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Timetable (VMVW03)

Flows, mappings and shapes

Monday 11th December 2017 to Friday 15th December 2017

Monday 11th December 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (INI Director)
10:00 to 11:00 Nir Sochen
Point correspondences in the functional map framework
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Olga Veksler
Convexity and Other Shape priors for Single and Multiple Object Segmentation
Shape is a useful regularization prior for image segmentation. First we will talk about convexity shape prior for single object segmentation. In the context of discrete optimization, object convexity is represented as a sum of 3-clique potentials penalizing any 1-0-1 configuration on all straight lines. We show that these nonsubmodular interactions can be efficiently optimized using a trust region approach. While the quadratic number of all 3-cliques is prohibitively high, we designed a dynamic programming technique for evaluating and approximating these cliques in linear time. Our experiments demonstrate general usefulness of the proposed convexity constraint on synthetic and real image segmentation examples. Unlike standard second order length regularization, our convexity prior is scale invariant, does not have shrinking bias, and is virtually parameter-free. Segmenting multiple objects with convex shape prior presents its own challenges as distinct objects interact in non-trivial manner. We extend our work on single convex object optimization by proposing a mutli-object convexity shape prior for multi-label image segmentation. Next we consider simple shape priors, i.e. priors that can be optimized exactly with a single graph cut in the context of single object segmentation. Segmenting multiple objects with such simple shape priors presents its own challenges. We propose a new class of energies for segmentation of multiple foreground objects with a common simple shape prior. Our energy involves infinity constraints. For such energies standard expansion algorithm has no optimality guarantees and in practice gets stuck in bad local minima. Therefore, we develop a new move making algorithm, we call double expansion. In contrast to expansion, the new move allows each pixel to choose a label from a pair of new labels or keep the old label. This results in an algorithm with optimality guarantees and robust performance in practice. We experiment with several types of shape prior such as star-shape, compactness and a novel symmetry prior, and empirically demonstrate the advantage of the double expansion.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Emanuele Rodolà
Spectral approaches to partial deformable 3D shape correspondence
In this talk we will present our recent line of work on (deformable) partial 3D shape correspondence in the spectral domain. We will first introduce Partial Functional Maps (PFM), showing how to robustly formulate the shape correspondence problem under missing geometry with the language of functional maps. We use perturbation analysis to show how removal of shape parts changes the Laplace-Beltrami eigenfunctions, and exploit it as a prior on the spectral representation of the correspondence. We will show further extensions to deal with the presence of clutter (deformable object-in-clutter) and multiple pieces (non-rigid puzzles). In the second part of the talk, we will introduce a novel approach to the same problem which operates completely in the spectral domain, avoiding the cumbersome alternating optimization used in the previous approaches. This allows matching shapes with constant complexity independent of the number of shape vertices, and yields state-of-the-art results on challenging correspondence benchmarks in the presence of partiality and topological noise. Authors: E. Rodola, L. Cosmo, O. Litany, J. Masci, A. Bronstein, M. Bronstein, A. Torsello, D. Cremers
14:30 to 15:30 Wei Zhu
Euler's elastica based segmentation models and the fast algorithms
In this talk, we will discuss two image segmentation models that employ L^1 and L^2 Euler's elastica respectively as the regularization of active contour. When compared with the conventional contour length based regularization, these high order regularizations lead to new features, including connecting broken parts of objects automatically and being well-suited for fine elongate structures. More interestingly, with the L^1 Euler's elastica as the contour regularization, the segmentation model is able to single out objects with convex shapes. We will also discuss the fast algorithms for dealing with these models by using augmented Lagrangian method. Numerical experiments will be presented to illustrate the features of these Euler's elastica based segmentation models.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Andrew Zisserman
3D Shape Inference from Images using Deep Learning
The talk will cover two approaches to obtaining 3D shape from images. First, we introduce a deep Convolutional Neural Network (ConvNet) architecture that can generate depth maps given a single or multiple images of an object. The ConvNet is trained using a prediction loss on both the depth map and the silhouette. Using a set of sculptures as our 3D objects, we show that the ConvNet is able to generalize to new objects, unseen during training, and that its performance improves given more input views of the object. This is joint work with Olivia Wiles. Second, we use ConvNets to infer 3D shape attributes, such as planarity, symmetry and occupied space, from a single image. For this we have assembled an annotated dataset of 150K images of over 2000 different sculptures. We show that 3D attributes can be learnt from these images and generalize to images of other (non-sculpture) object classes. This is joint work with Abhinav Gupta and David Fouhey.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 12th December 2017
09:00 to 10:00 Laurent Younes
Riemannian Diffeomorphic Mapping and Some Applications
We review a few applications of large deformation diffeomorphic metric mapping and some of its variants, within a sub-Riemannian framework in diffeomorphism groups and shape spaces. After describing the basic principles, the talk will focus on applications in the construction of laminar coordinates in the cortical ribbon, on the quantification of fine motor tasks in children through letter tracing and on the statistical estimation of a changepoint in brain shape evolution for Alzheimer’s disease.
10:00 to 11:00 Lok Ming Lui
Recent advances of Computational Quasiconformal Geometry in Imaging, Graphics and Visions
Computational quasiconformal geometry (CQC) has recently attracted much attention and found successful applications in various fields, such as imaging, computer graphics and visions. In this talk, I will give an overview on the recent advances of CQC. More specifically, I will talk about how quasiconformal structures can be efficiently and accurately computed on different surface representations, such as meshes and point clouds. Applications of CQC in medical imaging and visions will also be discussed. Finally, the possibility to extend CQC to higher dimensions will also be examined. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Lourdes Agapito
Capturing 3D models of deformable objects from monocular sequences
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Maks Ovsjanikov
Efficient regularization of functional map computations
In this talk, I will give a brief overview of the functional map framework and then describe some recent approaches that allow to incorporate both geometric and topological constraints into functional map computations. Namely, I will discuss a method to obtain functional maps that follow structural properties of pointwise correspondences, ways to encode embedding-dependent (second fundamental form) information and finally a technique to efficiently compute bi-directional correspondences using functional map adjoints.
14:30 to 15:30 Weihong Guo
Simultaneous Image Segmentation and Registration and Applications
Image segmentation and registration play active roles in machine vision and image analysis. In particular, image registration helps segmenting images when they have low contrast and/or partial missing information. We explore the joint problem of segmenting and registering a template (e.g. current) image given a reference (e.g. past) image. We solve the joint problem by minimizing a functional that integrates Geodesic Active Contours and Nonlinear Elastic registration. The template image is modeled as a hyper-elastic material (St. Venant-Kirchho model) which undergoes deformations under applied forces. To segment the deforming template, a two-phase level set based energy is introduced together with a weighted total variation term that depends on gradient features of the deforming template. This particular choice allows for fast solution using the dual formulation of the total variation. This allows the segmenting front to accurately track spontaneous changes in the shape of objects embedded in the template image as it deforms. To solve the underlying registration problem we use gradient descent and adopt an implicit-explicit method and use the Fast Fourier Transform. This is a joint work with former PhD student Thomas Atta-Fosu.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Carl Olsson
Compact Rank Models and Optimization
Wednesday 13th December 2017
09:00 to 10:00 Yuri Boykov
Low-order graphical models for shapes and hierarchies in segmentation
This talks discusses simple (low-order) graphical models imposing practically powerful constraints on shapes and hierarchical structure of segments in the context of binary and multi-object labeling of images. We discuss properties and optimization for generic shape priors like "star", geodesic star, hedgehog, as well as models for partially-ordered labeling of interacting objects. While the talk focuses on biomedical applications where structural constraints (shapes and hierarchy) come from anatomy, the discussed general graphical models are useful for semi-supervised computer vision problems. Related papers appeared at CVPR 2017, 2016, ICCV 2009, 2005, ECCV 2008.
10:00 to 11:00 Tieyong Zeng
Two-Stage/Three-Stage Method for Image Segmentation
The Mumford–Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this talk, we propose a two-stage segmentation method based on the Mumford–Shah model. The first stage of our method is to find a smooth solution to a convex variant of the Mumford–Shah model. In the second stage the segmentation is done by thresholding the previous image into different phases. Experimental results show the good performance of the proposed method. The idea is then generalized for image segmentation under non-Gaussian noise, color image segmentation and selective image segmentation for medical images.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Stephen Marsland
Langevin equations for landmark image registration with uncertainty
Co-author: Tony Shardlow (University of Bath)

