743 research outputs found
Functional Liftings of Vectorial Variational Problems with Laplacian Regularization
We propose a functional lifting-based convex relaxation of variational
problems with Laplacian-based second-order regularization. The approach rests
on ideas from the calibration method as well as from sublabel-accurate
continuous multilabeling approaches, and makes these approaches amenable for
variational problems with vectorial data and higher-order regularization, as is
common in image processing applications. We motivate the approach in the
function space setting and prove that, in the special case of absolute
Laplacian regularization, it encompasses the discretization-first
sublabel-accurate continuous multilabeling approach as a special case. We
present a mathematical connection between the lifted and original functional
and discuss possible interpretations of minimizers in the lifted function
space. Finally, we exemplarily apply the proposed approach to 2D image
registration problems.Comment: 12 pages, 3 figures; accepted at the conference "Scale Space and
Variational Methods" in Hofgeismar, Germany 201
Macroscopic contact angle and liquid drops on rough solid surfaces via homogenization and numerical simulations
We discuss a numerical formulation for the cell problem related to a homogenization
approach for the study of wetting on micro rough surfaces. Regularity properties of the solution are
described in details and it is shown that the problem is a convex one. Stability of the solution with
respect to small changes of the cell bottom surface allows for an estimate of the numerical error, at least
in two dimensions. Several benchmark experiments are presented and the reliability of the numerical
solution is assessed, whenever possible, by comparison with analytical one. Realistic three dimensional
simulations confirm several interesting features of the solution, improving the classical models of study
of wetting on roughness
On the estimation of the Wasserstein distance in generative models
Generative Adversarial Networks (GANs) have been used to model the underlying
probability distribution of sample based datasets. GANs are notoriuos for
training difficulties and their dependence on arbitrary hyperparameters. One
recent improvement in GAN literature is to use the Wasserstein distance as loss
function leading to Wasserstein Generative Adversarial Networks (WGANs). Using
this as a basis, we show various ways in which the Wasserstein distance is
estimated for the task of generative modelling. Additionally, the secrets in
training such models are shown and summarized at the end of this work. Where
applicable, we extend current works to different algorithms, different cost
functions, and different regularization schemes to improve generative models.Comment: Accepted and presented at GCPR 2019 (http://gcpr2019.tu-dortmund.de/
Revisiting energy release rates in brittle fracture
International audienceWe revisit in a 2d setting the notion of energy release rate, which plays a pivotal role in brittle fracture. Through a blow-up method, we extend that notion to crack patterns which are merely closed sets connected to the crack tip. As an application, we demonstrate that, modulo a simple meta-stability principle, a moving crack cannot generically kink while growing continuously in time. This last result potentially renders obsolete in our opinion a longstanding debate in fracture mechanics on the correct criterion for kinking
Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization
We present a practical implementation of an optimal first-order method, due
to Nesterov, for large-scale total variation regularization in tomographic
reconstruction, image deblurring, etc. The algorithm applies to -strongly
convex objective functions with -Lipschitz continuous gradient. In the
framework of Nesterov both and are assumed known -- an assumption
that is seldom satisfied in practice. We propose to incorporate mechanisms to
estimate locally sufficient and during the iterations. The mechanisms
also allow for the application to non-strongly convex functions. We discuss the
iteration complexity of several first-order methods, including the proposed
algorithm, and we use a 3D tomography problem to compare the performance of
these methods. The results show that for ill-conditioned problems solved to
high accuracy, the proposed method significantly outperforms state-of-the-art
first-order methods, as also suggested by theoretical results.Comment: 23 pages, 4 figure
Power calculation for gravitational radiation: oversimplification and the importance of time scale
A simplified formula for gravitational-radiation power is examined. It is
shown to give completely erroneous answers in three situations, making it
useless even for rough estimates. It is emphasized that short timescales, as
well as fast speeds, make classical approximations to relativistic calculations
untenable.Comment: Three pages, no figures, accepted for publication in Astronomische
Nachrichte
Korn and Poincare-Korn inequalities for functions with a small jump set
In this paper we prove a regularity and rigidity result for displacements in GSBDp, for every p > 1 and any dimension n ≥ 2. We show that a displacement in GSBDp with a small jump set coincides with a W1,p function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincar´e-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in GSBDp
A parametric level-set method for partially discrete tomography
This paper introduces a parametric level-set method for tomographic
reconstruction of partially discrete images. Such images consist of a
continuously varying background and an anomaly with a constant (known)
grey-value. We represent the geometry of the anomaly using a level-set
function, which we represent using radial basis functions. We pose the
reconstruction problem as a bi-level optimization problem in terms of the
background and coefficients for the level-set function. To constrain the
background reconstruction we impose smoothness through Tikhonov regularization.
The bi-level optimization problem is solved in an alternating fashion; in each
iteration we first reconstruct the background and consequently update the
level-set function. We test our method on numerical phantoms and show that we
can successfully reconstruct the geometry of the anomaly, even from limited
data. On these phantoms, our method outperforms Total Variation reconstruction,
DART and P-DART.Comment: Paper submitted to 20th International Conference on Discrete Geometry
for Computer Imager
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