4,695 research outputs found
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
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
A Riemannian View on Shape Optimization
Shape optimization based on the shape calculus is numerically mostly
performed by means of steepest descent methods. This paper provides a novel
framework to analyze shape-Newton optimization methods by exploiting a
Riemannian perspective. A Riemannian shape Hessian is defined yielding often
sought properties like symmetry and quadratic convergence for Newton
optimization methods.Comment: 15 pages, 1 figure, 1 table. Forschungsbericht / Universit\"at Trier,
Mathematik, Informatik 2012,
Dynamic Facial Expression Generation on Hilbert Hypersphere with Conditional Wasserstein Generative Adversarial Nets
In this work, we propose a novel approach for generating videos of the six
basic facial expressions given a neutral face image. We propose to exploit the
face geometry by modeling the facial landmarks motion as curves encoded as
points on a hypersphere. By proposing a conditional version of manifold-valued
Wasserstein generative adversarial network (GAN) for motion generation on the
hypersphere, we learn the distribution of facial expression dynamics of
different classes, from which we synthesize new facial expression motions. The
resulting motions can be transformed to sequences of landmarks and then to
images sequences by editing the texture information using another conditional
Generative Adversarial Network. To the best of our knowledge, this is the first
work that explores manifold-valued representations with GAN to address the
problem of dynamic facial expression generation. We evaluate our proposed
approach both quantitatively and qualitatively on two public datasets;
Oulu-CASIA and MUG Facial Expression. Our experimental results demonstrate the
effectiveness of our approach in generating realistic videos with continuous
motion, realistic appearance and identity preservation. We also show the
efficiency of our framework for dynamic facial expressions generation, dynamic
facial expression transfer and data augmentation for training improved emotion
recognition models
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
A relaxed approach for curve matching with elastic metrics
In this paper we study a class of Riemannian metrics on the space of
unparametrized curves and develop a method to compute geodesics with given
boundary conditions. It extends previous works on this topic in several
important ways. The model and resulting matching algorithm integrate within one
common setting both the family of -metrics with constant coefficients and
scale-invariant -metrics on both open and closed immersed curves. These
families include as particular cases the class of first-order elastic metrics.
An essential difference with prior approaches is the way that boundary
constraints are dealt with. By leveraging varifold-based similarity metrics we
propose a relaxed variational formulation for the matching problem that avoids
the necessity of optimizing over the reparametrization group. Furthermore, we
show that we can also quotient out finite-dimensional similarity groups such as
translation, rotation and scaling groups. The different properties and
advantages are illustrated through numerical examples in which we also provide
a comparison with related diffeomorphic methods used in shape registration.Comment: 27 page
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