Application and Optimisation of the Spatial Phase Shifting ...

106 Quantification of displacement-measurement errors
4.2 Noise quantification in this work
For a quantitative comparison of TPS and SPS, we will have to test different speckle sizes, fringe
densities, and experimental set-ups, which means that a universal method is needed to find the reference
data from which to calculate σ d . From the preceding discussion, it appears desirable to avoid estimating
∆ϕ ref(x,y) from the experiment, which means that the theoretical displacement function should be known.
Furthermore, unwrapping should be avoided because it involves additional, and sometimes unknown,
image processing by the unwrapping algorithm.
A concept fulfilling these requirements is fitting a synthetic, noise-free sawtooth image to the completely
unprocessed original one. This of course requires that we know very well what type of fringe pattern the
experiment should generate. We choose a linear phase course in x- and/or y-direction as displacement
function, which gives straight and equidistant sawtooth fringes with arbitrary density and direction. This
approach is sufficiently general for our purpose: provided the field of sensitivity is quasi-uniform, it
adapts to out-of-plane tilts, and in-plane rotations.
Since the global phase is not controlled in most of the experiments, the positions of the white-black edges
can vary considerably for otherwise identical displacements; therefore the synthetic fringe pattern has to
be given the correct phase offset as well.
Together, we have three parameters to optimise in order to obtain the best-matching synthetic image: (i) the
number of fringes per image width (1024 pixels) in x-direction, N x ; (ii) the number of fringes per image
height (768 pixels) in y-direction, N' y ; and (iii) the phase offset N 0 at some arbitrary point. For the latter, a
practical choice is the upper left corner of the images that is interpreted as (0,0) by computer graphics.
In the plots that follow in Chapters 5 and 5, N' y is multiplied by 4/3 to yield N y "fringes per 1024 pixels",
so that the fringe densities, not the actual fringe numbers in the image, are equal when N x =N y . Since we
are evaluating phase maps, the signs of N x and N y must match the respective phase gradient in the image.
Every triple (N x , N y , N 0 ) is a point in IR 3 from which a noise-free sawtooth image can be generated. Since
we are interested in the rms of the displacement-measurement error, σ d , first a least-squares fit must be
run to find that ∆ϕ ref(x,y) which minimises σ ∆ϕ , and then σ ∆ϕ must be converted to σ d via the
interferometric sensitivity vector. The quantity actually used for the fit are the pixels´ grey values in the 8-
bit phase map representations.
In multidimensional parameter spaces, it is generally not easy to implement fitting algorithms; most of
them are extensions of one-dimensional strategies. They tend to be mathematically complicated and
require some care to make them reasonably fail-safe. Apparently, there is only one genuinely
multidimensional fitting strategy, namely the "downhill simplex method" that is described in detail in
[Pre88]. It is easy to code and extend to more degrees of freedom, which is presumably why several
mathematics programs also include a "simplex" module. Although the simplex method is comparatively
slow, it has a high inherent robustness (indeed, it never failed to terminate correctly in thousands of runs
for this work).