Application and Optimisation of the Spatial Phase Shifting ...

4.1 Previous methods 103
4.1.2.2 Smoothing sine and cosine
To cope with the problem of edges, it has also been adopted to work on, in the mathematical sense,
continuous data: the sawtooth image (signifying the optical phase) can be decomposed a posteriori into
the sine and the cosine part from which it was originally generated [Lüh93] (cf. 3.4.5); this step has
recently been given the name of "trigonometric transform" [Sea98]. This gives two edge-free fringe
profiles that can be filtered with considerably larger filter windows, without affecting the 02π
transitions that appear again when the phase is re-calculated. However, too large a filter will attenuate the
contrast of the sine/cosine patterns or eliminate them completely, depending on their spatial frequency;
therefore the proper choice of filter size requires some care as well. Fig. 4.2 shows the improvement
brought about by this strategy when the same filter size as above is used.
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Fig. 4.2: Effect of image smoothing by decomposing into sine and cosine part, low-pass filtering each of them and
re-calculating the phase. As above, single image lines are shown, and the filtering was done in 2D.
Obviously, the edges and their heights are preserved in this case; but the fringe shape still remains noisy.
It improves a little when a median filter is used for the sine and cosine images: unlike the low-pass filter
that is simply an average formation, the median filter really eliminates outliers. Yet it is clear that the
ideal fringe profile will still not be restored by this type of filtering operation. Moreover, it is definitely
inappropriate for the case of deterministic large-scale distortions of the fringe profile, as Fig. 4.3 shows.
In this case, a severe phase-shift miscalibration resulted in a concentration of calculated phase values
around 0 and 180° (see Chapter 3.4.6), and the filtering does not even approximately restore the expected
fringe profile. While this is certainly an extreme example, it shows that filtering does not automatically
generate an ideal reference phase map ∆ϕ ref (x,y) where both random and deterministic errors ought to be
small or absent. Therefore, σ ∆ϕ will be underestimated when calculated with the black curve in Fig. 4.3 as
a reference.