Application and Optimisation of the Spatial Phase Shifting ...

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Appendix C: Derivation of intensity-correcting formulae
To include the influence of the speckle intensity, we can rewrite (3.68) as
I ( x , y , t) = O ( x , y ) + R + 2 O ( x , y ) R ⋅ cos( ϕ ( x , y , t) + α ( x , y )) , (C.1)
n k + n l i k + n l i k + n l O k + n l n k + n l
where R is assumed constant, O i (x k+n ,y l )+R=I b and 2 O ( x + , y ) R =M I ; we drop the spatial
i k n l
dependencies for convenience of notation. With D n I n –O n , we can write
D = R + 2⋅ O ⋅ R ⋅ cos( ϕ + α )
n n O n
R
: = a
+ 2 R cosϕ cosα O − 2 R sinϕ sinα
: = a
: = a
O n n O n n
0 1 2
O
, (C.2)
where the quantities of interest are a 1 and a 2 , since they contain ϕ O . Setting n ∈{-1, 0, 1}, thus assuming
phase steps of (–α, 0, α), the linear equation system is given by
⎛1
cosα
O 1 sinα
O ⎞⎛
1 a
⎜
⎟⎜
⎜1 O0
0 ⎟⎜
a
⎜
⎝1
cosα
O − sinα
O
⎟⎜
⎠⎝a
− − 0 −1
1 1
⎞ ⎛ D ⎞
⎟
⎟ D0
⎟ = ⎜ ⎟
⎜ ⎟
⎜ ⎟ ,
⎠ ⎝ D ⎠
1
2
1
(C.3)
which we abbreviate by Pa = D. As long as O -1 ,O 0 ,O 1 ≠0 and 0≠α≠180°, P is regular and
rank(P)=rank(P, D)=3 is valid; hence we can solve the equation system by inverting: a=P -1 D. This can be
carried out by Cramer's rule [Bro87, p. 159]. With the abbreviations
⎛1 C1 S1⎞
⎛a
⎜ ⎟⎜
⎜1 C2 0 ⎟⎜
a
⎜ ⎟⎜
⎝1 C3 S3⎠
⎝a
: = P
⎞ ⎛ D ⎞
−1
⎟
⎟ D0
⎟ = ⎜ ⎟
⎜ ⎟ ,
⎜ ⎟
⎠ ⎝ D ⎠
0
1
2
1
(C.4)