Title and Abstract

1David Mumford, Brown University

(a)Plenary talk:  The Partnership of Pure and Applied Mathematics

Abstract: Pure and applied math are Siamese (conjoined) twins -- they cannot live without each other. I want to start with some examples from early Chinese and early Western math to illustrate how vital each has been for the other. Then I will describe two areas where computer vision interacts with other areas of math, both pure and applied, to show how productive this interaction remains. But their partnership recently has not been easy and I want to end with some words of caution.

 

(b)Title: What is the 'right' Mathematical Model for an Image of the World?

Abstract: How can we analyze and write applications for images unless we first know 'what' an image is, mathematically speaking. An image is not white noise, a random function; it has layers of structure. In this talk I start with basic statistical properties already carve out a highly constrained subset of white noise. Then I will talk about the grammar of images, their phonemes and syntax and the challenges this poses which are still not fully resolved.

 

(c)Title: Warping Shapes with Geodesics and an Ancient Chinese Map

Title: Among all the features people use to describe an image, the 'shape' of an object is arguably the hardest to quantify mathematically. The polymath Albrecht Durer was the first to attempt this with human bodies and now medical imaging drives the search for more precise and informative ways to describe shape. As we indicated in my Plenary talk, one approach is to make an infinite dimensional manifold out of the set of all shapes (2 or 3 dimensional). I will develop two instances of this: (i) the use of conformal maps to make a wonderful negatively curved model for 2 dimensional shapes and (ii) the use of so called 'landmark' points to reduce shapes to a finite dimensional space whose geometry we can describe very explicitly. Finally we apply this landmark theory to analyze the Yujitu, a Song dynasty map of China, carved in stone and preserved in Xian.

 

2Martin Bauer, University of Vienna

TitleThe geometry of   the diffeomorphism   group

Abstract: In the field of computational anatomy the diffeomorphism group plays a central role. The space of medical images is acted upon by the diffeomorphism group and differences between images are encoded by diffeomorphisms in the spirit of Grenander’s pattern theory. But also Euler’s equations and other PDEs arising in hydrodynamics can be realized as geodesic equations on the diffeomorphism group with respect to suitable Riemannian metrics. This both motivates the study of the Riemannian geometry of the diffeomorphism group, a currently active area of research. The questions considered include properties of the curvature and the geodesic distance, well-posedness of the geodesic equation and the dynamics of singular solutions.

In my talk I will discuss Sobolev-type metrics of (fractional) order s≥ 0 on the group Diffc(M) of compactly supported diffeomorphisms of a manifold M. I will show that for the important special case M =S1 the geodesic distance on Diffc(S1) vanishes if and only if . For other manifolds I will obtain a partial characterization of the induced geodesic distance. Furthermore I will show well-posedness of the corresponding geodesic equation under certain assumptions on the orders.

In the second part I will discuss one particular metric in more detail, namely the homogenous Sobolev metric of order one. I will show that the space Diffc1(R) equipped with this metric is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat L2-metric. Here Diffc1(R) denotes the extension of the group of all compactly supported diffeomorphisms, that allows for a shift towards infinity. In particular this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter Saxton equation.

 

3Martins Bruveris, Imperial College, London

Title: Constructing reparametrization invariant metrics on spaces of plane curves

Abstract: Riemannian metrics on shape space are used to describe deformations that take one shape to another and to define a distance between them. I will present a family of metrics on the space of curves that can be isometrically embedded into a vector space, where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space Imm(S1, R2) of regular, parametrized plane curves and on the quotient space Imm(S1, R2)/Diff(S1) of unparame-trized curves. For the space of open parametrized curves it is possible to give explicit formulas for the geodesic distance and to show that the sectional curvature vanishes on the space of parametrized curves. The aforementioned embedding can also be used to derive a numerical algorithm that computes geodesics between unparametrized, closed curves.

 

4Lo-Bin Chang, National Chiao Tung University

Title: Generative Models for Image Analysis and Computations

Abstract: A probabilistic grammar for the grouping and labeling of parts and objects, when taken together with pose and part-dependent appearance models, constitutes a generative scene model and a Bayesian framework for image analysis. To the extent that the generative model generates features, as opposed to pixel intensities, the posterior distribution (i.e. the conditional distribution on part and object labels given the image) is based on incomplete information; feature vectors are generally insufficient to recover the original intensities. In this talk, I will propose several ways to learn pixel-level models for the appearances of parts. I will introduce a perturbation method for building non- Markovian hierarchical prior models, and discuss the issues in terms of computation through an asymptotic analysis.

