2020年河南大学反问题与图像处理网络研讨

会议手册

河南省人工智能理论及算法工程研究中心

河南大学数学研究中心

河南大学应用数学研究所

河南大学数学与统计学院

ZOOM ID: 210 089 8623

腾讯会议 ID: 286 488 733（备用）

2020年6月6日

因受新冠肺炎的影响，经学院领导和组委会商议，原定于2020年4月10日--4月12日举办的“2020年河南大学反问题与图像处理研讨会”改为线上研讨会（2020年6月6日，周六）。此次研讨会旨在促进反问题、优化算法与图像处理等领域专家学者之间的合作交流，推动该领域发展。

● 报告专家(姓氏拼音排序)

郭玉坤（哈尔滨工业大学）

韩德仁（北京航空航天大学）

李景治（南方科技大学）

刘继军（东南大学）

刘 歆（中国科学院）

陆 帅（复旦大学）

台雪成（香港浸会大学）

王泽文（东华理工大学）

文再文（北京大学）

杨俊锋（南京大学）

张 波（中国科学院）

●学术委员会（拼音排序）

孔德兴（浙江大学） 台雪成（香港浸会大学） 汪扬（香港科技大学）

王彦飞（中国科学院） 周 铁（北京大学）

● 组织委员会（拼音排序）

韩小森 罗守胜 庞志峰 肖运海 杨晓慧

● 时间

2020年6月6日

● 联系人信息

罗守胜 手机：18238240733 邮箱：sluo@henu.edu.cn

庞志峰 手机：18937836732 邮箱：zhifengpang@163.com

9:00 - 9:10	开幕式 （王彦飞，中国科学院）
时间	报告人	报告题目
主持人：王彦飞
9:10 - 9:40	刘继军 （东南大学）	On Fluorescence Imaging: Diffusion Equation Model and Recovery of Absorption Coefficient of Fluorophores
9:40 - 10:10	韩德仁 （北京航空航天大学）	A Parallel Operator Splitting Algorithm for Solving Constrained Total-variation Retinex
10:10 - 10:30	休息讨论
主持人：张波
10:30 - 11:00	台雪成 （香港浸会大学）	Deep Neural Networks (DNNs) with Shape Prior: Volume Preservation, Star-shapes and Convex Shape Representation by DNNs
11:00 - 11:30	刘歆 （中国科学院）	Exact Penalty Function for L21 Norm Minimization with Orthogonality Constraints
11:30 - 12:00	李景治 （南方科技大学）	Determining a Random Schröedinger Equation with Unknown Source and Potential
午休
主持人：韩德仁
14:00 - 14:30	陆帅 （复旦大学）	Linearised Inverse Schrödinger Potential Problem with a Large Wave Number
14:30 - 15:00	文再文 （北京大学）	An Exact Penalty Approach for Optimization with No-nnegative Orthogonality Constraints
15:00 - 15:15	休息讨论
主持人：陆帅
15:15 - 15:45	郭玉坤 （哈尔滨工业大学）	Fourier Method for the Multi-frequency Electromagnetic Inverse Source Problem
15:45 - 16:15	王泽文 （东华理工大学）	Some Novel Posteriori Selection Strategies of Regularization Parameters for Lanczos’Generalized Derivatives
16:15 - 16:30	休息 讨论
主持人：刘继军
16:30 - 17:00	杨俊锋 （南京大学）	Inertial Proximal ADMM
17:00 - 17:30	张波 （中国科学院）	Asymptotic Property of Phaseless Total-Field Operators and Phaseless Inverse Scattering at aFixedFrequency

On Fluorescence Imaging: Diffusion Equation Model and Recovery of Absorption Coefficient of Fluorophores

Jijun Liu

Southeast University

To quantify epifluorescence imaging of biological tissues under flat excitation and detection probes, we need to solve an inverse problem for the coupled radiative transfer equations which describe the excitation and emission fields in biological tissue. By transforming the radiative transfer system into coupled diffusion equations for the averaged fields, we have a nonlinear inverse problem to identify the absorption coefficient for a fluorophore by this system with impedance boundary condition, using some time resolved measurement data. For the linearized inverse problem obtained by ignoring the absorbing effect on the excitation field, we firstly establish the estimates of errors on the excitation field and the solution to the inverse problem, which ensures the reasonability of the model approximation qualitatively. Some numerical verifications are presented to show the validity of such a linearizing process quantitatively. Then, based on the analytic expressions of excitation and emission fields, the identifiability of the absorption coefficient from the linearized inverse problem is rigorously analyzed for absorption coefficient in special form, revealing the physical difficulty of the 3-dimensional imaging model by the back scattering diffusive system.

