Accurate reconstruction of multiple basis images directly from dual-energy data in CT

Abstract

Objective:

We develop optimization-based algorithms to accurately reconstruct multiple (>2) basis images directly from dual-energy (DE) data in CT.

Methods:

In medical and industrial CT imaging, some basis materials such as bone, metals, and contrast agents of interest are confined often spatially within regions in the image. Exploiting this observation, we develop an optimization-based algorithm to reconstruct, directly from DE data, basis-region images from which multiple (>2) basis images and virtual monochromatic images (VMIs) can be obtained over the entire image array.

Results:

We conduct experimental studies using simulated and real DE data in CT, and evaluate basis images and VMIs obtained in terms of visual inspection and quantitative metrics. The study results reveal that the algorithm developed can accurately and robustly reconstruct multiple (>2) basis images as well as VMIs over energy range of interest directly from DE data.

Conclusions:

The developed algorithm can yield accurate multiple (>2) basis images, VMIs, and physical quantities of interest from DE data in CT.

Significance:

The work may provide insights into the development of practical procedures for reconstructing multiple basis images, VMIs, and physical quantities from DE data in applications. The work can be extended to reconstruct multiple basis images in multi-spectral CT or/and photon-counting CT.

I. Introduction

In current dual-energy CT (DECT), two sets of data, referred to as dual-energy (DE) data, are collected with two distinct, effective low- and high-kV spectra [1]. The spatial distribution of the X-ray linear attenuation coefficient (LAC) within the subject imaged is a function of X-ray energy and spatial variables. In general, the LAC distribution is approximated by a linear combination of two basis images, representing often water and bone materials in medical imaging or two other basis materials in industrial imaging, and algorithms have been developed for reconstruction of two basis images from DE data [2]–[5].

In medical CT imaging, e.g., the subject scanned may be composed of multiple basis materials, including, in addition to water and bone, fat, contrast materials such as iodine and gadolinium, and metal implants or inserts such as titanium implants and stainless-steel biopsy needle. While it is possible to reconstruct two basis images from DE data, there remain significant theoretical and practical interests in obtaining multiple $K\ge 3$ basis images directly from DE data, for clinical applications such as virtual unenhanced images in contrast-enhanced DECT, liver-fat quantification for detecting and diagnosing liver disease, and detection of bone marrow edema [6].

DE data may be insufficient for accurate reconstruction of $K>2$ basis images without using some prior information (or constraints,) because a total number of $K\times I$ voxels (i.e., unknowns) in $K>2$ basis images of size $I$ is substantially larger than 2 × $I$ voxels in two basis images. Previous effort exists in reconstructing $K=3$ basis images from DE data by using constraints such as the basis-image volume conservation [6]–[10]; and works have also been reported exploiting additional constraints, including basis triplet library and voxel-value bound for reconstructing $K=4$ basis images [9], [10]. These constraints can be interpreted as data augmentation possibly for alleviating the ill-conditionedness issue in reconstruction of $K=3$ or 4 basis images from DE data. Image-domain algorithms have also been investigated for obtaining multi-basis images from low- and high-keV images reconstructed separately from DE data [6], [11].

We investigate instead a different approach to addressing the reconstruction of $K>2$ basis images directly from DE data. In medical or industrial CT imaging, some basis materials, e.g., bone, metals, and contrast agents of interest are confined often within spatial supports smaller than the image array. We thus partition the image array into a set of spatially complementary basis regions of sizes smaller than the image array and assume ≤ 2 basis materials in each basis region, thus reducing $K\times I$ unknowns in multiple ($K>2$) basis images to ≤ 2 ×$I$. Therefore, this reduction of unknowns to be comparable to that in two basis images allows for possibly achieving accurate reconstruction of multiple ($K>2$) basis images directly from DE data, as illustrated below in detail.

In the work, we first derive a data model based upon the basis regions and formulate the reconstruction problem of basis-region images directly from DE data as a constrained optimization program. We then tailor the non-convex primal-dual (NCPD) algorithm [12], [13] to solve the optimization program for achieving reconstruction of the basis-region images. Finally, the basis images and virtual monochromatic images (VMIs) on the entire image array can readily be obtained from basis-region images reconstructed.

