Estimation of Trabecular Bone in Gray-Scale: Comparison Between Cone Beam and Micro Computed Tomography Data

Fig. 1

Slices (left) and volume renderings (right) of one of the imaged specimens. Top: images acquired through micro-CT. Bottom: images acquired through CBCT

2.3 Methods

The tensors were computed through the generalized MIL tensor (GMIL) and the GST.

2.3.1 GMIL Tensor

Basically, the GMIL tensor is computed in three steps. The mirrored extended Gaussian image (EGI) [12] is computed from a robust estimation of the gradient. Second, the EGI is convolved with a kernel in order to obtain an orientation distribution function (ODF). Finally, a second-order fabric tensor is computed from the ODF. More formally, the generalized MIL tensor is computed as:

${\sf MIL}=\int_\Omega \frac{v\,v^T}{C(v)^2} \, d\Omega,$

(1)

where v are vectors on the unitary sphere Ω, and C is given by:

$C=H\ast E,$

(2)

that is, the angular convolution (*) of a kernel H with the mirrored EGI E. Thanks to the Funk-Hecke theorem [3,9], this convolution can be performed efficiently in the spherical harmonics domain when the kernel is positive and rotationally symmetric with respect to the north pole.

One of the advantages of the GMIL tensor is that different kernels can be used in order to improve the results. In this study, the half-cosine (HC) and von Mises-Fisher (vMF) kernels have been applied to the images. The HC has been selected since it makes equivalent the generalized and the original MIL tensor. The HC is given by:

$H(\phi) = \left\{\begin{array}{ll} \cos(\phi) &, \mathrm{if} \phi\le \pi/2\\0 &,\mathrm{otherwise},\end{array}\right.$

(3)

with φ being the polar angle in spherical coordinates. Moreover, the vMF kernel, which is given by [14]:

$H(\phi)=\frac{\kappa}{4\pi \sinh(\kappa)} \, e^{\kappa \cos(\phi)},$

(4)

has been selected since it has a parameter κ that can be used to control its smoothing action. In particular, the smoothing effect is reduced as the values of κ are increased [18].

Figure 2 shows different kernels that can be used with the GMIL tensor. As already mentioned, these kernels must be positive and symmetric with respect to the north pole. As shown in the figure, the HC kernel is too broad (it covers half of the sphere), which can result in excessive smoothing. On the contrary, the impulse kernel is the sharpest possible kernel. As shown in [18], the GST makes use of the impulse kernel. In turn, the size of the smoothing effect of the vMF kernel can be controlled through the parameter κ. As shown in the figure, vMF is broader than the HC for small values of κ and it converges to the impulse kernel in the limit when $\kappa \rightarrow \infty$ .

Fig. 2

Graphical representation of some kernels from the broadest to the narrowest, where zero and the largest values are depicted in blue and red respectively. Notice that the impulse kernel has been depicted as a single red dot in the north pole of the sphere

2.3.2 GST Tensor

On the other hand, the GST computes the fabric tensor by adding up the outer product of the local gradients with themselves [25], that is:

${\sf GST}= \int_{p\in I} \nabla I_p\, \nabla I_p^T\, dI,$

(5)

where I is the image and $\nabla I_p$ is the gradient.

Notice that GST related to the well-known local structure tensor (ST) which has been used in the computer vision community since 1980s [4]. There are different methods for computing ST, including quadrature filters [7], higher-order derivatives [15] or tensor voting [19]. However, the most used ST is given by:

${\sf ST}_\sigma(p)={G_\sigma \ast \nabla I_p {\nabla I_p}^T}$

(6)

where $G_\sigma$ is a Gaussian weighting function with zero mean and standard deviation σ. In fact, ST becomes the GST when $\sigma \rightarrow \infty$ . The main advantage of this structure tensor is that it is easy to code.

3 Results

As already mentioned, the focus in this chapter is the estimation of anisotropy. As a matter of fact, both the GMIL (and therefore the MIL tensor) and the GST tensors yield the same orientation information, since they have the same eigenvectors (cf. [18] for a detailed proof). This means that only the eigenvalues of the tensors are of interest for the purposes of this chapter.

The following three values have been computed for each tensor:

$\begin{aligned} E1^{\prime} &= E1/(E1+E2+E3)\\ E2^{\prime} &= E2/E1, \\ E3^{\prime} &= E3/E1,\end{aligned}$

where E1, E2 and E3 are the largest, intermediate and smallest eigenvalues of the tensor. These three values have been selected since they are directly related to the shape of the tensor.

Tables 1–3 show the mean and standard deviation of E1’, E2’ and E3’ computed on micro-CT and CBCT for the tested methods, and the mean difference and standard deviation between micro-CT and CBCT. As a general trend, the tested methods tend to overestimate E1’ and underestimate E2’ and E3’ in CBCT. As shown, the best performance is obtained by vMF with κ =1 with small differences between tensors computed in both modalities. However, the tensors computed with this broad kernel are almost isotropic (cf. Tables 2 and 3), which makes it not suitable for detecting anisotropies in trabecular bone. It is also worthwhile to notice that the standard deviation of the differences increases with narrower kernels, such as GST. This means that a mild smoothing effect from middle range kernels such as vMF with κ=10, have a positive effect in the estimation of fabric tensors, since the differences between micro-CT and CBCT are reduced while keeping the anisotropy of the tensors.

Table 1

Mean (SD) of E1’ for fabric tensors computed on CBCT and micro-CT and the mean difference (SD) between both values. HC and vMF refer to the generalized MIL tensor, with the HC, and vMF kernels respectively. Parameter κ for vMF is shown in parenthesis. Positive and negative values of the difference mean over- and under estimations of CBCT with respect to micro-CT. All values have been multiplied by 100

Tensor	micro-CT	CBCT	Difference
HC	44.65 (1.54)	42.38 (0.90)	2.25 (0.84)
vMF(1)	34.12 (0.29)	34.70 (0.18)	0.42 (0.15)
vMF(5)	51.55 (3.56)	47.07 (2.17)	4.51 (1.82)
vMF(10)	58.98 (4.63)	53.90 (3.21)	5.11 (2.13)
GST	45.69 (1.58)	44.79 (1.58)	0.90 (2.09)

Table 2

Mean (SD) of E2’ for fabric tensors computed on CBCT and micro-CT and the mean difference (SD) between both values. HC and vMF refer to the generalized MIL tensor, with the HC, and vMF kernels respectively. Parameter κ for vMF is shown in parenthesis. Positive and negative values of the difference mean over- and under estimations of CBCT with respect to micro-CT. All values have been multiplied by 100

Tensor

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