Medical Engineering & Physics 23 (2001) 369–380 www.elsevier.com/locate/medengphy Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images Geoffrey Dougherty a,* , Geoffrey M. Henebry b a Department of Radiologic Sciences, Faculty of Allied Health Sciences, Kuwait University, P.O. Box 31470, 90805 Sulaibikhat, Kuwait b Center for Advanced Land Management Information Technologies, University of Nebraska, Lincoln, NE 68588-0517, USA Received 14 February 2001; accepted 23 May 2001 Abstract Fractal analysis is a method of characterizing complex shapes such as the trabecular structure of bone. Numerous algorithms for estimating fractal dimension have been described, but the Fourier power spectrum method is particularly applicable to self-affine fractals, and facilitates corrections for the effects of noise and blurring in an image. We found that it provided accurate estimates of fractal dimension for synthesized fractal images. For natural texture images fractality is limited to a range of scales, and the fractal dimension as a function of spatial frequency presents as a fractal signature. We found that the fractal signature was more successful at discriminating between these textures than either the global fractal dimension or other metrics such as the mean width and root-mean-square width of the spectral density plots. Different natural textures were also readily distinguishable using lacunarity plots, which explicitly characterize the average size and spatial organization of structural sub-units within an image. The fractal signatures of small regions of interest (32×32 pixels), computed in the frequency domain after corrections for imaging system noise and MTF, were able to characterize the texture of vertebral trabecular bone in CT images. Even small differences in texture due to acquisition slice thickness resulted in measurably different fractal signatures. These differences were also readily apparent in lacunarity plots, which indicated that a slice thickness of 1 mm or less is necessary if essential architectural information is not to be lost. Since lacunarity measures gap size and is not predicated on fractality, it may be particularly useful for characterizing the texture of trabecular bone.  2001 IPEM. Published by Elsevier Science Ltd. All rights reserved. Keywords: Texture; Fractal dimension; Fractal signature; Lacunarity; Bone architecture 1. Introduction Fractal models have long been considered appropriate for modelling the texture in medical images, with fractal dimension commonly used as a compact descriptor. The fractal dimension describes how an object occupies space and is related to the complexity of its structure: it gives a numerical measure of the degree of boundary irregularity or surface roughness. Exact fractals have attractive properties, such as invariance to scale and pro- jection: but for real structures, fractality is present only in a statistical sense and only over a limited range of scales. The estimation of fractal dimension is a notori- ously difficult procedure, complicated by the fact that the values (both elevation and position) for real data are * Corresponding author. Tel.: +965-483-0929; fax: +965-483-3662. E-mail address: geoff@hsc.kuniv.edu.kw (G. Dougherty). 1350-4533/01/$ - see front matter  2001 IPEM. Published by Elsevier Science Ltd. All rights reserved. PII:S1350-4533(01)00057-1 digitized and are often sparse and cover only a relatively short range of dimensions. Numerous algorithms for estimating fractal dimension have been described [1–7]. They are all based on meas- uring an image characteristic, chosen heuristically, as a function of a scale parameter. Generally these two quan- tities are linearly regressed on a log–log scale, and the fractal dimension obtained from the resulting slope, although nonparametric estimation techniques have also been used [8]. However, the image characteristic of interest must be chosen with care if the resulting estimate is to be a valid, reliable and accurate indicator of fractal dimension [9,10]. Certain characteristics can be less robost when applied to digitized data, especially when these are sparse. Algorithms that implicitly assume an exactly self-similar fractal model are inappropriate for medical images, since in particular pixel intensity and position are different physical properties and cannot be expected to scale with the same ratio. Thus, methods that