Scaling calibration in region of interest reconstruction with the 1D and 2D ATRACT algorithm

Abstract

Purpose

   Recently, a reconstruction algorithm for region of interest (ROI) imaging in C-arm CT was published, named Approximate Truncation Robust Algorithm for Computed Tomography (ATRACT). Even in the presence of substantial data truncation, the algorithm is able to reconstruct images without the use of explicit extrapolation or prior knowledge. However, the method suffers from a scaling and offset artifact in the reconstruction. Hence, the reconstruction results are not quantitative. It is our goal to reduce the scaling and offset artifact so that Hounsfield unit (HU) values can be used for diagnosis.

Methods

   In this paper, we investigate two variants of the ATRACT method and present the analytical derivations of these algorithms in the Fourier domain. Then, we propose an empirical correction measure that can be applied to the ATRACT algorithm, to effectively compensate the scaling and offset issue. The proposed method is evaluated on ten clinical datasets in the presence of different degrees of artificial truncation.

Results

   With the proposed correction approach, we achieved an average relative root-mean-square error (rRMSE) of 2.81 % with respect to non-truncated Feldkamp, Davis, and Kress reconstruction, even for severely truncated data. The rRMSE is reduced to as little as 10 % of the image reconstructed without the scaling calibration.

Conclusions

   The reconstruction results show that ROI reconstruction of high accuracy can be achieved since the scaling and offset artifact are effectively eliminated by the proposed method. With this improvement, the HU values may be used for post-processing operations such as bone or soft tissue segmentation if some tolerance is accepted.

Appendix

Assume the 1D Fourier transform of $g_{1}\left( \lambda ,u,v\right)$ with respect to $u$ is ${\mathcal {F}}_{1}\left\{ g_{1}\right\} \left( \lambda ,\omega _{u},v\right)$, then we can obtain

$$\begin{aligned} g_{1}\left( \lambda ,u,v\right) =\int \limits _{0}^{\infty }{\mathcal {F}}_{1}\left\{ g_{1}\right\} \left( \lambda ,\omega _{u},v\right) e^\mathrm{i 2\pi \omega _{u}u}\mathrm d \omega _{u}. \end{aligned}$$

(21)

Applying the second-order derivative to both side of the equation above yields

$$\begin{aligned} \frac{\partial }{\partial u^{2}}g_{1}\left( \lambda ,u,v\right)&= \frac{\partial }{\partial u^{2}}\int \limits _{0}^{\infty }{\mathcal {F}}_{1}\left\{ g_{1}\right\} \left( \lambda ,\omega _{u},v\right) e^\mathrm{i 2\pi \omega _{u}u}\mathrm d \omega _{u}\nonumber \\&= \int \limits _{0}^{\infty }{\mathcal {F}}_{1}\left\{ g_{1}\right\} \left( \lambda ,\omega _{u},v\right) \frac{\partial }{\partial u^{2}}e^\mathrm{i 2\pi \omega _{u}u}\mathrm d \omega _{u}\nonumber \\&= \int \limits _{0}^{\infty }\left[ {\mathcal {F}}_{1}\left\{ g_{1}\right\} \left( \lambda ,\omega _{u},v\right) \left( 2\pi \mathrm i \omega _{u}\right) ^{2}\right] e^\mathrm{i 2\pi \omega _{u}u}\nonumber \\&\times \mathrm d \omega _{u} \end{aligned}$$

(22)

Then, the inverse Fourier transform of $\left( \mathrm i 2\pi \omega _{u}\right) ^{2}{\mathcal {F}}_{1}\left\{ g_{1}\right\}$ is equivalent to $\partial ^{2}g_{1}/\partial u^{2}$ has been shown.