Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 13
2.2.3.1 Intensity gradients
From (2.8) we get [Ebe79b; Gra94, formula 3.325]
1 ⎛ I ⎞
x
p( I x ) = exp −
,
I x
C I
⎜
⎝ C I
⎟ − ∞ < < ∞ , (2.9)
2 2 2 ⎠
0 0
which function is called Laplacian density. It is a negative exponential function for either sign of I x with a
mean value and standard deviation of
I x = 0 σ = 2 C0
I , (2.10)
I x
and has been experimentally verified in [Ebe79a]. The similarity between the distributions of the intensity
and its gradient has a simple and astonishing reason that has been found in [Fre96b]: speckles tend to be
"congruent", i.e. to have very similar intensity profiles, irrespective of their brightness. Hence, bright
spots are associated with large intensity gradients, while smaller gradients belong to dim speckles. The
speckles' congruence propagates the negative exponential intensity distribution to the gradients.
This observation implies that we find an interaction of the speckle intensity and its derivative in the
corresponding pdf. Indeed, the intensity and its gradient are not statistically independent since their joint
density
1 I
Ix
p( I , Ix
) = exp ⎛ ⎞ 1
⎜−
⎟ ⋅ exp
⎛ ⎜−
I ⎝ I ⎠ 2 2π
I C ⎝ 8IC
0
2
0
⎞
⎟ , (2.11)
⎠
found from (2.8) by integration, is not separable. This joint density function is plotted in Fig. 2.3.
¢ I £2¤2πC 0 ¥ p (I ,I x )
5
4
3
2
-0.375
1
0
0
0.5
I / I ¡
1
1.5
0.375
0
I x /8C 0
Fig. 2.3: Pseudo-3D plot of p(I, I x ).