Evaluating the fractal dimension of profiles

Abstract

There are many definitions of the fractal dimension of an object, including box dimension, Bouligand-Minkowski dimension, and intersection dimension. Although they are all equivalent in the continuous domain, they differ substantially when discretized and applied to digitized data. We show that the standard implementations of these definitions on self-aSne curves with known fractal dimension (Weierstrass-Mandelbrot, Kiesswetter, fractional Brownian motion) yield results with significant errors. An analysis of the source of these errors leads to a new algorithm in one dimension, called the variation method, which yields accurate results. The variation method uses the notion of e oscillation to measure the amplitude of the one-dimensiona1 function in an e neighborhood. The order of growth of the integral of the e oscillation (called the e variation), as e tends toward zero, is directly related to the fractal dimension. In this paper, we present the variation method for one-dimensional {1D) profiles and show that, in the limit, it is equivalent to the classical boxcounting method. The result is an algorithm for reliably estimating the fractal dimension of 1D profiles; i.e. , graphs of functions of a single variable. The algorithm is tested on profiles with known fractal dimension.

Measuring fractal dimension and complexity — an alternative approach with an application

“…The simulation also shows systematic deviations of both methods from the fractal dimension. The XCM typically overestimates for smaller fractal dimensions, and the BCM underestimates for larger fractal dimensions, a tendency of the BCM which was also recognized by Dubuc et al (1989).…”

Roughness characteristic length scales of micro-machined surfaces: A multi-scale modelling

“…Now, the same result will be computed by taking the origin in the Stage I, Stage II and another origin points in the Stage III (l ∈ {2, 10,20,60,200,500,1000} m). Fig.…”

“…The arithmetic average height (R a ) and the total amplitude (R t ) also called the "Peak to Valley" are very often used to characterize the surface roughness irregularities but, in most studies, the effect of the evaluation length is not taken into account. This is all the more unfortunate since these parameters depend on the observation scale [20,26]. Such a kind of dependence is integrated in the fractal concept, which basis is to find invariant scale parameters.…”

“…This clearly means that the scale relation R t (l) ∝ l 0.32 holds. To explain this relation, the determination of the fractal dimension by "Oscillation Method" from Dubuc et al [20,26] must be introduced. The τ-oscillation of the function f in x is defined as:…”

“…To characterise roughness surfaces, the Peak to Valley parameter [17,18] that represents the range of the roughness amplitude (called R t or PV) is of major interest in the case of dimensional tolerance characterisation for MEMS. However, this parameter depends on the evaluation length [19,20] and as the scanning area is often lower than the part area, the R t parameter of the part cannot be evaluated on the whole surface. As a consequence, a multi-scale modelling has to be constructing to extrapolate data from the sampling length to the whole sample.…”