CP-542
Calculation of Turbulence Degraded Point Spread Function of an
Imaging System
G. Saplakoglu, F. Erden and A. Altmtag
Bilkent University,
Department of Electrical and Electronics Engineering
Bilk& 06533,
Ankara, Turkey.
Summary
used. In section 3 we will formulate the problem that
will be dealt with, and sketch the method of analysis.
We will conclude in section 4, emphwizing our main
results.
In this paper the point spread function (PSF) of a turbulence degraded imaging system is statistically characterixed. In particular sufficient data and functional
fits are given for the calculation of long exposure average PSF and optical transfer function (OTF). Meth- 2
Modeling
The Turbulent
Atods are presented for the calculation of second order
mosp here
statistic of PSF, hc~weverspecific results pertaining
to this ewe are beyond the scope of this paper and
Intuitively, the turbulent atmosphere can be thought
are not presented here.
of as a collection of random air packets [referred to
as eddies] with varying refraction indices n(r). Such
variations in the index of refraction cause optical path
1 Introduction
differences for beams propagating in different parts of
the atmosphere. Consequen& when these beams are
Effects of atmospheric turbulence on the propaga- collected by an imaging system a degraded image is
tion of optical waves is an extensively studied topic. obtained.
There are several excellent works that summarize the
In this work we will need the power spectral density
rearch done in this ‘area [see for example [l], [2], of the refractive index vatiakions. Far this purpose the
[3] 1. However the results presented in these and re- well known van Karman spectrum[4] will be used;
lated works usually involve fairly complicated expre~
sions most of which can not be evaluated anaIytically.
Hence the reader has to content with either tabulated
or plotted results presented for specific atmospheric
In 2.1 C& (meterU2j3) is cailed the structure conconditions.
Our aim in this work is twofold. First, at the con- stant and is directly related to the strength of the
cluaion of this paper we will point out to OUFmain turbulence. Its magnitude can‘ range between lO-17
results in 8 manner such that they will be accessible (weak turbulence) to lo-l3 (strong turbulence). The
to those professional that are not directly involved wavenumber ti (i/meter) is a quantity that is inin investigating the effects of turbulence on the prop- versely proportional to the size of the turbulent edagation of optical fields, but require a suitable char- dies. At a particular cc, iP&) represents the relaacterization of such effects in their research. A good tive amount of eddies with sizes 2n/~. The numeriexample to this category is the signal processing cum- cal values of the constants ccm and ccg are given by;
munity who work in the restoration of turbulence de- Km zz 5.92/10 and ~0 = 27FILo respectively where IO
graded images. Second, we will introduce some new and Lo are the inner and outer scales of the turbutools whereby higher statistical order characteriaa- lence respectively [ minimum and maximum sizes of
tion of turbulence degraded images can be numeri- turbulent eddies]. Near the ground 10 is typically a
cally evaluated. However the detailed treatment of few millimeters and Lo z h/2 where h is the height
this latter subject is beyond the scope of this paper above ground.
Since we will be dealing with extended objects as
and will be published elsewhere.
We will start our treatment in section 2 by stating opposed to simple plane waves, Gaussian beams or
the turbulence and propagation models that will be point sources, we will use extended HuygensFresnel
40-Z
Using standard Fourier optics techniques the intenprinciple[5] which is a modified version of HuygensFresnel principle that takes into account the effects of sity of the image produced by the optical system in
:
the turbulent medium. In its paraxial form, extended figure 1 is given by[6],
Huygen~Fresnel principle specifies the complex field
distribution go [at plane 2 = 0] in terms of the cornplex field distribution g-d at plane z = -d taking into
account the diffraction and turbulence effects over the
where &(p) is the intensity distribution of the fipropagation path via,
nal image, rl?jJp) is the geometrical image of the
object and 1 h(p,p’) I2 is the point spread function
that represent the combined effects of turbulence and
diffraction,
where k = WX, IN?&P’) is a complex function that
represents the turbulence induced phase and amplitude perturbations [ at (p, 0) ] of the field of a point
source located at (p’, -d& The statistics of $(p,pl’)
is readily available in the literature which we will refer
to when necessary.
1n our notation we will use p to indicate transverse coordinates (x, y) and r to indicate 3 dimensions (2, v, t). We will use paraxial approximation
throughout this work, hence the propagation will be
assumed to be taking predominantly along the z axis.
Consequently objects and images wiIl be assumed to
be planar and will be represented by functions defined
over the transverse coordinate p,
Note that the object coordinate pl’ is scaled by the
magnification m = &/dl of the imaging system. In
other words, 1 h(p,J) I2 is the intensity at p in the
image plane due to a point source whose image is
located at p’ also in the image plane.
