MODELS OF IMAGES AND IMAGE OPERATIONS - Fundamentals of Three-Dimensional Digital Image Processing - page 21

2.2 Continuous and digitized images
2.2.1 Continuous images
We will begin with the modeling of a 2D image with a 2D continuous image
as a digitized image is obtained by digitizing a continuous one.
An image, specifically a 2D image, is defined as a scalar function of two
variables f ( x, y ), where f denotes a gray value at a point ( x, y )onanimage
plane. We call this a continuous image . The physical meaning of the value of
f varies according to the individual imaging process. A few examples are as
follows:
(1) drawings on a sheet of paper: reflectivity.
(2) photographic negative recording of an outdoor scene: the transmission co-
ecient is proportional to the amount of light energy exposed to the film.
This usually represents the intensity of the light reflected by the object in
the scene.
(3) medical X-ray image: attenuation coecient to X-ray of an object.
(4) ultrasound image: the intensity of the sound wave reflected by an object.
(5) thermogram: the temperature on the surface of an object.
We simply call the value of f image density or gray value .
Remark 2.1 (Density). In some cases the terms intensity and density are
clearly distinguished from each other. The intensity refers to the amount of
light energy exposed to a sensor, and the density represents the logarithm of
the intensity. It is said that the response to a light stimulus for the human vi-
sual system is approximately proportional to the logarithm of the light energy
exposed to the eye [Hall79].
2.2.2 Digitized images
A continuous image is digitized via a two-step procedure: sampling and quan-
tization .
(a) Sampling : There are two ways to understand sampling. The first is that
a function f ( x, y ) representing a continuous image is sampled at points
( sampling points ) of an ordered array (such as a rectangular array or
square lattice) or a triangular array (or triangular lattice). The second
approach is to consider an image plane as being divided into a set of small
cells of the same shape and size, called image cells , image elements ,or
pixels . The average of values of f over each cell is considered as the density
value of the image at the center point of the cell. Usually a square cell or a
hexagonal cell is adopted in practical applications. Two interpretations are
eventually possible if we consider the sampling point as the center point of
an image cell. Note that the square lattice corresponds to the square pixel
system and the triangular lattice to the hexagonal pixel system (Fig. 2.1).