2.1.1

Material decomposition

Consider a multi-bin system with B > 2 energy bins and, for simplicity, a 2-dimensional image space. The material decomposition starts with the ansatz that the X-ray linear attenuation coefficient μ(x, y; E) can be approximated by a linear combination of M basis materials

where a_m and τ_m(E) are the basis coefficients and basis functions, respectively. It is usually resolved in the sinogram domain and thus the target variables are the material line integrals

where R denotes the Radon transform operator. The expected number of photons in energy bin j follows the polychromatic Beer-Lambert law

This is our forward model. Finally, the measured data is the vector y := [y₁,…, y_B] where for each j we assume that

Hence, the (non-linear) inverse problem is to map the photon counts y to the material line integrals A := [A₁, …, A_M]. The most common approach to this problem is maximum likelihood.^10–12 Setting up the objective as the log likelihood and simplifying yields

This is subsequently solved using some iterative algorithm, e.g., the logarithmic barrier method.¹³

2.1.2

Data generation

After generating numerical basis material phantoms (soft tissue, bone and iodine) by segmenting CT images from the KiTS19 dataset,⁹ photon-counting imaging was simulated using the fanbeam function in Matlab and a spectral response model of a photon-counting silicon detector¹⁴ with 0.5 × 0.5 mm² pixels. The simulation was performed for 120 kVp and 200 mAs with 1579 detector pixels and 1600 view angles. After simulating Poisson noise, the maximum likelihood method was used for material decomposition of the simulated sinograms into bone and soft tissue basis sinograms, which were then reconstructed on a 583 × 583 pixel grid. To avoid streak artifacts due to photon starvation, a logarithmic barrier function was used to penalize large negative basis projection values. To simulate the effect of threshold variations, the simulation was performed with a random threshold shift (σ = 0.5 keV) applied independently to each of the eight thresholds of each detector pixel, and two material decompositions were performed: one with the actual bin thresholds that were used in the simulations, including the random shift, and one with the nominal bin thresholds, where the latter configuration yields images with ring artifacts.

2.2.1

Problem statement

We propose an image processing technique for ring artifact correction based on deep neural networks. More formally, let x ∈ ℝ^M×H×W denote the streak corrupted basis sinograms and y ∈ ℝ^M×H×W their streak artifact free counterpart, where M, H and W are the number of basis materials, view angles and detector pixels, respectively. Then our objective is to learn

We let f_θ be a neural network and learn the map (6) by learning parameters θ.

2.2.2

Network architecture

UNet is an widely utilized architecture for a range of different tasks in biomedical imaging. The defining feature is the encoder-decoder structure. We use a version of the original UNet¹⁵ shown in Fig. 1.

Figure 1.

Illustration of UNet.¹⁵

2.2.3

Loss functions

Mean square error (MSE) is perhaps the most commonly used loss function for applications of deep learning in biomedical imaging

Using MSE loss will encourage the output to match the target pixel-by-pixel. This low-level per-pixel comparison is well known to produce output that is overly-smooth and lacking fine details that affect the perceptual quality.^{16, 17}For several image transformation tasks, it has proved useful to instead employ a perceptual loss function which, instead of comparing pixel-by-pixel, compares high-level feature representations between the output and target. These feature representations are extracted from a pretrained convolutional neural network. We follow¹⁶ and use VGG16¹⁸ pretrained on ImageNet¹⁹ as feature extractor, or loss network. Let ϕ_j denote the j-th layer of VGG16, then our perceptual loss is defined as

where C_j is the number of channels in layer j. We will set j = 9 which corresponds to “relu2_2” in.¹⁶