8 research outputs found
Geodesics in Heat
We introduce the heat method for computing the shortest geodesic distance to
a specified subset (e.g., point or curve) of a given domain. The heat method is
robust, efficient, and simple to implement since it is based on solving a pair
of standard linear elliptic problems. The method represents a significant
breakthrough in the practical computation of distance on a wide variety of
geometric domains, since the resulting linear systems can be prefactored once
and subsequently solved in near-linear time. In practice, distance can be
updated via the heat method an order of magnitude faster than with
state-of-the-art methods while maintaining a comparable level of accuracy. We
provide numerical evidence that the method converges to the exact geodesic
distance in the limit of refinement; we also explore smoothed approximations of
distance suitable for applications where more regularity is required
Limits and consistency of non-local and graph approximations to the Eikonal equation
In this paper, we study a non-local approximation of the time-dependent
(local) Eikonal equation with Dirichlet-type boundary conditions, where the
kernel in the non-local problem is properly scaled. Based on the theory of
viscosity solutions, we prove existence and uniqueness of the viscosity
solutions of both the local and non-local problems, as well as regularity
properties of these solutions in time and space. We then derive error bounds
between the solution to the non-local problem and that of the local one, both
in continuous-time and Backward Euler time discretization. We then turn to
studying continuum limits of non-local problems defined on random weighted
graphs with vertices. In particular, we establish that if the kernel scale
parameter decreases at an appropriate rate as grows, then almost surely,
the solution of the problem on graphs converges uniformly to the viscosity
solution of the local problem as the time step vanishes and the number vertices
grows large
Riemannian Multi-Manifold Modeling
This paper advocates a novel framework for segmenting a dataset in a
Riemannian manifold into clusters lying around low-dimensional submanifolds
of . Important examples of , for which the proposed clustering algorithm
is computationally efficient, are the sphere, the set of positive definite
matrices, and the Grassmannian. The clustering problem with these examples of
is already useful for numerous application domains such as action
identification in video sequences, dynamic texture clustering, brain fiber
segmentation in medical imaging, and clustering of deformed images. The
proposed clustering algorithm constructs a data-affinity matrix by thoroughly
exploiting the intrinsic geometry and then applies spectral clustering. The
intrinsic local geometry is encoded by local sparse coding and more importantly
by directional information of local tangent spaces and geodesics. Theoretical
guarantees are established for a simplified variant of the algorithm even when
the clusters intersect. To avoid complication, these guarantees assume that the
underlying submanifolds are geodesic. Extensive validation on synthetic and
real data demonstrates the resiliency of the proposed method against deviations
from the theoretical model as well as its superior performance over
state-of-the-art techniques
Proceedings of the First International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'06) - Geometrical and Statistical Methods for Modelling Biological Shape Variability
International audienceNon-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult to relate the anatomical shape of different subjects. The goal of computational anatomy is to analyse and to statistically model this specific type of geometrical information. In the absence of any justified physical model, a natural attitude is to explore very general mathematical methods, for instance diffeomorphisms. However, working with such infinite dimensional space raises some deep computational and mathematical problems. In particular, one of the key problem is to do statistics. Likewise, modelling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed. The goal of the workshop was to foster interactions between researchers investigating the combination of geometry and statistics for modelling biological shape variability from image and surfaces. A special emphasis was put on theoretical developments, applications and results being welcomed as illustrations. Contributions were solicited in the following areas: * Riemannian and group theoretical methods on non-linear transformation spaces * Advanced statistics on deformations and shapes * Metrics for computational anatomy * Geometry and statistics of surfaces 26 submissions of very high quality were recieved and were reviewed by two members of the programm committee. 12 papers were finally selected for oral presentations and 8 for poster presentations. 16 of these papers are published in these proceedings, and 4 papers are published in the proceedings of MICCAI'06 (for copyright reasons, only extended abstracts are provided here)