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1.INTRODUCTIONLow-dose computed tomography (CT) is one of the most direct and effective ways to reduce the radiation dose to patients. However, a trade-off between image quality and patient dose always exists. Many efforts have been put into this area to find better ways of balancing the trade-off. Deep learning methods are one of the most recent and promising developments in reducing noise in CT imaging. When high-quality images are accessible for training, neural networks trained either in image domain1,2 or during the reconstruction process3,4 showed promising performance. On the other hand, there are several works attempting to tackle the problem without the presence of high-quality images by exploiting the Noise2Noise pipeline5. Wu et al6 showed that a denoising network with Noise2Noise training is equivalent to training with clean labels (high-quality images) when a few conditions are satisfied. One of the four conditions is that the network should have paired noisy data with zero-mean, independent noise. For Noise2Noise application in CT imaging, it is crucial to find such paired data. While such data could be acquired with two scans of the same patient, this exposes the patient to additional dose and will have misregistration artifacts. Pairing simulated low-dose images with the original normal-dose images does not satisfy this condition since some of the noise in the simulated low-dose image comes from the normal-dose image, so the two images do not have independent noise. In one Noise2Noise approach, Wu et al7 constructed the independent image pairs via random projection splitting. Yuan et al8 proposed a Noise2Noise based denoising method named ‘Half2Half’. In their training pair construction, binomial selection was applied to the projection data, splitting it into two pseudo half-dose scans. For the aforementioned methods, the dose allocation is fixed, and both of them split dose evenly to the paired images. In this paper, we propose a method to simulate arbitrary dose levels and independent noise from an existing CT scan. Paired images can be generated at any desired dose reduction level from a single CT scan, which provides more diversity in training data given the same normal-dose CT scans. 2.METHODSFor simplicity, the normal-dose projection domain measurements (raw data) PND can be modeled as the sum of a Poisson and Gaussian random variable8: where λ is the mean counts and σe is the standard deviation of electronic noise. If we denote the photon counts from the source as I0 and the object pathlength as l, the mean counts can be formulated as Hence, the expectation and variance of PND can be given by equations (3) and (4): For a specified dose level d (0 < d < 1), according to equation (1) the measured counts For the low-dose simulation process, we want to emulate the behavior of Table 1.Conventional noise insertion
The conventional noise insertion adds additional quantum noise Q and electronic noise E, which are scaled by a factor depending on the dose level. In practice, when generating Q, we use PND as a surrogate for variance λ since the true λ is unknown from a single realization. The result is defined as: which can be viewed as a synthetic projection acquired at dose level d as it shares the identical probability distribution (noise properties) as As is shown in the Appendix, we prove that which means that they are uncorrelated. On the assumption that We thus have 3.NUMERICAL SIMULATIONTo validate the proposed method, numerical simulation was carried out in MATLAB with the built-in Shepp-Logan phantom and Radon projection method for a monoenergetic source. In the simulation, the x-ray source flux was set to 5e4 photons per ray and 𝜎e = 5 counts, which is referred to as normal dose for the remainder of the paper. The projections were reconstructed with filtered backprojection (FBP), and the images are illustrated in Fig. 1. We include the ideal image with noiseless projections [using equation (2)] in Fig. 1(a), which is the ground-truth image. Fig. 1(b) is the reconstructed image under normal dose [using equation (1)]. By subtracting Fig. 1(a) from Fig. 1(b), we obtain the noise image as shown in Fig. 1(c). Fig. 1.Reconstructed image and noise image of Shepp-Logan phantom. (a) ideal image, noiseless ground-truth. (b) reconstructed image using normal dose, flux of source I0= 5e4 photons, 𝜎e = 5 counts. (c) noise image, subtracting (a) from (b). The display window for (a) and (b) is [0, 0.4] cm-1. The display window for (c) is [-0.02, 0.02] cm-1. ![]() From the normal dose projections, it is possible to synthesize projections at a specific dose level following the conventional noise insertion steps in Table 1. We can also simulate a real low-dose scan acquired at the same dose level. Fig. 2 displays the results of both reconstructed and noise images for normal dose, real low-dose Fig. 2.Noise insertion results. (a), (b) and (c) are reconstructed and noise images for normal dose, real low-dose, and synthetic low-dose, respectively. Noise images are determined by subtracting the ground-truth image in Fig. 1(a). The display window for the first row (a-c) is [0, 0.4] cm-1 and is [-0.02, 0.02] cm-1 for the second row (d-f). ![]() Equations (5) and (6) guide us in the generation of independent noise images from normal dose images. Fig. 3 shows a realization of the Fig. 3.Independent noise images at 30% dose level. Image ![]() For other dose levels, the processing can be easily repeated, which forms the curves in Fig. 4. The horizontal axis denotes the relative dose levels from 5% to 95% of normal dose. The vertical axis is the noise in image domain. The blue squares are the real low-dose images Fig. 4.Noise at different dose levels. The average correlation between images ![]() In general, the independent noise image We demonstrate this assertion with a simple test. For the 50% dose level, we plot the difference in noise between the In Fig. 6, the correlations between We also list the correlation coefficients under different electronic noise (σe) levels in Table 2. The correlation between synthetic image Table 2.Correlation coefficients
4.DISCUSSION AND CONCLUSIONSIn this paper, a simulation tool was demonstrated for simultaneously synthesizing low-dose images at arbitrary dose level and independent noisy images. The method extends the conventional noise insertion procedure and creates a byproduct image with independent noise along with the low-dose image at a specific dose level. Correlation between the synthetic and independent noise images was investigated both analytically and numerically, which verified that they are uncorrelated. Thus, they are independent under the assumption that they form a bivariate normal distribution. For now, we only carried out preliminary validation with a simple simulation in MATLAB. Future work will extend these concepts to a more accurate forward projection model with polychromatic spectrum and non-ideal detector response (energy integrating or photon counting). Also, we are using a linear FBP reconstruction algorithm so that projection domain analysis can be transferred directly to the image domain (although this does include a non-linear log step). Iterative reconstruction methods may violate our linearity assumptions in the image domain, even if the projection domain noise properties hold. Another challenge might be the accuracy of our noise models in severely attenuated areas with photon starvation, such as behind metal. Lastly, we plan to demonstrate the utility of our independent noise simulation on CT denoising networks by fully leveraging the Noise2Noise principle. Our belief is that training with a wide range of simulated dose levels paired with independent noise will outperform other training methods like Half2Half or pairing simulated low dose images with normal dose images. REFERENCESChen H, Zhang Y, Kalra MK, et al,
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