# CNRS Défi Imag’In Cavalieri

### CAlcul des Variations pour L'Imagerie, l'Edition et la Recherche
d'Images

Workshop
on Optimal Transport and Optimization in Imaging

### Tuesday 11th

*9h45-10h30: C.-B.
Schönlieb.**Combined
modelling of optimal transport and segmentation.*
Abstract

Coffee
break

* *

*11h00-11h45: ** **N.
Bonneel.**Title.*
Abstract

*
** 11h45-12h00: L.
Chizat. **Title.*
Abstract

Lunch
break

*14h00-14h45: ** F. Iutzeler. **Modified
fixed points iterations and applications to randomized and
accelerated optimization algorithms.*
Abstract

Monotone operators have shown to be very interesting mathematical
objects and enabled the derivation of efficient algorithms. In
this talk, we will rely on simple fixed points iterations of an
operator with some contraction property enabling monotone
convergence to some fixed point. Then, we will see how
deterministic and stochastic variations of these iterations enable
to derive randomized or accelerated versions of popular algorithms
such as the proximal gradient or ADMM. More precisely, we will
focus on i) randomization to develop stochastic optimization
algorithms; and ii) inertia to accelerate the convergence.

*14h45-15h30: **
C. Boyer. **Adapting
to unknown noise level in super-resolution**.*
Abstract

We study sparse spikes deconvolution over the space of
complex-valued measures when the input measure is a finite sum of
Dirac masses. We introduce a new procedure to handle the spike
deconvolution when the noise level is unknown. Prediction and
localization results will be presented for this approach. An
insight on the probabilistic tools used in the proofs could be
briefly given as well.

*15h30-16h00: **
C. Poon. * *Geometric
properties of solutions to the total variation denoising
problem.* Abstract

Total variation (TV) denoising has been extensively studied in
imaging sciences since its introduction by Rudin, Osher and Fatemi
in 1992. However, the majority of theoretical works TV
regularization are actually concerned with solutions to total
variation denoising in the absence of noise. For instance, it is
known that the jump set of the regularized solutions are contained
in the jump set of the original data, but this knowledge is fairly
meaningless in the presence of noise since the noisy function can
introduce arbitrarily many new discontinuities. Furthermore, works
that consider the impact of noise on TV regularized solutions
typically derive integral estimates such as L^2 error bounds or
bounds on the total variation of the solution. However, such
estimates do not inform on the proximity of the gradient support
of the regularized solution to that of the clean function.

This talk is concerned with the impact of noise on the regularized
solutions. In particular, we consider stability in the gradient
support of solutions and address the following question: Given a
small neighbourhood around the support of Df, when will the
gradient support of the regularized solution be contained inside
this neighbourhood if the noise function has sufficiently small
L^2 norm?

Coffee
break

*16h30-17h15: ** B. Wirth.**
Optimal design of transport networks.*
Abstract

*
*

Several applications in biology and engineering are concerned
with the optimal transportation of substances from source
locations to sink locations. As opposed to classical optimal
transport, models for transport networks take into account that
it is more efficient to transport material in bulk. The
resulting optimal transport networks typically have a branching
structure. We discuss different model formulations and their
equivalence as well as the geometry of (almost) optimal
networks, which can be analyzed by proving energy scaling laws
in the regime of small preference for bulk transport.

*
**17H15-17h45: ** E.
Cazelles. **Regularized
Wasserstein barycenter.* Abstract*
*

The concept of barycenter in the Wasserstein space allows the
dention of a notion of Frechet mean of a set of probability
measures. However, depending on the considered data, such
barycenters may be irregular. In this paper, we thus introduce a
convex regularization of Wasserstein barycenters for random
measures supported on ℝ^{d}.
We prove the existence and uniqueness of such barycenters for a
large class of regularizing functions. A stability result of
regularized barycenters in terms of Bregman divergence
associated to the convex regularization term is also given.
Hence, we study the case of data made of i.i.d. random
probability measures. In particular, we prove the convergence in
Bregman divergence of the regularized empirical barycenter of a
set of n random probability measures towards its population
counterpart, and we discuss its rate of convergence. This
approach is shown to be appropriate for the study of discrete or
absolutely continuous random measures. We then focus on the
analysis of probability measures supported on the real line. In
this setting, we propose an ecient minimization algorithm based
on accelerated gradient descent for the computation of
regularized Wasserstein barycenters. This approach is nally
applied to the statistical analysis of simulated and real data
sets.

*17H45-18h15: ** M.
Schmitz. **Not available.*
Abstract

Accounting for and correcting an instrument’s Point Spread
Function (PSF) is a key element in many astronomical missions,
including the upcoming Euclid survey where one of the key
objectives is to perform galaxy shape measurements for weak
lensing. Because of the non-linear way in which the PSF varies
across the field of view, and the way in which Optimal Transport
distances account for geometric warping, using them for
PSF-related problems is a very promising prospect. Using recent
numerical schemes to compute approximate Wasserstein distances,
and thus Wasserstein Barycentres (Sliced Wasserstein Barycentres
and entropic penalty), we perform feature learning on images,
and apply it in particular to PSFs and galaxy images.

