the [Delta]-typical set of trees to be the L-generation trees with log of their negative log probability within [Delta] of log p, then the typical set has probability equaling the nonextinction probability and log growth rate of [p.sub.L]. The combinatories of the set of all trees that can be generated from the context-free substitution rules is also studied. It is proven that for all context-free grammars that are strongly connected and have at least one substitution rule with two daughters or more, the combinatoric growth rate of the set of trees is also supergeometric and equals the largest growth rate of any random branching process with the same substitution rules. Instances of regular, pseudo-linear and context-free grammars are studied for demonstrating the theory, and as a particular example it is shown that the arithmetic expression language has log-number of unique L-generation programs growing at a rate [1.75488.sup.L]. Index Terms--Context-free languages, branching processes, trees, entropy, equipartition-theorem.

Neuromorphology is crucial to identifying neuronal subtypes and understanding learning. It is also implicated in neurological disease. However, standard morphological analysis focuses on macroscopic features such as branching frequency and connectivity between regions, and often neglects the internal geometry of neurons. In this work, we treat neuron trace points as a sampling of differentiable curves and fit them with a set of branching B-splines. We designed our representation with the Frenet-Serret formulas from differential geometry in mind. The Frenet-Serret formulas completely characterize smooth curves, and involve two parameters, curvature and torsion. Our representation makes it possible to compute these parameters from neuron traces in closed form. These parameters are defined continuously along the curve, in contrast to other parameters like tortuosity which depend on start and end points. We applied our method to a dataset of cortical projection neurons traced in two mouse brains, and found that the parameters are distributed differently between primary, collateral, and terminal axon branches, thus quantifying geometric differences between different components of an axonal arbor. The results agreed in both brains, further validating our representation. The code used in this work can be readily applied to neuron traces in SWC format and is available in our open-source Python package brainlit : http://brainlit.neurodata.io/. [ABSTRACT FROM AUTHOR]

Advances in the development of largely automated microscopy methods such as MERFISH for imaging cellular structures in mouse brains are providing spatial detection of micron resolution gene expression. While there has been tremendous progress made in the field of Computational Anatomy (CA) to perform diffeomorphic mapping technologies at the tissue scales for advanced neuroinformatic studies in common coordinates, integration of molecular- and cellular-scale populations through statistical averaging via common coordinates remains yet unattained. This paper describes the first set of algorithms for calculating geodesics in the space of diffeomorphisms, what we term space-feature-measure LDDMM, extending the family of large deformation diffeomorphic metric mapping (LDDMM) algorithms to accommodate a space-feature action on marked particles which extends consistently to the tissue scales. It leads to the derivation of a cross-modality alignment algorithm of transcriptomic data to common coordinate systems attached to standard atlases. We represent the brain data as geometric measures, termed as space-feature measures supported by a large number of unstructured points, each point representing a small volume in space and carrying a list of densities of features elements of a high-dimensional feature space. The shape of space-feature measure brain spaces is measured by transforming them by diffeomorphisms. The metric between these measures is obtained after embedding these objects in a linear space equipped with the norm, yielding a so-called "chordal metric". • Space-feature measures provide a flexible representation for the registration of recently developed molecular imaging modalities. • Their discretization on simplicial meshes induces a new variation on the large deformation diffeomorphic metric mapping family of algorithms. • This algorithm can be adapted to perform atlas-based registration. • They can be interpreted in terms of point process realizations. [ABSTRACT FROM AUTHOR]