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I.INTRODUCTIONThe motivation for using low-dose x-ray computed tomography (CT) is to reduce the patient radiation exposure [1-3]. Since x-ray radiation exposure may play a role in getting cancers, it is advised to reduce the x-ray exposure to an As-Low-As-Reasonably-Achievable (ALARA) level [4]. One way of low-dose imaging is the sparse view method, but sparse angular sampling frequently leads to characteristic streak artifacts. This under sampling situation is also a case of compressed sensing. Many researchers attempted to solve this compressed sensing problem. One method is the iterative image reconstruction method that minimizes the total-variation (TV) norm or other measures of the image [5]-[11]. Most recently, research activities are mainly in the deep learning area [12]-[19]. It is fair to say that deep learning methods are dominating the current publications and conferences. This paper investigates a nonlinear filter that is not deep learning based. Our filter is based on the BM3D denoising method, which was proposed by Dabov et al. [20][21]. The BM3D method uses block matching and aggregation strategies to obtain three-dimensional image blocks; its denoising uses Wiener filtering. The BM3D is currently the state-of-the-art in image denoising. The BM3D method requires two inputs: the noisy image and the noise power spectral density image. The original purpose of BM3D is for random noise reduction. In our application of sparse-view tomography, our main concern is the angular aliasing streak artifacts. These artifact patterns are deterministic and object dependent. These artifacts are usually more pronounced than the random noise. The strategy of this paper is to treat the deterministic artifacts as random noise when calculating the ‘noise’ power spectral density function (image). II.METHODSA.‘Noise’ power spectral densityFor a given CT image, G, resulted from sparse-view projection measurements, its associated artifact power spectral density function, P, is difficult to obtain. This is because the true image, T, is not available. In this paper, the artifact power spectral density function, P, is obtained by noiseless computer simulations, that simulate full-scan and sparse-scan projections of some random objects. A full scan has sufficient angular measurements. The reconstructed images from this full-scan data set are treated as (gold standard) true images, Tsimu. The artifact image, A, is the differences between the gold standard true image, T, and the given sparse-scan image, Gsimu: In this paper, we use 1000 random simulated objects. Therefore, we have 1000 2D artifact images, A’s. Let B be the 2D Fourier transform of image A defined in (1). For each element in B, we calculate its norm square and denote the resulting frequency-domain image be P. This resultant 2D image, P, has the same dimension as the image A, is real, and is nonnegative. Even in the noiseless cases, P is not zero due to the sparse-view streaking artifacts. In forming 1000 versions of P, no noise is added. Therefore, the image P is better referred to as the artifact spectral function (instead of the noise spectral function). Let B.The proposed algorithmIn the conventional BM3D algorithm, the noise is assumed to be stationary. The Wiener filter is used for denoising in the BM3D algorithm. The Wiener filter assumes stationary noise with a noise power spectral density function where GCT is a 2D given sparse-view CT image, We must point out that in calculating C.Computer simulationsWe generated 1000 noiseless random 256×256 phantoms, each of which had 2 random ellipses of random shapes, random locations, and random intensities. We generated 2 versions of projections for each computergenerated phantom: one with 60 views over 360° (sparse scan case); the other one with 180 views over 360° (full scan case). Images were reconstructed using the filtered backprojection (FBP) algorithm using all projections for both full scan and sparse scan cases. One averaged artifact power spectral density image, We then generated a new random 256×256 phantom and generated a sparse scan with 60 views (test case). The FBP reconstruction, GCT, was calculated from this new test case 60-view data. The proposed algorithm (2) was applied to this FBP image, GCT, to obtain the final image, H. D.Clinical dataHere we had one set of sparse-scan CT images for one patient. The set contained 512×512 2D images. The original projections were not available. We knew the imaging geometry. The number of views was 200 views over 360°. In order to use the proposed method to reduce the angular aliasing artifacts, we generated a new averaged artifact power spectral density image III.RESULTSA.Computer simulation resultsFig. 1 shows 2 (out of 1000) representative random phantoms. Their sparse-view versions using 60 views are shown in Fig. 2. Fig. 3 shows the average artifact power spectral density image by considering 1000 sparse/full pairs of the simulated images. Two new random phantoms sparse-scan images are shown in Fig. 4. These new phantoms are NOT among the 1000 phantoms used in estimating the artifact power spectral density image, because the new ones contain 3 ellipses while the old ones contain 2 ellipses. The results of the proposed method are shown in Fig. 5. B.Patient data resultsThree patient image pairs are shown in Figs. 6, 7 and 8, respectively. The images are sparse-scan images without and with the proposed BM3D processing. Fig. 9 shows the Figure 6.The sparse-scan patient image slice #160 before (upper) and after (lower) processing by the proposed method.  Figure 7.The sparse-scan patient image slice #140 before (upper) and after (lower) processing by the proposed method.  V.CONCLUSIONSWe have attempted a method to reduce the sparse-scan angular aliasing artifacts without using any patient training data. This method is a direct application of the BM3D filter by replacing the noise power spectral density function with the artifact power spectral density function. The BM3D filter assumes stationary noise that is characterized by the noise power spectral density function. Noise and artifacts are never the same. Noise is random, while artifacts are somewhat deterministic. Artifacts are not stationary. Strictly speaking, the artifact power spectral density function does not exist because it is not stationary. Our ad hoc method assumes the norm square of the Fourier transform of the error image as the artifact power spectral density function, which is calculated with computer simulations and depends on the imaging geometry only. Patient data is not used in finding the artifact power spectral density function. Our results indicate that the proposed method is not effective enough for practical applications. The artifacts are still present, and the images are over-smoothed after processing. More work needs to be done. However, insights our from this study suggest that some features can be obtained my simulations when there is no real data available. Another thing we observe is that the Wiener filter is not an effective method to remove artifacts, and a better approach should be considered. REFERENCESW. Yu, C. Wang, X. Nie, and D. Zeng,
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