Fourier phase estimation is based on the principle 
that the phase of the Fourier transform of an image is 
a linear function of the displacement. The reference 
image of the target (the entity function [11]) and the 
observed image are divided pixel by pixel. The phase 
of the resultant complex-valued image is a linear 
function of pixel coordinates with a slope 
corresponding to the position of the target. A least 
squares fit to the slope of the phase is used as the 
position estimate. As noise is added, the otherwise 
erratic performance curve of Fourier phase estimation 
is smoothed somewhat. The method does not perform 
well until the number of quantization levels gets 
quite large, at which point it surpasses the center of 
mass method of calculating the centroid. 
Centroid estimation is done using the "center of 
mass" calculation as was discussed in section 4. This 
estimate is straightforward to compute. Explicit 
calculations have shown that for the 3 by 3 window 
it exhibits a bound of 24 on its reciprocal RMS error 
in the absence of quantization. With coarse 
quantization, this bound is exceeded to a level of 
about 35 (with 14 quantization levels), illustrating 
the perversity of quantization effects. 
Position decoding is consistently and clearly superior 
to the other algorithms both with and without noise 
and at all quantization levels except possibly near- 
binary. Improvement with the number of 
quantization levels is extremely linear (up to an 
beyond 100 quantization levels), as predicted by the 
optimality criterion of the locales theory. Noise 
degrades performance somewhat but linearity is 
preserved. The gap between the lineal bound and 
optimal position estimation in the absence of noise 
can be attributed to the fact that the lineal bound is 
based on the limiting assumption that all of the 
locales have the same size. It is reasonable to 
speculate that the divergence of the two curves can be 
derived from knowledge of the distribution of locales 
sizes (or perhaps even the converse). 
62 
SUMMARY 
The possibility of acquiring very high quality digital 
imagery brings the analysis of quantization effects 
into the forefront of digital image metrology. Sub- 
pixel position estimation to very high levels of 
precision are possible in theory. A foundation of 
solid principles is needed for the effective 
accummulation of knowledge and experience in high 
precision measuring techniques. The theory of locales 
is potentially such a unifying basis. Locales are 
simply defined and easily generated. They serve to 
estimate position as well as the position uncertainty 
due to quantization. They lead to a natural and easily 
stated definition of optimal position estimation. The 
optimal position estimation algorithm called 
"position decoding" has been introduced here for the 
first time. Preliminary performance simulations have 
shown it be an effective tool for digital image 
metrology. 
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