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