10/6/2015
102-0627-00L Applied Radar Remote Sensing
for Environmental Parameter Estimation
SAR image statistics
Manuele Pichierri
Earth Observation and Remote Sensing,
Institute of Environmental Engineering, ETH Zürich
07 October, 2015
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Lecture Outline
•
A short introduction on random variables for SAR images
(speckle effect)
•
•
•
•
Distributions of SAR complex pixel
Distributions of magnitude
Distributions of phase
Distributions of intensity
•
Multiplicative and additive noise
•
Matlab exercise
• Read a SAR image
• Plot histograms of experimental data
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Random variables vs SAR
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Random Variables: reminder (1)
A Random Variable (RV) is a variable whose value is subject to statistical
variation.
Example: a RV is the result of throwing a die. For each throw, we can
assume 6 different possible values (1 to 6). Each time we throw the die, we
can’t exactly know the outcome (unless your die is loaded…)
Definitions:
• Realisation (or observed value): each single result of throwing the die
• Probability Density Function (PDF): a function that describes the
statistical behavior of a RV
• Mean value (or expected value): the central tendency of a RV
• Variance: a measure of how the observed values are spread out around
the expected value
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Random Variables: reminder (2)
Ideal Mathematical World
x: Random Variable

f X ( x) : PDF


f X ( x)dx  1
PDF has unitary area (i.e. the integral of 𝑓𝑋 over the range of observed values is equal to 1)
Mean value


E[ x]   xf X ( x)dx
Variance
VAR[ x]   ( x  E[ x]) 2 f X ( x)dx


In the real world, we do not have infinite realisations of our RV
Mean value and variance are estimated over a limited (finite) number of samples
Variance estimator
Mean value estimator

1
N
N
x
i 1
2 
i
1
N
N
 (x  )
2
i 1
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Random Variables and SAR
?
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Random Variables and SAR (2)
• Distributed target: a single resolution cell consists of a collection of
randomly-distributed scattering elements (as a “rule of thumb”, the
wave interacts strongly with objects whose dimensions are bigger or
on the scale of its wavelength).
• The resolution cells are of the order of meter(s). The wavelength is of
the order of (tens of) centimeters --> many scatterers in the same
resolution cell!
• The backscattered wave is the result of the superimposition of the
waves coming from each individual scatterer (due to the linearity of
Maxwell equations).
• If you have no precise knowledge of the locations of the scatterers
within the resolution cell, then the backscattered field has a strong
component of randomness.
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Random Variables and SAR (2)
unknowns
Resolution cell
𝐸𝑖 ∈ ℂ
Electric field
backscattered by the
i-th scatterer
observable
𝑁(𝑥,𝑦)
𝐸(𝑥,𝑦) =
𝐸𝑖
𝑖=1
Total electric field (coherent
sum of N scatterers)
𝐼𝑚{𝐸}
Random path in
the complex plane
𝒚
𝒙
i-th scatterer
𝑅𝑒{𝐸}
Definition: Speckle is a result of interference of the coherent echoes
produced by individual scatterers within a resolution cell.
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Random Variables and SAR (3)
• Speckle appears as a granular pattern in SAR images, due to pixel-bypixel variations of the measured intensities.
• Speckle is a deterministic electromagnetic effect. However, it must be
analysed statistically, due to the complexity of the imaging process and
the observed scenario.
• Each pixel of a distributed target is one realisation of a RV (i.e. the
random path is different).
• Fully-developed speckle: if the number of individual scatterers 𝑁𝑠
within the resolution cell is (deterministic and) sufficiently large (𝑁𝑠 ≫
1), then the complex SAR pixel can be modelled as a Circular Symmetric
Complex Gaussian.
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PDFs (1)
Hp: Fully-developed speckle (𝑁𝑠 ≫ 1)
𝑅𝑒 𝐸 and 𝐼𝑚 𝐸 are independently and identically Gaussian distributed
Mean value Standard deviation
𝑅𝑒 𝐸 , 𝐼𝑚 𝐸 ~ 𝑁 0, 𝜎
𝑓𝑅𝑒 𝑅𝑒 =
1
2𝜋𝜎
𝑒
−
𝑅𝑒 2
2𝜎2
𝑓𝐼𝑚 𝐼𝑚 =
1
2𝜋𝜎
𝑒
−
𝐼𝑚2
2𝜎 2
The PDF of the phase is Uniform (phase contains no information)
ϕ = arg 𝐸 ~ Π [−π, π]
E    (   ) 2  0
 1
for     ,  

f  ( )   2

0
VAR   
   
12
2

2
3
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PDFs (2)
The PDF of the (single-look) magnitude is Rayleigh
𝑀 = 𝐸 ~ 𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ 𝜎
f M ( m) 
m
2

