Application and Optimisation of the Spatial Phase Shifting ...

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Appendix A: Counting events
Intensity level crossings per unit length
In Chapter 2 we have encountered two occasions where probabilistic events had to be counted. The
derivation is similar for both of them. The level-crossing problem of (2.14) starts from the integral
[Bar80]
Nd ( It ) = ∫ δ ( I − It
) ∂ I / ∂ x dx ; (A.1)
d
here N d (I t ) is the number of times that the intensity crosses the value I t on a path d (the probability for the
point I=I t being an extremum has measure zero on a straight line). The δ function assures that the integral
responds only when I=I t ; to make each such contribution equal to one, i.e. to establish a counting function,
the integration over x must be undone by the derivative ∂I/∂x; the modulus signs ensure that +1 is being
counted for each event. However, since now I x appears, which is not independent of I, we need to know
its expectation value at a given I, which requires the joint pdf p(I, I x ) and changes the integral to
N ( I ) = δ ( I − I ) p( I, I ) I dI dx
d t t x x x
d Ix
=
∫ ∫
∫ ∫
d Ix
p( I , I ) I dI dx
t x x x
on integrating over unit length, one obtains the density of the level-crossings,
( )
ρ( I ) = ∫ p I , I I dI ,
t t x x x
I x
;
(A.2)
(A.3)
which is (2.14).
Intensity zero points per unit area
By the same line of argument as above, we can start from [Ber78, Bar81]
N ( A ) ( A ) ( A , A ) / ( x , y ) dxdy , (A.4)
= ∫∫ δ δ ∂ ∂
disl r i r i
S
where the dislocation is expressed by the vanishing of A r and A i , the integral is over an area S and the
quantity between the modulus signs is the Jacobian GJG=FA r,x A i,y – A r,y A i,x F. Obviously, we need
p(A r , A i , A r,x , A i,x , A r,y, A i,y ) to evaluate this integral, or, more specifically, p(0 , 0 , A r,x , A i,x , A r,y , A i,y ) after
the δ functions are accounted for. In analogy to above, we write
N
disl
( )
= ∫∫ ∫ ∫ ∫ ∫ p 0, 0, A r, x , A i, x , A r, y , A i, y ∂( Ar , Ai ) / ∂ ( x, y)
dA r, xdA i, xdA r, ydA i,
ydxdy
, (A.5)
S Ar , x Ai , x Ar , y Ai , x