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Irregular boundary lines can be characterized by fractal dimension, whichprovides important information for spatial analysis of complex geographicalphenomena such as cities. However, it is difficult to calculate fractaldimension of boundaries systematically when image data is limited. Anapproximation estimation formulae of boundary dimension based on square iswidely applied in urban and ecological studies. However, the boundary dimensionis sometimes overestimated. This paper is devoted to developing a series ofpracticable formulae for boundary dimension estimation using ideas fromfractals. A number of regular figures are employed as reference shapes, fromwhich the corresponding geometric measure relations are constructed; from thesemeasure relations, two sets of fractal dimension estimation formulae arederived for describing fractal-like boundaries. Correspondingly, a group ofshape indexes can be defined. A finding is that different formulae havedifferent merits and spheres of application, and the second set of boundarydimensions is a function of the shape indexes. Under condition of datashortage, these formulae can be utilized to estimate boundary dimension valuesrapidly. Moreover, the relationships between boundary dimension and shapeindexes are instructive to understand the association and differences betweencharacteristic scales and scaling. The formulae may be useful for thepre-fractal studies in geography, geomorphology, ecology, landscape science,and especially, urban science.

Two Sets of Simple Formulae to Estimating Fractal Dimension of Irregular Boundaries