Application and Optimisation of the Spatial Phase Shifting ...

4.2 Noise quantification in this work 109
As mentioned before, the fitting method can be easily extended to greater dimensionality. If, for instance,
a correlation fringe pattern is to be evaluated, two degrees of freedom, namely I b and M I , are added and
the algorithm can determine the fringe visibility in IR 5 . More complicated fringe structures could also be
treated. But every new variable increases the number of iteration steps as well as the time for a single
iteration, so that the issue of speed gains importance in such applications.
Since the resulting measurements of σ ∆ϕ will mostly appear converted to graphs of σ d in the following
chapters, it may be helpful at this point to provide the reader with a pictorial representation of the various
amounts of noise. The image parts grouped in Fig. 4.7 are taken from an out-of-plane TPS measurement
series with decreasing object illumination.
Fig. 4.7: Image segments from results of deformation measurements using TPS with varying, and rather weak,
object illumination. σ ∆ϕ as grey values: 13.3, 21.2, 28.0, 40.3, 51.9 and 63.1; as phase: 18.8°, 30.0°, 39.5°,
56.9°, 73.3° and 89.0°, in obvious order.
The last sawtooth image in the figure is hardly discernible as such and therefore raises the question
whether results like this are of any use at all. It turned out, however, that the filtering procedure described
in section 4.1.2.2 still improved the image sufficiently to enable correct unwrapping; but as explained
above, the phase error could be determined without doing so. Other examples of sawtooth images severely
degraded by synthetic Gaussian noise have been presented in [Kad97].
From the preceding overview of methods, it is clear that the approach to noise quantification presented
here is new only in that it avoids unwrapping before the best-plane fit; however, it is the only strategy
known to me that can generate noise-free data with no user interaction – except for the input of starting
parameters – even from the worst of results, and is hence free of arbitrariness. While this may not always
be necessary, it is desirable from a methodological point of view.