 

5Shu-Ming Chang and Mei-Heng Yueh, National Chiao-Tung University

Title: Discrete Surface Ricci Flows and Heat Flows with Applications

Abstract: All closed surfaces with Riemannian metric can be conformally embedded onto three canonical spaces: the sphere, the plane, and the hyperbolic space. In this talk first we focus on closed 2-dimensional surfaces with genus one to conformally embed onto the plane efficiently from a computational point of view. Given a mesh with genus one and then we propose a harmonic remeshing technique to get a Delaunay-like circle packing. In the discrete Euclidean surface Ricci flow we set suitable conditions on the boundary of the mesh, and solve a deflated solution in the algorithm of Newton's method for optimizing Ricci energy to obtain a fundamental domain with a rectangle shape. We act up- and right-reflections as the deck transformation group on the fundamental domain to obtain a universal covering space for the genus one surface. On the other hand in this talk we show some numerical methods of surface morphing by using the conformal mapping and the idea of homotopy. We uniformize the structure of each mesh by using the conformal parameterization, and construct the morphing function by using cubic spline homotopy. In order to control the morphing process as our desire, here comes a surface matching problem. From the Riemann mapping theorem, we know that there exists a unique conformal mapping from a simply connected surface into a unit disk. We may achieve the conformal mapping with the discrete heat flow. Therefore, we reduce a surface matching problem into a unit disk matching problem and solve this problem by using the weighted least square method.

 

6David Xianfeng Gu, Stony Brook University

Title: Computing Surface Shape Space and Mapping Space Using Ricci Flow

Abstract: All compact oriented metric surfaces can be classified by conformal (angle preserving) transformation group, each orbit is a Riemann surface. The quotient space is the Moduli space. From computational point view, the universal covering space of the Moduli space, Teichmuller space, is easier to manipulate. The map space between two Riemann surfaces is equivalent to the Beltrami differential space defined on the source.  The Teichmuller map, that minimizes the distortion of the conformal structure, gives the Riemannian metric of Teichmuller space.

In this talk, practical algorithms based on surface Ricci flow will be introduced, which can compute the Teichmuller coordinates of a surface, reconstruct the mapping from its Beltrami differential, and compute the Teichmuller distance between two shapes.

 

7Tan-Chi Ho, Shing-Tung Yau Center, National Chiao-Tung University

Title: Point Cloud Registration

Abstract: Modeling the real-world object by capturing the range images has drawn attention in recent years. The key ingredient is the registration of point cloud in which point-to-point correspondence is established for point clouds in different orientations or poses. In this talk, we will discuss the challenges in registration of static and dynamic point clouds as well as some recent progress in this field.

 

8Xiaoming Huo, Georgia Institute of Technology

Title: Detectability and Related Theorems

Abstract: We examine the problem in determining when certain type of underlying structures is detectable in noisy images. Certain testing methods will depend on the distribution of the length of the longest chains that connect locally significant hypotheses tests. I will describe some recent results in deriving the asymptotic distribution of these largest lengths. I will also describe some optimality guarantee of proposed detection methods.

 

9Eric Klassen, Florida State University

Title: A Complete Moduli Space for Absolutely Continuous Curves Modulo Reparameterization

Abstract: Using the square root velocity representation, we obtain a complete metric space structure on the space of absolutely continuous curves in R^n modulo reparameterization. For numerical purposes, the set of piecewise linear curves forms a useful dense subspace of this moduli space. We discuss methods of finding geodesics in this moduli space, its applications in shape theory and functional analysis, and its global geometry. We also discuss some advantages and disadvantages of this approach compared to other interesting approaches to this problem, especially those providing manifold structures on related moduli spaces. Finally, we discuss possible extensions of this method to absolutely continuous curves in Riemannian manifolds.

