Application and Optimisation of the Spatial Phase Shifting ...

4.1 Previous methods 105
all unwrapping algorithms substitute "bad" pixels by some "better" estimate and hence tend to suppress
errors without the user's request.
The attractive feature of this method is that, if the – continuous – theoretical displacement function is
qualitatively known, one can generate completely noise-free reference data, e.g. a best-fit plane. The
parameters for the displacement function are adjusted to match the measured values best, which will be
done by an iterative fitting process. An example of this is presented in Fig. 4.4: the sawtooth image
∆ϕ(x,y) whose fringe profile has been shown in Fig. 4.1 and Fig. 4.2 was unwrapped – without prior
filtering –, and a best-fit plane was subtracted from the resulting height data ∆d(x,y). Hence, the residual
displacement deviations δd(x,y) – scaled back to grey levels to allow a comparison with the previous
figures – could be directly evaluated for their rms, σ δd .
128
96
64
1/4 data
1/8 data
δ d [grey levels]
32
0
-32
-64
-96
-128
0 50 100 150 200 250
x -position/pixel
Fig. 4.4: Deviation δd between unwrapped sawtooth image and best-fit plane (δd=0). "1/4 data": average of 4 lines
of input image; "1/8 data": average of 8 lines of input image; see text.
Comparing the deviations δd in Fig. 4.4 with the deviations of the white curves in Fig. 4.1-Fig. 4.3 from
the expected linear fringe profile, it is evident that a substantial unintentional smoothing has occurred: the
spikes have been removed. This is in part due to the abovementioned pixel replacement during the
conversion of ∆ϕ to ∆d by the unwrapping algorithm; but the more important contribution comes from the
data reduction that could not be switched off [Ett97]: on unwrapping with the highest choosable
resolution, an image with, e.g., 1024768 pixels will be shrunk to 256192 averaged height values,
which, as known, reduces the spatial resolution and the noise. On testing a two times lower output
resolution, one finds however that the values for σ δd are almost the same for the corresponding image
lines out of a 256192 entry field (denoted by "1/4 data" in Fig. 4.4) and out of a 12896 entry field
(denoted by "1/8 data" in Fig. 4.4), respectively; this is, little further data smoothing takes place after the
unwrapping step. While the automatic noise suppression during unwrapping is certainly useful for
practical tasks, it runs counter to our intentions of quantitative error determination, and is therefore not
considered further.