A | 
representation be computationally efficient. The most 
efficient method of representation generally varies 
with the form of the target. In the case of a Gaussian 
dot for example, a small table of the radii of the 
concentric contours is both compact and efficient. 
More complex targets such as corners or crosses 
convolved with the imaging system's response 
function may require careful consideration to obtain 
efficient evaluation. 
The robustness of the algorithm, as implemented for 
the simulation studies, is acheived largely through 
the use of a tolerance threshold in the truth map. The 
threshold specifies how low the cell probability can 
go before the cell is disregarded from further 
processing. This involves more processing effort 
than simply retaining only the cells with the highest 
probability. The modification is necessary because 
the best choice of locale may well lie outside the 
"most feasible" region for some of the pixels. 
The position estimation algorithm was implemented 
using a 3 by 3 pixel window to locate a Gaussian 
dot. It typically executes in about 2 seconds per point 
and evaluates approximately 3000 cells to achieve an 
accuracy of better than one hundredth of a pixel. 
10. COMPARATIVE PERFORMANCE 
The Gaussian dot was selected for comparative 
performance tests for several reasons; it is easily 
processed by a number of algorithms, it's radial 
symmetry allows reduction of performance measures 
to one-axis parameters, and it is a fairly realistic 
representation of a target. The processing window 
was restricted to a 3 by 3 square. For the Gaussian 
dot as described in section 4, this modest processing 
window will span the non-zero portion of the target 
for any amplitude less than 54. The limited scope of 
the comparison is acknowledged and the need for 
further simulation and real-data performance 
evaluation is emphasised. Nevertheless, the results 
suggest that the algorithm will provide excellent 
performance under varied conditions and will function 
well with real data. 
The algorithms to which position decoding is 
compared are; linear interpolation, Fourier phase 
estimation and centroid estimation. The results of the 
comparison with and without noise are presented in 
figure 8. 
Linear interpolation takes advantage of the fact that 
the log of the ratio of the pixels on the left and right 
sides of the window is a linear function of the x-axis 
position of the Gaussian target. To reduce the bias in 
this estimate, quantization was first converted from 
integer truncation to rounding by adding 0.5 to the 
quantized value. This algorithm performed quite well, 
showing continued and rapid performance 
improvement with the number of quantization levels. 
Noise reduced its performance only slightly. 
61 
Reciprocal RMSE 
N 
eo 
Lineal Bound 
(a) 
© 
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Cc 
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JI 
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(d) (b) 
Ce 
o 
(c) 
— 
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o 
  
0 10 20 30 
Number of Quantization Levels 
FIGURE 8a 
Position estimation errors for the 
Gaussian dot. The lineal bound is 
determined by an estimate of the 
locale density. The reciprocal RMS 
position errors are shown for four 
estimators; (a) position decoding, 
(b) linear interpolation, 
(c) Fourier phase estimation and 
(d) centroid estimation. 
Lineal Bound 
a 
  
  
  
FIGURE 8b 
Position estimation errors for the 
Gaussian dot in the presence of +1 bit 
of noise. The reciprocal RMS position 
errors are shown for four estimators; 
(a') position decoding 
(b') linear interpolation, 
(c) Fourier phase estimation, and 
(d') centroid estimation. 
For comparison, the 'lineal bound' and 
(a) position decoding without noise are 
also shown.