 

10Ping Li, Cornell University

Title: BigData: Probabilistic Hashing for Efficient Search and Learning in High-Dimensional Data

Abstract: This talk will present a series of work on probabilistic hashing methods which typically transform a challenging (or infeasible) massive data computational problem into a probability and statistical estimation problem. For example, fitting a logistic regression (or SVM) model on a dataset with billion observations and billion (or billion square) variables would be difficult. Searching for similar documents (or images) in a repository of billion web pages (or images) is another challenging example. In certain important applications in the search industry, a web page is often represented as a binary (0/1) vector in billion square (2 to power 64) dimensions. For those data, both data reduction (i.e., reducing number of nonzero entries) and dimensionality reduction are crucial for achieving efficient search and statistical learning.

This talk will present two closely related probabilistic methods: (1) b-bit minwise hashing and (2) one permutation hashing, which simultaneously perform effective data reduction and dimensionality reduction on massive, high-dimensional, binary data. For example, training an SVM for classification on a text dataset of size 24GB took only 3 seconds after reducing the dataset to merely 70MB using our probabilistic methods. Experiments on close to 1TB data will also be presented. Several challenging probability problems still remain open.

 

11Zhouchen Lin, School of EECS, Peking University

Title: Learning Partial Differential Equations for Computer Vision and Image Processing

Abstract: Many computer vision and image processing problems can be posed as solving partial differential equations (PDEs). However, designing PDE system usually requires high mathematical skills and good insight into the problems. In this paper, we consider designing PDEs for various problems arising in computer vision and image processing in a lazy manner: learning PDEs from training data via optimal control approach. We first propose a general intelligent PDE system which holds the basic translational and rotational invariance rule for most vision problems. By introducing a PDE-constrained optimal control framework, it is possible to use the training data resulting from multiple ways (ground truth, results from other methods, and manual results from humans) to learn PDEs for different computer vision tasks. The proposed optimal control based training framework aims at learning a PDE-based regressor to approximate the unknown (and usually nonlinear) mapping of different vision tasks. The experimental results show that the learnt PDEs can solve different vision problems reasonably well. In particular, we can obtain PDEs not only for problems that traditional PDEs work well but also for problems that PDE-based methods have never been tried before, due to the difficulty in describing those problems in a mathematical way.

 

12Ronald Lok Ming Lui, the Chinese University of Hong Kong

Title: Teichmuller Extremal Mapping and its Applications

Abstract: Registration, which aims to find an optimal 1-1 correspondence between shapes (or images), is important in different areas such as in computer vision and medical imaging. Conformal mappings have been widely used to obtain a diffeomorphism between shapes that minimizes angular distortion. Conformal registrations are beneficial since it preserves the local geometry well. However, when extra constraints (such as landmark constraints) are enforced, conformal mappings generally do not exist. This motivates us to look for an optimal quasi-conformal registration, which satisfies the required constraints while minimizing the conformality distortion. Under suitable condition on the constraints, a unique diffeomporphism, called the Teichmuller extremal mapping between two surfaces can be obtained, which minimizes the maximal conformality distortion. In this talk, an efficient iterative algorithm, called the Quasi-conformal (QC) iterations, to compute the Teichmuller mapping will be presented. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs), and look for an optimal BC associated to the desired Teichmuller mapping. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the Linear Beltrami Solver(LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration. Using the proposed method, the Teichmuller mapping can be accurately and efficiently computed within 10 seconds. The obtained registration is guaranteed to be bijective. This proposed algorithm can also be practically applied to real applications. In the second part of my talk, I will present how Teichmuller extremal mapping can be used for brain landmark matching registration, constrained texture mapping and face recognition.

 

13Yi Ma, University of Illinois at Urbana-Champaign

Title: The Pursuit of Low-dimensional Structures in High-dimensional Data

Abstract: In this talk, we will discuss a new class of models and techniques that can effectively model and extract rich low-dimensional structures in high-dimensional data such as images and videos, despite nonlinear transformation, gross corruption, or severely compressed measurements. This work leverages recent advancements in convex optimization for recovering low-rank or sparse signals that provide both strong theoretical guarantees and efficient and scalable algorithms for solving such high-dimensional combinatorial problems. These results and tools actually generalize to a large family of low-complexity structures whose associated regularizers are decomposable. We illustrate how these new mathematical models and tools could bring disruptive changes to solutions to many challenging tasks in computer vision, image processing, and pattern recognition. We will also illustrate some emerging applications of these tools to other data types such as web documents, image tags, microarray data, audio/music analysis, and graphical models.

