variable @ , 
dimensional 
vely, by 
(>) the 
ch that it 
(1) 
dy 
))) dxdy 
2 are the 
'espectively, 
parameters. 
20. .. and 
PDF) of the 
tion. In this 
1e image in 
intensity 4; 
x) 
Q) 
on, the PDF 
distribution. 
x), u(X) is 
rpolation. If 
u(X) . Proof 
iges can be 
dxdy 
G) 
votion partial 
(16) | 
function. 
  
  
Next, our aim is to estimate the set of Gamma 
parameters Ó — (u;). We use maximum likelihood estimation, 
0, — arg max log p(y | 9) . Assuming that, in each region, the 
samples y;,j-—1l...N are independent and identically 
distributed, the log likelihood 
N N 
i logpo/|0) -log[ [76,10 - 9 18 767; 16) 
j=1 jai 
Taking the derivative of log p(y|0) with respect to 0 and 
setting them to zero, we obtain 
N, 
Hu eS y, N, (5) 
= 
with N; the number of pixels in C2; . 
2.2 Proof of Gamma distribution on scale image 
For a uniform and regular grid, bilinear interpolation averages 
the values of four neighboring pixels. For a given image 
ug (X), X = (x, y), let the decomposed image be u(X Y; vi 
nid 
in image U, u(X) = 2 HX), X; in image u,. 
i=l 
The Moment Generating Function (MGF) of a random variable 
Y is defined as @y(t)= E(e") = [4 aram eR, 
wherever this expectation exists. For the Gamma 
  
Ly 
A, : Hu VL C 
distribution I(44, L) , P,j(y) » ——C—)- e " . The 
di AG 4 
MGF of T(u,L) is nct r y 
—Ët 
L 
As described above, the value of a bilinear reduced image is 
derived by averaging the neighboring four pixels of original 
4 
; o4 ; b 
image, thus Y 2 — > Y, , where Y; is the original image 
4 izl 
pixel value and Y is the reduced image pixel value. 
On the assumption that Y; values are independently and 
identically distributed (i.i.d.), we obtain the MGF of y. 
4 4 
; l5 dL 
«ren s. «0-[[H0-C7 
i=l i= E 
L 
V 1 
nd ¥ = 47, 40 = dr (40) =(—)* 
lo: 
The MGF has the property that if two distributions have the 
same MGF, then they are identical at all points. We know that 
  
1 
y= , therefore 
Eu. 
AL 
the MGF of distribution L'(1454L) is ( 
1 
we get Y — DI(15AL). 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
     
2.3 OTSU algorithm used to initialize the level set function 
Otsu algorithm (1979) is a nonparametric and unsupervised 
method of automatic threshold selection for image segmentation, 
and it is a classical method for optical image segmentation. 
However, due to the strong speckle noise of SAR image, Otsu 
method could be invalidated. Thus, we applied it on the coarsest 
scale image, which has a poor speckle noise, to obtain the 
initialize segmentation result. 
Assume that f* is the threshold value calculated by Otsu 
algorithm (1979), and u is the image of coarsest scale, and 
then we simply initialize the level set function ¢ as follows: 
$-u-t* (6) 
It is easy to see that zero level set ¢ =0 is actually the set of 
image pixels that satisfies # = # * . This threshold segmentation 
method can be interpreted as follows: image data (regarded as a 
2-D function, with image gray level at each pixel representing 
the function value) subtracts a well-defined threshold; thus, the 
value of the level set function at each pixel denotes the 
difference between gray level and the zero level set. During 
evolution, image pixels with values far from the threshold are 
hard to move across the boundary, and vice versa, while in the 
classical approach, the level set function is initialized by a SDF, 
meaning that pixels far from the zero level set (decided by 
spatial distance) are hard to move across the boundary. This 
results in that objects far from the zero level set being hard to 
detect. With a proper threshold, we can get a better 
segmentation result after 10 or 20 iterations, saving time and 
also improving the accuracy. 
2.4 Post-processing 
When we obtained the segmentation result of proposed method, 
one can find that there are still some confused objects. Thus, 
post-processing is needed. In order to remove small segments, 
“connected component labelling” algorithm is used to label the 
binary image, thus, we can obtain several objects. Area size of 
object is defined to decide whether this object can be removed. 
In this paper, if area size of object less than 400 pixels, then 
those objects are removed. 
3. EXPERIMENT AND ANALYSIS 
In this section we demonstrate our method on real SAR image. 
We compare our method with the below methods: 
ALGI: Segmentation method using single-scale level set 
method with gamma model and we use several rectangular 
regions in the water region to initialize the contour. 
ALG2: Segmentation method using single-scale level set 
method with gamma model and we use OTSU algorithm to 
initialize the contour. 
ALG3: Water segmentation method proposed by Sliverira and 
Heleno (2009), the only difference is that we use OTSU 
algorithm to initialize the contour. 
All of our images are tested without a filter, the scale level is 2 
and the time step is 0.05. We estimate the computational time in 
seconds when our calculations are performed on a personal 
computer with an Intel Core(TM) i3 CPU, Quad-Core Processor