2.1

Dual-Energy Cone-Beam Projection Model

The polyenergetic forward model for dual-energy cone-beam projections is written as:

where l/h means physical quantities under low/high incident x-ray spectrum, p represents transmission photons, ω represents spectrum, μ stands for LAC. Due to low scanning dose and large volume coverage, CBCT projections suffer high noise level (n_l/h) and severe scatter contamination (s_l/h). Consequently, the actual measurements are modeled as:

The superscript n, s represent noise and scatter, respectively. After log normalization, the line integrals with and with scatter and noise are:

2.2

Projection-Domain Material Decomposition

According to the dual-energy CT theory [17], the LAC could be decomposed as:

ϕ_1/2 is energy-dependent basis function, which could be interpreted as mass attenuation coefficient of two basis materials, then ρ_1/2 is the density of the basis material correspondingly. Plug Eq. (5) into Eq. (3), we obtain:

where q₁ = ∫ρ₁(x)dl, q₂ = ∫ρ₂(x)dl. Given incident spectrum ω_l/h and two basis functions ϕ_1/2, the material decomposition is an inverse problem which recovers q_1/2 from q_l/h. Since no explicit expression for this inversion, A commonly used analytical decomposition method is polynomial approximation [18]:

where α_i,j, β_i,j are pre-calibrated coefficients using calibration phantom. Although this method could achieve high accuracy via increasing the polynomial order N, previous research revealed that this decomposition is sensitive to projection noise [19].

2.3

Residual W-shape Network

Figure 1 presents the ResWnet Architecture, which consists of scatter correction module, material decomposition module and decomposition denoising module. For simplicity, projections represent log-normalized projections hereafter.

Figure 1.

ResWnet architecture. Gray, blue and yellow boxes represent scatter correction, material decomposition, decomposition denoising module, respectively.

Scatter Correction Module

Scatter correction module aims to remove the noise and scatter signal in projections. In this work, scatter correction module 𝒮 employs a Residual U-shape network (ResUnet) [20]. The paired , are concatenated as two-channel input, which then passes six encoder blocks and five decoder blocks successively. To avoid gradient vanishing and exploding [21] in very deep networks, each encoder and decoder block adopts residual architecture [22] with a shortcut connection from input to output. Considering that and q_l/h share the same structure with only numerical difference, a global shortcut connection is applied to directly add raw projections to the output projections. Different from the conventional Unet, the last two encoder blocks in 𝒮 keep the same number of features to reduce model parameters without compromising the performance.

Material Decomposition Module(: q_l/h ↦ q_1/2)

Material decomposition module aims to recover the material composition projections q_1/2 from the dual-energy projections q_l/h. As discussed in Sec.II.B, this inversion could be approximated by polynomials functions, and the parameters α_i,j, β_i,j could be determined by pre-calibration. Thus, there is no need to train a sub network for decomposition. In proposed network, acts as depicted in Eqs. (7). N was set to 4 and there is no trainable parameter in this module.

Decomposition Denoising Module

Since the module 𝒮 cannot reduce the noise level to zero and the polynomial-based decomposition is sensitive to noise, the output of would be noisy. Decomposition denoising module 𝒟 was used for further suppress the noise. Same as the scatter correction module, 𝒟 also employs the ResUnet architecture

2.4

Data Generation and Network Training

To obtain the training, validation and testing datasets, we first download 22 head CT scans from public dataset in https://wiki.cancerimagmgarchive.net/pages/viewpage.action?pageid=39879146. These CT images were decomposed into four tissues: fat, muscle, 200mg/cc bone and 800mg/cc bone using multi thresholds segmentation. We performed polyenergetic forward projection of each volume using two spectrums generated by Spektr [23]. These are the label projections q_l/h for scatter correction. The input projections were then obtained via adding scatter and noise signals generated by Monte Carlo simulation. Muscle and 800mg/cc bone were selected as basis materials in this work, and fat and 200mg/cc bone were decomposed onto these two basis. The label projections of material decomposition q_1/2 were generated by forward projecting the muscle and 800mg/cc composition images. Each head produced 100 projections, and the projection angles equally distributed between 0 and 2π.

During the model training, network 𝒮 were firstly optimized by:

The second term was added because we hope 𝒮 could not only reduce the scatter signal, but also produce a projection noise distribution that minimizes the decomposition noise. In this work, w₁, w₂ were set to 0.9 and 0.1, respectively. Using the trained network 𝒮*, the network 𝒟 was finally trained by:

Parameters of both 𝒟 and 𝒮 were optimized by Adam optimizer with an initial learning rate of 0.0004 which decay 8% after each epoch. Batch size was set to 4 and training stopped after 100 epochs.