12,747 research outputs found
Conformal Wasserstein distances: comparing surfaces in polynomial time
We present a constructive approach to surface comparison realizable by a
polynomial-time algorithm. We determine the "similarity" of two given surfaces
by solving a mass-transportation problem between their conformal densities.
This mass transportation problem differs from the standard case in that we
require the solution to be invariant under global M\"{o}bius transformations.
We present in detail the case where the surfaces to compare are disk-like; we
also sketch how the approach can be generalized to other types of surfaces.Comment: 23 pages, 3 figure
Surface Comparison with Mass Transportation
We use mass-transportation as a tool to compare surfaces (2-manifolds). In
particular, we determine the "similarity" of two given surfaces by solving a
mass-transportation problem between their conformal densities. This mass
transportation problem differs from the standard case in that we require the
solution to be invariant under global M\"obius transformations. Our approach
provides a constructive way of defining a metric in the abstract space of
simply-connected smooth surfaces with boundary (i.e. surfaces of disk-type);
this metric can also be used to define meaningful intrinsic distances between
pairs of "patches" in the two surfaces, which allows automatic alignment of the
surfaces. We provide numerical experiments on "real-life" surfaces to
demonstrate possible applications in natural sciences
Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction
Given a tetrahedral mesh and objective functionals measuring the mesh quality
which take into account the shape, size, and orientation of the mesh elements,
our aim is to improve the mesh quality as much as possible. In this paper, we
combine the moving mesh smoothing, based on the integration of an ordinary
differential equation coming from a given functional, with the lazy flip
technique, a reversible edge removal algorithm to modify the mesh connectivity.
Moreover, we utilize radial basis function (RBF) surface reconstruction to
improve tetrahedral meshes with curved boundary surfaces. Numerical tests show
that the combination of these techniques into a mesh improvement framework
achieves results which are comparable and even better than the previously
reported ones.Comment: Revised and improved versio
Computerized Analysis of Magnetic Resonance Images to Study Cerebral Anatomy in Developing Neonates
The study of cerebral anatomy in developing neonates is of great importance for
the understanding of brain development during the early period of life. This
dissertation therefore focuses on three challenges in the modelling of cerebral
anatomy in neonates during brain development. The methods that have been
developed all use Magnetic Resonance Images (MRI) as source data.
To facilitate study of vascular development in the neonatal period, a set of image
analysis algorithms are developed to automatically extract and model cerebral
vessel trees. The whole process consists of cerebral vessel tracking from
automatically placed seed points, vessel tree generation, and vasculature
registration and matching. These algorithms have been tested on clinical Time-of-
Flight (TOF) MR angiographic datasets.
To facilitate study of the neonatal cortex a complete cerebral cortex segmentation
and reconstruction pipeline has been developed. Segmentation of the neonatal
cortex is not effectively done by existing algorithms designed for the adult brain
because the contrast between grey and white matter is reversed. This causes pixels
containing tissue mixtures to be incorrectly labelled by conventional methods. The
neonatal cortical segmentation method that has been developed is based on a novel
expectation-maximization (EM) method with explicit correction for mislabelled
partial volume voxels. Based on the resulting cortical segmentation, an implicit
surface evolution technique is adopted for the reconstruction of the cortex in
neonates. The performance of the method is investigated by performing a detailed
landmark study.
To facilitate study of cortical development, a cortical surface registration algorithm
for aligning the cortical surface is developed. The method first inflates extracted
cortical surfaces and then performs a non-rigid surface registration using free-form
deformations (FFDs) to remove residual alignment. Validation experiments using
data labelled by an expert observer demonstrate that the method can capture local
changes and follow the growth of specific sulcus
Gauge Invariant Framework for Shape Analysis of Surfaces
This paper describes a novel framework for computing geodesic paths in shape
spaces of spherical surfaces under an elastic Riemannian metric. The novelty
lies in defining this Riemannian metric directly on the quotient (shape) space,
rather than inheriting it from pre-shape space, and using it to formulate a
path energy that measures only the normal components of velocities along the
path. In other words, this paper defines and solves for geodesics directly on
the shape space and avoids complications resulting from the quotient operation.
This comprehensive framework is invariant to arbitrary parameterizations of
surfaces along paths, a phenomenon termed as gauge invariance. Additionally,
this paper makes a link between different elastic metrics used in the computer
science literature on one hand, and the mathematical literature on the other
hand, and provides a geometrical interpretation of the terms involved. Examples
using real and simulated 3D objects are provided to help illustrate the main
ideas.Comment: 15 pages, 11 Figures, to appear in IEEE Transactions on Pattern
Analysis and Machine Intelligence in a better resolutio
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
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