 

This is joint work with John Wright of Columbia, Emmanuel Candes of Stanford, Zhouchen Lin of Peking University, and my students Zhengdong Zhang, Xiao Liang of Tsinghua University, Arvind Ganesh, Zihan Zhou, Kerui Min and Hossein Mobahi of UIUC.

 

14Peter W. Michor, University of Vienna

Title: Overview on Geometries of Shape Spaces and Diffeomorphism groups.

Abstract: For a compact manifold M and a manifold N, we consider the spaces of embeddings and of immersions of M into N, the regular Lie groups of diffeomorphisms of M and of those diffeomorphisms on N which fall rapidly to the identity, and the spaces of Riemannian metrics on M and on N.

There are several actions of the groups; The space of immersions modulo the group of reparamerizations is an orbifold (with finite isotropy groups). There are several natural weak Riemannian metric of the spaces of metrics. Fixing a Riemannian metric g of bounded geometry on N induces several weak Riemannian metrics on the spaces of embeddings and of immersions and on the diffeomorphism group of N.

I will review their interplays, their geodesic equations, what is known about well-posedness for them, and what is known about curvatures.

 

15Long Quan, the Hong Kong University of Science and Technology

Title: Image-based Modeling

Abstract: Image-based modeling is the confluence of Computer Vision and Computer Graphics. In the first part of the talk, I will review the state of the art of the three dimensional reconstruction from images or structure from motion. I will focus on the uncalibrated framework and the quasi-dense approach that we have developed for many years. The quasi-dense approach is unique in that it is the most suitable for building up the object modeling applications. In the second part of the talk, I will present a series of modeling applications we have been pursuing in the recent years. I will start with smooth surface modeling from images, then move to the prior-based hair and tree modeling. I will finish with the most recent building modeling for digital city applications.

 

16Dimitris Samaras, Stony Brook University

Title: Dense Non-rigid Surface Registration Using High-Order Graph Matching and Quasi-Conformal Geometry

Abstract: I will present our most recent techniques for non-rigid surface matching and tracking combining quasi-conformal mappings with efficient global optimization methods. I will first describe a high-order graph matching formulation to address non-rigid surface matching. The singleton terms capture the geometric and appearance similarities (e.g., curvature and texture) while the high-order terms model the intrinsic embedding energy.

The method includes: 1) casting 3D surface registration into a graph matching problem that combines both geometric and appearance similarities and intrinsic embedding information, 2) the first implementation of high-order graph matching algorithm that solves a non-convex optimization problem, and 3) an efficient two-stage optimization approach to constrain the search space for dense surface registration. Furthermore by considering the set of all possible 3D surface matchings defined by specifying triplets of correspondences in the uniformization domain, we introduce a new matching cost between two 3D surfaces, which can be efficiently computed for surface tracking applications. Our method is validated through a series of experiments demonstrating its accuracy and efficiency, notably in challenging cases of large and/or non-isometric deformations, or meshes that are partially occluded, as well as dense, anisometric 3D surface tracking experiments.

 

17Jian Sun, Tsinghua University

Title: Detection and approximation of linear structures from data

Abstract: In many real-world applications data come as discrete metric spaces sampled around 1-dimensional linear structures (metric trees or graphs). Building on elementary tools from topology, we provide a framework in which one can measure how much data can be approximated by 1-dimensional metric structures and we provide algorithms to reconstruct such approximating structures. We also implement our algorithm and illustrate its performances on various data sets.

This is a joint work with Frederic Chazal.

 

18John Wright, Columbia University

Title: Provably Effective Algorithms for Learning Visual Data Representations

Abstract: Learning good representations for high-dimensional visual data is fundamental challenge in computer vision. In this talk, we discuss several recent results on guaranteed algorithms for constructing simple data representations. Motivated by problems in image compression and super-resolution, we consider the problem of learning sparsely used dictionaries. We describe recent progress in understanding this problem, including results on local recoverability and generalization. We then describe one situation in which provably effective learning is possible in a global sense. We consider an arbitrary square dictionary and a random, sparse coefficient matrix, and prove that O (n log n) samples are sufficient to uniquely determine the coefficient matrix. The analysis suggests a polynomial-time algorithm, called Exact Recovery of Sparsely-Used Dictionaries (ERSpUD), which provably recovers the dictionary and coefficient matrix when the coefficient matrix is sufficiently sparse. For applications in detection or recognition, additional structure beyond sparsity may be desirable. We describe how related tools from convex programming can be leveraged to construct provably good representations for certain object instance detection problems.

