Material-separating regularizer for multi-energy X-ray tomography

TL;DR

This work tackles dual-energy X-ray tomography for two materials by introducing a material-separating regularizer that couples nonnegativity with an inner-product penalty $\mathcal{S}(\mathbf{g})=2\langle \mathbf{g}^{(1)}, \mathbf{g}^{(2)}\rangle$. The reconstruction is posed as a constrained quadratic program and solved with a tailored preconditioned interior-point method, including a specialized block-diagonal preconditioner to keep spectral bounds stable across iterations. Numerical tests on 2D phantoms with sparse projections and simulated noise show that the proposed IP method yields fewer misclassified pixels in material decomposition than Joint Total Variation, while overall image-quality metrics are comparable. The approach generalizes to more materials and higher dimensions, offering a practical route for reliable material-specific tomography in low-dose dual-energy imaging. The combination of discretization-invariant continuum theory and efficient optimization makes this method appealing for practical material-discrimination tasks in nondestructive testing and cultural heritage applications.

Abstract

Dual-energy X-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two X-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials. A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional. Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, Joint Total Variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging.

Figures (11)

• Figure 1: Alternative imaging protocols. (a) Two projection images are recorded from each source location: one with low (L) and another with high (H) energy. In this case we have $A^H=A^L$. (b) Only one projection image is recorded at every source location, alternating between low and high energies. In this case we have $A^H\not=A^L$.
• Figure 2: (a) Magnitude of the elements of $A^TA$ for $N=32$. (b) Magnitude of the mean element along a specific diagonal against the distance from the main diagonal.
• Figure 3: Original phantoms. The four different phantoms which we used in our simulations are shown here in the resolution we actually used. First row shows material one (PVC in these simulations) and second row shows material 2 (iodine in these simulations). These images show the perfect separation of the materials into their own images, so they serve us as a ground truth, where the results of the other methods can be compared.
• Figure 4: Reconstruction results with JTV and IP regularizations for HY and Bone phantoms. The first row represents material 1 and the second row represents material 2. First column shows JTV reconstructions, second column shows IP-method reconstructions and third column is the ground truth.
• Figure 5: Reconstruction results with JTV and IP regularizations for Egypt phantom. The first row represents material 1 and the second row represents material 2. First column shows JTV reconstructions, second column shows IP reconstructions and third column is the ground truth.

Theorems & Definitions (14)

• Theorem 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Remark 1
• Remark 2
• Lemma 3.3
• proof
• Lemma 3.4