(D (D 
o 
rt 
the 
pk 
2. FORMAL MODEL OF GEOMETRIC PRECISION 
In this section a mathematical model is developed for geometric precision. 
Before proceeding with the development, we will discuss some of the ideas 
informally. The main concept is that of a 'LOCALE', which represents a set 
of indistiguishable positions for an image detail. The locales are 
positional equivalence classes which partition each pixel. The image 
detail under consideration is formalized as an 'ENTITY'. The entity 
incorporates the properties of the imaging system, including for example, 
the intensity response function of the scanner, the transfer function of 
the scanning aperture, and systematic distortions of the imaging system. 
The pixel values of the digital image are sampled values of the entity so 
that the entity corresponds in a way to a continuous image of the object. 
The pixel values are determined from the entity exclusively by the sampling 
interval of the scan. 
The development presumes, but is not dependent upon, a regular scanning 
grid. The term "aperture" will be used here to refer to the effective 
scanning aperture of the system and should not be confused with the 
interval of the scanning grid. The aperture size is implicit in the 
definition of the entity. The scanning system is assumed to provide a 
uniformly spaced square grid of pixel values. This spatial quantization 
will be referred to as a scanning or sampling pattern whereas intensity 
quantization will simply be referred to as quantization, provided no 
ambiguity arises. 
A formal definition of geometric precision is not explicitly 
presented, instead the discussion revolves around the definition of a 
locale. An example of a definition of geometric precision in terms of 
locales might be as follows: 
Definition: 
"The geometric precision of an entity is the reciprocal 
square root of the average area of the locales in a pixel; 
the geometric precision of a digital image is given by the 
geometric precision of the ''standard' entity defined 
as a bivariate Gaussian function scaled so as to fully 
utilize the dynamic range of the pixel and to have the 
scanning interval correspond to one standard deviation". 
This definition incorporates an entity which is about the same size as one 
square of the scanning grid. Anything much smaller than this is not 
resolvable in a digital image. Since the precision with which one can 
locate an entity generally improves as the entity gets larger, by selecting 
a small standard entity the definition gives geometric precision in terms 
of a 'worst-case' situation. 
If the scanning interval is the same as the aperture size then, unless the 
dynamic range is very large, only the immediately adjacent pixels will 
contain information on the position of such an entity. The locales will be 
determined by the pixel values in these neighbouring pixels. These pixel 
values are referred to collectively in this paper as the 'IMAGE FUNCTION', 
which is a set valued function of the position of the entity. À locale 
corresponds exactly with the set of all positions mapping into a given 
value of the image function. 
We now proceed with the development of the formal model of these concepts.