Using simulated and real DE data, we verify the NCPD algorithm’s accuracy in solving the data model based upon the basis regions derived and also evaluate its robustness in yielding $K>2$ basis images. Following visual inspection of the basis images and VMIs obtained, we analyze quantitatively reconstruction accuracy and also estimate physical quantities of interest, including LACs and iodine-contrast (IC) concentrations. The study results reveal that the NCPD algorithm can reconstruct $K>2$ basis images directly from DE data.

We develop in Sec. II the data model based upon basis regions, the optimization program, and the NCPD algorithm for yielding $K>2$ basis images from DE data, and present in Secs. III and IV basis images and VMIs and quantitative results obtained in simulated- and real-data studies, followed by discussions and conclusions made in Secs. V and VI.

II. Methods

A. The standard data model

In CT imaging with an effective polychromatic X-ray spectrum, the standard data model based upon the entire image array can be expressed in a discrete form as

$$g_{sj}\left(\left\{b_{ki}\right\}\right)=-\ln \sum _{m_e=1}^{M_e}{q}_{sj{m}_e}\exp \left(-\sum _{i=1}^I{a}_{sji}\sum _{k=1}^K{\mu }_{k{m}_e}{b}_{ki}\right)\text{,}$$ (1)

where $g_{sj}$ denotes model data for ray $j\left(j=1,2,\dots ,J^{(s)}\right)$ and spectrum $s$ ($s=1,2$ in DECT), $J^{(s)}$ the total number of rays for spectrum $s$; $b_{ki}$ (in unit of g/cm³) the basis image of material $k$ ($k=1,2,\dots ,K$) at voxel $i$ ($i=1,\cdots ,I$), $K$ the total number of basis images, and $I$ the total number of voxels in a basis image; $q_{sj{m}_e}$ normalized spectrum $s$, i.e., the product of the incident X-ray spectrum and detector response, at energy bin $m_e$ ($m_e=1,2,\dots ,M_e$) for ray $j$, $M_e$ the total number of energy bins; ${\mu }_{k{m}_e}$ the mass attenuation coefficient of material $k$ at energy $m_e$; and $a_{sji}$ the contribution weight of $b_{ki}$ at voxel $i$ to data of ray $j$ and spectrum $s$.

We use vector $b_k$ of size $I$ with elements $\left\{b_{ki}\right\}$ to denote basis image $k$ in a concatenated form in the order of $x-$, $y-$, and $z$-axis and subsequently form a VMI at energy $m$ as

$$f_m\left(\left\{b_k\right\}\right)=\sum _{k=1}^K{\mu }_{km}{b}_k,$$ (2)

where VMI $f_m$ is a vector of size $I$ with elements $\left\{f_{mi}\right\}$ denoting the LAC value at voxel $i$. In DECT or multi-spectral CT, the reconstruction task is to determine basis images $b_k$ from model data or measured data. With $b_k$ reconstructed, VMIs can readily be obtained by use of Eq. (2) in which ${\mu }_{km}$ can be looked up in the NIST database [14].

In general, it is considered that the reconstruction of $K$ basis images (i.e., determination of $K\times I$ unknowns in Eq. (1)) requires data sets acquired with $S=K$ distinct spectra. In DECT, algorithms exist for reconstruction of $K=2$ basis images and then VMIs from DE data. However, the accuracy of the existing algorithms diminish in reconstruction of $K>2$ basis images from DE data, as the total number of unknowns increases to $K\times I$ that can be substantially larger than 2×$I$.

B. The data model based upon basis regions

In medical or industrial CT imaging, it can be observed that some basis materials are confined often within portions of the image array. Exploiting this observation, we develop below a data model involving a number of unknowns considerably smaller than that in the standard data model in Eq. (1).

