Unlike run-length classifiers and co-occurrence classifiers, Laws’ energy metrics [45] are local neighborhood values obtained by convolving the image with two out of five one-dimensional convolution kernels to yield a feature vector with up to 25 elements. Laws proposes a three-step process where the image is first subjected to background removal (for example, a highpass filter step), subsequently convolved with the kernel pair, and finally lowpass filtered. The resulting image is referred to as the energy map. Typically, Laws’ energy maps result in a high-dimensional feature vector per pixel and advertises itself for classification or segmentation with highdimensional clustering techniques or artificial intelligence methods. In one example [48], classical statistical parameters were obtained from the texture energy maps, and the discrete first-order finite difference convolution kernel that approximates ∂ 2/∂ x∂ y showed some ability to discriminate between cases with and without fractures. However, Laws’ texture maps are not in common use for this special application of texture analysis. One possible reason is the fixed scale on which the texture maps operate. Laws’ convolution kernels are fixed to 5 by 5 pixels, and pixel noise dominates this scale. Thus, the method suffers from the same problems as the co-occurrence matrix with short displacements. However, it is conceivable that a combination of multiscale decomposition (such as a wavelet decomposition) with Laws’ texture energy maps provides a more meaningful basis to obtain a quantitative description of the texture.

In summary, texture analysis methods in the spatial domain provide, to some extent, information that is related to actual trabecular structure. Therefore, texture analysis methods have the potential to provide information on bone architecture that is independent from bone density. However, texture analysis methods are sensitive towards the image formation function (e.g., projection image versus tomography), towards the point-spread function, the pixel size relative to trabecular size, and towards image artifacts, most notably noise.

9.3.3 Frequency-Domain Methods

The Fourier transform decomposes an image into its periodic components. A regular, repeating pattern of texture elements (sometimes referred to as texels), causes distinct and narrow peaks in the Fourier transform. A comprehensive discussion of the Fourier transform and its application in image analysis can be found in the pertinent literature [39, 40]. Since the Fourier transform reveals periodic components, i.e., the distances at which a pattern repeats itself, operations acting on the Fourier-transform of an image are referred to as frequency-domain operations in contrast to the spatial-domain operations that were covered in the previous section. The magnitude of the Fourier transform is often referred to as the frequency spectrum. In two-dimensional images, the frequency spectrum is two-dimensional with two orthogonal frequency components u and v. It is possible to analyze those frequency components separately and include properties such as texture anisotropy.

Alternatively, the spectrum can be reduced to one dimension by averaging all frequency coefficients at the same spatial frequency ω = √u2 + v2. Intuitively, the frequency spectrum can be interpreted as a different representation of the spatialdomain image data that emphasizes a specific property, namely, the periodicity.

Trabecular bone does not exhibit any strict periodicity, because trabeculae are to some extent randomly oriented and have random size. The Fourier transform of random structures does not show distinct peaks. Rather, the frequency components decay more or less monotonically with increasing spatial frequency. This property is demonstrated in Fig. 9.7. The Fourier transform images (more precisely, the logtransformed magnitude of the Fourier transform) of a relatively regular texture and an irregular, bone-like structure are shown. Peaks that indicate the periodicity of the knit pattern do not exist in the Fourier transform of the bone-like structure. However, the decay behavior of the central maximum contains information about the texture. A fine, highly irregular texture, typically associated with healthy bone architecture, would show a broad peak with slow drop-off towards higher frequencies. Conversely, osteoporotic bone with large intratrabecular spacing would show a narrow peak with a faster drop-off towards higher frequencies.

In the analysis of trabecular bone texture, frequency-domain methods are often used to detect self-similar properties. These methods are described in the next section. Moreover, a number of single-value metrics can be derived from the Fourier transform. These include the root mean square variation and the first moment of the power spectrum (FMP, (9.8)):

Here, F(u, v) indicates the Fourier coefficients at the spatial frequencies u and v, and the summation takes place over all u, v. The computation of the FMP-value can be restricted to angular “wedges” of width θ , where the angle-dependent FMP (θi) is computed over all u, v with tan(θi) < v/u ≤ tan(θi + θ ). In this case, the minimum and maximum value of FMP (θi) provide additional information on the anisotropy of the texture. For example, if the texture has a preferredly horizontal and vertical orientation, FMP(θ ) shows very distinct maxima for θ around 0◦, ±90◦, and 180 ◦. Conversely, the FMP-index for a randomly oriented texture has less distinct maxima, and the coefficient of variation of FMP(θ ) is lower. Two recent representative studies describe the use of frequency-domain metrics in radiographic images of the femur in patients with osteoprosis [49] and osteolysis [50]. Special use of the anisotropy was made in a study by Chappard et al. [51] and Brunet-Imbault et al. [52].

Frequency-domain methods have two major advantages over spatial-domain methods for the analysis of bone structure and its texture representation in images. First, frequency-domain methods are less sensitive against background irregularities and inhomogeneous intensity distributions. Trend-like background inhomogeneities map to very low spatial frequencies, and the corresponding Fourier coefficients

Fig. 9.7 Frequency-domain representation of texture. (a): Photography of a knit texture that contains a somewhat regular repeating pattern. The regularity of the pattern is represented by distinct peaks in the Fourier transform (b). The slight off-vertical orientation of the knit texture finds a correspondence of the off-horizontal orientation of the Fourier-transform pattern. (c): Synthetic trabecular pattern from Fig. 9.5. Such a pattern does not have repeated texture elements, and the frequency coefficients decay mostly monotonically with increasing frequencies (d). The decay behavior can be used to characterize any irregular texture

are close to the center of the Fourier spectrum image. By omitting the F(u, v) from any single-value metric computation for small u,v, trends are automatically excluded from the metric. Second, frequency-domain methods are usually less noise-sensitive than spatial-domain methods. Many spatial-domain methods act on the pixel level (examples are Laws’ texture energies or the co-occurrence matrix with low displacements), and pixel noise directly influences the measured values. In the frequency domain, the energy of the noise is broadly distributed over the entire frequency spectrum. High spatial frequencies, i.e., those frequencies that are