Discussion

We have used small (25x25) image samples, in this comparison as only small samples are available in many real applications. The `goodness' of the estimators is assessed according to the correlation of generated and estimated fractal dimensions as this leads to biases in each estimator being ignored; this is appropriate as we are primarily interested in the discriminant power of each technique.

The spatial correlation estimator gave high correlation coefficients (fig. 1a, r_p=0.890, r_r=0.891) and on visual inspection appears to be the least biased of all the estimators (note the approximate clustering around the line D_estimated=D_generated); this may be expected as the approach is the most direct of all the measurement methods.

The Fourier estimator (fig. 1b , r_p=0.980, r_r=0.982) gave a very linear response with correlation coefficients close to unity. However, the response was not the expected D_estimated= D_generated. This may, in part, be due to distortion of the power spectrum due to the periodicity assumption of the Fourier transform.

The Fourier estimator gave significantly (p<0.01) higher correlation than the tapered Fourier estimator (fig. 1c , r_p=0.919, r_r=0.958). This indicates that high frequency terms introduced to the Fourier spectrum by the periodicity assumption do not add significant noise to the estimate although they may result in bias. Furthermore, tapering itself increases noise, possibly because the available data is less well utilised. Fitting a linear regression line to the response of the tapered Fourier estimator yielded D_estimated= 2.02D_generated-2.03, which again is a substantial deviation from the expected D_estimated=D_generated. The lack of significant bias in the spatial correlation estimator implies this is not a result of bias in the FBM generation algorithm. Moreover, the process of tapering rules out the possibility of the periodicity assumption distorting the estimate. These results therefore appear to be inconsistent with a power spectrum of the form f^-(b+1) where b=2H+1, and instead suggest a spectrum of the form f^-a where a=4H. The latter formulation is also consistent with uncorrelated white noise (which from equation has H=0) having a uniform power spectrum, whereas the original formulation incorrectly yields a spectrum of the form f^-2.

The Fourier estimator also yielded significantly (p<10^-10) higher correlation coefficients than the ring Fourier estimator (fig. 1d , r_p=0.868, r_r=0.873). The response of the ring estimate is similar to the Fourier estimator for generated D greater than 2.5, however below 2.5 the response floors at around 2.6; this effect is likely to be associated with equal weighting of low and high frequency rings in the curve fitting despite dependence upon substantially different numbers of Fourier coefficients.

The box counting estimator (fig. 1e , r_p=0.837, r_r=0.835) gave relatively low correlation coefficients and generated a small range of estimates, having a tendency to underestimate. The approach assumes a self-similar fractal - this result suggests considerable approximations are involved in the application of box counting to discrete FBM.

The surface estimator (fig. 1f , r_p=0.860, r_r=0.858) gave average correlation with little bias.

The modified blanket estimator (fig. 1g , r_p=0.834, r_r=0.836) gave significantly higher correlation (p<0.05) than the standard blanket estimator (fig. 1h , r_p=0.618, r_r=0.632). However, while the standard method showed only small biases, the modified method tended to generate underestimates leading to a reduced range. We conclude that the use of A(L) rather than dA(L) in the estimation of the fractal dimension leads to a more stable but also more biased estimate. Even with the modification the blanket estimator gave low correlation coefficients. Further, the method underestimates the fractal dimension, particularly for dimensions greater than 2.85 where we see the a `drop-off' of the response in figs. 1g and 1h .

The most striking feature of all of the plots in fig. is the degree of divergence from one-to-one matching of generated and estimated fractal dimensions; correlation is lower than might be expected and there are large biases in most of the estimators. These findings are not inconsistent with previous reports. For example, in [6] the box counting estimator, without a lower limit on box size, was applied to 10 (256x256) fractal textures and a range of generated dimensions of 2.0-2.9 gave estimated dimensions of 2.07-2.53. In [8] several estimators were applied to a (128x128) random noise texture (D=3.0) and dimensions in the range 2.4- 2.95 obtained. As these discrepancies were observed for large texture samples, substantial discrepancies may be expected for the (25x25) textures of this study.

The Fourier estimator gave significantly (p<0.01) higher correlation coefficients than all other estimators. Therefore, despite these results being inconsistent with the assumed form for spectral density, the Fourier estimator is the preferred approach for the characterisation of FBM.

For one-dimensional continuous FBM, self-similar estimation methods such as box counting have been shown to be valid only if the range of the dependent variable (in our case pixel intensity) is made large compared to the range of the independent variable (in our case position on the image matrix) [13]. We may therefore expect the self-similar methods to provide better estimates when applied to the textures with re-scaled intensity.

The box counting estimate gave D 0 for all samples - the large range and lack of continuity resulted in the number of boxes being fixed at the number of pixels irrespective of box size or fractal dimension.

The surface estimator gave increased correlation coefficients (r_p=0.886, r_r=0.889 whereas previously r_p=0.860, r_r=0.858) however the changes were not shown to be significant (p>0.05). An increased range and bias was also observed.

The modified blanket estimator gave decreased correlation coefficients (r_p=0.755, r_r=0.769 whereas previously r_p=0.834, r_r=0.836), however once again the changes were not significant (p>0.05). There was little bias and only modest change in the range.

For the surface and blanket estimators the results are inconclusive as to whether re- scaling pixel intensity to a large range improves correlation. However, irrespective of intensity scaling, the self-similar estimation methods were poor compared to the Fourier methods; correlation coefficients were substantially and significantly lower (p<0.001).