As described in Appendix I, we introduce a partition of the image array into $L$ spatially complementary basis regions characterized by region mask ${\mathcal{R}}_{l{I}_l}$ and set ${\text{Φ}}_{l{K}_l}$: ${\mathcal{R}}_{l{I}_l}$ is a diagonal matrix of size $I\times I$ with element $r_{lii}=1$, or 0, if voxel $i$ is within, or outside of, basis region $l$, and $I_l=\text{tr}\left({\mathcal{R}}_{l{I}_l}\right)\le I$ denotes the number of voxels within basis region $l$, whereas ${\text{Φ}}_{l{K}_l}$ of size $K_l$ includes elements that are indices $k^{\prime }\text{s}$ of the $K_l$ basis materials contained in basis region $l$, i.e., $K_l=\text{card}\left({\text{Φ}}_{l{K}_l}\right)\le K$, $l=1,2,\dots ,L$, and $k=1,2,\dots ,K$. We also construct matrix ${\mathcal{R}}_l^{\prime }$ of size $I_l\times I$ by removing all rows in ${\mathcal{R}}_{l{I}_l}$ containing only zeros.

Given basis images, we consider a special partition, referred to as the orthogonal basis-region set (OBRS) partition in which the number and types of basis materials within each basis region are known. We define basis-region image ${\mathbf{\text{b}}}_{lk}^{\prime }$ of size $I_l$ as ${\mathbf{\text{b}}}_{lk}^{\prime }={\mathcal{R}}_l^{\prime }{\mathbf{\text{b}}}_k$ of material $k$ in basis region $l$ in the OBRS partition, it is shown in Appendix I that

$${\mathbf{\text{f}}}_m(\mathbf{\text{b}})={\mathbf{\text{f}}}_m^{\prime }\left({\mathbf{\text{b}}}^{\prime }\right)=\sum _{l=1}^L\sum _{k\in {\text{Φ}}_{l{K}_l}}{\mu }_{km}{\mathcal{R}}_l^{\prime \top }{\mathbf{\text{b}}}_{lk}^{\prime },$$ (3)

where vector ${\mathbf{\text{f}}}_m^{\prime }\left({\mathbf{\text{b}}}^{\prime }\right)$ of size $I$ is referred to as the basis-region VMI at energy $m$, and that a data model based upon basis regions from Eq. (1) is given by

$$\begin{gathered} g_{sj}\left(\left\{b_{ki}\right\}\right)=g_{sj}^{\prime }\left(\left\{b_{lk{i}^{\prime }}^{\prime }\right\}\right) \\ =-\ln \sum _{m_e=1}^{M_e}{q}_{sj{m}_e}\exp \left(-\sum _{l=1}^L\sum _{k\in {\text{Φ}}_{l{K}_l}}{\mu }_{k{m}_e}\sum _{i^{\prime }=1}^{I_l}{a}_{sjl{i}^{\prime }}^{\prime }{b}_{lk{i}^{\prime }}^{\prime }\right), \end{gathered}$$ (4)

where $b_{lk{i}^{\prime }}^{\prime }$ is element $i^{\prime }$ of basis-region image ${\mathbf{\text{b}}}_{lk}^{\prime },a_{sjl{l}^{\prime }}^{\prime }={\sum }_{i=1}^I{a}_{sji}{r}_{li{i}^{\prime }}^{\prime }$,and $r_{li{i}^{\prime }}^{\prime }$ is element $\left(i,i^{\prime }\right)$ in matrix ${\mathcal{R}}_l^{\prime \top }$ of size $I\times I_l$. The total number of unknowns, i.e., $\left\{b_{lk{k}^{\prime }}^{\prime }\right\}$, in Eq. (4) is given by $N={\sum }_{l=1}^L{K}_l\times I_l$, as shown in Eq. (17) of Appendix. I. Therefore, if any $K_l$ is smaller than $K$, we have $N<K\times I$.

In this work, we focus on obtaining $K>2$ basis images under the condition that each basis region of the OBRS partition includes $K_l\le 2$ basis materials and thus have $N\le 2\times I$ substantially smaller than $K\times I$, i.e., the total number of unknowns in the standard data model in Eq. (1). It is well-known that two basis images (with a total number of 2×$I$ unknowns) can accurately be reconstructed from DE data. Therefore, it is possible for accurate reconstruction of basis-region images of $N\le 2\times I$ unknowns from DE data through solving the data model based upon basis regions in Eq. (4). Using Eq. (7) below, $K>2$ basis images ${\mathbf{\text{b}}}_k$ can then be obtained from basis-region images ${\mathbf{\text{b}}}_{lk}^{\prime }$ reconstructed.

