8,795 research outputs found
A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve
We derive a variational model to fit a composite B\'ezier curve to a set of
data points on a Riemannian manifold. The resulting curve is obtained in such a
way that its mean squared acceleration is minimal in addition to remaining
close the data points. We approximate the acceleration by discretizing the
squared second order derivative along the curve. We derive a closed-form,
numerically stable and efficient algorithm to compute the gradient of a
B\'ezier curve on manifolds with respect to its control points, expressed as a
concatenation of so-called adjoint Jacobi fields. Several examples illustrate
the capabilites and validity of this approach both for interpolation and
approximation. The examples also illustrate that the approach outperforms
previous works tackling this problem
Fitting subdivision surfaces to unorganized point data using SDM
We study the reconstruction of smooth surfaces from point clouds. We use a new squared distance error term in optimization to fit a subdivision surface to a set of unorganized points, which defines a closed target surface of arbitrary topology. The resulting method is based on the framework of squared distance minimization (SDM) proposed by Pottmann et al. Specifically, with an initial subdivision surface having a coarse control mesh as input, we adjust the control points by optimizing an objective function through iterative minimization of a quadratic approximant of the squared distance function of the target shape. Our experiments show that the new method (SDM) converges much faster than the commonly used optimization method using the point distance error function, which is known to have only linear convergence. This observation is further supported by our recent result that SDM can be derived from the Newton method with necessary modifications to make the Hessian positive definite and the fact that the Newton method has quadratic convergence. © 2004 IEEE.published_or_final_versio
A graph-spectral approach to shape-from-shading
In this paper, we explore how graph-spectral methods can be used to develop a new shape-from-shading algorithm. We characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction. Modeling the blocks of the weight matrix as distinct surface patches, we use a graph seriation method to find a surface integration path that maximizes the sum of curvature-dependent weights and that can be used for the purposes of height reconstruction. To smooth the reconstructed surface, we fit quadrics to the height data for each patch. The smoothed surface normal directions are updated ensuring compliance with Lambert's law. The processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained. We provide results on synthetic and real-world imagery
Parametric Regression on the Grassmannian
We address the problem of fitting parametric curves on the Grassmann manifold
for the purpose of intrinsic parametric regression. As customary in the
literature, we start from the energy minimization formulation of linear
least-squares in Euclidean spaces and generalize this concept to general
nonflat Riemannian manifolds, following an optimal-control point of view. We
then specialize this idea to the Grassmann manifold and demonstrate that it
yields a simple, extensible and easy-to-implement solution to the parametric
regression problem. In fact, it allows us to extend the basic geodesic model to
(1) a time-warped variant and (2) cubic splines. We demonstrate the utility of
the proposed solution on different vision problems, such as shape regression as
a function of age, traffic-speed estimation and crowd-counting from
surveillance video clips. Most notably, these problems can be conveniently
solved within the same framework without any specifically-tailored steps along
the processing pipeline.Comment: 14 pages, 11 figure
About the Algebraic Solutions of Smallest Enclosing Cylinders Problems
Given n points in Euclidean space E^d, we propose an algebraic algorithm to
compute the best fitting (d-1)-cylinder. This algorithm computes the unknown
direction of the axis of the cylinder. The location of the axis and the radius
of the cylinder are deduced analytically from this direction. Special attention
is paid to the case d=3 when n=4 and n=5. For the former, the minimal radius
enclosing cylinder is computed algebrically from constrained minimization of a
quartic form of the unknown direction of the axis. For the latter, an
analytical condition of existence of the circumscribed cylinder is given, and
the algorithm reduces to find the zeroes of an one unknown polynomial of degree
at most 6. In both cases, the other parameters of the cylinder are deduced
analytically. The minimal radius enclosing cylinder is computed analytically
for the regular tetrahedron and for a trigonal bipyramids family with a
symmetry axis of order 3.Comment: 13 pages, 0 figure; revised version submitted to publication
(previous version is a copy of the original one of 2010
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