953 research outputs found
Pre-Symmetry Sets of 3D shapes
The investigation of 3D euclidean symmetry sets (SS) and medial axis is an
important area, due in particular to their various important applications.
The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in
2D) is the set of pairs of points which contribute to the symmetry set, that
is, the closure of the set of pairs of distinct points p and q in M, for which
there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of
this paper is to address problems related to the smoothness and the
singularities of the pre-symmetry sets of 3D shapes.
We show that the pre-symmetry set of a smooth surface in 3-space has locally
the structure of the graph of a function from R^2 to R^2, in many cases of
interest.Comment: ACM-class: I.2; I.5; I.4; J.2. Latex, 3 grouped figures. The final
version will appear in the proceedings of the First International Workshop on
Deep Structure, Singularities and Computer Vision, Maastricht June 200
From segment to somite: segmentation to epithelialization analyzed within quantitative frameworks
One of the most visually striking patterns in the early developing embryo is somite segmentation. Somites form as repeated, periodic structures in pairs along nearly the entire caudal vertebrate axis. The morphological process involves short- and long-range signals that drive cell rearrangements and cell shaping to create discrete, epithelialized segments. Key to developing novel strategies to prevent somite birth defects that involve axial bone and skeletal muscle development is understanding how the molecular choreography is coordinated across multiple spatial scales and in a repeating temporal manner. Mathematical models have emerged as useful tools to integrate spatiotemporal data and simulate model mechanisms to provide unique insights into somite pattern formation. In this short review, we present two quantitative frameworks that address the morphogenesis from segment to somite and discuss recent data of segmentation and epithelialization
Homeostatic Elastic States and the Stability of Elastic Arteries
Vascular mechanics has undergone significant growth within the last 50 years owing to the rapid development of nonlinear continuum mechanics occurring roughly within the same period and motivated primarily by rubber materials. However, one important distinction of blood vessels, in contrast to typical engineering materials is that, through a variety of physiological mechanisms, they seek to maintain constant a preferred mechanical state (mechanical homeostasis) thereby exhibiting a remarkable mechanical stability in response to temporal evolution and alterations in blood pressure, vessel tethering forces and geometry and material properties. The mechanical state experienced by blood vessels plays a critical role in mechanical homeostasis and mechanical stability, and there remains a pressing need for mechanical/mathematical analysis to i) understand/predict the stretch/stress states within vessels and how they evolve with increasing blood pressure and tethering forces, ii) understand/predict the mechanical stability of arteries in response to diverse stimuli such as inhomogeneities in geometry and material properties. This dissertation seeks to add to this vibrant field by conducting a rigorous analysis of i) the mechanics of the homeostatic states of uniform circumferential stress and uniform stretch in an N-layer cylindrical artery subject to circumferential prestress, axial tethering force and the pressure of blood and ii) the local mechanical stability by imperfection growth in a solid body subject to inhomogeneities in geometry and material properties. In order to make these results relevant to a blood vessel, a micromechanics based constitutive relation is proposed based on the more or less regular architecture of a large elastic artery composed of collagen, elastin and vascular smooth muscle. Although the primary focus of the work is on the healthy artery, the effect on imperfection growth of diseased tissue constituents is accounted for in a simple model of damaged elastin and collagen
Non-affinity of liquid networks and bicontinuous mesophases
Amphiphiles self-assemble into a variety of bicontinuous mesophases whose
equilibrium structures take the form of high-symmetry cubic networks. Here, we
show that the symmetry-breaking distortions in these systems give rise to
anomalously large, non-affine collective deformations, which we argue to be a
generic consequence of mass equilibration within deformed networks. We propose
and study a minimal liquid network model of bicontinuous networks, in which
acubic distortions are modeled by the relaxation of residually-stressed
mechanical networks with constant-tension bonds. We show that non-affinity is
strongly dependent on the valency of the network as well as the degree of
strain-softening/stiffening force in the bonds. Taking diblock copolymer melts
as a model system, liquid network theory captures quantitative features of two
bicontinuous phases based on comparison with self-consistent field theory
predictions and direct experimental characterization of acubic distortions,
which are likely to be pronounced in soft amphiphilic systems more generally.Comment: 23 pages, 9 figure
Quantum spin liquid with a Majorana Fermi surface on the three-dimensional hyperoctagon lattice
Motivated by the recent synthesis of -LiIrO -- a spin-orbit
entangled Mott insulator with a three-dimensional lattice structure of
the Ir ions -- we consider generalizations of the Kitaev model believed
to capture some of the microscopic interactions between the Iridium moments on
various trivalent lattice structures in three spatial dimensions. Of particular
interest is the so-called hyperoctagon lattice -- the premedial lattice of the
hyperkagome lattice, for which the ground state is a gapless quantum spin
liquid where the gapless Majorana modes form an extended two-dimensional
Majorana Fermi surface. We demonstrate that this Majorana Fermi surface is
inherently protected by lattice symmetries and discuss possible instabilities.
