1.

INTRODUCTION

Image reconstruction in computed tomography (CT) depends heavily on the quality of the available raw data. However, in many instances the acquisition parameters lead to raw data that are insufficient for a conventional method such as filtered back-projection (FBP). Typical problems are a low number of projections, small angular range, or low tube current. This work focuses on the issue of truncated projections, i.e. the case where the patient does not fully fit in the maximum field of view (FOV) of the scanner. In clinical practice, truncation most often occurs with obese patients, patients that are not centered properly on the table, or when using C-arm systems, which generally feature small FOVs.

Sinogram truncation is twofold problematic. On the one hand, voxels inside of the FOV suffer from cupping artifacts. On the other hand, there is a demand of artificially extending the FOV. Voxels in the extended FOV (eFOV) are of interest for several algorithms, including metal artifact reduction and beam hardening correction in CT, and attenuation and scatter correction in PET.^{1, 2}

In order to reduce artifacts within the FOV, several data extrapolation methods have been developed.^3–5 These conventional algorithms are able to restore the CT values within the FOV to acceptable values, such that the cupping artifacts are removed. However, voxels in the eFOV are typically not reconstructed accurately. Recently, there have also been deep learning methods developed for the purpose of extending the FOV, showing promising results.^{6, 7}

Another method that has shown promise for reconstructing truncated raw data is the discrete algebraic reconstruction technique (DART).^8–10 DART is a well-known algorithm for reconstructing discrete CT volumes, e.g. in the context of non-destructive testing of homogeneous objects. It was specifically developed for scans with few projections or limited angular range.^11–13 Utilizing the assumption that the object consists of very few attenuation coefficients, DART is capable of image reconstruction with severely limited data.

In this work, we apply DART to clinical CT data to obtain a prior image for detruncation, i.e. to fill the missing projections. Although clinical CT volumes are not strictly homogeneous, a typical CT scan consists only of soft tissue, bone, and air. In addition, for the purposes of the projection completion, the patient is often approximated by a single attenuation value.^3–5 Although DART was developed for discrete CT, it internally computes real valued images, which we use for detruncation. To evaluate our method, it is compared to the established adaptive detruncation (ADT).⁴

2.

METHODS

We define the detruncation problem as follows. We have access to truncated parallel beam raw data p_t(ϑ, ξ) with M_t pixels of size Δξ. Let , such that the detector range is −ξ_t,max ≤ ξ ≤ ξ_t,max. In order to reconstruct an artifact free image, the full detector should have M_f > M_t pixels, corresponding to . Thus, the required extrapolation includes pixels within ξ_t,max < |ξ| ≤ ξ_f,max. In image domain, this expands the original FOV with radius R_M,t = ξ_t,max to the eFOV with radius R_M,f = ξ_f,max. Due to the limitations of the scanner, the eFOV cannot be larger than the bore size. Figure 1 shows our strategy to fill the missing projections: First, generate a prior image using the DART algorithm. Second, forward-project the prior image and fill in the missing data in the original sinogram using the prior sinogram.

Figure 1.

Scheme of the proposed DART detruncation. The dashed box highlights the DART reconstruction, the right side shows the data completion step.

2.1

Generating a DART Prior Image

DART employs a combination of discretization and real-valued algebraic reconstruction.^{11, 12} Figure 1 shows the basic DART scheme on the left. During each DART iteration, only a subset of pixels is changed, while the rest of the pixels are set to constant values. In order to determine the fixed pixels, we first segment the current image estimate into air and tissue by thresholding at −500 HU. Then, every pixel that is fully surrounded by pixels of the same class is fixed and set to a value of −1000 HU or 100 HU for air or tissue, respectively. Each pixel is given an additional probability of 65% to be classified as non-fixed, so that values inside homogeneous patches can be corrected. Subsequently, we perform five SART iterations where only the non-fixed pixels are considered.¹⁴ One SART iteration is defined as

where f is the current image estimate, f_new is the new estimate, p are the raw data, λ is a relaxation factor, X is the forward-projection and X^T is the back-projection.

Since the fixed pixels contain no noise, all projection noise will be distributed over the non-fixed pixels. Therefore, the last step is a smoothing with a Gaussian filter with a standard deviation of 0.5 pixels. This finishes one iteration and the algorithm continues with the segmentation step. Since there is no formal stopping criterion, we set the maximum number of iterations N_iter to 5000. In most cases, a few hundred iterations were sufficient. To initialize the DART algorithm, we perform a simple cosine detruncation followed by an FBP. Note that only the original raw data are used for the DART iterations.

