Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 15
We see that d + (I t ) is the distance over which I remains above the threshold I t , which is set to 1.5I in this
example. We find three above-threshold events and hence obtain three different measurements of
d + (1.5I). But instead of collecting events, it makes of course more sense to aim analytically for an
average of the above-level distance, d + (I t ), and fortunately its theoretical derivation is available.
The number of events per length unit that the signal crosses I t , the so-called level-crossing density, can be
calculated by means of a long-known formula by Rice, as detailed in [Ebe79b, Bar80]:
∞
8C0
It
It
.
ρ( It ) = Ix p( It , Ix ) dIx
= exp
⎛ ⎞ 122
∫
⎜−
⎟ =
π I ⎝ I ⎠ d
−∞
s
π
It
I
exp
⎛ ⎜−
⎝
It
I
⎞
⎟
⎠
(2.14)
(see also Appendix A), where we have used (2.2) and (2.43) to relate the expression to the speckle size d s
produced by a circular scattering spot; an example for a square spot is given in [Bah80]. The average
number of level crossings per speckle size d s is depicted in Fig. 2.5, and reveals that I/2 is being crossed
almost once per speckle size (for pixel-integrated speckle it should, and does, contract about I [Bar88]).
1
/ d s
¢ N(I t )
£
0.5
0
0 1 2 3 4 I t / I ¡ 5
Fig. 2.5: Expected number ¤N (I t per speckle size d )¥ s of crossings of intensity level I t as a function of normalised
threshold intensity I t /¤I¥. This curve follows simply from setting d s =1 in (2.14).
Being aware, however, that (2.14) accounts for both positive and negative crossings, we conclude that I(x)
goes beyond or below I/2 every other speckle size. Now we can answer the question over what distance
I remains above/below a certain I t , by evaluating the expressions for the average above- and below-level
dwell distances [Bar80 * ],
d
+
∫ p( I)
dI
I
2d
I
t
s
( I ) ; d ( I )
t
∞
= =
− t
=
1
I
ρ( I
t
t
)
122 . π
2
1−
1
2
∞
∫
It
p( I)
dI
ρ( I )
t
2ds
=
122 . π
I
I
t
⎛ ⎛
⎜exp⎜
⎝ ⎝
I
I
⎞
⎟ −
⎠
⎞
1⎟
, (2.15)
⎠
which are the total fractions of distance that the intensity spends beyond/below I t , divided by the mean
density of upward/downward level crossings. Of these latter, each contributes of course one half to the
* With a misprint in Eq. (16), ¦ where σ and µ must be swapped.