We thus provide the first example of an analytically tractable microscopic
model of interacting SU(2) spin-1/2 degrees of freedom in three spatial
dimensions that harbors a spin liquid with a two-dimensional spinon Fermi
surface
Doctor of Philosophy
dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges
Structural characterization and statistical-mechanical model of epidermal patterns
In proliferating epithelia of mammalian skin, cells of irregular
polygonal-like shapes pack into complex nearly flat two-dimensional structures
that are pliable to deformations. In this work, we employ various sensitive
correlation functions to quantitatively characterize structural features of
evolving packings of epithelial cells across length scales in mouse skin. We
find that the pair statistics in direct and Fourier spaces of the cell
centroids in the early stages of embryonic development show structural
directional dependence, while in the late stages the patterns tend towards
statistically isotropic states. We construct a minimalist four-component
statistical-mechanical model involving effective isotropic pair interactions
consisting of hard-core repulsion and extra short-ranged soft-core repulsion
beyond the hard core, whose length scale is roughly the same as the hard core.
The model parameters are optimized to match the sample pair statistics in both
direct and Fourier spaces. By doing this, the parameters are biologically
constrained. Our model predicts essentially the same polygonal shape
distribution and size disparity of cells found in experiments as measured by
Voronoi statistics. Moreover, our simulated equilibrium liquid-like
configurations are able to match other nontrivial unconstrained statistics,
which is a testament to the power and novelty of the model. We discuss ways in
which our model might be extended so as to better understand morphogenesis (in
particular the emergence of planar cell polarity), wound-healing, and disease
progression processes in skin, and how it could be applied to the design of
synthetic tissues
Shape analysis in shape space
This study aims to classify different deformations based on the shape space concept. A shape space is a quotient space in which each point corresponds to a class of shapes. The shapes of each class are transformed to each other by a transformation group preserving a geometrical property in which we are interested. Therefore, each deformation is a curve on the high dimensional shape space manifold, and one can classify the deformations by comparison of their corresponding deformation curves in shape space. Towards this end, two classification methods are proposed.
In the first method, a quasi conformal shape space is constructed based on a novel quasi-conformal metric, which preserves the curvature changes at each vertex during the deformation. Besides, a classification framework is introduced for deformation classification. The results on synthetic and real datasets show the effectiveness of the metric to estimate the intrinsic geometry of the shape space manifold, and its ability to classify and interpolate different deformations.
In the second method, we introduce the medial surface shape space which classifies the deformations based on the medial surface and thickness of the shape. This shape space is based on the log map and uses two novel measures, average of the normal vectors and mean of the positions, to determine the distance between each pair of shapes on shape space.
We applied these methods to classify the left ventricle deformations. The experimental results shows that the first method can remarkably classify the normal and abnormal subjects but this method cannot spot the location of the abnormality. In contrast, the second method can discriminate healthy subjects from patients with cardiomyopathy, and also can spot the abnormality on the left ventricle, which makes it a valuable assistant tool for diagnostic purposes
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