## 3 Methodology
### Preliminary  
Training a denoising neural network in a classical supervised manner can be performed by solving the following:  
\[\underset{\theta}{\text{argmin}}E\left[\parallel f_{\theta}(x)-y\parallel_{2}^{ 2}\right], \tag{1}\]  
where \(f_{\theta}\) is a denoising neural network parameterized by \(\theta\), \(x\) is the noisy image used as an input to the neural network, and \(y\) is the clean image with the same information as the noisy image \(x\). During training, \(f_{\theta}\) is optimized to map a noisy image to a clean image and is applied in self-supervised learning-based denoising methods in the same manner. As mentioned, self-supervised learning-based denoising methods can train a neural network without paired images. These methods are commonly based on the blind spot strategy, which has exhibited promising results in the natural image domain. However, it is difficult to apply the blind spot strategy to LDCT denoising due to the assumption that the noise is pixel-wise independent. In the rest of this subsection, we describe why the noise in the LDCT reconstructed image is not pixel-wise independent. Filtered back projection (FBP) is a widely used reconstruction algorithm in which a point of a 3D reconstructed image is reconstructed by integrating the points of filtered projections along the ray. For the convenience of explanation, in 2D parallel beam reconstruction, let us assume an object distribution function \(f(x,y)\). All of the Cartesian space we used for explanation in this subsection is illustrated in Figure 2.  
In the assumed tomographic system, a projection \(p\) can be achieved by a Radon transformation [30]. The formulation is below:  
\[p(\theta,\ t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\delta(ycos \theta-xsin\theta-t)dxdy, \tag{2}\]  
where \(\delta(\cdot)\) is [PERSON]'s delta function, \(\theta\) is the projection angle, and \(t\) is the distance between the detector and the center of the rotation. The projection p is the ray sum along the line  
Figure 2: Tomographic system assumed for descriptive purposes. **(Left)** Illustration for the Radon transform theorem. **(Right)** Illustration for the Fourier slice theorem. through the object. The noise in the reconstructed LDCT images is caused by the photon noise in the projection images [2]. Therefore, the formula considering the noise distribution function n can be formulated as follows:  
\[\hat{p}(\theta,\ t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\delta( ycos\theta-xsin\theta-t)dxdy+n(\theta,t), \tag{3}\]  
where \(\hat{p}\) is a parallel projection with photon noise and \(n\) is a random photon noise distribution function observed in the projection images. The projection with noise \(\hat{p}\) using the Fourier slice theorem is also defined as follows:  
\[F_{1}\hat{p}=F_{2}\hat{f}(-\rho sin\theta,\rho cos\theta), \tag{4}\]  
where \(F_{n}\) represents the \(n\)-dimensional Fourier transform and \(\hat{f}\) denotes the object distribution function reconstructed from the projections with the noise. Equation (4) describes a 1D Fourier transform of parallel projection equal to the radial line parallel to the projection in the 2D Fourier transform of the object. The object distribution function \(\hat{f}\) in the Fourier domain to obtain the reconstructed image from the projection \(\hat{p}\) is defined as follows:  
\[\hat{f}(x,y)=F_{2}{}^{-1}F_{2}\hat{f}(x,y)=\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}F_{2}\hat{f}(\rho_{x},\rho_{y})e^{j2\pi(x\rho_{x}+y\rho_{y})} d\rho_{x}d\rho_{y}. \tag{5}\]  
Equation (5) is achieved by the Fourier inversion theorem. By inserting Equation (4) into Equation (5) and changing the variables \(d\rho_{x}\) and \(d\rho_{y}\) to the variable \(|\rho|d\rho d\theta\), Equation (5) can be rewritten as follows (the theoretical details are proven in [31]):  
\[\hat{f}(x,y)=\int_{0}^{2\pi}(\hat{p}*g_{\infty})(\theta,ycos\theta-xsin\theta )d\theta, \tag{6}\]  
where \(g_{\infty}\) is the ramp filter.