EE

5356 DIGITAL IMAGE PROCESSING
IMAGE DEGRADATION BY ATMOSPHERIC
TURBULENCE
This Project illustrates the atmospheric turbulence model. The degradation
model proposed by Hufnagel and Stanley [4] is used. The Fourier transform
of the ima
ge is multiplied with the degradation function H (u, v) explained
in this report later. The effect of the atmospheric turbulence with different
values of ‘k’ is illustrated.
Theory of Atmospheric Turbulence
Atmospheric turbulence is caused by the random fl
uctuations of the
refraction index of the medium. It can lead to blurring in images acquired
from a long distance away. Since the degradation is often not completely
known, the problems are viewed as blind image deconvolution or blur
identification. Image
degradation associated with atmospheric turbulence
often occurs when viewing remote scenes: the objects of interest will appear
blurred, and the severity of this blurring will typically change over time. In
addition, the stationary scene may appear to wave
r spatially. A classic
example of this kind of distortion is the airplane on the tarmac at an airport
on a hot day: although the jet is not moving, we see dynamic optical
distortions that are generally most visible behind the running engines. The
degradati
on model proposed by Hufnagel and Stanley [4] is used. As ‘k’
increases in value, so does the degree of the blur. In the physical world, a
number of factors affect the blurring distortion that we observe, such as
temperature, humidity, elevation, and wind
speed. In most cases these
atmospheric conditions are not known, nor is there generally any external
information available to help specify the blur function.
Random fluctuations of the refraction index cause atmospheric turbulence
degradation. These phenom
ena have been observed in long

distance
surveillance imagery and astronomy. The fluctuations in atmospheric
turbulence can be modeled as a dynamic random process that perturbs the
phase of the incoming light. From the refraction index structure functions,
Hufnagel and Stanley [4] derived a long

exposure optical transfer function
(OTF),
H (u, v) =
, 0 ≤ u,v ≤ N/2
H (N

u, N

v) =
, otherwise
to model the long

term effect of turbulence in optical imaging. Here
u
and
v
are
the horizontal and vertical frequency variables and
¸ ‘k’
parameterizes
the severity of the blur.
The original image used is 680 x 680 which is shown below (dog) and the
Fourier transform magnitude spectrum of this image is shown.
ESTIMATING THE DEGRADA
TION FUCTION
There are three ways to estimate the degradation function for use in the
restoration. The process of restoring an image by using a degradation
function that has been estimated is called as ‘blind deconvolution’ as the
degradation function is n
ot known completely.
1.
Observation
2.
Experimentation
3.
Mathematical modeling
1.
Estimation by Image Observation
If any degraded image is given without knowing the degradation function
H, H can be estimated by gathering information from the image itself. If
the ima
ge is blurred, take a small section of the image containing simple
structures, like part of an object and the background and look for areas of
strong signal content to reduce the effect of noise. An unblurred image of
the same size can be constructed by us
ing the sample gray levels of the
object and background. The observed subimage be denoted by g
s
(x,y),
and let the constructed subimage (estimate of the original image in that
area) be
(x,y). Assume that effect of noise is negligible as a stron
g

signal area is chosen,
H
s
(u, v) =G
s
(u,v) /
s
(u,v) (1)
where G
s
(u,v) and
s
(u,v) are the 2D

DFTs of g
s
(x,y) and
(x,y)
respectively.
From this function,
the complete function H(u,v) can be deduced by
using the fact of position invariance. If H
s
(u,v) has a radial plot which
turns out to be a butterworth lowpass filter, this information can be used
to construct a function H(u,v) on a larger scale, having
the same shape.
2.
Estimation by Experimentation
Images similar to the degraded image can be obtained with various
system settings until they are degraded as closely as possible to the image
we want to restore. An impulse is simulated by a bright dot of ligh
t, as
bright as possible to reduce the effect of noise. The Fourier transform of
this image is a constant. We get the following equation.
H(u,v) = G(u,v)/A (2)
G(u,v) is the 2D

