Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 69
− 0. 259I0 − 0. 707I1 + 0.
966I2
− ° mod 2π
= arctan
0. 966I − 0. 707I − 0. 259I
(3.50)
( ϕ 15 )
O
0 1 2
with the transfer characteristics shown in Fig. 3.11, which are indeed very similar to those of Fig. 3.10.
Note the different normalisation of the frequency axis; here, 2ν N = 3ν 0x , and –(ϕ O –15°) is detected at
2ν 0x α=240°/d p (aliased to –120°/d p ).
2
3.14
1.57
1
amp( S
~ ( νx))
amp( C
~ ( ν x ))
0
0 1 2 3
-1.57
ν x /ν 0x -3.14
0
0 1 2 ν x /ν 0x 3
arg( S
~ ( νx))
arg( C
~ ( ν x ))
Fig. 3.11: Filter spectrum for 3-step-120° phase-sampling formula (3.50); left: amplitudes, right: phases.
But also for cyclical permutations of the intensity samples, which is equivalent to changing the offset by
integer multiples of α [Schmi95b], the transfer functions of our formulae change considerably. This
brings up the question whether a formula really can benefit from such an operation: generally speaking,
improving the matching of amp( S ~ ( ν )) and amp( C ~ ( ν )) worsens the quadrature properties, and vice versa,
so that we are in need of a method to account for both aspects simultaneously.
An interpretation of S
~ ( ν ) and C
~ ( ν ) as complex phasors, also suggested in [Mal97], is very helpful to
x
x
reach conclusions about this point. Therefore we introduce the auxiliary function
~ ~
( ( ) ) arg ( ( ) )
bsc( x ): arg C ~ ν = ( ν x ) + S ν x − C ν x , (3.51)
where bsc stands for the bisector between S
~ ( ν ) and C
~ ( ν ) . Of course, it is the bisector only when the
x
moduli of S
~ ( ν x ) and C
~ ( ν x ) are equal; its general range is -π/2bsc(ν x )π/2. At ν x =ν 0 , S
~ ( ν x ) and
~ C(
ν ) are in quadrature, and bsc(ν 0x ) = –45°, which is the value indicating correct phase calculation.
x
This is valid for all ν x , since arg ( C
~ ( ))
x
x
ν is being subtracted, so that the angle between the phasors always
has one side on the real axis. The advantage of bsc(ν x ) is that it responds to changes in both modulus and
phase of S
~ ( ν ) and C
~ ( ν ) . The ideal situation is sketched in Fig. 3.12 on the left, being the graphical
representation of (3.41).
x
x