A Parallel Operator Splitting Algorithm for Solving Constrained Total-variation Retinex

Deren Han

Beihang University

An ideal image is desirable to faithfully represent the real-world scene. However, the observed images from imaging system are typically involved in the illumination of light. As the human visual system (HVS) is capable of perceiving identical color with spatially varying illumination, retinex theory is established to probe the rationale of HVS on perceiving color. In the realm of image processing, the retinex models are devoted to diminishing illumination effects from observed images. In this paper, following the recent work by Ng and Wang (SIAM J. Imaging Sci. 4:345-356, 2011), we develop a parallel operator splitting algorithm tailored for the constrained total-variation retinex model, in which all the resulting subproblems admit closed form solutions or can be tractably solved by some subroutines without any internally nested iterations. The global convergence of the novel algorithm is analysed on the perspective of variational inequality in optimization community. Preliminary numerical simulations demonstrate the promising performance of the proposed algorithm.

Deep Neural Networks (DNNs) with Shape Prior: Volume Preservation, Star-shapes and Convex Shape Representation by DNNs

Xue-Cheng Tai,

Hong Kong Baptist University

Convex Shapes and star-shapes are common in our daily life. It is important to design proper techniques to represent these shapes. We use Deep Neural Convolution Networks (DCNN) to segment objects from images or videos. So far, it is still a problem to guarantee that the output objects from a DCNN are convex shapes or star-shapes. In this work, we propose a technique which can be easily integrated into the commonly used DCNNs for image segmentation and guarantee that outputs are convex/star shapes. We can also incorporate volume preservation into the networks. This idea can also be used for other shape representation applications include 3D surface and shape reconstructions. The method is flexible and it can handle multiple objects and even allow some of the objects to be non-convex/non-star-shape. Our method is based on the dual representation of the sigmoid activation function in DCNNs.

In the dual space, the shape priors can be guaranteed by a simple quadratic constraint on a binary representation of the shapes. Moreover, our method can also integrate spatial regularization and some other shape prior using a soft thresholding dynamics (STD) method. The regularization can make the boundary curves of the segmentation objects to be simultaneously smooth and convex-shape/star-shape. We design a very stable active set projection algorithm to numerically solve our model.This algorithm can form a new plug-and-play DCNN layer calledCS-STD whose outputs must be a nearly binary segmentation of convex-shape/star-shape objects.

As an application example, we apply the convex-shape/star-shape prior layer to the retinal images segmentation by taking the popular DeepLabV3+ as a backbone network. Experimental results on several public datasets show that our method is efficient and outperform the classical DCNN segmentation methods.

This talk is based on joint works with Jun Liu, S. Luo and X. Wang.

Exact Penalty Function for L21 Norm Minimization with Orthogonality Constraints.

Xin Liu

Chinese Academy of Sciences

L21 norm minimization with orthogonality constraints, feasible region of which is called Stiefel manifold, has wide applications in statistics and data science. The state-of-the-art approaches adopt proximal gradient technique on either Stiefel manifold or its tangent spaces. The consequent subproblem does not have closed-form solution and hence requires an iterative procedure to solve, which is usually time consuming. In this paper, we discover that the Lagrangian multipliers of the orthogonality constraints in this class of problems are of closed-form expressions. By using this closed-form expression, we introduce a penalty function for this type of problems. We theoretically demonstrate the equivalence between the penalty function and the original L21 norm minimization under mild assumptions. Based on the exact penalty function, we propose an inexact proximal gradient method in which the subproblem is of closed-form solution. The global convergence and the worst case complexity are established. Numerical experiments illustrate the efficiency of our new proposed algorithm, when comparing with the existing methods.

Determining a Random Schröedinger Equation with Unknown Source and Potential

Jingzhi Li

Southern University of Science and Technology

This talk studies the direct and inverse scattering problem associated with a time-harmonic random Schröedinger equation with a Gaussian white noise source term. We establish the well-posedness of the direct scattering problem and obtain three uniqueness results in determining the variance of the source term, the potential and the mean of the source term, sequentially, by the corresponding far-field measurements. The first one shows that a single realization of the passive scattering measurement can uniquely recover the variance of the source term, without knowing the other two unknowns. The second shows that if active scattering measurement is further used, then a single realization can uniquely recover the potential function without knowing the source term. The last one shows that if full measurements are used, then both the potential and the random source can be uniquely recovered.