C. Constrained-optimization-based reconstruction

For discussion convenience, we further introduce congregate basis-region image vector ${\mathbf{\text{b}}}^{\prime }$ of size $N$ formed by concatenating ${\mathbf{\text{b}}}_{lk}^{\prime }$ in the order of $k\in {\text{Φ}}_{l{K}_l}$ and $l=1,\cdots ,L$, and thus rewrite $g_{sj}^{\prime }\left(\left\{b_{lk{i}^{\prime }}^{\prime }\right\}\right)$ as $g_{sj}^{\prime }\left({\mathbf{\text{b}}}^{\prime }\right)$. Furthermore, for spectrum $S$, we introduce vector $g_s^{¢}\left({\text{b}}^{\prime }\right)$ of size $J^{(s)}$ with entries $g_{sj}^{\prime }\left({\text{b}}^{\prime }\right)$ and also congregate vector $g^{\prime }\left({\text{b}}^{\prime }\right)$ of size $J=J^{(1)}+J^{(2)}$ that is obtained by concatenating $g_s^{\prime }\left({\text{b}}^{\prime }\right)$ over $s=1$ and 2. Also, we use congregated vector $g^{[\mathcal{M}]}$ of size $J$ to denote DE data measured with $S=2$ distinct spectra.

a). Optimization program:

Taking the Maclaurin-series expansion of ${\mathbf{\text{g}}}^{\prime }\left({\mathbf{\text{b}}}^{\prime }\right)$ w.r.t. ${\mathbf{\text{b}}}^{\prime }$ at ${\mathbf{\text{b}}}^{\prime }=0$, we can re-express data model Eq. (4) in a matrix-vector form as [12], [15]

$$g^{\prime }\left({\text{b}}^{\prime }\right)=\mathcal{H}\prime {\text{b}}^{\prime }+\text{Δ}{g}^{\prime }\left({\text{b}}^{\prime }\right),$$ (5)

where matrix $\mathcal{H}\prime$ of size $J\times N$, with elements $h_{sjl{i}^{\prime }}^{\prime }=a_{sjl{i}^{\prime }}^{\prime }{\sum }_{m_e}{q}_{sj{m}_e}{\mu }_{m_{e}k}$ indicating the contribution weight of $b_{lk{i}^{\prime }}^{\prime }$ at voxel $i^{\prime }$ to ray $j$ of spectrum $s$, and $\text{Δ}{g}^{\prime }\left({\text{b}}^{\prime }\right)=g^{\prime }\left({\text{b}}^{\prime }\right)-\mathcal{H}\prime {\text{b}}^{\prime }$ is the non-linear component in Eq. (4).

In an attempt to invert Eq. (5) for reconstruction of ${\mathbf{\text{b}}}^{\prime }$, we formulate the constrained optimization program below and then use its solution $\mathbf{\text{b}}\prime \ast$ as the reconstruction of ${\mathbf{\text{b}}}^{\prime }$:

$$\begin{gathered} {\text{b}}^{\prime \ast }=\underset{{\mathbf{\text{b}}}^{\prime }}{\arg \min }\frac{1}{2}{\Vert \left[g[\mathcal{M}]-\text{Δ}{g}^{\prime }\left({\text{b}}^{\prime }\right)\right]-\mathcal{H}\prime {\text{b}}^{\prime }\Vert }_2^2 \\ \text{s.t.}{\Vert \left(\left|\nabla f_{m_c}^{\prime }\left({\text{b}}^{\prime }\right)\right|\right)\Vert }_1\le {\gamma }_{m_c},m_c=1,2 \\ \text{and}{\text{b}}^{\prime }\ge 0\text{,} \end{gathered}$$ (6)

where data-fidelity term $\frac{1}{2}\parallel \cdot \parallel \frac{2}{2}$ denotes the objective function to be minimized, $\parallel \cdot {\parallel }_2^2$ the square of ${\ell }_2$-norm of a vector, $\parallel (\mid \nabla .\mid ){\parallel }_1$ the ${\ell }_1$-norm of the gradient magnitude image, i.e., the total-variation (TV), of the image, and ${\text{f}}_{m_c}^{\prime }$ is the basis-region VMI at constraint-energy $m_c$ (see Eq. (3).) The total-variation (TV) constraints are enforced on two basis-region VMIs at two well-separated energies for yielding accurate VMIs over a broad energy range of interest.

