Talks

Title: Representing Objects by Binary Visual Concept Encoding

Speaker: Alan Yuille (Johns Hopkins University)

Abstract: This talk will update progress on a research program whose goal is to develop hierarchical architectures with the same strong performance abilities as deep networks but which are also able to model the flexibility and adaptiveness of biological visual systems. These architectures are intended to be simpler and more explainable, require little supervision and few training examples, and to be adaptive to situations/environments which they have not encountered. In particular, we give new results for unsupervised learning of objects and object viewpoints, one- and few-shot learning for discriminative tasks, and the ability to deal with adversarial attacks. We conclude by speculating on how vision algorithms should be evaluated given the increasingly complexity of visual tasks and the impracticality of getting sufficient data for training and testing.

Title: Optimal Mass Transportation Theory Applied for Machine Learning

Speaker: David Gu (Stony Brook University)

Abstract: Generative Adversarial Nets (GAN) learns probability distributions in the data by a competition between the generator and the discriminator. In this talk, we interpret the GAN model from the optimal mass transportation point of view, and use convex geometric method to solve the problem. We would like to discuss the following questions: does a deep neural network learn a distribution or a function or map? What is the dimension of the solution space? Is the competition necessary? Does the network learn or just memorize? Why sometimes a neural network is easily fooled? How to replace part of the blackbox of GAN model by a transparent model?

Title: Skeletal models and shape representation

Speaker: Kathryn Leonard (California State University Channel Islands)

Abstract: This talk will consider skeletal models for shapes in two and three dimensions from the perspective of complexity, compression, parts decomposition and recognition, including similarity and articulation. We will discuss mathematical skeletal constructions such as the Blum medial axis, and functions on those skeletons that provide important information about a shape. We will also present results from a massive user study on shape part perception that provides insight into human cognition about shape parts.

Title: Multiscale Diffeomorphic Deformation Based Shape and Image Analysis

Speaker: Yan Cao (The University of Texas at Dallas)

Abstract: Biological shapes are highly structured. Quantitative measures of anatomical structure differences in normal and diseased states, during growth and aging are very important in medical image understanding. We would like to develop mathematical models and computational tools to represent and quantify the variation of different shapes and structures associated with them. In our approach, variations are modeled by smooth evolutions of shapes. A geodesic path is computed on the manifold of diffeomorphisms connecting two Shapes. A multiscale representation of diffeomorphisms is developed, which can be used for sythesize random diffeomorphisms and shape analysis.

Title: Fast Low Rank Reconstruction via Optimization on Manifold: Algorithms, Theory and Applications

Speaker: Ke Wei (Fudan University)

Abstract: Low rank models exist in many applications, ranging from signal processing to data analysis. Typical examples include low rank matrix completion and spectrally sparse signal reconstruction. We will present a class of computationally efficient algorithms which are universally applicable for those low rank reconstruction problems. Theoretical recovery guarantees will be established for the proposed algorithms under different random models, showing that the sampling complexity is essentially proportional to the intrinsic dimension of the problems rather the ambient dimension. Extensive numerical experiments demonstrate the efficacy of the algorithms and extensions to phase retrieval and low rank demixing will be briefly discussed.

Title: Spare representation in vision: algorithms and models

Speaker: Chenglong Bao (Tsinghua University)

Abstract: In recent years, the concept of sparse representation has been widely used in many applications. Among extensive works in this direction, K-SVD is a typical method in the dictionary learning. However, the convergence of K-SVD is not clear. In this talk, I will introduce an efficient numerical algorithm for solving dictionary learning problem with convergence guarantee. Moreover, a variant of dictionary learning models are proposed to dynamic texture classification.

Title: Data Recovery on Manifolds: A Theoretical Framework

Speaker: Yang Wang (Hong Kong University of Science and Technology)

Abstract: Recovering data from compressed number of measurements is ubiquitous in applications today. Among the best know examples are compressed sensing and low rank matrix recovery. To some extend phase retrieval is another example. The general setup is that we would like to recover a data point lying on some manifold having a much lower dimension than the ambient dimension, and we are given a set of linear measurements. The number of measurements is typically much smaller than the ambient dimension. So the questions become: Under what conditions can we recover the data point from these linear measurements? If so, how? The problem has links to classic algebraic geometry as well as some classical problems on the embedding of projective spaces into Euclidean spaces and nonsingular bilinear forms. In this talk I'll give a brief overview and discuss some of the recent progresses.

Title: Should one use an educated or uneducated basis?

