International Archives of Photogrammetry and Remote Sensing,Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999 
fixing the second moments of the distribution. The covari 
ance function can be assumed to be a decaying function 
characterized by its value at zero Cj, (0,0) = a) being the 
variance of f and the curvature C'^(0,0) = H fi at zero 
being the Hessian of the covariance function. 
The real image function g(r,c) — f(r,c) -f n(r,c) is com 
posed of the true image signal /(r, c) and the noise n(r, c), 
which is assumed to be white, n ~ JV(0, <r 2 n ). 
The separation of the segments is based not only on their 
expected value m, as in classical segmentation procedures, 
but also on their covariance structure. 
For the characterization of the covariance function we use 
the negative Hessian of the autocovariance function 
of g. Due to the moment theorem (Papoulis, 1984), it is 
identical to the covariance matrix Tg of the gradient V s g of 
the image function g over a window G’ t . The filter kernels 
h(x) of the gradient V s g and the window G t have width s 
and t (Fuchs, 1998). For the estimation of the negative 
Hessian of the available image function g we thus obtain: 
r 3t = G t * {'Vsg V S 0 T ) = - H g (1) 
/ 9r gr£ c \ 
\ Mh 9c ) 
We need to specify two scale parameters: 
1. Differentiation scale «: The image resolution in terms 
of the sharpness of the edges defines the scale for the 
differentiation kernel. E. g. the 3 x 3-Sobel kernels 
correspond to the 3 x 3-Binomial kernel having scale 
5 = 1/72. 
2. Integration kernel t: The width of the expected edges 
motivates the window size of the integration kernel. 
Observe, that the edge extraction does not distinguish 
between step and bar edges, thus also the expected 
width of bar edges can be used for specifying the inte 
gration kernel t. 
4.2 Texture parameters 
The core of the representation is the characterization of the 
local autocorrelation function, represented by the squared 
gradient Tg of the autocovariance function. By analyzing 
the eigenvalues A* of the matrix Tg, we obtain three local 
features. These are (Forstner, 1991): 
• the strength a (amplitude) of the texture. It represents 
the local variation of the intensity function and is mea 
sured using the variance of gradients, being the trace 
of the squared gradient: Fg: 
a =\r Tg = gt + gl = \i + \ 2 
• the direction <p or the orientation of the texture. The 
squared gradient is large in the direction of large in 
tensity variations, but small in directions with small in 
tensity variations, which in case of directed textures is 
identical to the subjective perception of the orientation 
of the texture. Instead of determining Gabor filters with 
different orientations, thus sampling the radial variation 
of the gradient, we directly determine the direction of 
smallest variation, being the direction «¿> of the eigen 
vector of Tg corresponding to the smallest eigenvalue: 
1 , 2 Q v Q c 
(x = - arctan —- __ 
r 0 9 9 
z 9r - gz 
The orientation lies in the range 0 < y < n, thus at 
pure edges the information on the direction of the gra 
dient is lost. 
• the anisotropy q (quality of direction) of the texture. 
It indicates the angular variation of texture and edge 
characteristics. The anisotropy can be measured by 
the ratio v of eigenvalues Ai and A 2 or, equivalently by 
the form factor: 
4detT^f ^ /Ai -A 2 \ 2 
\r 2 Tg \ Ai 4- \ 2 ) 
Isotropic, i. e. non-oriented, texture is characterized by 
q = 1, whereas anisotropic, i. e. oriented, texture is 
characterized by q< I- 
These three characteristic texture features are quite infor 
mative as can be seen in Fig. 4 and Fig. 5 1 . In the figures 
the strength is coded from low values (white) to large val 
ues (black), the direction is coded from <f> = -n/2 (white) 
to 4> = 7r/2 (black) and the anisotropy is coded from q = 0 
(white) to q = 1 (black). 
As the squared gradient of the given image function g in 
stead of the true image function / is used, we can expect 
the properties of the texture to be recoverable only in im 
ages with low noise. An information preserving filtering cf. 
(Forstner, 1991, Weidner, 1994) is useful as preprocessing 
step to reduce noise without smoothing the edges. 
4.3 Laplacian pyramid 
The texture parameters discussed above describe texture 
properties of a small image region, defined by the scale 
parameters s and t of the Gaussian windows in eq. (1). 
However, textures contain interesting characteristics: they 
are hierarchically structured. The texture parameters vary 
within the local neighborhood. This gives rise to a scale 
space analysis, which determines the variation of the tex 
ture features depending on the size of the local neighbor 
hood. 
To use the complete information, contained in the image 
signal, we perform a spectral decomposition of our texture 
parameters. 
To obtain a spatial scale decomposition, we use a Laplacian 
image pyramid (Burt and Adelson, 1983) applied to all three 
texture features. It separates the different spectral bands of 
the texture image features. The basis of the generation of 
an image k(r, c) of the Laplacian pyramid is the difference 
formation between two subsequent levels i and * — 1 of the 
Gaussian pyramid g,(r, c) 
U(r,c) = gi(r,c) - fifi-i(r,c) 
^he used image ’’Avenches” belongs to the data set in 
www.geod.ethz.ch/p02/proj ect s/AMOBE/index.html