 

This talk is based on joint works with Dan Spielman and Huan Wang (Yale), and with Cun Mu, Yuqian Zhang and Henry Kuo (Columbia).

 

19Laurent Younes, Johns Hopkins University

Title: Constrained Optimal Control for Diffeomorphic Registration

Abstract: Diffeomorphic shape analysis has emerged as a powerful approach for understanding nonrigid shape variation, designing deterministic and stochastic shape evolution models, performing segmentation, tracking or registration tasks while guaranteeing that topology remains unchanged. The approach, which is rooted in Grenander's deformable template theory, consists in considering the whole space as an elastic or viscous medium to which shapes of interest are attached. Global deformations of this space induce shape deformation by restriction. The interest of this approach is that the modeling effort can be made only once, by considering the space of global deformations, in order to induce a variety of shape spaces, one for each type of deformed objects (landmarks, curves, surfaces, images. . .). This space of global deformations is well defined and well-studied, since it can be represented as a group of diffeomorphisms in a Euclidean space, on which a Lie group structure, or invariant Riemannian metrics, can be defined.

In this setting, the registration problem (mapping a shape onto another) can be recast into an optimal control problem, in which the control is a vector field, namely the Eulerian velocity of an underlying time-dependent diffeomorphism. The control cost is generally defined a priori (i.e., it does not depend on the object that is being deformed) and controls the smoothness of the velocity.

There are, however, several situations, both in medical imaging and computer vision, in which additional constraints on the deformation arise in a natural way. The talk will describe several such examples, and associated computational methods. It will review, in particular numerical procedures that restrict the velocity field to finite-dimensional distributions, which can be useful when designing multi-scale registration schemes. We will also describe multi-phase registration problems, in which objects move non-rigidly within a deformable background, that is stitched to them to induce a homeomorphism of the ambient space.

Joint work with Sylvain Arguillére, Emannuel Trélat and Alain Trouvé.

Partially Supported by the ONR, NSF and NIH.

 

20Tong Zhang, Rutgers University, NJ

Title: Mathematical Models for Coding based Image Classification Methods

Abstract: In recent years, biologically inspired vision models such as sparse coding have become popular due to their strong empirical performance in image classification tasks. However, for a long time there had been no mathematical theory to explain why these methods are effective. In this talk, I will describe our recent efforts on developing a mathematical framework for these coding based image classification methods, where each image is represented as a distribution over local descriptors.

Our mathematical framework has two essential components: 1. nonlinear function approximation using geometric structures of local descriptors; 2. functional learning on probability distributions motivated using a kernel method. The mathematical theory justifies why coding models are effective for image classification, and reveals its limitations.

 

21Song-Chun Zhu, University of California, Los Angeles

Title: Stochastic Sets and Regimes of Mathematical Models of Images

Abstract: In his lecture “The Dawning of the Age of Stochasticity” at the conference “Mathematics towards the Third Millennium” in 1999, Dr. Mumford argued that stochastic models and statistical inference are more relevant, than logics, for many parts of mathematics and to understanding human mind. He was not alone! For example, IBM research argues that we are entering a new Era of Cognitive Computing, citing major breakthroughs in Watson system beating human champions in “The Jeopardy!” and the success of Google’s developments of driverless cars. What can mathematicians and statisticians contribute to this “phase transition”?

In this talk, I’d like to introduce some statistical phenomena, such as information scaling, in natural images, and mapping major models: i) ensembles in statistical physics, ii) sparsity models; and iii) stochastic image grammar to the different regimes of the entropy axis. Then I will show how an integrate model leads to solving problems, such as i) automated recognition of objects, scenes and events in video; ii) reasoning causality and predicting human intents; and iii) answering natural language queries on what, who, where, when and why.