Exploiting X-ray measurements acquired at multiple energies, spectral CT has the ability to recover the concentration maps of the constituents of the tissues in a quantitative manner. While the principle of dual-energy CT has been known for more than 30 years, recent developments in energy selective photon counting detectors have boosted the research in this area.

* FIG. Principle of spectral CT acquisition *

We consider an object made of $M$ materials, each with a density $\boldsymbol{\rho}_m$, which is imaged by a photon counting detector having $I$ energy bins. At each projection angle $\theta$, the photon counting detector provides $I$ projection images $\mathbf{s}_{i,\theta}$. Repeating the acquisition for $\Theta$ projections angles, we have

.$\mathbf{s} = \mathcal{G}(\boldsymbol{\rho})\quad$ with $\quad \begin{cases} \mathbf{s} = \{\mathbf{s}_{i,\theta}\}_{1 \le i \le I,\; 1 \le \theta \le \Theta}\\ \boldsymbol{\rho} = \{\boldsymbol{\rho}_m\}_{1 \le m \le M} \end{cases}$

where $\mathcal{G}$ denotes the (nonlinear) forward model that takes into account the acquisition geometry and detector response function.

The spectral CT problem consists in recovering the material density $\boldsymbol{\rho}$ from the energy-resolved projection images $\mathbf{s}$.

We have introduced material-dependent regularization of projection images and proposed to solve the material decomposition problem using a Gauss-Newton algorithm that can benefit from fast linear solvers. Our method has been compared to the reference maximum likelihood Nelder-Mead algorithm and a 70x faster decomposition was obtained.

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*FIG. Our iterative decomposition algorithm in action in a thorax phantom. Top row: ground-truth material images. Bottom row: decomposed material images.*

**Citation: ** N. Ducros, J.F.P.J. Abascal, B. Sixou, S. Rit, and F. Peyrin, Regularization of Nonlinear Decomposition of Spectral X-ray Projection Images,
*Medical Physics*, 44(9), e174-e187, 2017.