6,288 research outputs found
Intrinsic Inference on the Mean Geodesic of Planar Shapes and Tree Discrimination by Leaf Growth
For planar landmark based shapes, taking into account the non-Euclidean
geometry of the shape space, a statistical test for a common mean first
geodesic principal component (GPC) is devised. It rests on one of two
asymptotic scenarios, both of which are identical in a Euclidean geometry. For
both scenarios, strong consistency and central limit theorems are established,
along with an algorithm for the computation of a Ziezold mean geodesic. In
application, this allows to verify the geodesic hypothesis for leaf growth of
Canadian black poplars and to discriminate genetically different trees by
observations of leaf shape growth over brief time intervals. With a test based
on Procrustes tangent space coordinates, not involving the shape space's
curvature, neither can be achieved.Comment: 28 pages, 4 figure
Numerical inversion of SRNFs for efficient elastic shape analysis of star-shaped objects.
The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics.
This approach is based on defining new mathematical representations of parameterized surfaces, including the square root normal field (SRNF), and then using the L2 norm to compare their shapes. Past work is based on using the pullback of the L2 metric to the space of surfaces, performing statistical analysis under this induced Riemannian metric. However, if one can estimate the inverse of the SRNF mapping, even approximately, a very efficient framework results: the surfaces, represented by their SRNFs, can be efficiently analyzed using standard Euclidean tools, and only the final results need be mapped back to the surface space. Here we describe a procedure for inverting SRNF maps of star-shaped surfaces, a special case for which analytic results can be obtained. We test our method via the classification of 34 cases of ADHD (Attention Deficit Hyperactivity Disorder), plus controls, in the Detroit Fetal Alcohol and Drug Exposure Cohort study. We obtain state-of-the-art results
Computing distances and geodesics between manifold-valued curves in the SRV framework
This paper focuses on the study of open curves in a Riemannian manifold M,
and proposes a reparametrization invariant metric on the space of such paths.
We use the square root velocity function (SRVF) introduced by Srivastava et al.
to define a Riemannian metric on the space of immersions M'=Imm([0,1],M) by
pullback of a natural metric on the tangent bundle TM'. This induces a
first-order Sobolev metric on M' and leads to a distance which takes into
account the distance between the origins in M and the L2-distance between the
SRV representations of the curves. The geodesic equations for this metric are
given and exploited to define an exponential map on M'. The optimal deformation
of one curve into another can then be constructed using geodesic shooting,
which requires to characterize the Jacobi fields of M'. The particular case of
curves lying in the hyperbolic half-plane is considered as an example, in the
setting of radar signal processing
Continuous image distortion by astrophysical thick lenses
Image distortion due to weak gravitational lensing is examined using a
non-perturbative method of integrating the geodesic deviation and optical
scalar equations along the null geodesics connecting the observer to a distant
source. The method we develop continuously changes the shape of the pencil of
rays from the source to the observer with no reference to lens planes in
astrophysically relevant scenarios. We compare the projected area and the ratio
of semi-major to semi-minor axes of the observed elliptical image shape for
circular sources from the continuous, thick-lens method with the commonly
assumed thin-lens approximation. We find that for truncated singular isothermal
sphere and NFW models of realistic galaxy clusters, the commonly used thin-lens
approximation is accurate to better than 1 part in 10^4 in predicting the image
area and axes ratios. For asymmetric thick lenses consisting of two massive
clusters separated along the line of sight in redshift up to \Delta z = 0.2, we
find that modeling the image distortion as two clusters in a single lens plane
does not produce relative errors in image area or axes ratio more than 0.5%Comment: accepted to GR
Empirical geodesic graphs and CAT(k) metrics for data analysis
A methodology is developed for data analysis based on empirically constructed
geodesic metric spaces. For a probability distribution, the length along a path
between two points can be defined as the amount of probability mass accumulated
along the path. The geodesic, then, is the shortest such path and defines a
geodesic metric. Such metrics are transformed in a number of ways to produce
parametrised families of geodesic metric spaces, empirical versions of which
allow computation of intrinsic means and associated measures of dispersion.
These reveal properties of the data, based on geometry, such as those that are
difficult to see from the raw Euclidean distances. Examples of application
include clustering and classification. For certain parameter ranges, the spaces
become CAT(0) spaces and the intrinsic means are unique. In one case, a minimal
spanning tree of a graph based on the data becomes CAT(0). In another, a
so-called "metric cone" construction allows extension to CAT() spaces. It is
shown how to empirically tune the parameters of the metrics, making it possible
to apply them to a number of real cases.Comment: Statistics and Computing, 201
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