Application and Optimisation of the Spatial Phase Shifting ...

144 Improvements on SPS
Fig. 6.7 tells us that the complete sampling window is now definitely larger than the mean speckle size of
d s =3 d p . But in addition to the phase-error compensation, (3.56) also constitutes stronger spatial
averaging. Here, the sine and cosine functions are averaged before phase retrieval, which has been shown
to be a better choice than averaging phase maps after the arctangent operation [Hun97].
Although the 3+3 averaging formula still calculates the phase separately for each pixel, there is a loss of
spatial resolution associated with the larger sampling window. But since our "resolution cell" has already
been 3 pixels wide before, the relative change is not significant; and up to (at least) N x =100, the phase
gradient of the object displacement is well resolved and shows less noise than with the standard phase
calculation.
6.2.3 Combined intensity and phase gradient compensation
Each of the error-suppression strategies proposed suffers from the drawback that its effectiveness to cope
with I x or ϕ O,x could be reduced by the fluctuations not accounted for, i.e. ϕ O,x or I x . Hence it is natural to
combine both of the approaches to obtain a formula that reduces the σ d caused by the speckle structure of
both object intensity and phase. The simplest way to construct such a phase calculation is to establish an
averaging formula for terms as in (6.4). With α=90°/sample, we rewrite (6.5) for the two "boxes" of Fig.
6.7 [Bur98a]:
tan ϕ
tan ϕ
O0
O1
=
=
O0 ( D1 − D−1)
O ( D − D ) − O ( D − D ) : =
1 −1 0 −1 0 1
O1 ( D2 − D0
)
O ( D D ) O ( D D ) : =
− − − K
2 0 1 0 1 2
K2
K − K
3 1
K5
− K
6 4
, (6.7)
with pixel indices according to Fig. 6.7, and numbering of the K n according to the order of indices of the
square roots at the beginning of each term. Now applying what we have learnt in Chapter 3.2.2.4, we can
easily compose these two intensity-corrected phase calculations according to (3.55) and arrive at
ϕ
O
mod π = arctan
K2 + K4 − K6
− K + K + K
1 3 5
, (6.8)
which is an averaging formula correcting for both intensity and phase fluctuations. As already indicated in
Fig. 6.6, the intensity correction works best for B=3; the contribution to σ d coming from speckle phase
gradients was assumed to be independent of B. Fig. 6.9 gives an overview of the best results from all
combinations of phase-calculation methods and B values tested in this subsection. The black curves are
repeated from Fig. 6.8 for comparison; for the intensity-correcting formulae, the underlying set of
interferograms is necessarily a different one, with B=3, but also d s =3 d p .