Estimation of jump curves and surfaces has broad applications. One important application is image processing, since the image intensity function of a monochrome image can be regarded as a jump surface with jumps at the outlines of objects. Because of this connection, this book also introduces some image processing techniques, mainly for edge detection and image restoration, and discusses the similarities and differences between these methods and the related methods on estimating jump curves and surfaces in the statistical literature.
I started my research in nonparametric regression when I was a graduate student in China. At that time, most existing nonparametric regression methods assumed that the curves or surfaces to estimate were continuous. In my opinion, this assumption was faulty as a general rule because curves or surfaces could be discontinuous in some applications, and thus I decided to investigate this problem in my Master's thesis at Fudan University in China. After coming to the United States in 1991, I realized that this topic was closely related to image processing in computer science. I then went to the computer science departments at University of Georgia and University of Wisconsin to take courses on computer graphics and vision, and I also read hundreds of research papers in computer science journals at that time.
As I began my studies here in the United States during the early 1990s, several procedures were proposed in the statistical literature for estimating jump curves or surfaces. However, these procedures often imposed restrictive assumptions on the model, making the methods unavailable for many applications. Another limitation of the methods was that extensive computation was required. In addition, the existing procedures in the image processing literature did not have much theory to support them. A direct consequence was that, for a specific application problem, it was often difficult to choose one from dozens of existing procedures to handle the problem properly. Therefore, it was imperative to suggest some procedures that could work well in applications and have some necessary theory to support them; this became the goal of my Ph.D. thesis research at Wisconsin, and I have been working in the area since then. Part of this book summarizes my own research in this area.
This book has seven chapters. The first chapter introduces some basic concepts and terminologies in the areas of computer image processing and statistical regression analysis, along with presenting the overall scope of the book. Chapter 2 consists of two parts: the first part introduces some basic statistical concepts and terminologies, for the convenience of those readers who do not know or remember them well; and the second part introduces some conventional smoothing procedures in the statistical literature. These first two chapters constitute the prerequisite for the remaining chapters. Chapters 3-5 discuss some recent methodologies for fitting one-dimensional jump regression models, estimating the jump location curves of two-dimensional jump surfaces, and reconstructing two-dimensional jump surfaces with jumps preserved, respectively. Chapters 6 and 7 introduce some fundamental edge detection and image restoration procedures in the image processing literature. At the end of each chapter, some exercise problems are provided.
This book is intended for statisticians, computer scientists, and other researchers or general readers who are interested in curve/surface estimation, nonparametric regression, change-point estimation, computer vision and graphics, medical imaging, and other related areas.
The mathematical level required is intentionally low. Readers with some background in basic linear algebra, calculus through integration and differentiation, and an introductory level of statistics can easily understand most parts of the book. This book can be used as a primary text book for a one-semester course on nonparametric regression analysis and image processing or can be used as a supplemental text book for a course on computer vision and graphics. Some datasets used in this book can be downloaded from the following Wiley ftp site:
I thank Chooichiro Asano and Xianping Li for their help and support during the initial stage of my research on estimating jump curves or surfaces at Fudan University in China. With the precious help from Hubert Chen, Naihua Duan, Paul Switzer, and Bob Taylor, I had the chance to do my research in a better environment. It would have been impossible to have this book without their selfless support and help. Encouragement and help from Peter Hall and Steve Marron have had a great impact on my research as well. It was Peter who first told me the connection between jump curve/surface estimation and image processing. I am grateful to my Ph.D. thesis adviser Brian Yandell for his advice, encouragement, and the enormous amount of time spent on my thesis research during my graduate study at Wisconsin and for his continuing support since my graduation. Iréne Gijbels, Alexandre Lambert, and Jörg Polzehl read parts of the manuscript and provided many constructive suggestions and comments. The manuscript was used as lecture notes in my recent advanced topic course offered at the School of Statistics of University of Minnesota in the Spring of 2004; students from that class corrected a number of typos and mistakes in the manuscript. Mr. Jingran Sun kindly made Figure 6.14 used in Section 6.6. An anonymous reviewer assigned by Wiley reviewed the first five chapters and provided a very detailed review report, which much improved the presentation. I am fortunate to have had Jessica Kraker read the entire manuscript. She provided a great amount of constructive comments and suggestions.
Most of my research included in the book was carried out during my graduate study or work at Fudan University, University of Georgia, University of Wisconsin at Madison, Ohio State University, and University of Minnesota. I am indebted to all faculty, staff members, and graduate students of the related departments at these universities. Part of my research was finished during several short research visits to the Center for Mathematics and its Applications of Australian National University and to the Institut de Statistique of Université catholique de Louvain in Belgium. This book project was partially supported by a Grant-in-Aid of Research, Artistry and Scholarship at University of Minnesota, a National Security Agency grant, and a National Science Foundation grant.
Special thanks to my family for their love and constant support.