Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images
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 projection: 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 notoriously difficult procedure, complicated by the fact that the values (both elevation and position) for real data are digitized and are often sparse and cover only a relatively short range of dimensions.
Numerous algorithms for estimating fractal dimension have been described [1], [2], [3], [4], [5], [6], [7]. They are all based on measuring an image characteristic, chosen heuristically, as a function of a scale parameter. Generally these two quantities 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 do not meet the intensity scale independency requirement [11], such as the blanket [2], box-counting [6] and area [7] algorithms, were not considered. In contrast, the Fourier power spectrum method conveniently represents the statistical nature of real images by describing them in terms of a fractional Brownian motion model [12]. It has been shown to estimate the fractal dimension of self-affine fractals reliably and accurately [13], [14]. The variation of fractal dimension with scale can be considered as the fractal signature. The concept of a fractal signature has previously been used in the spatial domain to distinguish Brodatz textures [2] and textures in conventional radiographs of osteoarthritic knees [15] and lumbar vertebrae [16].
Lacunarity is a multi-scale measure of texture describing the complex intermingling of the shape and distribution of gaps within an image; specifically, it quantifies the deviation of a geometric shape from translational invariance. It is not predicated on self-similarity (i.e., fractality), and has been used most successfully with binarized images [17], [18]. A plot of a lacunarity against window size contains significant information about the spatial structure of an image at different scales. In particular, it can distinguish varying degrees of heterogeneity within an image, and in the case of a homogeneous image it can identify the size of a characteristic sub-structure. Lacunarity has been used previously to characterize landscape texture in binarized optical [17] and SAR (synthetic aperture radar) [18] images.
Osteoporosis is a prevalent bone disease characterized by a debilitating loss of bone strength and, consequently, fracture risk and the spine is a useful site for predicting osteoporotic failures. Although the relative contributions of trabecular and cortical bone to overall bone strength are unclear [19], most studies have concentrated on trabecular bone since it is the metabolically more active as evidenced by remodelling data [20]. It has been increasingly recognized that quantifying the structural quality of vertebral trabecular bone (assessed in terms of the integrity of its internal architecture) may assist in an earlier and more accurate diagnosis of osteoporosis than assessing bone quantity [in terms of bone mineral density (BMD)] alone [21], [22]. Indeed, fractal analysis of high-resolution images has been used to characterize bone microarchitecture [23]: and the fractal dimension of cubes of vertebral trabecular bone has been shown to be highly correlated (r=0.84) with their elastic modulus [24]. A major advantage of computed tomography (CT) imaging over other modalities such as dual energy X-ray absorption (DEXA) is its ability to isolate and measure trabecular bone separately from cortical bone. Since this is a prerequisite for texture studies CT imaging would be required for patient studies, even though the patient dose is higher than for a DEXA examination [25]. Although the limited resolution of commercial CT scanners precludes proper resolution of the trabecular structure 2-D axial images of thin vertebral slices contain some of this architectural information, albeit degraded by the inadequate modulation transfer function (MTF) of the imaging system, which we will refer to as texture.
The purpose of this study was to investigate the potential usefulness of fractal signature and lacunarity in quantifying the texture of trabecular bone in clinical CT images. Because of the coarse raster of clinical CT scanners the trabecular bone images used were small, typically 32×32 pixels. We used synthetic fractal images to assess the accuracy of our algorithm for estimating the fractal dimension of data sets of limited resolution. Natural texture images were used to determine the extent of fractality in real images, and to assess the efficacy of a fractal signature in discriminating between different textures. These images were also used to validate the lacunarity methdology and subsequent interpretation of lacunarity plots. The two methods, fractal signature and lacunarity, were then applied to small trabecular bone CT images to assess their utility and sensitivity in quantifying texture therein. The significance of correcting for the limited resolution and noise of the imaging system was also addressed.
Section snippets
Test images
Two sets of test images were used to test the accuracy of our protocols and their subsequent interpretation.
- 1.
Synthetic fractal images (64×64 pixels) were generated with fractal dimensions, D, of 2.25 to 4.0 in steps of 0.25, by filtering a white noise image in a method similar to that described by Saupe [4].
- 2.
Photographic images of texture from an album by Brodatz [26] have become a de facto standard for testing texture algorithms. We used a subset of the textures used in earlier studies [2], [5]:
Test images
The radial power spectral densities for the synthesized fractal images are linear (not shown), with statistical imprecision due to the finite size of the data sets [38]. Linear regression was used to obtain the gradients, and hence the measured fractal dimensions of the images. The close agreement of the measured and theoretical values of the fractal dimension of these images (Table 1) confirms the accuracy of the estimation algorithm. Both estimates of the width of the power spectrum, the mean
Discussion
Although macroradiographs of lumbar vertebrae [16], contact radiographs [23], [39], and high-resolution CT [40] and MRI [41] images show a fractal nature to the trabecular bone, whether images of vertebral trabecular bone acquired at the much lower spatial resolution of CT are fractal over any significant range has been disputed. A preliminary study of CT images of vertebral trabecular bone concluded that the images were fractal and that osteoporotic patients could be distinguished from normal
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