Joint Material Decomposition and Scatter Estimation for Spectral CT

Abstract

Accurate scatter correction is essential to obtain highquality reconstructions in computed tomography. While many correction strategies for this longstanding issue have been developed, additional efforts may be required for spectral CT imaging - which is particularly sensitive to unmodeled biases. In this work we explore a joint estimation approach within a one-step model-based material decomposition framework to simultaneously estimate material densities and scatter profiles in spectral CT. The method is applied to simulated phantom data obtained using a parametric additive scatter mode, and compared to the unmodeled scatter scenario. In these preliminary experiments, We find that this joint estimation approach has the potential to significantly reduce artifacts associated with unmodeled scatter and to improve material density estimates.

I. Introduction

XRAY scatter can pose significant challenges to image quality in computed tomography (CT) and, particularly, conebeam CT (CBCT), as the extensive amount of scatter generated by the large cone angle leads to loss of contrast, reduced contrast-to-noise ratio and the emergence of cupping and streak artifacts. The problem of scattered radiation has been addressed in several ways: for a comprehensive review of scatter correction methods refer to [1][2]. Hardware techniques for scatter rejection include anti-scatter grids and collimation approaches; whereas software-based scatter correction approaches typically involve the subtraction of a scatter estimate from the projection data - often achieved by experimental measurement or numerical estimation. Among the latter, common strategies utilize scatter-kernel superposition (SKS) and Monte Carlo (MC) simulations. SKS methods [3][4] involve the deconvolution of the projection data with kernels derived from scatter point-spread functions, obtained from MC simulations [5][6] or measurements [7]. MC simulations can provide very accurate scatter images, as they include complex stochastic processes avoiding the simplifications typical of other numerical methods. While SKS methods are known to provide good computational speed, MC simulations are in general more computationally demanding [8], although several acceleration techniques have been developed [9][10]. Both methods are subject to uncertainties with regard to modeling of the object, the imaging system, and/or other approximations (e.g. limited number of MC events, approximate kernel estimates, etc.).

While the problem of scattered radiation is well known with many proposed corrections methods, the challenges remain in the field of spectral CT [11], where basis material and density estimation can be more sensitive to unmodeled biases. In this work, we explore a joint estimation approach in the framework of one-step model-based material decomposition (MBMD), aiming to simultaneously estimate multiple quantities.

In the context of spectral CT, the joint estimation method was proposed to directly estimate the basis material densities from spectral data [12][13], avoiding the complications that emerge from the separation between the reconstruction and the decomposition steps. We want to extend this approach to model and estimate an additional unknown component represented by the scattered radiation. Specifically, we focus on determining both the densities of the material basis and the scatter profile within the projection data simultaneously. This is achieved by incorporating the latter into the forward model for the spectral measurements and jointly estimating all parameters. Such simultaneous estimation is similar, in spirit, to related work by Ha et al.[14] where one-step material decomposition is conducted in conjunction with estimation of unknown spectral calibrations. In following sections, we detail the model used by our approach and the estimation strategy including different regularization schemes. The estimator is applied to simulated phantom data with different additive scatter profiles. Material density estimates for estimated and un-estimated scatter conditions are compared, illustrating the potential of the joint estimation technique to improve spectral CT imaging.

II. Methods

A. Forward Model

For the energy-dependent spectral measurements of an object made of $N_m$ material basis we assume the following forward model:

$${\bar{y}}_i={\int }_E{S}_i\left(E\right)\text{exp}\left[-\sum _{m=1}^{N_m}{\mu }^m\left(E\right)\sum _{j=1}^{N_v}{A}_{ij}{\rho }_j^m\right]+s_i$$ (1)

where ${\rho }_j^m$ is the density of the $m$-th material in the $j$-th voxel, with $1\le j\le N_v$, being $N_v$ the total number of voxels; $A_{ij}$ represents the path length of the $i$-th projection through the $j$-th voxel; ${\mu }^m\left(E\right)$ denotes the energy-dependent mass attenuation coefficients and $S_i\left(E\right)$ indicates the energy-dependent system response, a general function that includes the model of the source spectra, filters, and the detector response. Finally, $s_i$ is the term that accounts for scatter.

