Workshop 2: Morphogenesis, Regeneration, and the Analysis of Shape
Anatomical trees appear as transportation systems distributing blood, water or air. Due to their critical role, statistics on populations of trees is essential to understanding disease. This is difficult due to anatomical variation in branching structure across subjects. I will present a geometric tree-space framework for leaf-labeled trees, discuss how statistics can be defined and performed in tree-space, and present applications to anatomical labeling of airway trees and large-scale statistics on the effect of Chronic Obstructive Pulmonary Disease on airway trees.
Shape spaces have emerged as a natural mathematical setting to think about shapes as a structured space. In that setting, group actions of diffeomorphisms provide nice vehicles to build a full processing framework called here diffeomorphometry. This talks will present a recently developed framework embedding the situation of geometrical shapes carrying a functional information called here f-shapes.
Shape analysis of objects intrinsically involves registration of points across objects. While registration is historically treated as a pre-processing step, a more recent trend is to incorporate it as an integral step in shape comparison. There are two distinct approaches for shape registration -- model based and metric based.
(1) In the first case, one assumes a model upfront that is generally of the type:
observation = deformed template & noise.
The goal then is to estimate the template and individual deformations given multiple observations.
(2) The second approach does not start with a model but uses a metric that forms an objective function for both: registration of points across objects and quantification of difference in shapes. This metric facilitates a consistent framework where registration and ensuing shape analysis (summarization, modes of variations, etc) are all performed under the same metric (and not as either pre- and/or post-processing).
It is the second approach that I will elaborate upon in this talk. The key property of valid metrics for registration is that the re-parameterization group acts by isometries under this metric. While such metrics are beginning to emerge for shape analysis of curves, surfaces, and even images, they are often too difficult to allow efficient solutions. In some cases a simple change of variable converts these invariant metrics into standard Euclidean metrics, the classical computational solutions apply.
Examples include square-root velocity functions for curves and square-root normal fields for surfaces.
I will demonstrate these ideas using different types of objects: curves in Euclidean spaces, trajectories on Riemannian manifolds, 2D surfaces, and vector-valued images.
Gut patterning, brain gyrification and wing shape.
Data Driven Methods for Shape Analysis and Visualization
Probabilistic Modeling of Shape
Xiaolei (Sharon) Huang
Fluorescence microscopy is frequently used to study two and three dimensional network structures formed by cytoskeletal polymer fibers such as actin filaments and actin cables. While these cytoskeletal structures are often dilute enough to allow imaging of individual filaments or bundles of them, quantitative analysis of these images is challenging. To facilitate quantitative, reproducible and objective analysis of the image data, we present an automated method to extract actin networks and retrieve their topology in 3D. Our method uses multiple Stretching Open Active Contours (SOACs) that are automatically initialized at image intensity ridges and then evolve along the centerlines of filaments in the network. SOACs can merge, stop at junctions, and reconfigure with others to allow smooth crossing at junctions of filaments. The proposed approach is generally applicable to images of curvilinear networks with low SNR. We demonstrate its potential by extracting the centerlines of synthetic meshwork images, actin networks in 2D Total Internal Reflection Fluorescence Microscopy images, and 3D actin cable meshworks of live fission yeast cells imaged by spinning disk confocal microscopy.
This talk will give an introduction to geometric algorithms for comparing and matching discrete geometric shapes such as point sets, polygonal curves, and graphs. We will study distance measures for shapes, approaches for matching shapes under transformations, and algorithms for reconciling sets of shapes by constructing simpler representative shapes. We will consider theoretical results as well as real-world applications including biomedical imaging and GPS trajectory analysis.
To set the stage I will introduce the phenotypic screening and the relevance of this approach to drug discovery. The talk will highlight a number of image analysis techniques that play an increasingly important role in phenotypic screening. In particular it will review algorithms for cell tracking and cell cycle estimation as well as image analysis based approaches for tissue mapping. Apart from discussing the image analysis algorithms the presentation will also outline what work will be necessary to integrated the high-content information in the overall workflow. The ongoing work at the newly established Target Discovery Institute (TDI) at the University of Oxford will also be presented. The overall goal and the research objectives of the different groups at TDI will be discussed.
Shape and function are intricately related in biology. We present three biological case studies where the goal is to quantify shape change in order to analyze how shape informs function. The biologists have specific questions they are interested in answering, and have domain knowledge that should be incorporated into the the shape correspondence algorithm. We show how we incorporate these constraints into the shape matching algorithms in order to provide our collaborators with biologically-meaningful shape correspondences.
Case study 1: Using strain to track ferret brain development.
Case study 2: Using geodesic distances and an approximate medial axis to track an in-vivo beating chicken hearts at an early stage of development (peristaltic motion).
Case study 3: Shape space based on natural neighbor coordinates for bat pinnae and noseleaves.
Mapping neural networks in brain, retina and spinal cord requires (1) comprehensive parts lists (vertex types), (2) nanometer scale connection detection (edge types), and (3) millimeter scale network tracing. Connectomics based on high-resolution automated transmission electron microscope imaging merges these operations and allows discovery of network modules and motifs as well as the their geometric patterning cell shapes.
The mammalian retina contains â‰ˆ 70 classes of neurons assembled into â‰ˆ 15 different network modules, with significant motif overlap. We analyzed mammalian retinal connectome RC1 to unpack this mesh of neurons. A key question that emerges in tracing networks is how neurons and patterns are regulated. I will describe the scales and modes of patterning (packing, tiling, covering) adopted by different neurons and discuss the implications for developmental regulation.
Kishore Rao Mosaliganti
How do animals develop similar organ sizes and shapes despite large fluctuations in initial growth conditions? How is size and shape control achieved across molecular-cellular-tissue scales? We answer this question in the context of early ear development that exhibits a highly stereotyped pattern of assembly and growth. Using in toto imaging technologies in the zebrafish embryo, we reconstructed morphogenetic patterns of cellular movements, cell number and shape changes, and tissue topology changes. We show that otic vesicle growth and regeneration is characterized by endolymph pressure and tissue stretching forces that provide feedback to circuits responsible for generating endolymph fluid. To systematically investigate how otic vesicle growth is controlled, we developed a minimal mathematical model linking tissue geometry and mechanics to tissue stretching forces, thus illuminating how size control to stage-specific volumes is accomplished. Because ear development shares many features with other developmental (eye, heart, kidney) and disease processes (tissue tumor formation), our results and mathematical model will inform understanding of the morphogenesis of other organs.
Julie Thierot's talk onCell Shape Determination in Single-Cell Motility
We construct a geometric method to directly quantify the difference between curves. That is, we construct a distance for parametrized curves in R^n modulo Euclidean transformations. This distance measures the local dissimilarity of k-jets (Taylor polynomials) of the curves. The distance is obtained from a variational principle, and can be constructed by solving a boundary value problem for a second-order ODE. As such it may prove to be computationally less expensive than EPdiff methods if one is only interested in a distance measure.