Pairs of images can be brought into alignment (registered) by finding corresponding points on the two images and deforming one of them so that the points match. This can be carried out as a Hamiltonian boundary-value problem, and then provides a diffeomorphic registration between images. However, small changes in the positions of the landmarks can produce large changes in the resulting diffeomorphism. We formulate a Langevin equation for looking at small random perturbations of this registration. The Langevin equation and three computationally convenient approximations are introduced and used as prior distributions. A Bayesian framework is then used to compute a posterior distribution for the registration, and also to formulate an average of multiple sets of landmarks. 
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Christ's College
Thursday 14th December 2017
09:00 to 10:00 Peter Michor
Soliton solutions for the elastic metric on spaces of curves
Joint work with: Martin Bauer (Florida State University), Martins Bruveris (Brunel University London), Philipp Harms (University of Freiburg). Abstract: Some first order Sobolev metrics on spaces of curves admit soliton-like geodesics, i.e., geodesics whose momenta are sums of delta distributions. It turns out that these geodesics can be found within the submanifold of piecewise linear curves, which is totally geodesic for these metrics. Consequently, the geodesic equation reduces to a finite-dimensional ordinary differential equation for a dense set of initial conditions.
10:00 to 11:00 Gabriel Peyre
Optimal Transport and Deep Generative Models
Co-authors: Marco Cuturi (ENSAE), Aude Genevay (ENS)