DFT of the observed image and A is a co
nstant
describing the strength of the impulse.
For spatially invariant systems , OTF is the normalized frequency
response
H (
ξ
1
, ξ
2
) / H (0,0) OTF : Optical transfer function. See p. 21 in [7].
3.
Estimation by Modeling
Degradation modeling gives an insight into the restoration problem. It
takes into account the environmental conditions that cause degradation.
For example, a de
gradation model proposed by Hufnagel and Stanley
[1964] is based on the physical characteristics of atmospheric turbulence.
This model has the following form, as described earlier.
H (u, v) =
, 0 ≤ u,v ≤ N/2
H (N

u, N

v) =
, otherwise
(3)
where ‘k’ is a constant that depends on the nature of the turbulence. This
equation is same as a Gaussian lowpass filter except for the 5/6 power in
the exponent.
This project shows examples of simula
ting blurring an image using (3)
with values k=0.0025 (severe turbulence), k=0.001 (mild turbulence), and
k=0.00025 (low turbulence). All images are of size 680 x 680.
follows.
The degradation function for different values of ‘k’ is as follow
s.
The degradation function obtained is multiplied with the Fourier transform
of the original image. The following figures illustrate the atmospheric
turbulence model for different values of ‘k’.
Degradation Function for k=0.0025
Degradation Function for k=0.001
Degradation Function for k=0.00025
Origin
al Image
Filtered image with k=0.0025 (Image corrupted by
the atmospheric turbulence)
Filtered image with k=0.001
Filtered image with k=0.00025
Image Corruption by Atmos
pheric Turbulence on different images
1.
Original Image
Fourier Transform of the image
Degradation Function for k=0.0025
Degradation Function for k=0.001
Degradation Function for k=0.00025
Original Image
Filtered image with k=0.0025
Filtered image with k=0.001
Filtered image with k=0.00025
2.
Original Image
Fourier Transform of the image
Degradation Function for k=0.0025
Degradation Function for k=0.001
Degradation Function for k=0.00025
Original Image
Filtered image with k=0.0025
Filtered image with k=0.001
Filtered image with k=0.00025
3.
Original Image
Fourier Transform of the image
Degradation Function for k=0.0025
Degradation Function for k=0.001
Degradation Function for k=0.00025
Original Image
Filtered image with k=0.0025
Filtered image with k=0.001
Filtered image with k=0.00025
4.
Original Image
Fourier Transform of the image
Degradation Function for k=0.0025
Degradation Function for k=0.001
Degradation Function for k=0.00025
Original Image
Filtered image with k=0.0025
Filtered image with k=0.001
Filtered image with k=0.00025
5.
Original Image
Fourier Transform of the image
Degradation Function for k=0.0025
Degradation Function for k=0.001
Degradation Function for k=0.00025
Original Image
Filtered image with k=0.0025
Filtered image with k=0.001
Filtered image with k=0.00025
Results
: It can be seen that for severe turbulence k=0.0025, the image is
very much blurred. For a mild turbulence, k= 0.001,
the image is blurred but
less than the effect of severe turbulence. For low turbulence, k=0.00025, the
image is slightly blurred.
References:
1.
D. Li, R. M. Mersereau and S. Simske
,
“Atmospheric Turbulence

Degraded Image Restoration Using
Principal C
omponents Analysis”
.
IEEE
geoscience and remote sensing
letters, vol. 4, no. 3, pp. 340

344,July 2007
2.
R C. Gonzalez and R E. Woods, “
Digital
Image Processing
III edition
”, Prentice

Hall, 2008,pp.356

359.
3.
D. Li and S. Simske,
“
Atmospheric
Turbulence Degra
ded Image Restoration by Kurtosis Minimization
”.
IEEE geoscience and remote sensing letters, vol. 6, no. 2, pp. 244

247,
April 2009.
4.
R. E. Hufnagel and N. R. Stanley,
“
Modulation Transfer Function Associated with Image Transmission
through Turbulence Medi
a
,”
Optical Society of America Journal A
,
vol. 54, pp. 52
–
61, 1964.
5.
http://www

ee.uta.edu/dip/Courses/EE5355/FFT.pdf
gives an insight of FFT and
to implement the filters.
6.
http://www

ee.uta.edu/dip/courses/ee5356/ee_5356.htm
for the images.
7.
A.K. Jain, “ Fundamentals of digital image
processing”, Upper Saddle River, NJ:Prentice Hall,1989. (page 258).
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