Abstract of Fractal analysis of surface roughness using spatial data

Research Report SRR96-008

Fractal analysis of surface roughness using spatial data

Steve Davies and Peter Hall

Abstract: We develop fractal models and methodology for data taking the form of surfaces. An advantage of fractal analysis is that it partitions roughness characteristics of a surface into a scale-free component (\fd) and properties that depend purely on scale. Particular emphasis is given to the issue of anisotropy where we show that, for many surfaces, the \fd\ of line transects across a surface must either be constant in every direction, or be constant in each direction except one. This virtual direction-invariance of \fd\ provides another canonical feature of fractal analysis, complementing its scale-invariance properties and enhancing its attractiveness as a method for summarising properties of roughness. The dependence of roughness on direction may be explained in terms of scale rather than dimension, and can vary with orientation. Scale may be described by a smooth, periodic function, and estimated nonparametrically. Our results and techniques are applied to analyse data on the surfaces of soil and plastic food wrap. For the soil data, interest centres on the effect of surface roughness on retention of rain water, and data are recorded as series of digital images over time. Our analysis captures the way in which both fractal dimension and scale change with rainfall, or equivalently with time. The food wrap data are on a much finer scale than the soil data, and are particularly anisotropic. The analysis allows us to determine the manufacturing process which produces the smoothest wrap, with least tendency for microorganisms to adhere.

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