b). The non-convex primal-dual (NCPD) algorithm:

As discussed in Appendix II, we derive the NCPD algorithm to solve the optimization program in Eq. (6) for reconstructing basis-region image $\mathbf{\text{b}}\prime \ast$, along with its pseudo-codes and necessary convergence conditions [12], [13]. In each of the numerical studies performed, $\mathbf{\text{b}}\prime \ast$ is obtained when the necessary convergence conditions described in Appendix II are achieved in terms of the computer floating precision used.

Once converged solution $\mathbf{\text{b}}\prime \ast$ (i.e., basis-region images ${\text{b}}_{lk}^{\prime \ast }$) are achieved, converged basis image ${\mathbf{\text{b}}}^{\ast }$ (i.e., ${\text{b}}_k^{\ast }$) on the image array can, as demonstrated in Appendix I, be obtained as

$${\text{b}}_k^{\ast }=\sum _{l\in {\text{Ψ}}_{k{L}_k}}{\mathcal{R}}_l^{\prime \top }{\text{b}}_{lk}^{\prime \ast },$$ (7)

where set ${\text{Ψ}}_{k{L}_k}$ of size $L_k$ encompasses elements that are indices $l\text{'s}$ of the $L_k$ basis regions containing basis material $k$ with $L_k=\text{card}\left({\text{Ψ}}_{k{L}_k}\right)\le L$. A recipe thus is provided in Eq. (7) for obtaining accurate basis images from the reconstructed basis-region images of the OBRS partition.

D. Evaluation of the NCPD algorithm

In addition to visual inspection of the basis images and VMIs obtained, we also compute metrics normalized root-mean-square error (nRMSE) and Pearson correlation coefficient (PCC) [16], and estimate LACs and IC concentrations, to evaluate quantitatively the accuracy of the basis images and VMIs against their respective truth or reference. In the simulated- and real-data studies in Secs. III and IV, the impact of the basis-region set (BRS) of a partition on the robustness of the NCPD algorithm is investigated by using two partitions with two distinctly different BRS’s, referred to as BRS 1 and BRS 2.

In the simulated- and real-data studies, results obtained for $K=3$ basis materials by use of the NCPD algorithm with BRS 1 and BRS 2 are referred to as “3MDE-BRS1” and “3MDE-BRS2” in Secs. III and IV, where 3MDE indicates 3 basis materials (i.e., 3M) considered. Also, in the simulated-data study with $K=6$ basis materials, results obtained by use of the NCPD algorithm with BRS 1 and BRS 2 are referred to as “6MDE-BRS1” and “6MDE-BRS2” in Sec. III, where 6MDE indicates 6 basis materials (i.e., 6M) involved.

In the simulated-data studies in Sec. III, BRS 1 is selected as the OBRS and then used in the data model based upon basis regions in Eq. (4) to generate DE data, which are identical to that generated with the standard data model in Eq. (1). As a result, the simulated-data study with BRS 1 (i.e., the OBRS) also serves to verify that the NCPD algorithm with the OBRS can numerically accurately solve optimization program in Eq. (6) and invert the data model based upon basis regions of the OBRS partition in Eq. (4) and thus equivalently the standard data model in Eq. (1).

In the real-data studies in Sec. IV, no knowledge about the OBRS is available, and two distinct basis-region partitions, i.e., BRS1 and BRS2, thus are used to demonstrate the performance stability of the NCPD algorithm in the studies involving real data that also contain noise and other physical factors inconsistent with data models Eqs. (1) and (4).

