1.

INTRODUCTION

Low-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 domain^1,2 or during the reconstruction process^3,4 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.

METHODS

For simplicity, the normal-dose projection domain measurements (raw data) P_ND can be modeled as the sum of a Poisson and Gaussian random variable⁸:

where λ is the mean counts and σ_e 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 P_ND can be given by equations (3) and (4):

For a specified dose level d (0 < d < 1), according to equation (1) the measured counts + Gaussian. Similarly, the expectation and variance of are and , respectively.

For the low-dose simulation process, we want to emulate the behavior of at dose level d from normal-dose scan P_ND. For conventional low-dose simulation, this is a well-known process to insert noise in the projection data. Detailed steps are listed in Table 1.

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 P_ND 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 . While this enables a simulated low-dose image, we still need a paired zero-mean, independent noise realization for Noise2Noise training. To this end, we define as:

As is shown in the Appendix, we prove that

which means that they are uncorrelated. On the assumption that form a bivariate normal distribution, they are independent if they are uncorrelated. This is a reasonable assumption for any modest number of photon counts, where the Poisson distribution is approximately Gaussian, and the other noise (electronic, added noise Q, E) are all Gaussian. Importantly, both and use the same noise realizations of Q and E, but they are scaled inversely and subtracted in as compared to , which leads to the uncorrelated property.

We thus have , which has zero-mean, independent noise of In the Noise2Noise conditions, independent noisy image pairs are required. The proposed method can generate such paired data where simulates data acquired at arbitrary dose level d. Note that does not correspond to any specific dose level, but rather is designed to satisfy the Noise2Noise conditions. This enables a diversity of dose levels, which may be beneficial to training CT denoising networks.

3.

NUMERICAL SIMULATION

To 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 𝜎_e = 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, 𝜎_e = 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 , the reconstructed image from , and synthetic low-dose , the reconstructed image from respectively, for dose level d=0.3, or 30% of the normal dose. Standard deviations of the phantom region are labeled on the noise images, where we find good correspondence between the synthetic low-dose image and the real low-dose image, as well as the increased noise in the low-dose images compared with the normal 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 pair at 30% dose level. Correlation between the noise images of and was calculated across all pixels in the phantom and across 10 realizations and was found to be near zero, which supports the independence we desire. As expected when d < 0.5, the noise magnitude in is lower than that in since more noise is added into than is added to according to the inverse scaling of the added quantum and electronic noise.

Fig. 3.

Independent noise images at 30% dose level. Image is at simulated low dose d =0.3, and has noise that is independent of although it has a different noise magnitude. The display window is [0, 0.4] cm^-1 for the first row and [-0.02, 0.02] cm^-1 for the second row.

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 at corresponding dose levels. The red dots are noise levels of synthetic low-dose images from the normal dose image R_ND. Again, they fit the blue squares very well at all dose levels. The orange dots are noise levels of images with independent noise from the synthetic low-dose images At lower dose, the independent image tends to have lower noise level, showing different noise behaviors to real or synthetic low-dose images.

Fig. 4.

Noise at different dose levels. The average correlation between images and across all dose levels is

In general, the independent noise image does not correspond to a specific dose level, even though the noise levels appear approximately symmetric to that of the low-dose images about d = 0.5. For example, the noise level in at 80% dose level is generally not equal to that in at 20%. However, for the special case of no electronic noise (σ_e = 0), it can be shown that this is the case, and the image represents a dose level of 1 – d.

We demonstrate this assertion with a simple test. For the 50% dose level, we plot the difference in noise between the and images (Fig. 5). When there is no electronic noise (σ_e = 0), the noise levels are indeed identical, but for σ_e > 0, more electronic noise is added to the synthetic 50% dose image than the independent noise image . Therefore, in general we are not splitting dose or creating another low-dose image. Instead, we have created an additional image with independent noise, which satisfies the Noise2Noise conditions.

Fig. 5.

Noise level difference between and at 50% dose level

In Fig. 6, the correlations between and are plotted in blue and red. As expected, the correlations between at different dose levels are close to 0. On the contrary, the correlations between increases with higher dose (red curve) since more of the noise in the synthetic low-dose image comes from the original normal-dose image. At lower dose, the increased amount of inserted noise leads to lower correlations with the original image.

Fig. 6.

Correlation at different relative dose levels

We also list the correlation coefficients under different electronic noise (σ_e) levels in Table 2. The correlation between synthetic image and independent noise image is generally near zero, which agrees with the theoretical analysis in equation (7).

Table 2.

Correlation coefficients

4.

DISCUSSION AND CONCLUSIONS

In 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.