Application and Optimisation of the Spatial Phase Shifting ...

5.4 Out-of-plane displacements 119
observe a distinctly reduced error. Further reduction of the speckle size does not greatly improve the
performance for this and other high N x .
On the other end of the scale, at d s = 1.5 d p , σ d from SPS consists chiefly of bias noise (i.e. σ d is already at
0.08 λ for N x =0) until decorrelation sets in. At moderate fringe densities, i.e. up to some 30 fringes over
the image width, we observe σ d to increase steeply for a speckle size below some 2.5 d p , which shows that
the SPS method is not very tolerant of low spatial coherence of the data points. In general, the SPS
experiments confirm a speckle size of about 3 d p to be most suitable. Since the available amount of object
light grows as 1/d s ², we will not stop here and try to further reduce d s without increasing σ d in Chapter 6.4.
In the TPS experiments, a speckle size around 1 d p turns out to yield the best results for low fringe
densities; yet at larger tilts, we obtain better measurements with larger speckles. This is due to imageplane
speckle displacement: the same lateral speckle displacement introduces less noise when the speckles
are larger, although the pattern in itself decorrelates faster.
With large speckles, the TPS measurements are worse than those from SPS as soon as the object is
moved. For high fringe densities and d s =10 d p , some entries are missing from the curves because
decorrelation had advanced in such a way that no trace of fringes was left (of course, the fitting algorithm
did find a minimum in the coarse random phase map; but it always does). In this case, reducing the
speckle size brings about a larger improvement of performance.
For N x 40, SPS performs better than TPS for any speckle size. This demonstrates a peculiarity of SPS:
because of the spatially extended phase-sampling window (see 3.4.4), some smoothing of the phase values
takes place as they are determined. The sampling window has an extent of 3 pixels in the x direction only,
which could introduce anisotropy; but the errors from the N y measurements agree with Fig. 5.5 quite well,
so that the one-dimensional phase sampling has no detectable effect.
The drastic increase of σ d for the speckle size of 0.5 d p is somewhat surprising, since it has been proven in
[Leh98] that very good TPS measurements remain possible even with much smaller speckles. In our case
however, there are also slight random in-plane shifts of the object that accompany the tilts. They do not
show up in the left-hand graph of Fig. 5.3 because of the larger speckles used there; but at d s = 0.5 d p , the
accuracy suffers noticeably from this minor effect.
To get an impression of what the obtained sawtooth images look like, Fig. 5.6 provides some example
results; the corresponding σ d values may be found from Fig. 5.5.