Application and Optimisation of the Spatial Phase Shifting ...

200 Appendix C: Derivation of intensity-correcting formulae
we have
⎛1 C1 S1⎞
⎜ ⎟
P = ⎜1 C2 0 ⎟ , det ( P)
= C2S3− C1S 3+ S1C3 − S1C2
⎜ ⎟
⎝1 C3 S3⎠
P
P
P
0
1
2
⎛ D ⎞
−1
C1 S1
⎜
⎟
= ⎜ D0
C2 0 ⎟ ,det ( P ) = D−
C2S3− C1D S3+ S1D C3−
S1C2
D
⎜
⎟
⎝ D C3 S3⎠
1
0 1 0 0 1
⎛1 D ⎞
−1
S1
⎜ ⎟
= ⎜1 D0
0 ⎟ , det ( P1
) = D0S3 − D−1S 3+ S1D1 − S1D0
⎜ ⎟
⎝1 D S3⎠
1
⎛1 C1
⎜
= ⎜1 C2
⎜
⎝1 C3
D ⎞
−1
⎟
D0
⎟ , det ( P2 ) = C2D1 − D0C3 + C1D0 − C1D1 + D−1C3−
D−1C2
,
⎟
D ⎠
1
(C.5)
from which we get a 0 , a 1 , a 2 by
det( P0
)
1
2
a0
= a1
= a2
=
det( ) , det( P )
det( ) , det( P )
P
P det( P ) .
(C.6)
For the quotient a 2 /a 1 = tan ϕ O , we obtain
a
a
2
1
det( A 2 ) C1( D0 − D1 ) + C2( D1 − D−1) + C3( D−1 − D0
)
= =
det( A ) S1( D − D ) + S3( D − D )
1
1 0 0 −1
=
α( )
( O−1 D1 − D0 + O1 D−1 − D0
)
O ( D − D ) + cos O ( D − D ) + O ( D − D )
0 1 −1 −1 0 1 1 −1 0
sin α ( ) ( )
,
(C.7)
which is (6.4).