Application and Optimisation of the Spatial Phase Shifting ...

6.3 Modified phase shifting geometry 147
A phase shift of (α x , α y )=(90°, 90°) yields carrier fringes slanted by 45° and permits arranging the
evaluated pixels in various ways. This is shown in Fig. 6.11: the target pixel of the phase calculation is I 3 ,
with some arbitrary phase of ϕ O , and the surrounding pixels have nominal phase shifts of ϕ O 90° as
indicated. Note that the pixel numbering can no longer indicate relative phase shifts (e.g., α 1 =α 2 ); besides,
we will identify the intensities I n simply by pixel numbers n where appropriate for simplicity of notation.
The geometry in Fig. 6.11 on the left suffices to use (3.19) for phase retrieval in both x- and y-direction; I 3
will be assigned the average of the calculations. To the right, some additional pixels with ϕ O +180° give
the possibility to use 3+3 averaging formulae; both of the methods will be explained below.
ϕ O −90° ϕO
ϕ O +90°
I 2 I 4 I 2 I 4
I 3
I 3 I4
5
I 1
I 1
I 8
ϕ O +180°
I 6 I 6 I 7
Fig. 6.11: Pixel clusters for phase calculation from oblique carrier fringes; orientation and spacing indicated by
black bars. For simplicity, pixels are numbered consecutively. Left-hand side: pixels usable for 3-point
formulae; right-hand side: pixels usable for 3+3-point formulae.
At this point it should be noted that the reduction in M I now needs to be determined by (3.67), since the
carrier frequency has an x- and a y-component. Therefore, for α x =α y =90°, we get (sin(π/4)/(π/4)) 2 = 0.81.
Consequently, another 10% of modulation is lost, which will lower the optimum value of B somewhat.
Now, using (3.19), a double phase determination is possible for pixel 3, according to
I − I
tan ϕ O =
I − I
4 3
2 3
=
I
I
− I
− I
6 3
1 3
, (6.10)
which corresponds to the x- and y-direction, respectively. The phase is then determined from
( 4 − 3) + ( 6 − 3)
( I − I ) + ( I − I )
I I I I
tanϕ O =
2 3 1 3
I − 2I + I
=
I − 2I + I
4 3 6
1 3 2
; (6.11)
this is neither a new phase-shifting formula, nor an extended averaging scheme in the sense of [Schmi95].
Instead we get an average that, despite being spatial, does not reduce the resolution of the measurement. It
serves to decrease σ d , although the phase measurements involved are not completely statistically
independent, since the central pixel 3 is used twice. Note also that the other two possibilities of
calculating the phase, with pixels {1, 3, 4} and {2, 3, 6}, would just double the numerator and
denominator in (6.11), which has no effect.