JSover: Joint Spectrum Estimation and Multi-Material Decomposition from Single-Energy CT Projections

TL;DR

JSover tackles the ill-posed problem of performing multi-material decomposition (MMD) from single-energy CT by formulating a one-step optimization that jointly estimates the X-ray spectrum $η(E)$ and material volume fractions $α(x)$ from SECT projections via a physics-informed forward model $ρ(r) = -\ln\left[∫_E η(E) e^{-\,∑_{i=1}^M μ_i(E) ∫_r α_i(x) dx} dE\right]$. It introduces two key representations: an unconstrained spectrum model using SoftMax over a library of spectra and an implicit neural representation (INR) to map 3D coordinates to material fractions, enabling differentiable end-to-end optimization. A differentiable forward model $ ilde{H}$ and a data-consistency loss drive joint estimation, with an INR prior imposing a low-frequency bias to regularize the ill-posed problem. Empirical results on simulated and real data show JSover outperforms state-of-the-art SEMMD methods in accuracy and computational efficiency, and remains robust under undersampling thanks to INR, representing the first unsupervised DL approach for both SEMMD and spectrum estimation with clear clinical potential.

Abstract

Multi-material decomposition (MMD) enables quantitative reconstruction of tissue compositions in the human body, supporting a wide range of clinical applications. However, traditional MMD typically requires spectral CT scanners and pre-measured X-ray energy spectra, significantly limiting clinical applicability. To this end, various methods have been developed to perform MMD using conventional (i.e., single-energy, SE) CT systems, commonly referred to as SEMMD. Despite promising progress, most SEMMD methods follow a two-step image decomposition pipeline, which first reconstructs monochromatic CT images using algorithms such as FBP, and then performs decomposition on these images. The initial reconstruction step, however, neglects the energy-dependent attenuation of human tissues, introducing severe nonlinear beam hardening artifacts and noise into the subsequent decomposition. This paper proposes JSover, a fundamentally reformulated one-step SEMMD framework that jointly reconstructs multi-material compositions and estimates the energy spectrum directly from SECT projections. By explicitly incorporating physics-informed spectral priors into the SEMMD process, JSover accurately simulates a virtual spectral CT system from SE acquisitions, thereby improving the reliability and accuracy of decomposition. Furthermore, we introduce implicit neural representation (INR) as an unsupervised deep learning solver for representing the underlying material maps. The inductive bias of INR toward continuous image patterns constrains the solution space and further enhances estimation quality. Extensive experiments on both simulated and real CT datasets show that JSover outperforms state-of-the-art SEMMD methods in accuracy and computational efficiency.

Figures (13)

• Figure 1: Two spectrum libraries with 10 different thickness Al tube filters at 120 kVP and 80 kVP generated by the SPEKTR toolkit punnoose2016technical.
• Figure 2: Optimization pipeline of the proposed JSover. Given SECT projection $\rho(\mathbf{r}), \forall \mathbf{r} \in \mathbf{\Pi}$, an MLP network $\mathcal{F}_\Phi$ maps multiple coordinates $\mathbf{x}$ along its X-ray $\mathbf{r}$ to the corresponding volume fractions $\boldsymbol{\alpha}(\mathbf{x}) = \mathcal{F}_{\Phi}(\mathbf{x})$. Concurrently, the spectrum $\eta$ is generated via a spectra model $\mathcal{S}$ (Eq. \ref{['eq:spectra_model']}), controlled by learnable parameters $\boldsymbol{\gamma}$. The spectrum $\eta$ and the volume fractions $\boldsymbol{\alpha}(\mathbf{x}), \forall \mathbf{x} \in \mathbf{r}$ are then used to generate predicted SECT measurements $\hat{\rho}(\mathbf{r})$ via a discrete SEMMD model $\tilde{\mathcal{H}}$ (Eq. \ref{['eq:semmd']}). Finally, the MLP network $\mathcal{F}_\Phi$ and spectrum parameters $\boldsymbol{\gamma}$ are optimized by minimizing a data consistency loss $\mathcal{L}_\text{DC}$ (Eq. \ref{['eq:loss-dc']}), calculating the discrepancy between the predicted $\hat{\rho}(\mathbf{r})$ and the acquired SECT projections ${\rho}(\mathbf{r})$.
• Figure 3: Two simulated digital XCAT phantoms. Here, the numbers indicate different regions of interest (ROI), listed in Table \ref{['tab:composition']}.
• Figure 4: The experimental setup for the real-world solution phantoms: A) Four solution phantoms (10 ml), B) Material compositions of the four solutions, C) The cone-beam CT scanner in our lab, and D) The reconstructed SECT image of 128$\times$128 size using the FBP fbp algorithm.
• Figure 5: Qualitative comparison of SEMMD reconstructions by TMA xue2020image, MSC xue2021multi, JSover-TV, and JSover-INR on simulated XCAT phantom A.