Linearised Inverse Schrödinger Potential Problem with a Large Wave Number

Shuai Lu

Fudan University

We investigate recovery of the (Schrödinger) potential function from many boundary measurements at a large wave number. By considering such a linearised form, we obtain a Hölder type stability which is a big improvement over a logarithmic stability in low wave numbers. Furthermore we extend the discussion to the linearised inverse Schrödinger potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate.

Based on the linearised problem, a reconstruction algorithm is proposed aiming at the recovery of the Fourier modes of the potential function. By choosing the large wave number appropriately, we verify the efficiency of the proposed algorithm by several numerical examples.

It is a joint work with Victor Isakov (Wichita) and Boxi Xu (SUFE).

An Exact Penalty Approach for Optimization with Nonnegative Orthogonality Constraints

Zaiwen Wen

Peking University

Optimization with nonnegative orthogonality constraints has wide applications in machine learning and data sciences. It is NP-hard due to some combinatorialproperties of the constraints. We first propose an equivalent optimization formulation withnonnegativeandmultiplesphericalconstraintsandanadditional single nonlinear constraint. Various constraint qualifications, the first- and second-order optimality conditions of the equivalent formulation are discussed. We design a class of (smooth) exact penalty models by keeping the nonnegative and multiple spherical constraints. The penalty models are exact if the penalty parameter is sufficient large other than going to infinity. A practical penalty algorithm with rounding technique is then developed. It uses a second-order method to approximately solve a series of subproblems with nonnegative and multiple spherical constraints. Extensive numerical results on the projection problem, orthogonal nonnegative matrix factorization problems and the K-indicators model show the effectiveness of our proposed approach.

Fourier Method for the Multi-frequency Electromagnetic Inverse Source Problem

Yukun Guo

Harbin Institute of Technology

This talk is concerned with an inverse problem of identifying the current source distribution of the time-harmonic Maxwell’s equations from multi-frequency measurements. Motivated by the Fourier method for the scalar Helmholtz equation and the polarization vector decomposition, we propose a novel method for determining the source function in the full vector Maxwell’s system. Relevant mathematical justifications of the method will be presented and numerical examples will be provided to demonstrate the feasibility and effectiveness of the method.

Some Novel Posteriori Selection Strategies of Regularization Parameters for Lanczos’Generalized Derivatives

Zewen Wang

East China University of Technology

Numerical derivatives aim to estimate the derivatives of unknown functions from their approximate values, and they are important in scientific research and engineering disciplines, such as in the edge detection of an image, source identification and magnetic resonance electrical impedance tomography. The problem of numerical derivatives is ill-posed, which means that small errors in the approximate values will induce large errors in the numerical derivatives.

In this talk，we mainly studies how to select reasonable stepsizes (regularization parameters) in Lanczos' generalized derivative, which is an stable approach of numerical derivatives. Four posterior selection strategies, in which two strategies need to know the noise level while the other two do not, are proposed with the convergence estimates of regularization solutions (Lanczos' generalized derivatives). Numerical experiments show that the last three posterior selection strategies, i.e. Posterior Selection Strategies II - IV, are feasible for Lanczos' generalized derivatives.

By the way, we will briefly introduce another recent work：Determination of an unknown time-dependent heat source from a nonlocal measurement by finite difference method.

Inertial proximal ADMM

Junfeng Yang

Nanjing University

We propose an inertial version of the proximal ADMM for linearly constrained separableconvex optimization. Convergence and rate of convergence results are established, and some numerical results are given to illustrate the benefits gained by the inertial extrapolation.

Joint with Caihua Chen, Raymond Chan, Shiqian Ma.

Asymptotic Property of Phaseless Total-Field Operators and Phaseless Inverse Scattering at a Fixed Frequency

Bo Zhang

Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences

Beijing 100190, China

Inverse scattering with full data have been studied extensively in the past decades. However,in many practical applications,the phase of the measured data can not be obtained accurately compared with its modulus andsometimes is even impossible to be obtained. Thus it is often desirable to consider inverse scattering with the modulus or intensity of the measured data (called phaseless data). In recent years, many work has been done on inverse scattering with phaseless far-field and near-field data, including uniqueness issues and numerical algorithms. In this talk, we consider inverse scattering with phaseless near-field data at a fixed frequency. We will propose a new framework to study this problem, including the uniqueness issue and a factorization-type imaging algorithm. Precisely, we will establish the asymptotic property in an operator norm of the phaseless total-field operator defined in terms of the phaseless total-field data measured on a large enough ball, employing the theory of oscillatory integrals. As far as we know, this is the first result on the asymptotic behavior in an operator norm of the phaseless total-field operator though pointwise asymptotic results of the phased scattered field have been previously obtained. This asymptotic property of the phaseless total-field operators is then used to establish uniqueness results and an approximate factorization method for inverse scattering with phaseless near-field data at a fixed frequency.