Speaker: Hongkai Zhao (University of California at Irvine)

Abstract: A choice of good basis is important for representation/approximation, analysis and interpretation of quantities of interest. A common difficult balance in real applications is between universality and specificity. This is a really application dependent question. We will show examples for which educated basis, i.e., designing problem specific basis or learning basis, are effective, as well as examples for which uneducated basis, i.e., using simple random basis and bless of dimensions, can also be effective.

Title: Recent advances of Computational Quasiconformal Geometry in Imaging, Graphics and Visions

Speaker: Ronald Lui (The Chinese University of Hong Kong)

Abstract: 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.

Title: Fast Algorithms for Euler´s Elastica energy minimization and applications

Speaker: Xuecheng Tai (Hong Kong Baptist University)

Abstract: This talk is divided into three parts. In the first part, we will introduce the essential ideas in using Augmented Lagrangian/operator-splitting techniques for fast numerical algorithms for minimizing Euler's Elastica energy.

In the 2nd part, we consider an Euler's elastica based image segmentation model. An interesting feature of this model lies in its preference of convex segmentation contour. However, due to the high order and non-differentiable term, it is often nontrivial to minimize the associated functional. In this work, we propose using augmented Lagrangian method to tackle the minimization problem. Especially, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently as those ones in the previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully.

Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.

In the 3rd part, we will introduce some recent fast algorithms for minimizing Euler's elastica energy for interface problems. The method combines level set and binary representations of interfaces. The algorithm only needs to solve a Rodin-Osher-Fatemi problem and a re-distance of the level set function to minimize the elastica energy. The algorithm is easy to implement and fast with efficiency.

The content of this talk is based joint works with Egil Bae, Tony Chan, Jinming Duan and Wei Zhu.

Related links:

1) ftp://ftp.math.ucla.edu/pub/camreport/cam17-36.pdf

2) AUGMENTED LAGRANGIAN METHOD FOR AN EULER’S ELASTICA BASED SEGMENTATION MODEL THAT PROMOTES CONVEX CONTOURS

3) Image segmentation using Euler’s elastica as the regularization

Title: Geodesic equation on the Universal Teichmuller space, Teichons and Imaging

Speaker: Sergey Kushnarev (Singapore University of Technology and Design)

Abstract: In this talk I will describe a way to parametrize a space of planar shapes by members of a coset space $PSL2(Rackslash Diff(S^1)$. These functions are known as fingerprints of the shape, or welding maps in the Teichmuller theory. Imposing Weil-Petersson metric on this shape space endows it with wonderful mathematical properties: unique geodesics between any two shapes. The geodesic equation on this space is the Euler-Poincare equation on the group of diffeomorphisms of the circle $S^1$ or EPDiff($S^1$). It admits a class of soliton-like solutions, Teichons. The resulting simpler geodesic equation is more tractable from the numerical point of view. Applications of image matching with Teichons on the database of hippocampi will be demonstrated.

Title: Solving Geometric PDEs on Manifolds Represented as Point Clouds and Applications

Speaker: Rongjie Lai (Rensselaer Polytechnic Institute)

Abstract: Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. It is a challenge to extract global geometric information hidden in the point clouds due to the lack of global connectivity. In our recent work, systematical numerical methods are proposed to solving PDEs on manifolds sampled as point clouds. These methods can achieve high order accuracy and enjoy flexibility of solving different type of equations on manifolds with possible high co-dimesion. We use the proposed methods to consider special designed geometric PDEs on point clouds, which provides us a bridge to link local and global information. Based on this method, I will discuss several applications including nonrigid point clouds registration, understanding manifold-structured data represented as incomplete inter-point distance by combining with low-rank matrix completion theory as well as calculating committor functions for overdamped dynamic system.

Title: Convex non-convex optimization in image processing.

Speaker: Tieyong Zeng (Hong Kong Baptist University)

Title: A Three-stage Approach for Segmenting Degraded Color Images: Smoothing, Lifting and Thresholding (SLaT)

Speaker: Raymond Chan (The Chinese University of Hong Kong)

Abstract: In this talk, we introduce a SLaT (Smoothing, Lifting and Thresholding) method with three stages for multiphase segmentation of color images corrupted by different degradations: noise, information loss and blur. At the first stage, a convex variant of the Mumford-Shah model is applied to each channel to obtain a smooth image. We show that the model has unique solution under different degradations. In order to handle the color information properly, the second stage is dimension lifting where we consider a new vector-valued image composed of the restored image and its transform in a secondary color space to provide additional information. This ensures that even if the first color space has highly correlated channels, we can still have enough information to give good segmentation results. In the last stage, we apply multichannel thresholding to the combined vector-valued image to find the segmentation. The number of phases is only required in the last stage, so users can modify it without the need of solving the previous stages again. Experiments demonstrate that our SLaT method gives excellent results in terms of segmentation quality and CPU time in comparison with other state-of-the-art segmentation methods.