To facilitate digital estimation, it is convenient to introduce the following discretization in energy,

$${\bar{y}}_i=\sum _{k=1}^{N_E}{S}_i\left(E_k\right)\Delta E\text{exp}\left[-\sum _{m=1}^{N_m}{\mu }^m\left(E_k\right)\sum _{j=1}^{N_v}{A}_{ij}{\rho }_j^m\right]+s_i$$ (2)

where $N_E$ is the number of bins of the energy spectra sampling and $\Delta E$ the width of the energy bins: $E_k=k\Delta E,k=1,\dots ,N_E$

The forward model (2) can be represented compactly using matrix notation:

$$\stackrel{-}{\mathbf{y}}=\mathbf{S}\text{exp}[-\mathbf{Q}\mathbf{A}\mathit{\rho }]+\mathbf{s}$$ (3)

where $\stackrel{-}{\mathbf{y}}$ is the $N$-dimensional vectorized stack of all projections and all the spectral channels, $\rho$ is the $N_m{N}_v$ vectorized stack of all the density images with $N_v$ voxels in $N_m$ material bases, $\mathbf{A}$ the $N{N}_m\times N_m{N}_v$ system matrix and $\mathbf{Q}$ is the $N{N}_E\times N{N}_m$ energy-dependent mass attenuation coefficient matrix. Here, $\mathbf{S}$ represents the $N\times N{N}_E$ matrix with spectral sensitivities. Finally, $\mathbf{s}$ is the vectorized stack of the scatter contribution for each measurement. In this work, two models have been considered for s, illustrated in Section II-C.

B. Model Based Material Decomposition

Assuming the spectral measurements follow a multivariate Gaussian distribution with mean given by Equation (3) and covariance $\Sigma$

$$y_i\sim N\left({\bar{y}}_i,\mathbf{\Sigma }\right)$$

the corresponding penalized weighted least-square objective function can be written as:

$$\psi =\frac{1}{2}{\left(\mathbf{y}-\mathbf{S}{e}^{-\mathbf{Q}\mathbf{A}\mathit{\rho }}-\mathbf{s}\right)}^T\mathbf{W}\left(\mathbf{y}-\mathbf{S}{e}^{-\mathbf{Q}\mathbf{A}\mathit{\rho }}-\mathbf{s}\right)+R\left(\mathit{\rho }\right)$$ (4)

Here, $\mathbf{W}={\mathbf{\Sigma }}^{-1}$. We further presume uncorrelated measurement noise, so the covariance matrix is diagonal, and we approximate the variance with the measurements. Hence $\mathbf{W}=\mathrm{D}\{1/\mathbf{y}\}$, with $\mathrm{D}\{\cdot \}$ representing a diagonal matrix. The term $R\left(\mathit{\rho }\right)$ is a regularizer controlling the smoothness of the estimated images, independently for each material basis. In this work, we utilize both a penalty function based on the L1 norm of the difference in density of neighboring pixels, or:

$$R\left(\mathit{\rho }\right)=\frac{1}{4}\sum _{m=1}^{M_m}{\beta }_m\sum _{j=1}^{N_v}\sum _{j^{\prime }\in {𝒩}_j}\left|{\rho }_j^m-{\rho }_{j^{\prime }}^m\right|$$ (5)

and a penalty function based on L2 norm:

$$R\left(\mathit{\rho }\right)=\frac{1}{4}\sum _{m=1}^{M_m}{\beta }_m\sum _{j=1}^{N_v}\sum _{j^{\prime }\in {𝒩}_j}{\left({\rho }_j^m-{\rho }_{j^{\prime }}^m\right)}^2$$ (6)

where ${\beta }_m$ is the regularization strength for the $m$-th material basis and ${𝒩}_j$ identifies the set of four pixels in the neighborhood of pixel $j$.

The minimization of the objective function (4) yields the material decomposition estimates $\hat{\rho }$ and it is achieved iteratively with a stochastic gradient descent algorithm.

