4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data.

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the [Formula: see text] metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

Versatile method for quantifying and analyzing morphological differences in experimentally obtained images.

Analyzing high-resolution images to gain insight into anatomical properties is an essential tool for investigation in many scientific fields. In plant biology, studying plant phenotypes from micrographs is often used to build hypotheses on gene function. In this report, we discuss a bespoke method for inspecting the significance in the differences between the morphologies of several plant mutants at cellular level. By examining a specific example in the literature, we will detail the approach previously used to quantify the effects of two gene families on the vascular development of hypocotyls in Arabidopsis thaliana. The method incorporates a MATLAB algorithm and statistical tools which can be modified and enhanced to tailor to different research questions in future studies.

Organ-specific genetic interactions between paralogues of the PXY and ER receptor kinases enforce radial patterning in Arabidopsis vascular tissue.

In plants, cells do not migrate. Tissues are frequently arranged in concentric rings; thus, expansion of inner layers is coordinated with cell division and/or expansion of cells in outer layers. In Arabidopsis stems, receptor kinases, PXY and ER, genetically interact to coordinate vascular proliferation and organisation via inter-tissue signalling. The contribution of PXY and ER paralogues to stem patterning is not known, nor is their function understood in hypocotyls, which undergo considerable radial expansion. Here, we show that removal of all PXY and ER gene-family members results in profound cell division and organisation defects. In hypocotyls, these plants failed to transition to true radial growth. Gene expression analysis suggested that PXY and ER cross- and inter-family transcriptional regulation occurs, but it differs between stem and hypocotyl. Thus, PXY and ER signalling interact to coordinate development in a distinct manner in different organs. We anticipate that such specialised local regulatory relationships, where tissue growth is controlled via signals moving across tissue layers, may coordinate tissue layer expansion throughout the plant body.

Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces.

Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides a comprehensive framework for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into euclidean metrics, allowing for the application of standard algorithms and statistical tools. For analyzing shapes of embeddings of in , Jermyn et al. [1] introduced square-root normal fields (SRNFs), which transform an elastic metric, with desirable invariant properties, into the metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is a method for inverting solutions (deformations, averages, modes of variations, etc.) computed in SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNF maps and their inverses, we take a numerical approach, and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found to be effective even for complex shapes that undergo significant deformations including bending and stretching, e.g., human bodies and animals. We use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces, and analysis of brain structures.

Accurate Morphology Preserving Segmentation of Overlapping Cells based on Active Contours.

The identification of fluorescently stained cell nuclei is the basis of cell detection, segmentation, and feature extraction in high content microscopy experiments. The nuclear morphology of single cells is also one of the essential indicators of phenotypic variation. However, the cells used in experiments can lose their contact inhibition, and can therefore pile up on top of each other, making the detection of single cells extremely challenging using current segmentation methods. The model we present here can detect cell nuclei and their morphology even in high-confluency cell cultures with many overlapping cell nuclei. We combine the "gas of near circles" active contour model, which favors circular shapes but allows slight variations around them, with a new data model. This captures a common property of many microscopic imaging techniques: the intensities from superposed nuclei are additive, so that two overlapping nuclei, for example, have a total intensity that is approximately double the intensity of a single nucleus. We demonstrate the power of our method on microscopic images of cells, comparing the results with those obtained from a widely used approach, and with manual image segmentations by experts.

Shape Analysis of Elastic Curves in Euclidean Spaces.

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL(2) metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.

Looking for shapes in two-dimensional cluttered point clouds.

We study the problem of identifying shape classes in point clouds. These clouds contain sampled points along contours and are corrupted by clutter and observation noise. Taking an analysis-by-synthesis approach, we simulate high-probability configurations of sampled contours using models learned from training data to evaluate the given test data. To facilitate simulations, we develop statistical models for sources of (nuisance) variability: 1) shape variations within classes, 2) variability in sampling continuous curves, 3) pose and scale variability, 4) observation noise, and 5) points introduced by clutter. The variability in sampling closed curves into finite points is represented by positive diffeomorphisms of a unit circle. We derive probability models on these functions using their square-root forms and the Fisher-Rao metric. Using a Monte Carlo approach, we simulate configurations from a joint prior on the shape-sample space and compare them to the data using a likelihood function. Average likelihoods of simulated configurations lead to estimates of posterior probabilities of different classes and, hence, Bayesian classification.