Kube [10] has shown that images of fractal surfaces (commonly occurring in nature) are also fractal, belonging to the class of fractional Brownian motion (FBM) fractals. Field [11] has also shown that many natural textures have linear log power spectra - a characteristic of FBM. These observations suggest many textures approximate FBM. An FBM texture is characterised by the expected squared difference in intensity between any two pixels being proportional to the distance between the pixels raised to the power 2H; where H is the Hurst coefficient and the fractal dimension, D, is given by D=3-H. Hence if I(x,y) represents the pixel intensity at position (x,y) on the image, then
For H=0.5 we obtain standard Brownian motion in which the mean square displacement (in pixel intensity) is proportional to the distance moved in (x,y). By analogy with Brownian motion, this relationship may be considered to be the compound effect of a large number of independent random increments in I(x,y) along the line joining the points (x 1 ,y 1 ) and (x 2 ,y 2 ). Such increments would be positively correlated for H>0.5 (D<2.5) and negatively for H<0.5 (D>2.5). For H=0 (D=3) pixel intensities (not increments) are independent and I(x,y) is white noise.
Fractals can be grouped into the general classes of self-similar and self-affine. Self-similar fractals have properties which are invariant where equal scaling ratios are applied to each dimension, whereas self-affine fractals require different scaling ratios in each dimension to maintain invariance. FBM is self-affine - a scaling ratio r in x and y requires scaling of r H in I(x,y).
In this chapter we use an FBM texture model for the comparison of texture measurement methods. We suggest that this is the only appropriate model as 1) for natural textures it is supported by the evidence of Kube [10] and Field [11] , 2) its properties are invariant to changes in pixel intensity scales which are often arbitrary, and 3) pixel intensity and position are different physical quantities and cannot be expected to scale with the same ratio. Some authors [9] [7] [6] [5] , however, have implicitly assumed a self-similar fractal model.
Several methods have been proposed for the generation of fractional
Brownian motion on a grid, these include fast Fourier transform
filtering, random midpoint displacement, successive random additions and
random cuts; these methods are described by Voss
[12]
. Fast Fourier transform filtering is rapid but introduces additional
boundary constraints through the periodicity of the transform. Random
midpoint displacement is rapid, but is an approximation which is not
stationary for all H. Successive random addition is an adaptation of
random midpoint displacement which generates a stationary result,
however we suggest that neither of these methods accurately generalise
from one- to two-dimensional fractals (the grid based generalisation
results in pairs of grid points on the same row or column conforming to
equation (1)
, while those on different rows and columns fail to). The random cuts
method accurately generates fractional Brownian motion but is expensive
computationally.
For the comparison of measurement methods accuracy rather than speed of
generation is required, and so we adopt the random cuts method. The
procedure is iterative. Firstly, the image grid is initialised to all
zeroes. In each iteration a line with a random angle (0-360 degrees from
a uniform distribution) and a random displacement (also from a uniform
distribution) from the centre of the grid is defined. All grid points on
the original `upper' side of the line are incremented by an amount d
(H-0.5)
where d is the distance from the grid point to the line (note we use
`upper' such that rotations of
and
+180 degrees generate the same line with opposite `upper' sides). A
large number of iterations are required to simulate FBM.
