Application and Optimisation of the Spatial Phase Shifting ...

6.7 Extensions of SPS by temporal unwrapping 167
maps were generated by a look-up table for α x =120°/sample (cf. 3.2 and Appendix B). The frame rate of
the image processing system (a Data Translation DT3852 frame grabber connected to an Alacron FT200
processor board with two 50-MHz i860 processors) was 0.5 Hz for an image size of 800600 pixels.
The method of filtering was tested by running the temporal unwrapping for some 30000 frames without
disturbing the system. At the end, the fringe counting procedure reported some 1.5 fringes; this error
suppression is sufficient for our purpose. However, the fluctuations in Φ(x, y) do not appear to be
perfectly random, since they do not vanish even in such a long averaging process. Their structure may be
seen in Fig. 6.24, where the displacement information corresponding to a range of 1.5 fringes has been
converted to grey values and expanded to the whole grey scale for better visibility of the effect.
Fig. 6.24: Errors in Φ(x, y) related to "random" noise, accumulated during
without actual object displacement.
30000 temporal unwrapping runs
Another problem occurs in the observation of real displacements. While noisy pixels are not necessarily
detected as such in every frame, their calculated Φ(x, y, t) will not follow the true course; instead, for most
of the noisy pixels it will hover around zero. If such a Φ(x, y, t) happens to be included in the averaging
operation (6.26) before it is re-aligned with its neighbours, its error will propagate into the surrounding
pixels. In the long run, this will lead to pixel clusters whose Φ(x, y, t) is dragged behind, i.e. will be
somewhere between zero – from where all observations start – and the true value. An example is given in
Fig. 6.25, where an out-of-plane tilt about the y axis has been tracked. The tilt was controlled by applying
a linear voltage ramp to a PZT which rotated the object holder slowly enough to satisfy the temporal
sampling requirement for all pixels in the image.
y
Fig. 6.25: Errors in Φ(x, y) related to object motion; "slow" pixels due to imperfect error rejection.
The line of zero displacement is marked by the white line and the accumulated displacement range Φ(x, y)
is +5 π at the left and -5 π at the right edge. The bright/dark backgrounds tend to deceive the eye; indeed,
the "slow" pixels on the left are brighter than those on the right, which means that the sign of motion is
correctly determined for all of them, but the measured displacement is underestimated. When spatial
averaging takes place in every frame, this behaviour can be suppressed by pixel weighting [Cog99]. In
this subsection, the faulty pixel clusters are selectively removed a posteriori; they must not be included in
the displacement computation [Hun93a], for they will generate a systematic error that is proportional to
the absolute displacement.