Download Speckle Noise and Imperfections in Coherent Imaging Systems: A

Speckle Noise and Imperfections in Coherent Imaging
Systems: A Simulation Study
By Alex Shefler
Abstract: Coherent Imaging Systems (CIS) such as optical and acoustical holograms and Synthetic Aperture
Radars (SAR) suffer from the phenomenon of speckle noise. It has been believed that the main reason of this
noise is the interference of the returned signals from different scatterers in each resolution cell. It is much less
known that speckle noise always appears due to any imperfections in measuring and reconstructing the
wavefield used for the imaging. The project implements a computer model that simulates such wavefield
distortions as limited resolving power of sensors, limitation of sensor’s dynamic range and measured signal
quantization to study statistical characteristics of speckle noise. The simulation has proved that speckle is not
always pure multiplicative and allowed the development of empirical analytical formulas for the statistical
parameters of the speckle noise as functions of the specific imaging system imperfections and of the properties
of the input signal.
Key-Words: Speckle noise, Synthetic Aperture Radar, Monte Carlo methods
1 Introduction. Limited sensor’s resolving power theory
Speckle is a granular noise that inherently exists in all types of coherent imaging systems (CIS) that
record both the amplitude and the phase of the back-scattered radiation. Speckle noise imposes fundamental
limitation on image quality and it reduces the ability of a human observer to resolve fine details and to detect
targets.
The most well known theory of speckle noise is that of limited sensor’s resolving power theory,
developed by J. Goodman [1]. Coherent imaging systems can be mathematically modeled as:
2
∞ ∞
Outimg( x , y ) = =
∫ ∫ Ampl (ξ , η ) exp[iPhase(ξ , η )]h( x, y; ξ , η )dξdη
+ n( x , y )
(1)
−∞−∞
where, Ampl(ξ,η)exp[iPhase(ξ,η)] is complex amplitude of the wavefront in the object plane that describes
object reflectance/transmittance properties, h( x , y;ξ ,η ) is point spread function of the system and n(x,y) is
signal independent random process that models sensor’s noise. For diffuse scattering objects, Phase(ξ,η) can be
modeled as a non-correlated or correlated random process that describes radiation scattering caused by object’s
“optical” inhomogeneities depending on the ratio of “typical” size of inhomogeneity and wave length. In the
assumption that the object has uniformly painted surface and by virtue of the central limit theorem of the
1
probability theory, one can compute probability density, the mean and standard deviation of the real and
imaginary part of the output image. The results reveale that the magnitude of the resultant speckle field follows
a Rayleigh probability distribution, the phases are uniformly distributed between [-π, π] and the speckle
intensities distribution can be described by a negative exponential distribution. It follows from this that speckle
can be regarded as a multiplicative noise and “speckle contrast” (Speckle standard deviation / mean) equals 1.
2 Speckle simulation in holography and SAR
2.1 Holography and SAR transfer function and the Fresnel transform
It was shown in [2] that the raw input signal in the radar h(x', r') can be obtained by superimposing all
the elementary returns from the illuminated surface and it can be approximated as a two-dimensional (2D)
convolution. The 2D Fourier Transform (FT) of the raw input signal in the radar was found to be:
H (ξ , η ) = ∫∫ dxdr γ ( x , r ) exp( − jξ x ) exp( − jη r )G (ξ , η ) = Γ (ξ , η )G (ξ , η )
where Γ(.) is the Fourier Transform of the object reflectivity pattern and G(.) is the FT of the impulse
response of the system. The evaluation of the system transfer function shows that G(.) is a band limited function
and the recorded raw signal, h(.), is the Fresnel transform of the object’s reflectivity pattern, γ(.):
∞
h( x' , r ' ) = ∫ γ ( x , y ) exp[− iπαβ (( x − x' ) 2 + ( y − r ' )) 2 ]dxdy
−∞
where α and β are real parameters.
It is known also that the same relationship holds also for holograms recorded in “near” diffraction zone.
In “far” (Fraunhofer) diffraction zone this relationship reduces to Fourier Transform.
2.2 Speckle noise simulation
From the above, it follows that the reconstruction of the SAR and holographic images is reduced to the
inverse Fresnel/Fourier transforms of input raw signals. The simulation method ([3,4]) implemented in the
project was used to study the effect of different distortions in the SAR and holographic instruments on the
statistical characteristics of speckle noise in the resulting images. The checked distortions were the antenna’s
(hologram) size limitation, the limitation of the dynamic range in measuring the orthogonal components of the
wavefield and the quantization of orthogonal components.
