Influence of a discontinuity on the spectral and fractal analysis of one-dimensional data

The analysis of a data area or segment containing steep transitions between regions with different textures (for example a cloud and its background) leads to addressing the problem of discontinuities and their impact on texture analysis. In that purpose, an original one-dimensional analytical model of spectrum and roughness function has been worked out, with a discontinuity between two fractal regions, each one specified by its average µ, standard deviation σ, spectral index β and Hurst exponent H. This has the advantage of not needing the generation of a fractal structure with a particular algorithm or random functions and clearly puts into evidence the role played by the average in generating spectral poles and side lobes. After validation of the model calibration, a parametric study is carried out in order to understand the influence of this discontinuity on the estimation of the spectral index β and the Hurst parameter H. It shows that for a pure µ-gap, H is well estimated everywhere, though overestimated, and β is overestimated in the anti-correlation range and saturates in the correlation range. For a pure σ-gap the retrieval of H is excellent everywhere and the behaviour of β is better than for a µ-gap, leading to less overestimation in the anti-correlation range. For a pure β-gap, saturation degrades measurements in the case of raw data and the medium with smaller spectral index is predominant in the case of trend-corrected data. For a pure H-gap, there is also dominance of the medium with smaller fractal exponent.