METHODS 24, 347–358 (2001) doi:10.1006/ meth.2001.1205, available online at http:/ / www.idealibrary.com on Wavelet Packet Fractal Analysis of Neuronal Morphology Cameron L. Jones* and Herbert F. Jelinek† ,1 *Centre for Mathematical Modelling, School of Mathematical Sciences, Swinburne University of Technology, Hawthorn 3122, Australia; and †School of Community Health, Charles Sturt University, New South Wales Albury, N.S.W., Australia cells to a high degree of precision (3, 4). One particu- An image analysis method called two-dimensional wavelet larly interesting technique involves measurement of packet analysis (2D WPA) is introduced to quantify branching the fractal dimension, D f (5–8). Fractal geometry of- complexity of neurons. Both binary silhouettes and contour fers a method for describing complex patterns using profiles of neurons were analyzed to determine accuracy and precision of the fractal dimension in cell classification tasks. noninteger dimensions. It provides an objective, Two-dimensional WPA plotted the slope of decay for a sorted mathematical formalism necessary to characterize list of discrete wavelet packet coefficients belonging to the such stochastic (fractal) images (10). In contrast, Eu- adapted wavelet best basis to obtain the fractal dimension clidean geometry deals with ideal objects and integer for test images and binary representations of neurons. Two- dimensions similar to the topological dimension. A dimensional WPA was compared with box counting and mass– fractal is generally considered to be composed of cop- radius algorithms. The results for 2D WPA showed that it could ies of selected features at different resolution scales. differentiate between neural branching complexity in cells of Such objects are then described as being statistically different type in agreement with accepted methods. The impor- tance of the 2D WPA method is that it performs multiresolution self-similar or scale invariant. Since nerve cells dif- decomposition in the horizontal, vertical, and diagonal orienta- ferentiate by repeated branching it seems appro- tions.  2001 Academic Press priate to examine neural morphogenesis from a topo- logical viewpoint using fractal geometry (11, 12). In this article fractal analysis is used as a classification tool rather than for establishing whether an image The appearance of complex branching patterns is representative of a fractal. during growth is readily seen in neurons. Naturally Wavelet image analysis has been successfully ap- occurring objects and many mathematically derived patterns cannot be described adequately using Eu- plied to many types of data and includes the charac- clidean geometry, since natural or stochastic objects terization of fractal and multifractal signals (13–17). fall between topological dimensions. Investigations This study focuses on the application of two-dimen- into spatial differentiation, pattern formation, and sional wavelet packet analysis (2D WPA) to quantify branching behavior of biological objects have fasci- the global fractal dimension (D f ) as an objective mea- nated researchers for decades (1, 2) and numerous sure to evaluate feature complexity in 2D images. In studies have looked for methods to objectively de- this context D f was measured for binary images of scribe neural branching. Computerized image analy- neurons as well as just those border pixels of the cell sis has provided new techniques to quantify morpho- contour and compared with a box counting method logical parameters such as complex boundaries of (18) as well as the mass–radius method (8, 19). Two-dimensional WPA is introduced as a new 1 To whom correspondence should be addressed. E-mail: hjeli- nek@csu.edu.au. method to measure the fractal dimension of 1046-2023/01 $35.00 347 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.