Application and Optimisation of the Spatial Phase Shifting ...

204 Appendix D: Alternative error-compensating formulae
respect, the schemes developed in 6.3 may be somewhat more suitable to preserve spatial resolution. The
relative pixel weights for (6.11) are visualised in Fig. D.7; it can be seen that the target pixel, in the centre
of the cross shape, contributes the largest part to phase calculation.
ϕ Ο
−90° ϕΟ
0
ϕ Ο
+90°
1
0
-2
1
1
-2
0
1
0
S xy
(n)
C xy
(n)
Fig. D.7: Relative pixel weights for spatial intensity sampling by (6.11).
When the sampling pixel cluster is enlarged to enable the application of (6.16), we get the weighting
windows depicted in Fig. D.8.
ϕ −90° Ο ϕΟ
ϕ Ο
+90°
0
0
8
–4
0
ϕ Ο
+180°
2
2
–4 –2 41
–4
0
–2
2
0
1
S xy
(n)
C xy
(n)
Fig. D.8: Relative pixel weights for spatial intensity sampling by (6.16).
Also in this case, the intensity sample from the target pixel enters the phase calculation with the greatest
weight; however, for C xy (n) some more remote pixels must be included, fortunately with small
contributions.
Comparing the sampling windows shown in Fig. D.7 and Fig. D.8 with those from Fig. D.2 and Fig. D.4,
it gets apparent that the 5-sample formulae are associated with significant low-pass filtering of the
resulting phase maps. In particular, the central pixel has zero weight in the implementations of the S xy (n),
this being a necessity in symmetrical 5-sample formulae.
On the other hand, since spatial resolution is generally a small problem in practical ESPI, the formulae
that have been briefly investigated here should prove useful as well.