In conventional applications, the DART result will again be segmented at the end of the algorithm, as a discrete volume is desired. In this work, DART is used with clinical data. Therefore, we omit the final thresholding and instead use the real valued image as the prior image.

2.4

Analysis

To quantify how well the algorithm can correct CT values inside the original FOV, we calculate the root mean squared error (RMSE) within the FOV with respect to the image reconstructed from the full detector. The RMSE for an image f(i, j) with respect to the ground truth image GT is defined as

Furthermore, we compute the RMSE within the eFOV and Dice score of the whole image to compare the ability of the detruncation to restore the CT values and patient outline outside of the original FOV. For the latter, the images are segmented into air and soft tissue with a threshold of −500 HU. The Dice score is defined as

where TP are the true positives, FP the false positives and FN the false negatives. To compare the DART detruncation with a conventional method, we also apply the adaptive detruncation (ADT)⁴ to the data.

3.

RESULTS

Figure 2 shows the first patient, including detruncation results from cosine detruncation, ADT, DART, and our method. In the left column, the scan suffers from light truncation. This causes minor cupping artifacts at the left and right side of the patient in the uncorrected FBP reconstruction. All methods have have successfully removed the cupping and restored CT values to voxels in the eFOV. After cosine detruncation, artifacts remain at the edge of the FOV and the tissue in the eFOV is too dark. In the ADT result, voxels on the right side are slightly brighter than in the other ground truth. For the case of strong truncation, displayed in the right column of Figure 2, there is a considerable difference between ADT and our method. While ADT is able to restore CT values inside the original FOV very well, the tissue outside of the FOV is too bright and the patient outline does not agree with the ground truth. In contrast, the DART-detruncated image is very close to the full-view reconstruction, although fat has been converted into water at the bottom of the image and some bone anatomy is incorrect. These findings are supported by the numerical evaluation in Table 1. In all cases, our method outperforms ADT in terms of RMSE and Dice score. Notably, the DART prior image has relatively high RMSE values (above 100 HU in all cases) but good Dice scores, indicating high image noise. Figure 4 shows the projections for the case of strong truncation. Although the ADT is able to estimate the projections approximately, the DART prior and DART detruncation are noticeably smoother and more accurate.

Figure 2.

Detruncation results for Patient 1. Left column shows mild truncation (M_t = 682), right column strong truncation (M_t = 372). Top to bottom: ground truth, truncated FBP, cosine detruncation, ADT, DART prior image, DART detruncation. C = 0 HU, W = 1000 HU.

Table 1.

RMSE and Dice values for both patients and levels of truncation. RMSE is given for the original and for the extended FOV.

Figure 3 displays the results of the second patient. Again, both ADT and DART detruncation perform well when the data are only mildly truncated. As for the first case, the ADT image features slightly too high CT values on the right side. The DART prior image and DART-detruncated image show a dark streak along the edge of the FOV. For strong truncation, the ADT again yields CT values that are too high on the left and right side of the FOV and does not provide a clear patient outline. Still, ADT clearly outperforms the cosine detruncation. While the DART reconstruction gives more accurate CT values in general, there are gaps in the soft tissue of the patient. This artifact is also evident in the RMSE of the eFOV and the Dice score, which are lower than ADT for both the DART and DART-detruncated image. In the case of mild truncation, our method still yields superior numerical results.

Figure 3.

Detruncation results for Patient 2. Left column shows mild truncation (M_t = 682), right column strong truncation (M_t = 372). Top to bottom: ground truth, truncated FBP, cosine detruncation, ADT, DART prior image, DART detruncation. C = 0 HU, W = 1000 HU.

Figure 4.

Projections for Patient 1 with strong truncation after detruncation. Top to bottom: ground truth, truncated FBP, cosine detruncation, ADT, DART prior image, DART detruncation.

4.

DISCUSSION & CONCLUSION

In this work, we propose a method of CT raw data detruncation based on the DART reconstruction. The DART detruncation was capable of reducing cupping artifacts in the FOV, as well as restoring CT values in the eFOV, for two levels of truncation. Compared to the conventional ADT, our method produced superior visual and numerical results. However, for patient 2 with strong truncation, some anatomy was incorrectly reconstructed as air, yielding worse numerical results than ADT. These artifacts would likely be prevented with a better initial estimate of the image, along with additional iterations of DART.

This work uses the original DART algorithm^{11, 12} with a single, manually determined threshold. In the future, a more refined version of DART, e.g. with automatic parameter optimization or improved performance for noisy projections^{13, 15, 16} should be preferred, in order to reduce computation time and image noise.