C. Simulation Study

The 2D digital phantom presented in Figure 1 was used to test the joint estimation of scatter and material densities with the model illustrated in Section II-A. This phantom was created with $0.5\times 0.5{\mathrm{m}\mathrm{m}}^2$ voxels. Mass attenuation coefficients shown in Figure 2a were obtained from Spektr. We simulated an imaging system including a 1024 pixel triple-layer indirect flat-panel detector (FPD) with a 0.3 mm pitch, assuming a source-to-axis distance of 80 cm and a source-to-detector distance of 120 mm. CsI scintillation layers were 0.20, 0.55 and 0.55 mm thick, for the top, middle, and bottom layers, respectively. A 120 kVp X-ray spectrum was generated with Spektr, including a 10 mm-thick aluminum filter. The bare-beam intensity is set to 4⋅10⁵ photons/pixel. The spectral response of the detector is shown in Figure 2b. The imaging geometry used 600 projections with a 0.6 degree sampling, and Poisson noise was added to the data.

Fig. 1.

Digital phantom with material bases: water, Calcium and Gadolinium.

Fig. 2.

(a) Mass attenuation coefficients for water, Calcium and Gadolinium. (b) Spectral response of the simulated triple-layer FPD imaging system.

Two scenarios were considered for scatter simulation:

1. Idealized Quadratic Case, where the profile $s$ is given by quadratic functions of the lateral detector position $u$:

   $$s_l\left(u\right)=-\frac{2}{r_l{N}_u}{u}^2+\frac{2}{r_l}u+m_l-\frac{N_u}{2r_l}\forall l=LT,LM,LB$$ (7)

   where $N_u$ is the total number of pixels, and the labels $LB,LM,LT$ refer to the top, middle and bottom layer of the detector, respectively. The functions are parameterized in terms of $m_l$ and $r_l$, controlling the maximum and the curvature of the profile, respectively. In particular, the parameters $m_l$ are chosen such that the maximum of the air normalized scatter is 10%.

2. Monte Carlo Case, where the profile $s$ is generated for each energy channel with a fast MC simulation approach previously developed [15] and studied for dual energy applications [16]. The air normalized scatter profiles are shown in Figure 3.

Given the symmetry of the simulated object, in both cases the scatter profile is assumed to be the same in each angular projection of the same energy channel.

Fig. 3.

Air normalized scatter profile obtained by fast MC simulation of a water phantom, in each layer of the simulated imaging system.

In the Quadratic case, data were fitted with the forward model of Equation (3) which incorporates a quadratic scatter profile expressed in Equation (7), so that the estimation model matches the data. For comparison, MBMD was performed also with a forward model that does not include the scatter term. In the Monte Carlo case, the material densities and scatter are also estimated assuming a quadratic profile for the scatter term, as in Equation (7). It’s important to note that in this instance, the assumption of a quadratic profile represents a simplification, considering that the scatter exhibits a more complex shape.

The regularization strength parameters ${\beta }_m$ were defined as follows:

• ${\beta }_w=10,{\beta }_{Ca}=10,{\beta }_{Gd}=200$ for water, Calcium and Gadolinium, respectively, when the L1 penalty of Equation (5) is used;

• ${\beta }_w=0.01,{\beta }_{Ca}=0.01,{\beta }_{Gd}=1$ for water, Calcium and Gadolinium, respectively, when the L2 penalty of Equation (6) is used.

In each decomposition, the water density image is initialized to its ground truth value, while all the other parameters are initialized to zero; 20000 iterations are used to achieve a good convergence.

III. Results

A. Quadratic Scatter Profile

Figure 4 summarizes the results of the MBMD of data simulated in the hypothesis of a quadratic scatter profile. These results were obtained using a forward model that does not include the scatter component. Specifically, the estimator is formulated as Equation (3) but without the term s. As can be seen from the figure, the reconstructions are dominated by artifacts that can be traced back to the un-modeled scatter component. Figure 5 shows that better results can be achieved in the three-material decomposition images when the scatter is included in the forward model: these results were obtained with the forward model of Equation (3) where the scatter has a quadratic profile as in Equation (7). Here the unregularized case appears noisy with some streaking artifacts. We observe that incorporating regularization with the L1 penalty enhances image quality compared to the unregularized scenario. However, there is a minor bias observed in the water and Gd density images corresponding to positions of other materials. Conversely, the reconstruction with the L2 penalty shows limited improvement compared to the unregularized case (though a more comprehensive parameter sweep may be used to fine-tune this reconstruction).

Fig. 4.

From top to bottom, the reconstructions of water, Calcium and Gadolinium concentrations from projection data incorporating a quadratic scatter profile, obtained with a forward model that neglects scatter, are directly compared with the phantom concentrations (on the leftmost column). Then, from left to right, the reconstructions are obtained without any regularization function in the objective function, with a penalty based on L1 norm (5) and with a penalty function based on L2 norm (6).

