Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 59
ϕ
ϕ
O,
i
O,
f
I3i
( x, y) − I1i
( x, y)
( x, y) mod 2π
= arctan
I ( x, y) − I ( x, y)
0i
2i
I3 f ( x, y) − I1f
( x, y)
( x, y) mod 2π
= arctan
I ( x, y) − I ( x, y)
0 f 2 f
, (3.22)
and then determine the phase change
( O, f
O,
i )
∆ϕ ( x, y) mod 2 π = ϕ ( x, y) mod 2π − ϕ ( x, y)
mod 2π mod 2π
. (3.23)
Admittedly, this requires more information than the phase-of-difference approach – 8 images with (3.16),
and 6 with (3.17) –, but eliminates all the problems brought about by the ambiguity of intensity
differences. Also, the pixels are truly regarded as independent entities, which accounts appropriately for
the speckle nature of the wavefront to determine. In this case, the phase calculation reproduces the
expected fringe profile rather well, as the white curves in Fig. 3.3 demonstrate. The displayed sawtooth
edges are somewhat blurred by the averaging over the residual speckle noise; but the corresponding
measured phase map, shown in Fig. 3.7, is of excellent quality when we compare it with the other
unfiltered results obtained so far. In that case, σ ∆ϕ = 18.2° without any low-pass filtering. Also, the pdf of
measured phases is now uniform, which shows that computational biases are negligible for this method.
Fig. 3.7: Result of calculating ∆ϕ with the difference-of-phases method.
This confrontation clearly indicates that it is necessary to genuinely measure the speckle phases twice to
get the best sawtooth image. While an approximate recovery of information from reduced data sets is
possible, the performance of this approach remains restricted. Therefore, the performance data given in
chapters 5 and 6 are based on sawtooth images from the difference-of-phases method without exception.
A practical merit of keeping ready the speckle phase distributions for every recorded object state is that
one need not compare all data to the initial state anymore. In other words, it becomes possible to track
phase differences incrementally even if the first and last state show decorrelated speckle patterns [Flo93].
We will come back to this issue in Chapter 6.7.