An improved dilation method for the measurement of fractal dimension

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Abstract

According to mathematical foundations of fractal dimensions (D) several methods were introduced for the measurement of D, first as manual techniques, later as computer algorithms. An image analysis environment may be especially appropriate for the automatic measurement of fractal dimensions. With regard to efficiency and accuracy dilation seems to be the most advantageous approach among the known methods (yard stick, box counting, dilation etc.). Unfortunately, there remains one misleading point: while analysing lineal objects with open ends (in contrast to closed contours) the measured D-values are systematically underestimated. The degree of bias depends on the number of end points and the complexity of a given structure. A method will be presented to avoid this bias by using direction controlled vectorial structuring elements in combination with specially designed anisotropic morphological filters. Applied to a given neuro-anatomical structure (processes of astrocytes) D is corrected from 1.56 to 1.68. A further application demonstrates the impact of using projection images on measured D-values.

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Over : Siegfried Eins