Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging: Mathematical Imaging and Vision

This handbook gathers together the state of the art on mathematical models and algorithms for imaging and vision. Its emphasis lies on rigorous mathematical methods, which represent the optimal solutions to a class of imaging and vision problems, and on effective algorithms, which are necessary for the methods to be translated to practical use in various applications. Viewing discrete images as data sampled from functional surfaces enables the use of advanced tools from calculus, functions and calculus of variations, and nonlinear optimization, and provides the basis of high-resolution imaging through geometry and variational models. Besides, optimization naturally connects traditional model-driven approaches to the emerging data-driven approaches of machine and deep learning. No other framework can provide comparable accuracy and precision to imaging and vision.

Written by leading researchers in imaging and vision, the chapters in this handbook all start with gentle introductions, which make this work accessible to graduate students. For newcomers to the field, the book provides a comprehensive and fast-track introduction to the content, to save time and get on with tackling new and emerging challenges. For researchers, exposure to the state of the art of research works leads to an overall view of the entire field so as to guide new research directions and avoid pitfalls in moving the field forward and looking into the next decades of imaging and information services. This work can greatly benefit graduate students, researchers, and practitioners in imaging and vision; applied mathematicians; medical imagers; engineers; and computer scientists.

本手冊匯集了關於影像和視覺的數學模型和算法的最新成果。它強調嚴謹的數學方法，這些方法代表了一類影像和視覺問題的最佳解決方案，以及有效的算法，這些算法對於將方法轉化為各種應用中的實際使用是必要的。將離散影像視為從功能表面採樣的數據，可以利用微積分、函數和變分微積分以及非線性優化等先進工具，並通過幾何和變分模型提供高分辨率成像的基礎。此外，優化自然地將傳統的基於模型的方法與機器和深度學習的新興基於數據的方法相連接。沒有其他框架能夠提供與影像和視覺相當的準確性和精度。

本手冊的章節由影像和視覺領域的領先研究人員撰寫，所有章節都以淺顯易懂的介紹開始，使本書對研究生學生易於理解。對於新來者，本書提供了全面且快速的內容介紹，以節省時間並開始應對新興挑戰。對於研究人員來說，接觸到研究工作的最新成果可以獲得整個領域的全面觀點，以指導新的研究方向，避免在推動領域發展並展望未來幾十年的影像和信息服務時遇到困境。本書對於影像和視覺的研究生、研究人員和從業人員；應用數學家；醫學影像師；工程師；以及計算機科學家都有很大的益處。

Ke Chen received his B.Sc., M.Sc. and Ph.D. degrees in Applied Mathematics, respectively, from the Dalian University of Technology (China), University of Manchester (UK) and University of Plymouth (UK). Dr. Chen is a computational mathematician specialised in developing novel and fast numerical algorithms for various scientific computing (especially imaging) applications. He has been the Director of a Multidisciplinary Research Centre for Mathematical Imaging Techniques (CMIT) since 2007, and the Director of the EPSRC Liverpool Centre of Mathematics in Healthcare (LCMH) since 2015. He heads a large group of computational imagers, tackling novel analysis of real-life images. His group's imaging work in variational modelling and algorithmic development is mostly interdisciplinary, strongly motivated by emerging real-life problems and their challenges: image restoration, image inpainting, tomography, image segmentation and registration.
Carola-Bibiane Schönlieb is Professor of Applied Mathematics at the University of Cambridge. There, she is head of the Cambridge Image Analysis group and co-Director of the EPSRC Cambridge Mathematics of Information in Healthcare Hub. Since 2011 she is a fellow of Jesus College Cambridge and since 2016 a fellow of the Alan Turing Institute, London. She also holds the Chair of the Committee for Applications and Interdisciplinary Relations (CAIR) of the EMS. Her current research interests focus on variational methods, partial differential equations and machine learning for image analysis, image processing and inverse imaging problems. She has active interdisciplinary collaborations with clinicians, biologists and physicists on biomedical imaging topics, chemical engineers and plant scientists on image sensing, as well as collaborations with artists and art conservators on digital art restoration.

Her research has been acknowledged by scientific prizes, among them the LMS Whitehead Prize 2016, the Philip Leverhulme Prize in 2017, the Calderon Prize 2019, a Royal Society Wolfson fellowship in 2020, a doctorate honoris causa from the University of Klagenfurt in 2022, and by invitations to give plenary lectures at several renowned applied mathematics conferences, among them the SIAM conference on Imaging Science in 2014, the SIAM conference on Partial Differential Equations in 2015, the SIAM annual meeting in 2017, the Applied Inverse Problems Conference in 2019, the FOCM 2020 and the GAMM 2021.