We also reconstruct basis images by use of a widely-adopted two-step method [4] based upon two materials of water and bone and use them as a reference in terms of revealing possible reconstruction errors and artifacts. The two-step method, referred to as the data-domain decomposition (DDD) method [4], estimates basis-image sinograms from which two basis images are reconstructed by use of the FBP algorithm. Results obtained with the DDD method in Secs. III and IV are referred to as the two-material-dual-energy-DDD (in short, “2MDE-DDD”) results.

III. Results of Simulated-Data Studies

We perform numerical studies using simulated DE data to verify that the NCPD algorithm can yield multiple basis images by numerically accurately solving the optimization program in Eq. (6) and also to reveal its robustness relative to the basis-region selection. The robustness of the NCPD algorithm in the presence of noise and additional physical factors is investigated in the real-data studies in Sec. IV below.

In the simulated-data studies, two digital torso phantoms, respectively, contain $K=3$ and $K=6$ basis materials, as water shown in Figs. 1 and 9, are used to generate DE data by use of Eq. (1) in DECT. Specifically, digital torso phantom one consists of a total of $K=3$ basis materials: water, bone, and 20-mg/ml iodine, whereas digital torso phantom 2 includes a total of $K=6$ basis materials: water, bone, 20-mg/ml iodine, 8-mg/ml gadolinium, titanium, and stainless steel. Iodine and gadolinium, titanium, and stainless steel are used, respectively, as contrast agents, metal implants, and biopsy needles. In both digital phantoms, while the water basis material is included in the entire image array, the other basis materials, including bone, iodine, gadolinium, titanium, and stainless steel, are confined within spatially non-overlapping regions, which are smaller than the image array, allowing for an OBRS partition, with each basis region containing 2 basis materials, i.e., water and one of the other basis materials of bone, iodine, gadolinium, titanium, or stainless steel.

Fig. 1:

Truth water (column 1), bone (column 2), and 20-mg/ml iodine (column 4) basis images in digital torso phantom one are shown in display window [0, 1.2], whereas the bone basis image (column 3) is shown also with a narrow display window [0, 0.01] to reveal its spatial support.

Fig. 9:

Row 1: truth basis images of water (column 1), bone (column 2), 20-mg/ml contrast iodine (column 3); and row 2: 8-mg/ml contrast gadolinium (column 1), titanium screws (column 2), and stainless-steel biopsy needle (column 3) in digital torso phantom two. Display windows: [0, 1.2].

We consider a DECT system of circular fan-beam geometry with source-to-rotation-center (SOR) and source-to-detector (SOD) distances of 100 and 150 cm. A linear detector of 70-cm length consisting of 896 detector elements of size 0.08 cm is used to collect DE data at 1440 views uniformly distributed over 2π. The 80- and 140-kV, referred to as the low- and high-kV, spectra are created using the TASMIC model [17]. A 2-mm Al filter is used for both low- and high-kV spectra, along with an additional 1-mm Cu filter for the high-kV spectrum only. Mass attenuation coefficients of the basis materials are either looked up from the NIST database [14] or generated using the NIST XCOM tool [18].

Using Eq. (1) for the DECT system described, DE data are generated from each of the two digital torso phantoms with $K=3$ and $K=6$ basis materials. For image reconstruction with the NCPD algorithm, we consider two partitions with basis-region sets, BRS 1 and BRS 2, which include basis regions of significantly different shapes and sizes. In particular, based on the truth basis images, BRS 1 is devised as the OBRS so that DE data generated with Eq. (1) are identical to those generated with Eq. (4) employing BRS 1 (i.e., the OBRS.) Conversely, BRS 2, different than BRS 1, is designed to introduce an inconsistency with DE data generated with BRS 1 (i.e., the OBRS.)

In each of the simulated-data studies, we first apply the NCPD algorithm with BRS 1 (i.e., the OBRS) to verifying that it can accurately reconstruct basis-region images by solving Eq. (4) with BRS 1 from which the $K=3$, or $K=6$, basis images are obtained by using Eq. (7). Furthermore, in an attempt to study the robustness of the NCPD algorithm relative the partition selection, we apply the NCPD algorithm with BRS 2, which is significantly different than BRS 1, to reconstructing the basis-region images and then the basis images from the same DE data generated by use of Eq. (1), or equivalently Eq. (4) with BRS 1 (i.e., the OBRS.)