Joint work with: X.H. Cai, M. Nikolova and T.Y. Zeng

Title: Low dimensional manifold model for image processing

Speaker: Zuoqiang Shi (Tsinghua University)

Abstract: In this talk, I will introduce a novel low dimensional manifold model for image processing problem. This model is based on the observation that for many natural images, the patch manifold usually has low dimension structure. Then, we use the dimension of the patch manifold as a regularization to recover the original image. Using some formula in differential geometry, this problem is reduced to solve Laplace-Beltrami equation on manifold. The Laplace-Beltrami equation is solved by the point integral method. Numerical tests show that this method gives very good results in image inpainting, denoising and super-resolution problem.

Title: Computation of crowded geodesics on the universal Teichmüller space

Speaker: Akil Narayan (University of Utah)

Abstract: The problems of geometric shape classification, identification, and cliquing are important considerations in pattern theory and computer vision. A popular mathematical approach is to consider the space of shapes as a metrized Riemannian manifold and to subsequently compute distances between points (i.e., shapes) on the manifold. We consider a particular metrization of planar shapes that has the attractive properties of scale and translation invariance: the Weil-Peterson metric on the universal Teichmueller space. This metrization reduces the Riemannian structure to anaylsis of univariate functions. However, sufficiently complicated shapes induce "crowding" in the univariate functional representations: extremely fine structure that must be resolved in order to correctly model the topological shape space. We present a numerical technique that is effective at resolving shapes with reasonable amounts of crowding, and provide associated computational results. Finally, we identify current shortcomings and discuss in-development solutions to these problems.

Title: Dictionary learning for seismic data processing

Speaker: Jianwei Ma (Harbin Institute of Technology)

Title: Manifold Learning for Brain Morphological Shapes

Speaker: Anqi Qiu (National University of Singapore)

Abstract: We present the algorithm, Locally Linear Diffeomorphic Metric Embedding (LLDME), for constructing efficient and compact representations of surface-based brain shapes whose variations are characterized using Large Deformation Diffeomorphic Metric Mapping (LDDMM). Our hypothesis is that the shape variations in the infinite-dimensional diffeomorphic metric space can be captured by a low-dimensional space. To do so, traditional Locally Linear Embedding (LLE) that reconstructs a data point from its neighbors in Euclidean space is extended to LLDME that requires interpolating a shape from its neighbors in the infinite-dimensional diffeomorphic metric space. This is made possible through the conservation law of momentum derived from LDDMM. It indicates that initial momentum a linear transformation of the initial velocity of diffeomorphic flows, at a fixed template shape determines the geodesic connecting the template to a subject's shape in the diffeomorphic metric space and becomes the shape signature of an individual subject. This leads to the compact linear representation of the nonlinear diffeomorphisms in terms of the initial momentum. Since the initial momentum is in a linear space, a shape can be approximated by a linear combination of its neighbors in the diffeomorphic metric space. In addition, we provide efficient computations for the metric distance between two shapes through the first order approximation of the geodesic using the initial momentum as well as for the reconstruction of a shape given its low-dimensional Euclidean coordinates using the geodesic shooting with the initial momentum as the initial condition. Experiments are performed on the hippocampal shapes of 302 normal subjects across the whole life span (18~94 years). Compared with Principal Component Analysis and ISOMAP, LLDME provides the most compact and efficient representation of the age-related hippocampal shape. Even though the hippocampal volumes among young adults are as variable as those in older adults, LLDME disentangles the hippocampal local shape variation from the hippocampal size and thus reveals the nonlinear relationship of the hippocampal morphometry with age.

This idea can be extended to learning of brain functional networks.

Title: Numerical Methods for Partial Differential Equations on Manifolds and Point Clouds

Speaker: Shingyu Leung (The Hong Kong University of Science and Technology)

Abstract: We present recent numerical methods for solving partial differential equations on manifolds and point clouds. In the first part of the talk, we introduce a new and simple discretization, named the Modified Virtual Grid Difference (MVGD), for numerical approximation of the Laplace-Beltrami operator on manifolds sampled by point clouds. We first introduce a local virtual grid with a scale adapted to the sampling density centered at each point. Then we propose a modified finite difference scheme on the virtual grid to discretize the LB operator. The new discretization provides more diagonal dominance to the resulting linear system and improves its conditioning. In the second part, we present a local regularized least squares radial basis function (RLS-RBF) method for solving partial differential equations on irregular domains or on manifolds. The idea extends the standard RBF method by replacing the interpolation in the reconstruction with the least squares fitting approximation. These two are joint work with Prof. Hongkai Zhao, Dr. Meng Wang and Dr. Ningchen Ying.