In deriving 3.3 and 3.4 we have used the fact that
for incoherent illumination,
< S-dl(YL)Sld,(P~~
>=<I h&44) Ia>qPX-343)
(3 5)
where < v > indicates time or enmmble avera&g
which are assumed to be interchangeable+ That ia,
the final image is obtained by averaging several short
exposure images, consequently, averages can be evaluated either by using time averaging [ which is physically
done when the exposure time T ie much greater
The b&c imaging system that will be analyzed is
than
about l/100 seconds] or averaging individual
shown in figure 1. The object is located at z = 41
members
of an ensemble, each member of which repplane. The medium between z = 41 and z = 0
plan- represent the turbulent medium that the rays resents a short exposure [T < l/l00 seconds] result.
Now that we have an expression for the PSF, we
have to propagate before reaching the lens. It is a~sumed that the distance dl is suffi&ntly large so that can obtain its statistics. The mean of PSF is given
paraxial approximations can be used. Note that this by,
condition is easily met in practice since the effects of
turbulence is a major problem only for those optical
systems that perform imaging over fairly long propagation paths.
At z = 0 plane we have 8 thin lens of focal length
F with pupil function a(p). There is no generality and its autocorrelation by,
lost in representing an imaging system with a thin
lens of finite size since a well corrected diffraction limited imaging system mimics a simple thin lens. The
medium between the lens and the image plane located
at x = & is assumed to be turbulence free, The object
distance, image distance and the focal length of the
lens are related by the well known imaging condition
l/d1 + l/da = l/F.
Our main goal in this work is to statistically characterize the long exposure point spread function (PSF)
The problem now is to evaluate the expect&i mls
that appear inside the integrals of 3.6 and 3.7. In
of the optical system shown in figure 1.
3
Imaging
Extended
Objects
Through
the Turbulent
Atmosphere
reference [7], it is shown using first and second order
Rytov appruximations[8], and assuming a statistically
atationzny turbulence that,
For a given K&Q-J in 3.14 we propose the following
fit,
(3.16)
where$1(P,P’) is
the first order Rytov approximation. Note that the mean irradiance of $1 is independent of spatial coordinates as implied by the stationarity Mumption and is given by,
Using similar methods, the expectation that appears
in 3.7 can be written as,
The expectation E{ $1 (PA, pa)+i(pB,
uated in [7],
E{~l(PArPk)~I(PS~P~)}
(2+2dl
jm
0
do K: a&)
pb)}
is Ed-
evaluated analytically.
Using the above functional
fit, 3.6 and 3.7 can be
evaluated. For example, for a circular pupil of diameter D, it is straight forward to show that 3.6 is shift
invariant an is given by,
Similarly the average optical transfer function
(OTF) can be written as,
=
I1
0
dt
Jo(l
7 = t(PA - PJJ + (1 - t)(pi
y 1 FE)
- p’B>
(3.11)
(3.12)
Using similar me thods the following expectation
can be calculated as
E{~l(PA,Pk)~l(PB~PIB)}
The values of the coefficients ai, bi and ci are given
in table 1. Maximum relative error, i.e. I 9(Y) mt(r) I h(r) is around 1%. Note that the functional
form of gEt is such that the i integral in 3.11 can be
where
CYl(P)=
1, Ms+
0,
otherwise
(3.19)
= -(W2L*
Specific plots corresponding to the above two equations are rare in the literature. One such example
is the OTF plots given in [9] which also refers to a
series
of reported experimental results. For easy cornConsequently now we have all the necessary forparison
those plots are reproduced in fig 2 together
mulas to numerically calculate equations 3.6 and 3.7.
with
equation
3,18, It can be observed from the figUnfortunately the integrals that appear in 3.11 and
ure
that
our
results
are in very good agreement with
3.13 can nut be evaluated analytically. For this purthe repor ted experiments.
pose we will first present a functional fit to the integral
expression,
4
(3.14)
definition
we have,
Conclusion
The main results of this paper are equations 3,17 and
3.18 which can be easily evaluated using the function
g&)
defineri in 3.14. As mentioned in the introduction, those readers that are not interested in the the-
ory of propagation in turbulence can directly use the
above equations in their work.
Although in this paper we have primarily dealt with
long exposure characteristics of turbulence degraded
imaging system, it should be pointed out that the
second order statistics of the PSF are rekated to the
short exposure characteristics of the optical system.
