Error Bounds on the Estimation of Fractal Dimension

Dubuc, B.; Dubuc, Serge

doi:10.1137/0733032

Cited by 28 publications

(22 citation statements)

References 12 publications

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“…Each and every one of these subjective choices might cause the resulting fractal dimension to wander outside the range of physically plausible values and may lead different observers to obtain very different estimates of the fractal dimension of a given stain pattern in a soil. This situation is in no way unique to the application of fractal geometry to preferential flow processes, as stressed recently by Dubuc and Dubuc [1996]:…”

Section: Introductionmentioning

confidence: 99%

Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils

Baveye¹,

Boast²,

Ogawa³

et al. 1998

Water Resources Research

109

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Abstract. Preferential flow is ubiquitous in field soils, where it has important practical implications for water and contaminant transport. Dyes are frequently used to visualize preferential flow pathways. The fact that stain patterns in pictures of soil profiles often exhibit convoluted geometries, reminiscent of fractals, has encouraged a number of authors to use the principles of fractal geometry to describe stain patterns. This description typically involves two numbers, a mass and a surface fractal dimension. The evaluation of either one via image analysis requires numerous subjective choices to be made, including choices regarding image resolution, the definition adopted for the "fractal" dimension, and the thresholding algorithm used to generate binary images. The present article analyzes in detail the influence of these various choices on the mass fractal dimension of stained preferential flow patterns. A theoretical framework in which to envisage these choices is developed, using the classical quadratic von Koch island as an example. This framework is then applied to a set of pictures of an actual stain pattern in an orchard soil. The results suggest that the (apparent) mass fractal dimension of the stain pattern varies between 1.56 and 1.88, depending on choices made at different stages in the evaluation of the fractal dimension. In each case considered, the dimension, determined by a straight line fit in a log-log plot, has extremely high statistical significance, with R > 0.999. Of the various parameters subject to choice, image resolution seems to have the most pronounced influence on the value of the fractal dimension, which increases markedly at higher resolution (smaller-pixel size). By analogy with the case of the quadratic von Koch island, this dependence on image resolution, as well as the fact that the surface area of the stain pattern does not decrease with pixel size, suggests that the stain pattern is not a mass fractal; that is, there is no reason to believe that its various dimensions differ from 2. The approach adopted in the present article could be useful whenever fractal dimensions are evaluated via image analysis.

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Section: Introductionmentioning

confidence: 99%

Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils

Baveye¹,

Boast²,

Ogawa³

et al. 1998

Water Resources Research

109

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“…In order to find the fractal dimension, the rate at which the area, V(e), decreases as e tends to 0 is calculated. It turns out that a log-log plot of V(e)/e 2 versus 1/e gives the fractal dimension as its slope [22,23]. …”

Section: Variation Methodsmentioning

confidence: 99%

Temporal variation of velocity components in a turbulent open channel flow: Identification of fractal dimensions

Rakhshandehroo

Shaghaghian

Keshavarzi

et al. 2009

Applied Mathematical Modelling

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a b s t r a c tFractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst's rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold's numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold's number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold's numbers.

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“…79] (also see Chapter 3). Dubuc and Dubuc [80] summarized some of these issues in the following manner:…”

Section: Operational Aspects Of the Evaluation Of The Fractal Charactmentioning

confidence: 99%

Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures: Theory and Operational Aspects of their Application to Natural Systems

Baveye

Boast

Gaspard³

et al. 2008

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems

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Error Bounds on the Estimation of Fractal Dimension

Cited by 28 publications

References 12 publications

Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils

Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils

Temporal variation of velocity components in a turbulent open channel flow: Identification of fractal dimensions

Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures: Theory and Operational Aspects of their Application to Natural Systems

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