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,

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
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 = -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