TABLE OF CONTENTS In this chapter, we help readers review several major tools in modern mathematical image and vision analysis, whose fundamental roles will manifest in later chapters. These topics reveal the geometric, stochastic, real analysis, and harmonic analysis aspects of this emerging field. In addition to providing useful background mathematical knowledge for later chapters, the present chapter can also serve as a valuable reference source for researchers in the field.
Curves and surfaces are basic geometric elements in image and vision analysis, and computer graphics. For example, they define and reveal the information of an automobile (in automatic traffic control), a star or planet (in astronomic imaging), a human body (in video surveillance), and an internal organ (in medical imaging and 3-D reconstruction). In this section, we review some basic theories on curves and surfaces in two or three dimensions. Readers are also referred to the classical geometry book by do Carmo [103] or some other monographs on geometric image analysis and computation, e.g., Romeny [256] and Kimmel [173].
A parametric curve is often denoted by x( t), with t varying on an interval and x = ( x 1, x 2) or x = ( x 1 , x 2, x 3) depending on the dimension. Differentiation with respect to t will be denoted by an overhead dot. We shall assume that x( t) is smooth enough,...
