The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
2. METHOD 
2.1 First Fundamental Form 
The vector-field way to look at gradient in multispectral image 
has been described from Zenzo (1986) to Scheunders (2002). 
The idea is the following. Let I (x, y): R"—>R N be a 
multispectral image with bands I¡(x,y): R 2 —>R, i = 1,..., N. The 
value of I at a given point (x 0 , yo) is a TV-dimensional vector in 
R\ then the multispectral image can be seen as a vector field. 
The difference of image values at two points P = (pc 0 , yo) and Q 
= (Xj, y,) is given by 
AI = 1(F)-1(0. (0 
When the Euclidean distance d(P, Q) between P and Q tends to 
zero, the difference becomes the arc element 
31 , 31 , 
dl= ^ dx + ^r dy ( 2 ) 
ox dy 
and its squared norm is given by 
dl 2 = 
(!) 
ai ai 
dy dx 
si ai ^ 
dx dy 
(I)’, 
flfcYr G xx G xy' 
dy) l G vx Gyy, 
(3) 
This quadratic form is called the first fundamental form. It 
allows the measurement of changes in a multispectral image. 
The extreme of the quadratic form (3) are obtained in the 
directions of the eigenvectors of the 2x2 matrix 
G = 
\pyx GyyJ 
(4) 
and the values attained there are the corresponding eigenvalues. 
Simple algebra shows that the eigenvalues are 
/L = 
Q„+G„±,/(G„-C„) i +4G’ 
(5) 
and the eigenvectors are (cos#±, sin#±), where the angles 6 ± are 
given by 
1 2 G 
0.=— arctan — 
2 - G„ 
0 =0 + +-. 
' + 2 
(6) 
Thus, the eigenvectors provide the direction of maximal and 
minimal changes at a given point in the multispectral image, 
and the eigenvalues are the corresponding rates of change. 
The multispectral discontinuities can be detected by defining a 
function/=/(2+, A_) that measures the dissimilarity between 2+ 
and 2_. A possible choice, the one adopted in the research, is 
f = yjk-*- (V 
which has the nice property of reducing to dl 2 for the one 
dimensional case. 
For multispectral image, a single-valued approach can be 
adopted by segmenting each band separately, or by first 
combining the bands into a single grey image. The concept of 
the first fundamental form however allows to access gradient 
information from all bands simultaneously (Scheunders, 2002). 
Furthermore, the first fundamental form can be used in image 
fusion (e.g. Scheunders, 2001). So it is implemented in the 
research to fuse the texture features of all bands. 
2.2 Log Gabor Bank Filtering 
For solving the over-segmentation problem of watershed 
transform, texture features are considered to mark edge features. 
In order to produce texture features, it is need to characterise 
the texture content of the image at each pixel. One of the most 
popular techniques is the use of a bank of differently scaled and 
orientated complex Gabor filters (Jain and Farrokhnia, 1991). 
Gabor filter has the capability of reaching the minimum bound 
for simultaneous localization in the space and frequency 
domains. However, the Gabor filter is mathematically pure in 
only the Cartesian coordinates where all the Gabor channels are 
the same size in frequency and hence have sensors that are all 
the same size in space. An objective of the filter design might 
be to obtain as broad as possible spectral information with 
maximal spatial localization in the research. One cannot 
construct Gabor function of arbitrarily wide bandwidth and still 
maintain a reasonably small Direct Centre (DC) component in 
the even-symmetric filter (Kovesi, 1996). 
Based on the characteristics of annular-distribution of Fourier 
spectrum in logarithmic coordinates, an alternative to the Gabor 
function is the log Gabor function proposed first by Field 
(1987). On the linear frequency scale the log Gabor function 
has a transfer function of the form 
G(f) = exp- 
-[log(///o)] 2 | 
2[log(a// 0 )] 2 J 
(8) 
where f 0 is the filter’s centre frequency. To obtain constant 
shape ratio filters the term a!f 0 must also be held constant for 
varying f 0 . There are two important characters for log Gabor 
function. Firstly, log Gabor function, by definition, always has 
no DC component, and secondly, the transfer function of the 
log Gabor function has an extended tail at the high frequency 
end, which conquers the over-representation in low frequency 
components (Kovesi, 1996). 
The filter is constructed directly in the frequency domain as 
polar-separable functions: a logarithmic Gaussian function in 
the radial direction and a Gaussian in the angular direction. The 
ratio between the angular spacing of the filters and angular 
standard deviation of the Gaussians is 1.2. A log Gabor filter 
bank comprising filters with different parameters of log Gabor 
functions provides a complete cover of spatial frequency 
domain so that it can generate a versatile model for texture 
description. 
Applying the complex-valued log Gabor filter to remotely 
sensed imagery I(x, y) yields a complex response R(x, y) with 
respective real and imaginary components 
R(x, y) = R r (x, y) +jR,(x, y). (9) 
The response R(x, y) can be computed either by calculating 
Rfx, y) and Rj(x, y) separately by 2-D convolution or directly