Application and Optimisation of the Spatial Phase Shifting ...

3.4 Spatial phase shifting 93
although such a test is not possible with the three-step 90° formulae, the findings thus far strongly indicate
a similar behaviour.
In the error-compensating formula (3.56), S(n) requires only two samples but C(n) uses four, whilst in
(3.57), both terms include four samples; but also for these methods, the performances are virtually
identical. Therefore we will not compare these in detail, but we do investigate the performance of (3.56)
against that of (3.58); these results are shown in Fig. 3.37.
ν N
ν N
ν y
0
ν y
0
–ν N –ν N
2 3 4|0 1 2 ν x /ν 0x 2 3 4|0 1 2 ν x /ν 0x
1.57
1.57
0.785
0.785
0
0 1 2 3 4
0
0 1 2 ν x /ν 0 3
-0.785
ν x /ν 0
-0.785
-1.57
-1.57
Fig. 3.37: Top row: bsc(ν x ,ν y ) for (3.56) (left) and (3.58) (right); bottom row: bsc(ν x ) for (3.56) (left) and (3.58)
(right).
From the images as well as from the plots, we can see that (3.56) operates with greater stability in the
whole spatial frequency range, including the signal regions. Also for the 3-sample formulae investigated
before, the tendency was recognisable that α=90° gives slightly safer phase determination than α=120°.
For the reasons mentioned above in 3.2.2.4, this difference is even more pronounced when we attempt to
correct phase-shifting errors. After our numerous considerations of spatial frequencies, the reasons for this
are clear: setting ν c,geom to ν N /2 assures best utilisation of the frequency plane and best suppression of
detuning errors.
Our scrutiny of the effect of different sampling pixel clusters yields the interesting result that the
representation of the used formula can be chosen at convenience. This facilitates a simple general strategy
for placing the signal sidebands optimally: given the invariant course of bsc(ν x ), or bsc(ν x ,ν y ) for
composite x- and y-phase shift, one can refer to that representation of the phase-extraction formula which
gives equal frequency responses of S(n) and C(n), and maximise the signal utilisation (in which the
system MTF will also play a role) while minimising the phase-shifting errors. Once this is done, one can