This talk is based on the recent results with Xu Xiaoxu and ZhangHaiwen.

学院简介

河南大学数学与统计学院是河南大学设立较早的院系之一，其前身为创建于1923年的原中州大学数理系。后历经算学系、数学系、数学与信息科学学院等阶段，2014年更名为数学与统计学院，2018年获“河南省教育系统先进集体”荣誉称号。

近百年来，学院严守“明德新民、止于至善”的校训，在黄际遇、陈作钧、樊映川、黄敦慈、杜孟模、刘亚星等先后在此执教的著名数学家、教育家的引领带动下，经过几代学人的接力耕耘，形成了严谨的治学精神，积累了深厚的学术底蕴，为学院持续健康发展奠定了坚实基础。目前，学院拥有数学和统计学两个一级学科博士学位授权点和博士后科研流动站，学科教学（数学）和应用统计两个专业硕士学位授权点。数学和统计学均为河南省重点一级学科。学院开设有数学与应用数学、信息与计算科学、统计学、金融数学等四个本科专业。其中，数学与应用数学专业为国家一流建设专业、河南省专业综合改革试点专业，信息与计算科学专业为河南省一流建设专业、河南省特色专业。

学院现有教职工121人，其中专任教师104人，行政教辅人员17人，博士学位获得者81人。在专任教师队伍中，有教授、副教授68人，博士、硕士生导师78人；双聘院士1人，河南省特聘教授1人，河南省讲座讲授1人；河南大学特聘教授5人，河南大学讲座教授5人，河南大学外籍“拔尖人才项目”全职特聘教授2人，河南大学“青年英才”1人；河南省教育厅学术技术带头人6人，河南省高校青年骨干教师14人，河南省杰出青年基金获得者6人；河南省教学标兵14人，河南省教育厅优秀教育管理人才1人，河南省文明教师2人。形成了一支师德高尚、业务精湛、结构合理、充满活力的高水平师资队伍。

学院建有河南大学数学研究中心、河南省人工智能理论及算法工程研究中心、数学建模实验室、金融统计实验室和数据分析技术实验室等五个教学科研平台，拥有现代数学研究所、应用数学研究所、教学方法研究室、非线性科学研究室等四个科研机构，办有全英文专业学术刊物《数学季刊》。学院注重发挥学科带头人的引领作用，积极参与国内外学术交流，不断凝练学科研究方向，持续加强学术团队建设，在代数与数论、偏微分方程与数学物理、复分析与复几何、概率论与数理统计、科学计算、数据分析与智能处理等方向形成了比较优势和特色，取得了一批优秀科研成果。据统计，2015年—2019年，学院教师发表论文500余篇，多项成果在Advances in Mathematics，Communications in Mathematical Physics, Bulletin of the London Mathematical Society等国际著名学术刊物上发表；获批国家自然科学基金项目51项，出版学术专著6部，荣获科研奖励114项。

学院下设数学系、应用数学系、信息与计算科学系、统计系、金融数学系、数学教育系和继续教育中心七个教学单位，现有本科生1329人，研究生198人。近年来，学院不断深化教育教学改革，探索实施分类教学、分类指导，推动完善本科生导师制度，数学与应用数学专业开设“明德计划”拔尖人才实验班，培养了一大批思想品德优良、专业知识牢固、实践能力突出、富有创新精神的高素质人才。先后有340余人次在全国大学生数学竞赛、丘成桐大学生数学竞赛和全国大学生数学建模竞赛等大型专业赛事中获奖；本科生考研录取率保持在30%以上，研究生和本科生一次性就业率均在95%以上。

展望未来，学院将全面贯彻落实党的教育方针，围绕立德树人根本任务，坚持走内涵式发展道路，加快建设成为省内领先、国内知名、有特色、高水平的研究教学型学院；在大众化教育的基础上，培养具有竞争力的拔尖创新人才和各类专门人才，为学校一流学科大学建设和地方经济发展做出应有贡献。