4.2 SPECKLE NOISE 
Speckle noise results from the overlay of phase-incoherent sig- 
nals within each resolution cell. The incoherence is caused by 
different distances between the sensor and the earth surface. 
This overlay of signals may be calculated as a vector sum 
of complex signal vectors, where the phase of the vectors is 
randomly distributed. It can be shown that the signal created 
by the sum, which may be thought of as a random walk in a 
two-dimensional vector space, is exponentially distributed for 
an intensity image and has a Rayleigh distribution for the am- 
plitude of the signal (Ulaby et al., 1982b). From this point of 
view it also clear, that the speckle noise is of a multiplicative 
nature, since the variations are generated by the signal itself 
using the random phase. 
In practical applications the noise is often reduced by multi 
look processing which is done by averaging independent sam- 
ples of several images. With an increasing number of samples 
averaged, the Rayleigh distribution of a signal approximates 
a Gaussian distribution. In the case of the ERS-1 GTC and 
GEC geocoded products three looks of an amplitude image 
are averaged and therefore we decided for simplicity to use 
a Gaussian distribution for the tests. From theory we know, 
that the coefficient of variation is 
0.523 
VN 
for N — 3 looks, where c denotes the standard deviation, p 
the mean of the signal. In practice we got a value of approx- 
imately 0.17 for the coefficient of variation within the ERS-1 
images. This discrepancy may be explained by the averag- 
ing done by the geocoding process and the resampling from a 
12.5 to a 25 meter pixel size. Both procedures calculate a new 
pixel value from neighboring pixels within the original image, 
thus a kind of averaging is done which reduces the speckle 
caused variance of the image. To obtain realistic results for 
practical applications we use the value of C — 0.17 obtained 
from the SAR images also within the synthetic image. 
C= = 0.302 
= 19 
Several filters need the noise level of the multiplicative speckle 
noise contained in the image. Some implementations use the 
equivalent number of looks (ENL) instead of the coefficient 
of variation. The ENL is defined by the noise of a one look 
intensity image as 
ENL(I) = (1/0)? 
For the test image we obtain ENL(I) = 34.6 as input for 
the algorithms. 
4.3 GEOMETRIC CRITERIA 
On the basis of a similar smoothing capability using the filter 
parameters mentioned above, we may now establish criteria 
for the appraisal of the geometric quality of the filters. To es- 
tablish a measurement for the quality, the use of synthetically 
generated data is helpful. This enables the calculation of the 
RMS-error between the filtered and the original, unspeckled 
image. To get results as objective as possible, several ge- 
ometric arrangements have to be observed, since different 
filters may prefer some type of geometry. Thus we estab- 
lished a test image containing points of different size, lines 
of different orientation and thickness as well as areas limited 
by edges intersecting at different angles of 180, 135, 90 and 
45 degrees. The areas are also arranged in different orienta- 
tions. The test image is shown in Figure 1. The highlighted 
  
Figure 1: Test image with highlighted measurement areas 
squares all over the image denote those areas, the RMS-error 
is calculated from. In order to determine just the geometric 
distortions and not the remaining variation all over the image, 
it is necessary to limit the measurement area to those pixels 
where the distortions occur. The RMS-error is calculated 
from each square by the formula 
RMS = 
  
where N is the number of pixels within the square, x; is the 
value of pixel i within the original image and &; that from 
the filtered image. Since no substantial difference was found 
between different orientations and different angles between 
the area edges, a mean value for all points, lines and area 
features within the test image was calculated to reduce the 
amount of data and to get more reliable results. 
Since adaptive filtering algorithms are sensitive also to the 
radiometry of an image, we duplicate the test image four 
times and use four different contrast values within the im- 
ages, representing four different signal to noise ratios. The 
contrast values used are 100, 80, 60 and 40 greylevels dif- 
ference between the dark and the light areas, equidistant 
from a level of 100. Calculating the sum of the 20 dis- 
tances of the Gaussian distributions results in a value of 
ds = 201 + 202 = 2(p1 + p2)C = 68, where C = 0.17 
denotes the coefficient of variation, c and p the standard 
deviation and mean of both distributions contained in an im- 
age. Thus the distributions intersect at their 20 distance for 
a contrast value of 68. The first two contrast values don't 
have a significant overlay of the distributions, the last one 
results in an intersection near the lo distance. 
44 OTHER CRITERIA 
Another criterion which was valued is the retention of the 
mean value in homogeneous areas which is a need for several 
applications including image classification. Furthermore the 
computation time was measured in order to detect algorithms 
which are not applicable in practice due to the exhausting of 
computer resources. The results are presented in the follow- 
ing section. 
138 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B2. Vienna 1996 
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