4.1 INTRODUCTION
The theoretical and experimental investigations which form this study rely on the ability to manipulate or process data, in the form of two-dimensional images, and thereby extract information which would not otherwise be available. The basis of this approach is founded in the branch of science known as signal processing.
A signal can be defined as "a function that conveys information, generally about the state or behaviour of a physical system" (Oppenheim and Shafer, 1989). Signals occur in two basic forms: continuous or discrete. Continuous signals, sometimes known as analogue signals, vary continuously as a function of time or space. An example of a continuous signal is the voltage produced by a microphone in response to sound. On the other hand, a discrete signal is one which takes only a finite number of values or exists only at discrete points in space or time. An example of a discrete signal is a coded message produced by a flashing lamp. The signal can have only one of two values; on or off. The two types of signal are shown diagrammatically in Figure 4.1.
The processing of signals may be performed by several methods but most processing and analysis is accomplished electronically by computer. However, it is usually very difficult to perform precise operations on continuous signals. Consequently, continuous signals are usually converted to discrete signals which are then processed on a digital computer.
B
Figure 4.1 Examples of (A) continuous and (B) discrete signals.
Signals occur in a wide variety of forms but a particularly important class of signals are those which occur in a two-dimensional format. In general, any two- dimensional function that bears information can be regarded as an image (Jain, 1989). When considered in this way an image is no longer simply the familiar photographic print or transparency, but is also a two-dimensional array of data. Such an ordered collection of data might represent, for example, the radar cross-section of a military target (radar imaging), the gravitational field over the earth’s surface (geophysical imaging) or the x-ray attenuation of a region of living tissue (radiographic imaging). In all cases the image carries information which is contained in some pattern of variations across the array.
4.2 TH E DISCRETE TW O-DIMENSIONAL SIGNAL - TH E DIGITAL IM AGE.
Digital images consist of discrete picture elements (Figure 4.2), usually abbreviated to pixels or pels, derived by sampling a continuous image. Associated with each pixel is a number which represents the value of the image function at that point. For example, this may be the brightness of a scene, as in a simple photograph, or the
F ig u r e 4 .2 A digital im age show ing the individual pixels in a m agnified portion o f the im age.
visible in the image or more precisely, the spatial resolution. The data, which the pixels represent, may be acquired directly from a digital scanner, as in the case of television pictures, or they may be derived from analogue images, such as radiographs, that have been digitised in the laboratory. The process of acquiring the digital image is often referred to as image capture.
4.2.1 Spatial digitization and quantization
Digitization is the name given to the analogue to digital conversion necessary to produce a digital image. It involves two distinct procedures: spatial digitization in which the image is sampled over the region of each pixel; and quantization where the energy levels (or "brightness") of each pixel is given one of a set of discrete levels (commonly
16, 64 or 256 grey-levels).
The conversion of a continuous image to a digital image suitable for processing by digital computer is accomplished by sampling the continuous image at several discrete points. The most common method of sampling is to scan the image in horizontal rows and determine the brightness at each sample point. This may be achieved in many different ways dependent upon the speed, cost and accuracy required. For most image
processing operations sampling is performed by a television camera with a vidicon tube or image dissector, where the image is converted to an electron image on a photosensitive plate called the target. A fine electron beam scans the target and generates a current proportional to the light falling on the target. This form of sampling is referred to as scan-in digitizing. The spatial resolution and quantization accuracy is limited with this method. Higher spatial resolution can be achieved with a scan-out digitizer where the object or continuous image is scanned directly with a fine collimated beam of laser light. The greatest accuracy may be achieved with a flat-bed scanning microdensitometer which employs a high precision optical system coupled to a mechanical scanning arm.
4.2.2 Two-dimensional sampling theory
The conversion of a continuous, analogue image to a discrete, digital format appears to imply a loss of information, because the samples are taken only at discrete points. In theory, this is true but only for images with infinite resolution. All real images, however, have finite resolution or more formally, they are band-limited. The sampling frequency necessary to recover all the information contained in a continuous band-limited image, is known as the Nyquist rate or Nyquist frequency (Dainty and Shaw, 1974). If the image is sampled below this rate ("under sampled") not only is information irretrievably lost but errors are introduced into the sampled image. These errors are known as aliasing (Pratt, 1991) and are not usually detectable except where the original image consists of a periodic pattern. Thus aliasing is occasionally seen as the offensive beating or vibrating pattern on a stripped shirt of a television newscaster.
4.2.3 Digital image storage
The binary numbers derived from sampling the original analogue image or scene can be stored in an electronic memory for subsequent analysis and display. The initial electronic memory is referred to as a frame-buffer or frames tore. Because digital images are two-dimensional arrays or matrices of pixels whose quantized levels are powers of 2, it is convenient to arrange the framestore memory into separate planes which correspond to these levels. Each plane in the memory is formed by an array of binary digits (bits) and corresponds to a slice through the image one bit deep. These planes
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