2,184 research outputs found
Curve Reconstruction via the Global Statistics of Natural Curves
Reconstructing the missing parts of a curve has been the subject of much
computational research, with applications in image inpainting, object
synthesis, etc. Different approaches for solving that problem are typically
based on processes that seek visually pleasing or perceptually plausible
completions. In this work we focus on reconstructing the underlying physically
likely shape by utilizing the global statistics of natural curves. More
specifically, we develop a reconstruction model that seeks the mean physical
curve for a given inducer configuration. This simple model is both
straightforward to compute and it is receptive to diverse additional
information, but it requires enough samples for all curve configurations, a
practical requirement that limits its effective utilization. To address this
practical issue we explore and exploit statistical geometrical properties of
natural curves, and in particular, we show that in many cases the mean curve is
scale invariant and oftentimes it is extensible. This, in turn, allows to boost
the number of examples and thus the robustness of the statistics and its
applicability. The reconstruction results are not only more physically
plausible but they also lead to important insights on the reconstruction
problem, including an elegant explanation why certain inducer configurations
are more likely to yield consistent perceptual completions than others.Comment: CVPR versio
The shape of hyperbolic Dehn surgery space
In this paper we develop a new theory of infinitesimal harmonic deformations
for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular,
this applies to complements of tubes of radius at least R_0 =
\arctanh(1/\sqrt{3}) \approx 0.65848 around the singular set of hyperbolic
cone manifolds, removing the previous restrictions on cone angles.
We then apply this to obtain a new quantitative version of Thurston's
hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery
coefficients outside a disc of ``uniform'' size yield hyperbolic structures.
Here the size of a surgery coefficient is measured using the Euclidean metric
on a horospherical cross section to a cusp in the complete hyperbolic metric,
rescaled to have area 1. We also obtain good estimates on the change in
geometry (e.g. volumes and core geodesic lengths) during hyperbolic Dehn
filling.
This new harmonic deformation theory has also been used by Bromberg and his
coworkers in their proofs of the Bers Density Conjecture for Kleinian groups.Comment: 46 pages, 3 figure
Weil-Petersson perspectives
We highlight recent progresses in the study of the Weil-Petersson (WP)
geometry of finite dimensional Teichm\"{u}ller spaces. For recent progress on
and the understanding of infinite dimensional Teichm\"{u}ller spaces the reader
is directed to the recent work of Teo-Takhtajan. As part of the highlight, we
also present possible directions for future investigations.Comment: 18 page
Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D
We propose an efficient approach for the grouping of local orientations
(points on vessels) via nilpotent approximations of sub-Riemannian distances in
the 2D and 3D roto-translation groups and . In our distance
approximations we consider homogeneous norms on nilpotent groups that locally
approximate , and which are obtained via the exponential and logarithmic
map on . In a qualitative validation we show that the norms provide
accurate approximations of the true sub-Riemannian distances, and we discuss
their relations to the fundamental solution of the sub-Laplacian on .
The quantitative experiments further confirm the accuracy of the
approximations. Quantitative results are obtained by evaluating perceptual
grouping performance of retinal blood vessels in 2D images and curves in
challenging 3D synthetic volumes. The results show that 1) sub-Riemannian
geometry is essential in achieving top performance and 2) that grouping via the
fast analytic approximations performs almost equally, or better, than
data-adaptive fast marching approaches on and .Comment: 18 pages, 9 figures, 3 tables, in review at JMI
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