MODELS OF IMAGES AND IMAGE OPERATIONS - Fundamentals of Three-Dimensional Digital Image Processing - page 32

Fig. 2.9. Unary operation of an image.

In 2.3.4 we discuss an operation among image operations which is a process
to generate a new image operation by combining two or more operations. Two
operations introduced here - a parallel composition and a serial composition -
are widely used in the following chapters. We will also present basic properties
of image operations such as inverse, iterative operation, and commutative and
distributive laws.
Finally in Subsection 2.3.5 we introduce several important operators in-
cluding a shift operator and position (shift) invariant operators.

2.3.1 Formulation of image operations

We will give a formal definition of image processing as a basis of theoretical
study in subsequent chapters. A process that generates a new image from an
input image is theoretically formulated as a unary operator on an image space
as follows.

Definition 2.5 (Unary operator). A process to derive a new image from a
given image is defined as a unary operator on an image space or as a mapping
from an image space

P 1 onto P 2 ,thatis,

mapping O : P 1 →P 2 (2.10)

where ( = the set of all images ) which are called

domain and range of the mapping O , respectively. The set of all operators is
called an operator space and is denoted by O . The image that is obtained by
applying an operator O to an image

P 1 ,and P 2 are subsets of P

F is expressed by O ( F ) (Fig. 2.9).

An operator in which the domain is a set of binary images is called a binary
image operator .
An operator I such that

I ( F )= F , ∀ F ∈P 1 , (2.11)

is called an identity operator with the domain P 1 . The identity operator does

not cause any effect on an image in its domain.