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### Wednesday 12th

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* *

*9h45-10h30: B.
Levy. **
**Semi-discrete
Optimal Transport in 3D: A Numerical Algorithm and Some
Applications.* Abstract

Semi-discrete algorithms seem to be a promising avenue for
developing efficient numerical methods for solving the
Monge-Ampère equation: Optimal Transport between a
piecewise linear density and a sum of Dirac masses can be
directly translated into a numerical algorithm, that maximizes a
concave function [Alexandrov, Aurenhammer, Mc Cann & Gangbo,
Merigot]. The gradient and Hessian of this objective function
involve integrals over the intersection between a Laguerre
diagram and the tetrahedral mesh that supports the input source
piecewise linear density. I will present an efficient algorithm
to compute these integrals, as well as its use by a Newton-type
solver [Kitagawa,Merigot,Thibert], and some applications in
fluid dynamics [Gallouet Merigot].

Coffee break

*11h00-11h45: **A.
Gramfort. **Optimal
transport for statistics problem in neuroimaging.*
Abstract

*
*

Knowing how the Human brain is anatomically and functionally
organized at the level of a group of healthy individuals or
patients is the primary goal of neuroimaging research. Yet a
number of statistical challenges emerge when processing
functional brain imaging data such as functional MRI (fMRI),
electroencephalography (EEG) or magnetoencephalography
(MEG).

One of them is computing an average of brain imaging data
defined over a voxel grid or a triangulation in order to
estimate a prototypical brain activation map for the
population. The problem is that data are large, the geometry
of the brain is complex and the between subjects variability
leads to spatially or temporally non-overlapping activation
foci. To address the problem of variability, data are
commonly smoothed before averaging, but this goes against
the objective of brain research to achieve better
spatiotemporal resolution. In this talk, I will show how
using optimal transport and Wasserstein barycenters one can
average fMRI and MEG brain activations without blurring the
results. A specifity of the approach is to be able to handle
non-normalized data.

In a second part of the talk, I will present some work in
progress where we use optimal transport with entropic
smoothing and some capacity constraint to address the
problem of domain adaptation (DA) in machine learning. In
biomedical applications shifts in distributions between
train data and test data are quite common. For example test
data could be coming from a different machine or from a very
different cohort of patients. This is the problem that DA
aims to solve and we propose to use optimal transport for
this.

*
**11h45-12h15: ** K.
Modin*.*The
polar decomposition as the limit of a lifted entropy
gradient flow.* Abstract

The work of Yann Brenier shows that optimal mass transport
(OMT) in the L^{2} sense is intrinsically tied to
polar decompositions. As a toy problem, one may consider OMT
in the category of multivariate Gaussian densities and the
associated polar decomposition A=PQ of real matrices. In
this talk I show that the P component can be obtained as the
limit of a gradient flow with respect to a lifted relative
entropy functional, thereby providing a new, geometric proof
of the polar decomposition of matrices. I shall also present
a corresponding infinite-dimensional lifted entropy gradient
flow on the space of convex functions. Here, the first
component in the polar decomposition of maps is an
equilibrium. Furthermore, I give a result hinting that the
infinite-dimensional flow has a unique limit in the smooth
category if the destination marginal is log-convex.

Lunch
break

*14h00-14h45: ** E.
Chouzenoux. * *Convergence
analysis of a stochastic majorize-minimize memory gradient
algorithm.* Abstract

Stochastic approximation techniques play a prominent role in
solving many large scale problems encountered in machine
learning or image/signal processing. In these contexts, the
statistics of the data are often unknown a priori or their
direct computation is too intensive, and they have thus to
be estimated online from the observations. For batch
optimization of an objective function being the sum of a
data fidelity term and a penalization (e.g. a sparsity
promoting function), Majorize-Minimize (MM) methods have
recently attracted much interest since they are fast, highly
flexible, and effective in ensuring convergence. The goal of
this work is to show how these methods can be successfully
extended to the case when the data fidelity term corresponds
to a least squares criterion and the cost function is
replaced by a sequence of stochastic approximations of it.
In this context, we propose an online version of an MM
subspace algorithm and we establish its convergence by using
suitable probabilistic tools. We also provide new results on
the convergence rate of such kind of algorithm. Numerical
results illustrate the good practical performance of the
proposed algorithm associated with a memory gradient
subspace, when applied to both non-adaptive and adaptive
linear system identification scenarios.

*
**14h45-15h15: **
P. Tan. * *Accelerated
Alternating Descent Methods for Dystra-like Problems.*
Abstract

In this talk, we present an alternating descent scheme for
problems involving nonsmooth objectives with a quadratic
coupling of the variables. Our algorithm performs in each
variable several descent steps, which improves the
performances compared to a single step. Thanks to a
FISTA-like trick, this scheme can also be accelerated, An
application of this work is the implementation of a fast
parallel solver for the proximity operator of the Total
Variation for color images. In this case, this method can be
interpreted as an inexact scheme.

*15h15-15h45: **
Q. Denoyelle. * *Title.*
Abstract

Coffee
break and conclusion of the Workshop

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