In this talk, I will review some recent advances on deep generative models through the prism of Optimal Transport (OT). OT provides a way to define robust loss functions to perform high dimensional density fitting using generative models. This defines so called Minimum Kantorovitch Estimators (MKE) [1]. This approach is especially useful to recast several unsupervised deep learning methods in a unifying framework. Most notably, as shown respectively in [2,3] (and reviewed in [4]) Variational Autoencoders (VAE) and Generative Adversarial Networks (GAN) can be interpreted as (respectively primal and and dual) approximate MKE. This is a joint work with Aude Genevay and Marco Cuturi.

References: [1] Federico Bassetti, Antonella Bodini, and Eugenio Regazzini. On minimum Kantorovich distance estimators. Statistics & probability letters, 76(12):1298–1302, 2006. [2] Olivier Bousquet, Sylvain Gelly, Ilya Tolstikhin, Carl-Johann Simon-Gabriel, and Bernhard Schoelkopf. From optimal transport to generative modeling: the VEGAN cookbook. Arxiv:1705.07642, 2017. [3] Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein GAN. Arxiv:1701.07875, 2017. [4] Aude Genevay, Gabriel Peyré, Marco Cuturi, GAN and VAE from an Optimal Transport Point of View, Arxiv:1706.01807, 2017

Related Links
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Alex Bronstein
Geometry and learning in 3D correspondence problems
The need to compute correspondence between three-dimensional objects is a fundamental ingredient in numerous computer vision and graphics tasks. In this talk, I will show how several geometric notions related to the Laplacian spectrum provide a set of tools for efficiently calculating correspondence between deformable shapes. I will also show how this framework combined with recent ideas in deep learning promises to bring correspondence problems to new levels of accuracy.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 John Ball
Nonlinear elasticity and image processing
A survey of nonlinear elasticity will be given with a view to possible applications in image processing. Then some particular image processing issues arising from recent experiments on low hysteresis alloys will be described.
14:30 to 15:30 Christopher Zach
When to lift (a function to higher dimensions) and when not
In the first part of my talk I will describe several instances where reformulating a difficult optimization problem into higher dimensions (i.e. enlarge the set of minimized variables) is beneficial. My particular interest are robust cost functions e.g. utilized for correspondence search, which serve as a prototype for general difficult minimization problems. In the second part I will describe problem instances of relevance especially in 3D computer vision, where reducing the set of involved variables (i.e. the opposite of lifting) is highly beneficial. In particular, I will clarify the relationship between variable projection methods and the Schur complement often employed in Gauss-Newton based algorithms. Joint work with Je Hyeong Hong and Andrew Fitzgibbon.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Darryl Holm
Stochastic Metamorphosis in Imaging Science
In the pattern matching approach to imaging science, the process of metamorphosis in template matching with dynamical templates was introduced in [7]. In [5] the metamorphosis equations of [7] were recast into the Euler-Poincar ́e variational framework of [4] and shown to contain the equations for a perfect complex fluid [3].   

This result related the data structure underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids [2]. In particular, it cast the concept of Lagrangian paths in imaging science as deterministically evolving curves in the space of diffeomorphisms acting on image data structure, expressed in Eulerian space. In contrast, the landmarks in the standard LDDMM approach are Lagrangian.  

For the sake of introducing an Eulerian uncertainty quantification approach in imaging science, we extend the method of metamorphosis to apply to image matching along stochastically evolving time dependent curves on the space of diffeomorphisms. The approach IS guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in [6, 1].  