e
E  m  
m2
2 2
u ( m)
VAR  m  

2
4  2

2
The PDF of the intensity (or power, or energy) is Exponential
1
2𝜎 2
𝑊 = 𝐸 2 ~ 𝐸𝑥𝑝
fW ( w) 
1
2

e
2
E  w   1  2 2
w
2 2
u ( w)
VAR  w   2  4 4
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Multiplicative VS Additive Noise (1)
• Under some circumstances, speckle may cause difficulties for image
interpretation and compromise the accuracy of feature
classification/parameter estimation algorithms.
• It is also for these reasons that speckle is often described as noise…
but (formally) it’s not, as it is a repeatable phenomenon (unlike e.g.
thermal noise)!
• It can be demonstrated that the intensity of a SAR image may be
parametrized by:
𝑊 = 𝑚𝑊0
where 𝑊0 is the expected value of the intensity and 𝑚~𝐸𝑥𝑝 1 is an
exponential RV with unitary mean. As we are multiplying the actual value
by a random variable, the “noise” is defined “multiplicative”.
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Multiplicative VS Additive Noise (2)
When dealing with real SAR measurements, we must also consider the
existence of additive noise due to e.g. the circuitry or the antennas
𝑁(𝑥,𝑦)
𝐸(𝑥,𝑦) =
𝐸𝑖 + 𝑛
, where 𝑛 ∈ ℂ
𝑖=1
Mean value
This additive (thermal) noise is modelled as a
Circular Symmetric Complex Gaussian distribution
𝑛 ~ 𝑁 0, 𝜎𝑛
Standard deviation
The Signal to Noise Ratio (SNR) can be calculated as:
SNR 
E
n
2
2
When the SNR is high, the contribution of the thermal noise can be
neglected. On the other hand, if the power of the backscattered signal is
low (e.g. comparable to the noise floor of the instrument), the additive
noise must be taken into account.
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Noise applied to an optical image
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Example of additive noise on sound
Original signal (no additive noise)
Frequency change
The original signal is corrupted by additive noise…
SNR = 1
SNR = 0.1
The smaller the SNR, the more difficult is the detection of the original signal
(in the second case, we hardly distinguish the change of frequency…).
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Summary
• SAR images may be affected by large statistical variation due to
speckle and additive noise.
• Speckle is the coherent sum (interference) of the echoes generated by
scatterers in the same resolution cell.
• Speckle is a deterministic phenomenon, but it is analyzed statistically
due to the complexity of the SAR imaging process.
• Speckle is often described as «noise» as it complicates image
interpretation.
• A single SAR pixel does not contain any significant information about
the (distributed) target. We must use statistical moments to
completely characterize the target.
• We need some mathematical tools to mitigate the speckle effect (stay
tuned…).
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SAR statistics in Matlab
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Read a SAR image in Matlab
The data that will be used can be freely downloaded from:
http://earth.eo.esa.int/polsarpro/datasets.html#ESAR
ESAR, Oberpfaffenhofen, DE
2x2 Complex Sinclair format [S2] without header: 2616 rows x 1540 cols
Once you have downloaded the data, save them in a folder dedicated to
this practical.
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Visualize the SAR image
Each pixel of the SAR image is
a complex number: i.e. it has a
magnitude value and a phase:
'Image 1 HH loading...'
F1='D:\data\opairfield1pre12_l_hh.bin';
Fhh=fopen(F1,'r','l');
dimr = missing;
dima = missing;
c  a  jb  Ae j  C
Rowhh=fread(Fhh,[2, dimr*dima],'float');
fclose('all');
Dhh=Rowhh(1,:)+1i*Rowhh(2,:);
HH=zeros(dimr,dima);
HH(:)=Dhh;
clear HH1 Dhh Rowhh
j  1
f = missing;
figure (1), imshow(abs(HH),[0,
f*mean(mean(abs(HH)))])...
, title('Magnitude of SAR image');
min_phase = missing;
max_phase = missing;
figure (2), imshow(angle(HH),[min_phase,
max_phase]), title('Phase of the SAR image');
This part of the code reads
the SAR image as a matrix of
complex
numbers
and
visualize the magnitude and
phase of each element.
Which value of f gives you the
best “contrast”?
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Visualize the SAR image
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Visualize the SAR image
Phase contains NO information!
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Select a test area
HH_vis = HH;
dr = missing;
da = missing;
cr = 680; % Example: Forest
ca = 1100;
HH_zoom = HH(cr-dr/2:cr+dr/2,ca-da/2:ca+da/2);
figure (3), ia = imresize(HH_zoom,[5*dr 5*da]);
imshow(abs(ia),[0, f*mean(mean(abs(HH_zoom)))])...
, title('Crop of SAR image: Forest');
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Plot of distributions: Real and Imaginary part
𝑅𝑒 𝐸 , 𝐼𝑚 𝐸 ~ 𝑁 0, 𝜎
xSLC = ((0:dim)./dim0.5)*2*max(max(real(HH_zoom)));
Hist_Real = real(HH_zoom(:));
H_Real = hist(Hist_Real,xSLC);
Real part
Imaginary part
Hist_Im = imag(HH_zoom(:));
H_Im = hist(Hist_Im,xSLC);
figure (4),
plot(xSLC,H_Real),title('Histogram
of Real and Imaginary part')...
,
xlabel('Real/Imaginary part'),
ylabel('Occurrences');
hold on
plot(xSLC,H_Im,'g');
hold off
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Plot of distributions: Magnitude and Phase
𝐸 ~ 𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ 𝜎
arg 𝐸 ~ Π [−π, π]
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Plot of distributions: Intensity
𝐸 2 ~ 𝐸𝑥𝑝
1
2𝜎 2
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Try it yourself
•
Complete the remaining part of the Matlab code
•
Select other areas for your analysis (e.g. bare soil areas, agricultural
fields)… be careful to take areas with distributed targets.
• Are the shapes of the histograms different for different areas?
• Is their mean value changing?
•
Try to compute the histograms over the whole image (instead of a
small area). The histograms will represent now a very heterogeneous
area.
• Do the histograms look like the previous ones?
The exercise can be completed at home. If you like (it is not compulsory),
you can send me a short report with the outcomes (e.g. images,
histograms) by email at pichierri@ifu.baug.ethz.ch... I’d be happy to give
you a feedback!
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