For exampIe the variance of the PSF specifies an envelope around the mean PSF within which the majcwity of the particular realizations of the short exposure
PSF’a must Le. Such information cm &so be used
in & membership based restoration algorithms[ lo].
However, 88 mentioned previously, the calculation and
interpretation of higher order statistics, in particular
the correlation function 3.7 is beyond the scope of this
paper. Nevertheless sufficient material is presented
here to enable interested readers to aid in calculating
some second order statistic.
References
Strohbehn, ed., Law- Beam Propaguiion
in the Aimosphare, (Springer-Verlag, New York,
1978).
J.W.
A. Ishimaru, Wawe Prulpaguir’on atali Scaikting
in Randurn Media, Vuls. 1 and 2, (Academic
Press, New York, 1978).
R.M. Manning, Sfoc~asfic EIcciromagnetic 1lnage Pmpagation and Adaptive Compensation,
(McGraw-Hill, Inc., 1993).
J.W. Goodman, Siatisiical
Intcrsience, 1985).
Optics,
(Wiley-
RF.
Lutomirski
and H.T, Yura, Propugdion
of a Fini#e Opiical Bea7?a in an mh2u~e?2eous
Me&m, Applied Optics, Vol. 10, No, 7, July
1971, pp+ 18524658.
J .H. Shapiro, D#nx#ion Limikd Ahwspheric
haging of Ex#ended Objects, J. Opt. Sex. Am.,
Vol. 66, No. 5, May 1976, pp, 469-477.
H.T. Yura, Mti#~al Coherence Ftcnction of UK
niie Cmws Section Optical Beam Pmpagafing in
a %&len# &x&m, Applied Optics, Vol. 11I
No. 6, June 1972, pp. 1399 - 1406.
H.T. Yura, Opkal Pqagaik
Thmgh a Twbdent Medium, J. Opt. Sot. Am, Vol. 59, January 1969, pp+ 111412.
R.E. Hufnagel and N.R. Stanley, &buUkn
7hnsfer Fmclion Associated wiih hiage Z-hnsmission #Arough Turldeni Media, J. Opt- Sot.
Am., Vol. 54, No. 1, January 1964, pp. 5%6L
MI. Sezan and H.J. Trussed, Pro#olype Image
Cunsfminis for Set- Theurettc huge Reshmztiun,
IEEE Trans. on Signal Proc., Vol. 39, No. 10, October 1991, ppw 2275-2285.
40-5
Turbulent Medium
Figure 1: The imaging system analysed in text.
L52587223557296e-01
2.22~40403416090e-01
L53560049849163e-04
5.3 1100749264345e-05
3,11912509978245e-03
1*78898283405542e-03
8.44463086599324~02
1.29122982863946e+OO
4.32127386703938e-02
4.62634914234235~~02
9.22111409451529e-03
3.50105359537060e-01
3.07229641581417~01
9*03728380407168e-0I
L69647810~48367e01
2.65879456665575~~01
4.12$73188094264e-06
6.31667100284521e07
3.51919439483754~~02
3+56802359344033e-02
l.O7767909208223e-04
2.7962087$098983e-05
2.24475783616493e-03
1.125474#3604572&-03
2.37864517196957e-01
1.3519866990?389e+UU
1.54938390693092e-01
4.93556986787747~01
Table 1: Values of the coeeficients in 3.16
40-6
1
09
l
08
l
07+
06
l
E
0
05
I
04
l
03+
02
l
01
l
0
Figure 2: Theoretical and experimental OTF curves for dl = 1.1 x H14na,F = 0,9Hnt, ~0 = 0,2513m-‘, X =
O.SSpm,Ci = 4.85 x 10-15m-2~3, L? = lna . Solid curve is the theoretical OTF given in reference [9].
The dashed and dash-dotted curves are the OTF’s calculated by the methods presented in this work for
h/W = lo3 and lo4 respectively, The remaining curves represent experimental data obtained at different
dates as presented in [9].
Discussion
Comment/Question
:
Do you hxe ir&mnation in the long time integration processof the image?
AuthodPresenter’s
reply :
Indeed, informdon is lost when the integration time is extended, Averaging, which is the
result of increasing the exposure time, is a low-pass operation that blurs the image, in fact
reducing thereby the available information+
D. Hiihn, GE
Comment/Question
:
Does your model also offer information on a possible phase function of the atmospheric
OTF?
Author/Presenter%
reply :
Since long exposure imaging involves a significant amount of averaging,phase information
is lost, Intuitively this is the reason behind the fact that equation (3- 18) is a purely real
function.