Computed tomography image representation using the Legendre polynomial and spherical harmonics functions

Abstract

The representation of computed tomography (CT) images using the Legendre polynomial (LPF) and spherical harmonics (SHF) functions was investigated. We selected 100 two-dimensional (2D) CT images of 10 lung cancer patients and 33 three-dimensional (3D) CT images of head and neck cancer patients. The reproducibility of these special functions was evaluated in terms of the normalized cross-correlation (NCC). For the 2D images, the NCC was 0.990 ± 0.002 (1sd) with an LPF of order 70, whereas for the 3D images, the NCC was 0.971 ± 0.004 (1sd) with an SHF of degree 70. The results showed that the LPF was more efficient than the Fourier series. As the thoracic and head areas are cylindrical and spherical, respectively, expansions with the LPF and SHF achieved an efficient representation of the human body. CT image representation with analytical functions can be potentially beneficial, such as in X-ray scattering estimation.

Appendix

In this appendix, we show the method of the X-ray scattering estimation using the CT image represented with spherical harmonics. The X-ray scattering model employed here is the following three-dimensional (3D) convolution integral,

$$D\left(\overrightarrow{r}\right)= \int K(\overrightarrow{r}-\overrightarrow{r}{^{\prime}}) \rho (\overrightarrow{r}{^{\prime}}) \psi (\overrightarrow{r}{^{\prime}})\mathrm{d}\overrightarrow{r}{^{\prime}}, $$

(A1)

where $D\left(\overrightarrow{r}\right)$ means the scattered fluence of X-ray at detector position $\overrightarrow{r}$, $K(\overrightarrow{r}-\overrightarrow{r}{^{\prime}})$ means the scattering kernel, $\psi (\overrightarrow{r}{^{\prime}})$ means the fluence at the scatter position $\overrightarrow{r}{^{\prime}}$, and $\rho (\overrightarrow{r}{^{\prime}})$ means the electron density at the scatter position $\overrightarrow{r}{^{\prime}}$. $K(\overrightarrow{r}-\overrightarrow{r}{^{\prime}})$ has the fixed form, whereas $\psi (\overrightarrow{r}{^{\prime}})$ and $\rho (\overrightarrow{r}{^{\prime}})$ depends on the scatter object. Namely, one can simulate any X-ray scattering via Eq. (A1) by changing the object affecting $\psi (\overrightarrow{r}{^{\prime}})$ and $\rho (\overrightarrow{r}{^{\prime}})$. Further, $\psi (\overrightarrow{r}{^{\prime}})$ is decreased as passing the object, and at the detector position, this is approximately independent on $\overrightarrow{r}{^{\prime}}$ by assuming that the path length of the object is constant [5, 6]. This approximation yields,

$$D\left(\overrightarrow{r}\right) \sim \int K(\overrightarrow{r}-\overrightarrow{r}{^{\prime}}) \rho (\overrightarrow{r}{^{\prime}}) \mathrm{d}\overrightarrow{r}{^{\prime}}, $$

(A2)

to evaluate the scattered fluence at detector position. Then, the kernel is expanded with the spherical harmonics as,

$$K(\overrightarrow{r}-\overrightarrow{r}{^{\prime}})=\sum_{\mathrm{lm}}{k}_{\mathrm{lm}}\left(r,{r}{^{\prime}}\right){Y}_{\mathrm{lm}}\left(\theta ,\phi \right){Y}_{\mathrm{lm}}^{*}\left({\theta }{^{\prime}},{\phi }{^{\prime}}\right) , $$

(A3)

where ${k}_{\mathrm{lm}}\left(r,{r}{^{\prime}}\right)$ is the radial part of the kernel, and ${Y}_{lm}\left(\theta ,\phi \right)$ and ${Y}_{lm}^{*}\left({\theta }{^{\prime}},{\phi }{^{\prime}}\right)$ are the spherical harmonics and its complex conjugate, respectively. Now, assuming that the physical density of the object (namely CBCT images) can be expressed by the spherical harmonics, we have,

$$D\left(\overrightarrow{r}\right) \sim \sum_{lm}{Y}_{lm}\left(\theta ,\phi \right)\int {k}_{lm}\left(r,{r}{^{\prime}}\right){R}_{lm}\left(r{^{\prime}}\right){{r}{^{\prime}}}^{2}d{r}{^{\prime}}, $$

(A4)

where the orthogonal relation of the spherical harmonics Eq. (7) is used. Here, ${R}_{lm}\left(r{^{\prime}}\right)$ is the coefficient of the radial part obtained in the image expansions with the SHF (see Eq. (6)). The efficiency by the use of Eq. (A4) is obvious; the integrals of angler parts in 3D convolution integral Eq. (A2) are dropped by means of its analytical form. The residual integral for radial part can be numerically performed. For this, ${k}_{lm}\left(r,{r}{^{\prime}}\right)$ is required to be evaluate in advance. This is done by the following integral,

$${k}_{lm}\left(r,{r}{^{\prime}}\right) = \int K\left(r-{r}{^{\prime}}\right){Y}_{lm}\left(\theta ,\phi \right){Y}_{lm}^{*}\left({\theta }{^{\prime}},{\phi }{^{\prime}}\right)\mathrm{dcos}\theta \mathrm{d}\phi \mathrm{dcos}\theta {^{\prime}}\mathrm{d}\phi {^{\prime}}. $$

(A5)

Here, the relatively high computational effort is necessary to evaluate this integral. Once ${k}_{lm}\left(r,{r}{^{\prime}}\right)$ is evaluated, however, same values can be available even if the scattering object (namely, patient) is changed. This means that the expansion of the actual object (patient) is important in the practical application.