Application and Optimisation of the Spatial Phase Shifting ...

166 Improvements on SPS
This procedure requires a storage interval ∆t for the phase maps ϕ O (x, y, t) which is matched to the
possibly varying velocity of object deformation and displacement. The implicit fringe counting capability
of TPU lends itself for driving such a matched course of ∆t automatically. As far as I know, this issue has
only once been dealt with before on the basis of speckle decorrelation analysis [Gül93]; here however, the
quantity of interest that we want to limit is the number of fringes in ∆ϕ(x, y) instead of the speckle
decorrelation [Bur00b].
In practice, ESPI often deals with objects consisting of several independent parts that may undergo
different displacements and deformations. However, sawtooth fringes do not allow to determine rigid
body movements or the sign of the deformation itself, unless the fringe orders are tracked by additional
devices like, for instance, a phase stabilisation unit [Bro00]. We will see that temporal unwrapping
delivers these data for each object part automatically.
6.7.1 Temporal unwrapping of speckle phases
The use of temporal unwrapping is not entirely straightforward in speckle interferometry; we will
therefore briefly consider the cumulative impact of speckle noise on displacement data.
Not surprisingly, badly modulated pixels cause problems also in this application. The statistical
fluctuations of the calculated phase should yield a displacement of zero when monitored over a sufficient
number of frames. It was however observed that even for longer observation sequences with hundreds of
frames, some of these pixels seemed to change their phase constantly in one direction; both signs of
displacement were present. In a fringe counting procedure, these pixels would trigger data storage even
when no actual displacement has occurred. Therefore such outliers have to be suppressed; and as usual in
speckle interferometry, a low-pass filter can serve to do so. This may be objectionable in TPS, because it
impairs the spatial resolution; but for larger speckle sizes, as used in SPS, the resolution will not suffer
greatly.
There are sophisticated and well-founded filtering schemes [Hun97] that give excellent rejection of noise
in the displacement map over long, albeit not infinite, times of observation [Cog99]. For reasons of
processing speed, a simpler filtering scheme is used here. The accumulated phase Φ(x, y, t) of a pixel at
time t is considered faulty when it differs by more than π from the accumulated phase of at least one out
of its nearest neighbours. In that case,
( )
( x y t) = ( x y − t) + ( x − y t) + ( x + y t) + ( x y + t)
Φ , , : Φ , 1, Φ 1, , Φ 1, , Φ , 1, / 4 (6.26)
and the outlier is eliminated.
By the selection criterion, filtering takes place only when necessary, and processing time is saved. This
helps to obtain a high frame rate, which is very important since also temporal unwrapping relies on the
sampling theorem, as detailed above; and due to the cumulative nature of the process, errors due to missed
fringes (violation of the sampling condition) will last in the map of Φ(x, y, t) until it is cleared. The phase