115 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
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
Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores
We provide the explicit solutions of linear, left-invariant,
(convection)-diffusion equations and the corresponding resolvent equations on
the 2D-Euclidean motion group SE(2). These diffusion equations are forward
Kolmogorov equations for stochastic processes for contour enhancement and
completion. The solutions are group-convolutions with the corresponding Green's
function, which we derive in explicit form. We mainly focus on the Kolmogorov
equations for contour enhancement processes which, in contrast to the
Kolmogorov equations for contour completion, do not include convection. The
Green's functions of these left-invariant partial differential equations
coincide with the heat-kernels on SE(2), which we explicitly derive. Then we
compute completion distributions on SE(2) which are the product of a forward
and a backward resolvent evolved from resp. source and sink distribution on
SE(2). On the one hand, the modes of Mumford's direction process for contour
completion coincide with elastica curves minimizing , related to zero-crossings of 2 left-invariant derivatives of the
completion distribution. On the other hand, the completion measure for the
contour enhancement concentrates on geodesics minimizing . This motivates a comparison between geodesics and elastica,
which are quite similar. However, we derive more practical analytic solutions
for the geodesics. The theory is motivated by medical image analysis
applications where enhancement of elongated structures in noisy images is
required. We use left-invariant (non)-linear evolution processes for automated
contour enhancement on invertible orientation scores, obtained from an image by
means of a special type of unitary wavelet transform
Cortical Functional architectures as contact and sub-Riemannian geometry
In a joint paper, Jean Petitot together with the authors of the present paper
described the functional geometry of the visual cortex as the symplectization
of a contact form to describe the family of cells sensitive to position,
orientation and scale. In the present paper, as a "homage" to the enormous
contribution of Jean Petitot to neurogeometry, we will extend this approach to
more complex functional architectures built as a sequence of contactization or
a symplectization process, able to extend the dimension of the space. We will
also outline a few examples where a sub-Riemannian lifting is needed
Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance
We consider a two player game, where a first player has to install a
surveillance system within an admissible region. The second player needs to
enter the the monitored area, visit a target region, and then leave the area,
while minimizing his overall probability of detection. Both players know the
target region, and the second player knows the surveillance installation
details.Optimal trajectories for the second player are computed using a
recently developed variant of the fast marching algorithm, which takes into
account curvature constraints modeling the second player vehicle
maneuverability. The surveillance system optimization leverages a reverse-mode
semi-automatic differentiation procedure, estimating the gradient of the value
function related to the sensor location in time N log N
Riemannian cubics on the group of diffeomorphisms and the Fisher-Rao metric
We study a second-order variational problem on the group of diffeomorphisms
of the interval [0, 1] endowed with a right-invariant Sobolev metric of order
2, which consists in the minimization of the acceleration. We compute the
relaxation of the problem which involves the so-called Fisher-Rao functional a
convex functional on the space of measures. This relaxation enables the
derivation of several optimality conditions and, in particular, a sufficient
condition which guarantees that a given path of the initial problem is also a
minimizer of the relaxed one. This sufficient condition is related to the
existence of a solution to a Riccati equation involving the path acceleration.Comment: 34 pages, comments welcom
- …