In the simulation, a Monte Carlo method was used for objects with pseudo-random phase (both noncorrelated and correlated). Flow diagram of the model is shown in Fig. 1. After the computation (of the object’s
scattered wave field by Fresnel or Fourier transforms, distortions are introduced and the reconstruction of object
magnitude is performed by the inverse transforms. Finally, the statistical characteristics of the speckle are
measured.
Fig 1:
The simulation process – one experiment
As a test object, brightness distribution in a form of a step wedge was chosen. Its rough surface is modeled by a
pseudo-random diffuser with statistically independent and uniformly distributed in the range (-π, +π) samples of
the phase. The limitation of the antenna’s (hologram) area, which is equivalent to limited resolution, is specified
in terms of the fraction of the area that contains the entire field. The limitation of the sensor’s dynamic range is
specified in units of the standard deviation σ of the hologram signal obtained from a uniformly painted object..
Quantization of orthogonal components is carried out in the range (-4σ, +4σ) in a nonlinear scale defined as Pth law quantization.
3 Simulation results
For each of the mentioned distortions, the simulation checks several levels of distortions. The output image is
formed of bands of images, each band corresponding to a specific level of distortion. In the illustrative images
showb below , the first band in the output images is the result of the reconstruction from undistorted
instruments.
3.1 The effect of the antenna’s (hologram) size limitation
The simulation confirmed that, in the case of antenna’s (hologram) size limitation, speckle noise appears and
increases as the limitation is becoming tighter. The speckle distribution function measurement revieled that the
distribution functions differ substantially between the low limitation and the high limitation. It was shown, that
for size limitations till 95% of the entire size, the distribution is close to Gaussian. For limitations from 87% to
95%, the distribution is perfectly Rayleigh and for limitations of 70% to 84% it changes from the Rayleigh
distribution to the negative exponential distribution. For tighter limitations, the noise has the negative
exponential distribution, as it predicted by the limited sensor’s resolving power theory. The parameters for all
the distributions, including Gaussian, Rayleigh and negative exponential, were calculated and empirical
formulas were suggested for each of them. Other statistical properties of the noise, such as speckle contrast,
were checked also and corresponding empirical mathematical formulas were developed. It was shown that the
speckle mean value is a parabolic function of the brightness and can be described by the same formula for all
limitation values. The speckle standard deviation depends also in a parabolic way on the brightness, with a
parabola factor that depends on the limitation degree. The speckle contrast was found to be independent on the
brightness for all limitations which confirms that the noise is multiplicative. The speckle contrast depends on
the antenna’s (hologram) size limitation by a fifth degree polynom and they asymptotically tend to the
theoretical limit of 1 for severe limitations. The effect was also checked on “real” object (optical sattelite
images) and the results were similar for relative “flat” areas. The results are illustrated in Fig. 2
Speckle contrast versus OUTmn, Niter=30
Speckle contrast versus size limitation, Niter=30
1
1
0.9
0.9
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SzLim 1
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Gray level
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Simulated step wedge and results
Speckle contrast vs. brightness
Input image
Antenna size limitation: Output image
0
0.2
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Fraction of antenna
0.8
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Speckle contrast vs. limitation
Speckle contrast versus size limitation, Niter=20
1.6
Fraction of antenna area
1.4
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100
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Area no. 1
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Input “real” image
100
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Input image brightness
Output image for limit = 50%
250
Speckle contrast for the different areas
Fig. 2. Speckle noise and speckle contrast for limitation of the antenna’s (hologram) size
3.2 The effect of the sensor’s dynamic range limitation
When only sensor’s dynamic range limitation occurs, speckle-like noise appears in the reconstructed
images as well and, as the limitation is tighter, the noise level is higher. The simulation demonstrated that the
noise distribution is a Gaussian one for low limitations (limit > σ). For tighter limitations, the speckle noise
distribution become Rayleigh one and it tends to negative exponential for dark gray levels. The statistical
properties of the noise were checked and empirical mathematical formulas were suggested on the base of the
simulation results: the mean versus the brightness can be described by a second degree polynom, for each
limitation, and the stdev versus the brightness is almost linear for each limitation.
It was found that the speckle contrast increases as the saturation limit is tighter. Also, for the same
saturation limit, the speckle contrast is higher for darker gray levels, i.e. it highly depends on the brightness of
the input object. In this case, the resulting noise is not purely multiplicative (speckle contrast is not independent
on the brightness), but it is also not purely additive (the noise stdev and mean depend on the brightness).