Carola graduated from the Institute for Mathematics, University of Salzburg (Austria) in 2004. From 2004 to 2005 she held a teaching position in Salzburg. She received her PhD degree from the University of Cambridge (UK) in 2009. After one year of postdoctoral activity at the University of Göttingen (Germany), she became a Lecturer at Cambridge in 2010, promoted to Reader in 2015 and promoted to Professor in 2018.
Prof. Xue-Cheng Tai is a Chief Research Scientist and Executive Program Director at Hong Kong Center for Cerebro-cardiovascular Health Engineering (COCHE), Hong Kong Science Park. He was a Professor and Head at the Department of Mathematics at Hong Kong Baptist University (China) since 2017. Before 2017, hr served as Professor at the Department of Mathematics at Bergen University (Norway). His research interests include Numerical PDEs, optimization techniques, inverse problems and image processing. He has done significant research work his research areas and published over 250 top quality international conference and journal papers. He is the winner of the 8th Feng Kang Prize for scientific computing. He served as organizing and program committee members for a number of international conferences and has been often invited for international conferences. He has served as referee and reviewers for many premier conferences and journals.
Prof. Laurent Younes is a professor in the Department Applied Mathematics and Statistics, Johns Hopkins University (USA), that he joined in 2003, after ten years as a researcher for the CNRS in France. He is a former student of the Ecole Normale Supérieure (Paris) and of the University of Paris 11 from which he received his Ph.D. in 1988. His work includes contributions to applied probability, statistics, graphical models, shape analysis and computational medicine. He is a fellow of the IMS and of the AMS.

Ke Chen（陳科）分別在中國大連理工大學、英國曼徹斯特大學和英國普利茅斯大學獲得應用數學的學士、碩士和博士學位。陳博士是一位計算數學家，專門開發新穎且快速的數值算法，應用於各種科學計算（尤其是影像）應用。自2007年以來，他一直擔任多學科研究中心數學影像技術（CMIT）的主任，並自2015年起擔任EPSRC利物浦醫療數學中心（LCMH）的主任。他領導一個大型的計算影像團隊，致力於對現實生活影像的新穎分析。他的團隊在變分建模和算法開發方面的影像工作主要是跨學科的，受到新興現實生活問題及其挑戰的強烈推動：影像恢復、影像修補、斷層掃描、影像分割和配準。

Carola-Bibiane Schönlieb（卡羅拉-比比安娜·舒恩利布）是劍橋大學應用數學教授。她是劍橋影像分析小組的負責人，也是EPSRC劍橋醫療信息數學中心的聯合主任。自2011年以來，她是劍橋耶穌學院的院士，自2016年起是倫敦阿倫·圖靈研究所的院士。她還擔任歐洲數學學會應用和跨學科關係委員會（CAIR）的主席。她目前的研究興趣集中在變分方法、偏微分方程和機器學習在影像分析、影像處理和逆向成像問題上的應用。她與臨床醫生、生物學家和物理學家在生物醫學影像領域進行著積極的跨學科合作，與化學工程師和植物科學家在影像感測方面進行合作，並與藝術家和藝術修復師合作進行數字藝術修復。

她的研究獲得了多個科學獎項的肯定，其中包括2016年LMS Whitehead Prize、2017年Philip Leverhulme Prize、2019年Calderon Prize、2020年皇家學會Wolfson獎學金、2022年克拉根福大學榮譽博士學位，並應邀在多個著名應用數學會議上發表主題演講，包括2014年SIAM影像科學會議、2015年SIAM偏微分方程會議、2017年SIAM年會、2019年應用逆問題會議、2020年FOCM和2021年GAMM。

Carola於2004年畢業於奧地利薩爾茨堡大學數學研究所。2004年至2005年期間，她在薩爾茨堡擔任教職。她於2009年獲得劍橋大學的博士學位。在德國哥廷根大學進行了一年的博士後研究後，她於2010年成為劍橋大學的講師，並於2015年晉升為讀者，2018年晉升為教授。

Prof. Xue-Cheng Tai（戴學成）是香港科學園香港腦心血管健康工程中心（COCHE）的首席研究科學家和執行計劃總監。自2017年以來，他一直擔任中國香港浸會大學數學系的教授和系主任。在2017年之前，他曾在挪威卑爾根大學數學系擔任教授。他的研究興趣包括數值偏微分方程、優化技術、逆問題和影像處理。他在這些研究領域做出了重要的研究工作，並發表了250多篇國際頂級會議和期刊論文。他是第八屆馮康科學計算獎的獲獎者。他曾擔任多個國際會議的組織和程序委員會成員，並經常應邀參加國際會議。他還擔任許多頂級會議和期刊的審稿人。

Prof. Laurent Younes（羅蘭·尤內斯）是約翰霍普金斯大學應用數學和統計學系的教授。他畢業於奧地利薩爾茨堡大學數學和統計學系，並於2004年獲得博士學位。他的研究興趣包括計算解剖學、圖像分析和形狀統計學。他在這些領域的研究工作獲得了廣泛的認可，並發表了許多高質量的國際會議和期刊論文。他是多個國際會議的組織和程序委員會成員，並經常應邀發表主題演講。