Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 63
∞
~ ~
S'( ) I ( )
*
0 = ν S ( ν ) dν
∫
−∞
∞
x x x
~ ~
C'( ) I ( )
*
0 = ν C ( ν ) dν ,
∫
−∞
x x x
(3.35)
where tilde denotes the Fourier transforms and the sign convention [Bro87]
∞
~ f ( ν x ): = ∫ f ( x)exp( + 2 π i ν x x)
dx
(3.36)
−∞
is adopted, i.e. the phase runs forward in the Fourier transform.
It is seen from (3.35) that the spectrum of I(x) is weighted, or filtered, by the spectra of S(x) and C(x),
which is why we have called them filter functions. We will therefore refer to S
~ ( ν ) and C
~ ( ν ) as filter
spectra. Since I(x), S(x) and C(x) are real functions with Hermitian Fourier transforms, we can simplify
the integrals (3.35) to [Fre90a]
x
x
∞
∫
*
x x x
S' ( 0) = 2 Re I ~ ( ν ) S ~ ( ν ) dν
0
∞
∫
*
x x x
C' ( 0) = 2 Re I ~ ( ν ) C ~ ( ν ) dν .
0
(3.37)
This is, the filter outputs are indeed composed of all input spatial frequencies that may be present in I(x),
with weights determined by the moduli of the filter spectra, S
~ ( ν ) and C
~ ( ν )
latter also as filter responses. With our initial choice of S(x) and C(x), we have
∞
∫
S'( 0) = I ( x) ⋅ − sin 2πν
xdx
−∞
∞
∫
0x
C'( 0)
= I ( x) cos2πν xdx ,
−∞
0x
x
x
; we will refer to these
(3.38)
being proportional to the Fourier sine and cosine transforms [Bra87, p. 17] of I(x) at the frequency ν 0x ,
and the spectral descriptions read:
∞
∫
S'( 0) = 2Re I ~ ( ν ) iδ ( ν −ν ) dν
0
∞
∫
x x 0x x
C'( 0) = 2Re I ~ ( ν ) δ ( ν −ν ) dν .
0
x x 0x x
(3.39)
This models the ideal case that we can evaluate the signal over an infinite amount of space, which leads to
unity filter responses at the nominal frequency ν 0x and perfect suppression of all other ν x .