where b is the number of bits used for the pixel. It can be easily seen 
that no matter what the support of the entity is (i.e., how many terms 
there are in the summation of equation 3-6), the bound on the number of 
locales will still vary in an exponential fashion with the number of bits 
per pixel. 
  
Now consider how the geometric precision varies with the sampling interval. 
The 'standard entity' mentioned in the definition of geometric precision 
(section 2) will produce a pattern of locales which will depend upon the 
amplitude and spread of the Gaussian function. Since the spread of the 
entity function is determined by the sampling interval, the size but not the 
pattern of locales will change as the sampling interval is changed. A result 
of this is that the number of locales along a straight line of fixed length 
and the root mean locale area will both vary directly with the sampling 
interval. Since the number of pixels in the image varies as the square of 
the sampling interval, the number of locales will vary as the square root of 
the number of pixels. This is a much weaker dependance than the exponential 
variation with the number of bits discussed above. 
Since rhe storage required for a digital image varies directly as the number 
of bits per pixel times the number of pixels, it follows that if the number 
of locales is the exclusive determinant of the available geometric precision 
in a digital image then it would be most efficient to trade off a 
smaller sampling interval for more bits per pixel. Resolution requirements 
and noise limit this tradeoff. 
5. MAXIMIZING AVAILABLE GEOMETRIC PRECISION 
Once the form of the entity has been established by such factors as the shape 
of the object, the aperture shape, and the response of the imaging system, 
there remains the sampling frequency and quantization level to determine the 
limits of geometric precision in the digital image. We will presume that the 
aperture size 1s commensurate with the scanner sampling interval so that the 
unit raster square, as determined by the sampling interval, is about the same 
size as the pixel (determined by aperture size). It is also presumed that 
the quantization levels are uniformly spaced and may be fuliv.utilized.. The 
presence of noise and the requirement that the object be recognizable will 
jointly determine an optimal scanning interval and number of bits per pixel. 
To achieve object recognition in the digital image there must be adequate 
resolution. Although dynamic range and effective pi: 
determining the available resolution, the prima 
sampling interval be matched to the spatial ext 
essentially a Nyquist criterion. 
pixel size play a part in 
ry criteron should be that the 
ent of the object, This is 
Once an adequate sampling interval has been established, one may address the 
question of the number of bits per pixel. Apart from data volume 
constraints, the determining factor is that the dynamic range must be 
maximized. The dynamic range is the number of distinquishable levels of 
Pixel values in the presence of noise. If the range of pixel values span the 
2^ values provided by b bits per pixel and if the noise is within 
v levels then the dynamic range D is 
b 
Du= 2 /v (v>1) (5-1) 
In the absence of noise (v=1), the bound on the number of locales for an 
entity increases without bound as the number of pixel values increases. This 
is not the case in the presence of noise. The bound on the number of locales