Application and Optimisation of the Spatial Phase Shifting ...

102 Quantification of displacement-measurement errors
particular pixel array. Fig. 4.1 illustrates the effect of median filtering by data from a measurement of a
mere out-of-plane tilt that should give a linear phase profile.
grey value
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128
96
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32
0
raw data
9x9 median
0 50 100 150 200 250
x -position/pixel
Fig. 4.1: Effect of image smoothing by a median window. A single image row is displayed for both raw and
filtered data, but the median filtering was done, as usual, in 2D.
While the spikes in the raw data could be removed with a 33 pixel filter, the distortions of the fringe
profile continued to distinctly calm down until the filter kernel size of 99 was reached. Even with so
large a filter window, there are significant deviations from the expected linear course of the phase. The
0255 transitions where the phase is "wrapped" (white-black edges in the image) remain sharp, but the
fringe profile nearby gets rounded off. Hence the raw data set will in fact be more accurate in those
regions despite the higher noise. With fringe densities as low as in the figure (some 5 fringes over 1024
pixels), it would be possible and desirable to use very large median windows; but due to the edge
falsification, this must be ruled out.
There have been successful attempts to eliminate the edge falsification by generating a second sawtooth
image ∆ϕ meas (x,y)+π, where the wrap edges are shifted a posteriori by half a fringe width. Then both
∆ϕ meas (x,y) and ∆ϕ meas (x,y)+π are filtered and only the wrap-free regions from both images reassembled,
where, of course, the phase shift by π must be undone in the second image [Vik90]; this is perfectly
permissible because the fringe offset in sawtooth images is arbitrary. It was found that the edge
degradation is very efficiently suppressed by this technique.
The so-called classification filtering method described in [Own91c] exceeds the performance of the
median filter: it is edge-preserving, much faster than the median processing – that almost always involves
pixel sorting – and also yields the best noise tolerance of all filtering routines studied in [Own91c].
Another high-performance sawtooth-image filter is the partially recursive window described in [Pfi93]; it
proceeds line by line and stores the smoothed data back to their original addresses, so that the filter
window will operate on both smoothed and raw data in subsequent image lines. The performance of this
filter has been compared with other filters recently in [Aebi99].