[1] D. O. Crisan, F. Flandoli, and D. D. Holm. Solution properties of a 3D stochastic Euler fluid equation. arXiv preprint arXiv:1704.06989, 2017. URL
[2] F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu. Geometric dynamics of optimization. Comm. in Math. Sciences, 11(1):163–231, 2013. [3] D. D. Holm. Euler-Poincaré dynamics of perfect complex fluids. In P. Newton, P. Holmes, and A. Weinstein, editors, Geometry, Mechanics, and Dynamics: in honor of the 60th birthday of Jerrold E. Marsden, pages 113–167. Springer, 2002. [4] D. D. Holm, J. E. Marsden, and T. S. Ratiu. The Euler–Poincar ́e equations and semidirect products with applications to continuum theories. Adv. in Math., 137:1–81, 1998. [5] D. D. Holm, A. Trouvé, and L. Younes. The Euler-Poincar ́e theory of metamorphosis. Quarterly of Applied Mathematics, 67:661–685, 2009. [6] Darryl D Holm. Variational principles for stochastic fluid dynamics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2176):20140963, 2015. [7] A. Trouvé and L. Younes. Metamorphoses through Lie group action. Found. Comp. Math., 173–198, 2005.
Friday 15th December 2017
09:00 to 10:00 Zachary Boyd
Formulations of community detection in terms of total variation and surface tension
Network data structures arise in numerous applications, e.g. in image segmentation when graph cut methods are used or in the form of a similarity graph on the pixels in certain clustering methods. Networks also occur as social, biological, technological, and transportation networks, for instance, all of which are receiving a lot of attention right now. "Community detection" is a body of techniques for extracting large- and medium-scale structure from such graphs. Most community detection formalizations turn out to be NP-hard and in practice are horrendously nonconvex. Practitioners from many fields are struggling to find formulations that (1) helpfully summarize the network data and (2) are computationally tractable. Most formulations have neither property. In my talk, I will give two examples of how existing community detection models can be understood in terms of objects familiar in image processing. The first example casts the popular modularity heuristic as a graph total variation problem with a soft area balance constraint. The second views the more flexible stochastic block model as a discrete surface tension minimization problem, which in the two-community case is exactly equivalent to the first example. These equivalences can potentially benefit both the network science community and the image processing community by allowing tools from one domain to be applied to the other. As an example, I show how mean curvature flow, phase field, and threshold dynamics approaches to continuum total variation minimization can be adapted to community detection in graphs, including nonlocal means graphs for hyperspectral images and videos. The positive results hint that the methods commonly used in image processing can be readily applied to much more general problems involving arbitrary graph structures. I will also mention some possible future work in the reverse direction, where I would like to bring methods from the network science literature into image processing. This is joint work with Egil Bae, Andrea Bertozzi, Mason Porter, and Xue-Cheng Tai.
10:00 to 11:00 Elaine Crooks
Compensated convexity, multiscale medial axis maps, and sharp regularity of the squared distance function
Co-authors: Kewei Zhang (University of Nottingham, UK), Antonio Orlando (Universidad Nacional de Tucuman, Argentina)

Compensated convex transforms enjoy tight-approximation and locality properties that can be exploited to develop multiscale, parametrised methods for identifying singularities in functions. When applied to the squared distance function to a closed subset of Euclidean space, these ideas yield a new tool for locating and analyzing the medial axis of geometric objects, called the multiscale medial axis map. This consists of a parametrised family of nonnegative functions that provides a Hausdorff-stable multiscale representation of the medial axis, in particular producing a hierarchy of heights between different parts of the medial axis depending on the distance between the generating points of that part of the medial axis. Such a hierarchy enables subsets of the medial axis to be selected by simple thresholding, which tackles the well-known stability issue that small perturbations in an object can produce large variations in the corresponding medial axis. A sharp regularity resu lt for the squared distance function is obtained as a by-product of the analysis of this multiscale medial axis map.

This is joint work with Kewei Zhang (Nottingham) and Antonio Orlando (Tucuman). 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 François-Xavier Vialard
An interpolating distance between Wasserstein and Fisher-Rao
In this talk, we present the natural extension of the Wasserstein metric to the space of positive Radon measures. We present the dynamic formulation and we show its associated static formulation. Then, we relate this new metric to the Camassa-Holm equation and show that this Camassa-Holm equation is actually an incompressible Euler equation in higher dimensions. We also present some applications of this new metric as a similarity measure in inverse problems in imaging.
12:30 to 13:30 Lunch @ Wolfson Court
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons