Alfredo Dubra's PhD thesis - Imperial College London

4. Wavefront reconstruction from shear phase maps
and subdivision methods (using Matlab 6.1 on a computer with a Pentium 4 processor
at 2.2 GHz and 500 MB RAM memory). Figure 4.3 b) shows the measured
time for an individual reconstruction against the number of reconstruction points N.
The trends of the plot are in agreement with those of the figure 4.3 a) for large N.
Differences between the methods are attributable to the exact details of the computational
implementations. If we compare the performance of the DFT methods
against reconstruction using the pseudoinverse for the case N = 3 × 10 3 , the speed
improvement factor is around 50 (which is comparable to that predicted by the ratio
of the number of multiplications, 2 N 2 /3 N log 2 N ≈ 170). If we try to put this in
the context of our tear topography estimation problem, we have that each slope map
is around 400 samples across, that is N ≈ 1 × 10 5 . This would imply that if we
were using a pseudoinverse reconstruction method, each topography map calculation
would take 3 orders of magnitude longer than with either of the DFT algorithms (i.e.
≈ 200 s/reconstruction). However, this was not the main reason for not using a pseudoinverse
method. The actual calculation of the pseudoinverse itself was not possible
at all with current technology in reasonable times (≈ 100 years with the available
computing power).
4.4 Noise performance
We now compare the DFT methods against the least square pseudoinverse in terms
of their associated noise coefficient[104], which quantifies the effect of white noise in
the data on the reconstructed phase,
C = 1 ∑
R ω R ω , (4.9)
N pupil
where N pupil is the number of wavefront samples inside the pupil of interest P , ω is an
index that runs through all the elements of the reconstruction matrix R. Figures 4.4
(a) and (b) show the noise coefficients for different pupil sizes and unit shear, without
any data extension. For unit shear no subdivision or data extension is necessary
for the DFT reconstruction. Note that the pseudoinverse has a lower (better) noise
coefficient for both pupil geometries.
We now investigate the effect of extending the data by enlarging the original grid
as previously explained. Figures 4.5 a) and b) show the noise coefficients versus
the number of extra rows/columns used for circular and square pupils respectively,
ω
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