Fig. 5.

From top to bottom, the reconstructions of water, Calcium and Gadolinium concentrations from projection data incorporating a quadratic scatter profile, obtained with the forward model (3) that includes scatter with a quadratic profile (7), are directly compared with the phantom concentrations (on the leftmost column). Then, from left to right, the reconstructions are obtained without any regularization function in the objective function, with a penalty based on L1 norm (5) and with a penalty function based on L2 norm (6).

In Figure 6 we show the scatter profiles that were jointly estimated with the material densities. We observe that the scatter estimates do not match the true profiles exactly and in most cases the maximum in underestimated. This likely leads to the residual artifact in Figure 5. However the estimated scatter profiles reproduce the scatter level in the detector region outside the phantom shadow well enough to allow for a satisfactory material decomposition.

Fig. 6.

From left to right, estimates of the scatter quadratic profile in layer $LT,LM$ and $LB$. The profile maximum is initialized to zero

We have observed some variability in these scatter estimation results. Consider the results shown in Figure 7. In constrast to the previous case, here the scatter profile parameters $m_l$ were not initialized with zeros but with an arbitrary value, higher than the ground truth. The scatter profiles are now in most cases overestimated. Whether this phenomenon is due to the initialization, the stochastic nature of the optimization process, or is inherent to the estimation problem is the subject of ongoing studies.

Fig. 7.

From left to right, estimates of the scatter quadratic profile in layer $LT,LM$ and $LB$. The profile maximum is initialized to an arbitrary higher value.

B. MC Scatter Profile

The projection data with the addition of the Monte Carlo scatter profile were jointly processed with a quadratic scatter profile estimation model. The results of the MBMD are shown in Figure 8. As in the results discussed in the previous Section, the utilization of the L1 penalty brings clear advantages in the noise reduction over the unregularized case, while further fine-tuning may be required for the L2 penalty regularization strength. Unlike the other cases, where the density images appears substantially biased, the reconstruction with L1 penalty presents only a slight underestimation of the water and Calcium concentrations. The Gadolinium concentration is instead slightly overestimated, but in the latter image the shape of the water and Calcium inserts are also visible.

Fig. 8.

From top to bottom, the reconstructions of water, Calcium and Gadolinium concentrations from projection data incorporating a MC simulated scatter profile, obtained with the forward model (3) that includes scatter with the quadratic profile (7), are directly compared with the phantom concentrations (on the leftmost column). Then, from left to right, the reconstructions are obtained without any regularization function in the objective function, with a penalty based on L1 norm (5) and with a penalty function based on L2 norm (6).

Figure 9 shows the scatter profile that is jointly estimated with material densities for each energy channel. While there is an evident data-model mismatch, the reconstruction of the profile appears to be more accurate in the air projection region, i.e. towards the extremities of the detector, as observed also in Figure 6 of the previous Section.

Fig. 9.

From left to right, the MC simulated scatter profile of is estimated as a quadratic function (7) in layer $LT,LM$ and $LB$.

IV. Conclusion

In this work, we proposed a model-based method to jointly estimates material densities and scatter in spectral CT data. We simulated spectral CBCT data of a three-material basis phantom assuming a triple layer FPD imaging system. Specifically, two scenarios were explored for the simulation of scatter. In the first case, a simplified model assuming a quadratic function for the scatter profile was adopted. The material density estimates were compared when obtained from a forward model that neglects scatter and from our proposed forward model, showing the advantages of jointly estimating the scatter from the projection data. Additionally, we compared reconstructions using an objective function with various regularization strategies, including no regularization, L1 norm, and L2 norm penalty functions. In the second case, the scatter profiles of each spectral channel are obtained with a fast scatter MC simulation. The material density estimates are obtained using a forward model incorporating a quadratic scatter profile.

Preliminary findings reveal the need for fine-tuning some geometry parameters (e.g., FOV) and estimator parameters (e.g., L2 regularization strengths). We observed an instability in the scatter estimation with different initialization values, prompting further investigation into whether this is initialization-related or stems from the stochastic nature of the optimization algorithm. However, these preliminary results suggest that the joint estimation technique has the potential to significantly enhance density estimates in spectral CT reconstructions.