Introduction

Modern digital technology has made it possible to manipulate multi-dimensional signals

with systems that range from simple digital circuits to advanced parallel computers. The
goal of this manipulation can be divided into three categories:

* Image Processing image in -> image out

* Image Analysis image in -> measurements out

* Image Understanding image in -> high-level description out

We will focus on the fundamental concepts of image processing. Space does not permit
us to make more than a few introductory remarks about image analysis. Image
understanding requires an approach that differs fundamentally from the theme of this
book. Further, we will restrict ourselves to two-dimensional (2D) image processing
although most of the concepts and techniques that are to be described can be extended
easily to three or more dimensions. Readers interested in either greater detail than
presented here or in other aspects of image processing are referred to

We begin with certain basic definitions. An image defined in the "real world" is
considered to be a function of two real variables, for example, a(x,y) with a as the
amplitude (e.g. brightness) of the image at the real coordinate position (x,y). An image
may be considered to contain sub-images sometimes referred to as regions-of-interest,
ROIs, or simply regions. This concept reflects the fact that images frequently contain
collections of objects each of which can be the basis for a region. In a sophisticated
image processing system it should be possible to apply specific image processing
operations to selected regions. Thus one part of an image (region) might be processed
to suppress motion blur while another part might be processed to improve color
rendition.

The amplitudes of a given image will almost always be either real numbers or integer
numbers. The latter is usually a result of a quantization process that converts a
continuous range (say, between 0 and 100%) to a discrete number of levels. In certain
image-forming processes, however, the signal may involve photon counting which
implies that the amplitude would be inherently quantized. In other image forming
procedures, such as magnetic resonance imaging, the direct physical measurement
yields a complex number in the form of a real magnitude and a real phase. For the
remainder of this book we will consider amplitudes as reals or integers unless otherwise
indicated.
Digital Image Definitions
A digital image a[m,n] described in a 2D discrete space is derived from an analog image
a(x,y) in a 2D continuous space through a sampling process that is frequently referred
to as digitization. The mathematics of that sampling process will be described in Section
5. For now we will look at some basic definitions associated with the digital image. The
effect of digitization is shown in Figure 1.

The 2D continuous image a(x,y) is divided into N rows and M columns. The intersection
of a row and a column is termed a pixel. The value assigned to the integer coordinates
[m,n] with {m=0,1,2,...,M-1} and {n=0,1,2,...,N-1} is a[m,n]. In fact, in most cases a(x,y)--
which we might consider to be the physical signal that impinges on the face of a 2D
sensor--is actually a function of many variables including depth (z), color ( ), and time
(t). Unless otherwise stated, we will consider the case of 2D, monochromatic, static
images in this chapter.

Figure 1: Digitization of a continuous image. The pixel at coordinates [m=10, n=3] has
the integer brightness value 110.

The image shown in Figure 1 has been divided into N = 16 rows and M = 16 columns.
The value assigned to every pixel is the average brightness in the pixel rounded to the
nearest integer value. The process of representing the amplitude of the 2D signal at a
given coordinate as an integer value with L different gray levels is usually referred to as
amplitude quantization or simply quantization.
Common Values
There are standard values for the various parameters encountered in digital image processing. These
values can be caused by video standards, by algorithmic requirements, or by the desire to keep digital
circuitry simple. Table 1 gives some commonly encountered values.

Symbol Typical values

Parameter
Rows N 256,512,525,625,1024,1035

Columns M 256,512,768,1024,1320

Gray Levels L 2,64,256,1024,4096,16384

Table 1: Common values of digital image parameters

Quite frequently we see cases of M=N=2K where {K = 8,9,10}. This can be motivated by
digital circuitry or by the use of certain algorithms such as the (fast) Fourier transform
(see Section 3.3).

The number of distinct gray levels is usually a power of 2, that is, L=2B where B is the
number of bits in the binary representation of the brightness levels. When B>1 we speak
of a gray-level image; when B=1 we speak of a binary image. In a binary image there
are just two gray levels which can be referred to, for example, as "black" and "white" or
"0" and "1".

Characteristics of Image Operations

 Types of operations
 Types of neighborhoods

There is a variety of ways to classify and characterize image operations. The reason for
doing so is to understand what type of results we might expect to achieve with a given
type of operation or what might be the computational burden associated with a given
operation.
Types of operations
The types of operations that can be applied to digital images to transform an input image a[m,n]
into an output image b[m,n] (or another representation) can be classified into three categories as
shown in Table 2.

Generic
Characterization
Complexity/Pixel
Operation

- the output value at a specific coordinate is dependent only on the

* Point constant
input value at that same coordinate.

- the output value at a specific coordinate is dependent on the input

* Local P2
values in the neighborhood of that same coordinate.

- the output value at a specific coordinate is dependent on all the

* Global N2
values in the input image.

Table 2: Types of image operations. Image size = N x N; neighborhood size = P x P. Note that
the complexity is specified in operations per pixel.

This is shown graphically in Figure 2.

Figure 2: Illustration of various types of image operations

Types of neighborhoods
Neighborhood operations play a key role in modern digital image processing. It is therefore
important to understand how images can be sampled and how that relates to the various
neighborhoods that can be used to process an image.

* Rectangular sampling - In most cases, images are sampled by laying a rectangular

grid over an image as illustrated in Figure 1. This results in the type of sampling shown
in Figure 3ab.
* exagonal sampling - An alternative sampling scheme is shown in Figure 3c and is
termed hexagonal sampling.

Both sampling schemes have been studied extensively and both represent a possible
periodic tiling of the continuous image space. We will restrict our attention, however, to
only rectangular sampling as it remains, due to hardware and software considerations,
the method of choice.

Local operations produce an output pixel value b[m=mo,n=no] based upon the pixel
values in the neighborhood of a[m=mo,n=no]. Some of the most common neighborhoods
are the 4-connected neighborhood and the 8-connected neighborhood in the case of
rectangular sampling and the 6-connected neighborhood in the case of hexagonal
sampling illustrated in Figure 3.

Figure 3a Figure 3b Figure 3c

Rectangular sampling Rectangular sampling exagonal sampling 4-connected 8-connected 6-

connected