Regression analysis provides a major statistical tool for estimating curves and
surfaces from data with noise, using various smoothing techniques. From linear 
regression to parametric regression to nonparametric regression, this area has 
experienced more than 100 years of development. Conventional nonparametric 
regression analysis was developed only to estimate {\it continuous} curves or 
surfaces. When a conventional nonparametric regression procedure is used for 
estimating a jump curve or surface, the estimator does not converge to the 
truth at the jump locations as the sample size increases. Therefore, new 
statistical tools are necessary for estimating curves and surfaces with jumps. 
This book introduces some recent statistical methodologies for this purpose.

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 {\sf ftp} site: 
\begin{center}
{\sf ftp://ftp.wiley.com/public/sci\_tech\_med/image\_processing\_jra}
\end{center}

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. Ir\'{e}ne Gijbels, Alexandre Lambert, and J\"{o}rg 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 Universit\'{e} 
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.