The effect was also checked, with similar results, on “real” object (optical sattelite images).
OUTs tdev versus Dynamic Range limit, Niter=20
Output NORMALIZED image
GrLv (SQRT) 1/8
2/8
3/8
4/8
5/8
6/8
7/8
8/8
9
8
50
7
Range limitation interval
Mean versus Dynamic Range limit, Niter=20
10
6
100
5
16
GrLv (SQRT) 1/8
2/8
3/8
4/8
5/8
6/8
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Input image brightness
1
1.5
2
2.5
Dynamic range limit
3
3.5
0.5
1
1.5
2
2.5
Dynamic range limit
3
3.5
250
Simulated step wedge and results
Speckle contrast vs. brightness
Speckle contrast vs. limitation
3.3 The effect of the Quantization
As in the case of the sensor’s dynamic range limitation case, it was found that quantization of wave field
measurements also causes speckle-like noise and, as less quantization levels are used, the noise level grows.
The speckle noise distribution was empirically proved to be Gaussian with variance that depends on the
object’s gray level and on the number of quantization levels and with the mean that almost does not depend on
the number of quantization levels. The speckle contrast was found to be inversely proportional to the
brightness:
Speckle contrast = const/brightness,
where const that depends on the number of quantization levels (its values were also calculated from
simulation results).
Yet another observation is that, for the same number of quantization levels, the speckle contrast is
higher for darker gray levels, i.e. it depends on the brightness of the input object. In this case, the resulting
noise in the intensity image is not purely multiplicative (speckle contrast is not independent of the brightness),
but is not additive either (the noise standard deviation and mean depend on the brightness).
In order to understand the nature of the speckle noise due to quantization, the magnitude image was
checked also. The results have shown that the noise in the magnitude image is purely Gaussian and is additive
for the number of quantization levels larger than 16. The mean of the noise tends to zero and the stdev does not
depend on the image magnitude. Moreover, the stdev level was found to be inversely proportional to the
number of quantization levels. The results are illustrated in Fig. 3
The simulation of P-th quantization has shown that one can optimize the non-uniform quantization
scale in order to reduce the intensity of the speckle noise. Different nonlinearity indexes P were checked and
the conclusion is that the optimal value of the nonlinearity index P is about 0.5.
The above results proved to be true also for rough surfaces (non-correlated phases) and also for diffusers with a
non-uniform directivity pattern (correlative phases) of the scattering.
-3
Stdev versus OUTmn, Niter=20
Noise mean for quant.
x 10
Quantization of Orth.Comp., P=0.5: Output image
NoQuan
Q=64
56
48
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32
24
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Number of quantization levels
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0.01
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Gray level
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250
Simulated step wedge and results
Mean of magnitude image vs. brightness
Stdev of magnitude image vs. brightness
Fig. 3 Speckle noise due to quantization of hologram orthogonal components
4 Conclusions
A simulation method, implemented in the project, allowed studying speckle noise statistical properties in SAR
and holographic images. The research showed that the appearance of the speckle noise in SAR and holographic
images should be associated not only with the properties of physical objects to diffusely scatter irradiation but
also with the fact that real coherent imaging systems are incapable of distortion less recording the object whole
wave field. The scattering properties alone, without the instrument distortions, will not cause the effect of
speckle. In the project, statistical properties of the speckle noise were investigated and empirical analytical
formulas were suggested, describing the dependence of speckle noise statistical parameters on the parameters
of distortions. These results can be a useful tool for designing the input stages of a SAR and holographic
instruments.
References:
[1]. Goodman, J.W., Statistical Properties of Laser Speckle Patterns In Laser Speckle and Related Phenomena,
J.C. Dainty, ed. Springer Verlag, Berlin, 1975.
[2]. G. Franceschetti, R. Lanari: Synthetic Aperture Radar Processing, CRC Press, 1999.
[3]. L. Yaroslavsky, M. Eden: Fundamentals of Digital Optics, Birkhauser, Boston, 1996
[4]. L. Yaroslavsky, Digital holography: 30 years later, IS&T / SPIE’s 14’th Annual Symposium Electronic
Imaging 2002, Science and Technology Conf. 4659A, “Practical Holography XVI”, San Jose, CA, 22-23
Jan. 2002, Proc. of SPIE, v. 4659