A. Results of three-basis-image reconstruction

1). Study conditions:

Digital torso phantom one of 200×256 square pixels of size 0.14 cm is composed of three truth basis images, shown in Fig. 1, from which we generate DE data by using Eq. (1) along with the scanning parameters described above that are used also in image reconstruction below.

Partitions BRS 1 (row 1) and BRS 2 (row 2) in Fig. 2 are chosen each of which encompasses two spatially complementary basis regions (columns 1 & 2), and two basis materials, water & bone (column 1) or water & iodine (column 2), are contained in each of the two basis regions. Therefore, there are 2×$I$ unknowns in basis-region images to be reconstructed.

Fig. 2:

BRS 1 (row 1) and BRS 2 (row 2) partitions in which the basis regions highlighted in white contain bone & water (column 1) and iodine & water (column 2) basis materials.

As illustrated, BRS 1 is the OBRS for verifying the NCPD algorithm, whereas BRS 2, significantly different than BRS 1, is designed for evaluating the robustness of the NCPD algorithm with BRS 2 when applied to data generated by using Eq. (4) with BRS 1 (or equivalently Eq. (1).) We use the TV values of the truth VMIs at 50 keV and 90 keV as the values of TV-constraint parameters ${\gamma }_1$ and ${\gamma }_2$ in optimization program Eq. (6) in the study.

2). Visual inspection:

Using the NCPD algorithm with BRS 1, we reconstruct basis-region images ${\text{b}}_k^{\prime \ast }$ from DE data and then obtain basis images ${\text{b}}_k^{\ast }$ on the entire image array by using Eq. (7), as shown in column 1 of Fig. 3. It can be observed that the basis images obtained are visually identical to their truth counterparts shown in Fig. 1, suggesting that the NCPD algorithm with BRS 1 can accurately yield $K=3$ basis images from DE data. Additionally, the DDD method can reconstruct only water and bone basis images as shown in column 3 of Fig. 3.

Fig. 3:

Basis images of water (row 1), bone (row 2), and iodine (row 3) obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3). Display windows: [0, 1.2].

Using the basis images obtained, we subsequently create VMIs at 33 keV and 100 keV, which are shown in Fig. 4, along with their truth in column 4 of Fig. 4, with the tissue (rows 1 & 3) and lung (rows 2 & 4) display windows. Furthermore, as shown in the truth VMI at 33 keV in Fig. 4, we select two red-box regions of interest (ROIs) containing iodine contrast and vertebral bone, and then compute the difference images between truth and obtained VMIs at 33 keV and 100 keV within the ROIs, which are shown in Fig. 5. It can be observed that the VMIs obtained by using the NCPD algorithm with BRS 1 are visually identical to their respective truth. However, while the DDD method reconstructs a visually reasonable VMI at 100 keV, as expected, it yields a VMI at 33 keV with the iodine contrast and vertebral bone ROIs considerably different from the corresponding truth.

Fig. 4:

VMIs at 33 keV (rows 1 & 2) and 100 keV (rows 3 & 4) obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3), along with truth VMIs (column 4). VMIs in rows 1 & 3 and 2 & 4 are with display windows [−1000, 1000] HU and [−1000, −500] HU for revealing tissue and lung details.

Fig. 5:

Zoomed-in views of differences of the truth and VMIs at 33 keV (rows 1 & 2) and 100 keV (rows 3 & 4) in two ROIs, enclosed by the red boxes in Fig. 4, containing IC (rows 1 & 3) and vertebral bone (rows 2 & 4). The VMIs are obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3). Display window: [−200, 200] HU.

3). Quantitative analysis:

We compute nRMSE and PCC to evaluate quantitatively VMIs obtained. In Fig. 6, results reveal that the NCPD algorithm with BRS 1 yields VMIs with the lowest nRMSEs and highest PCCs∼ 1 in energy range 25 keV∼120 keV as compared to those obtained with the DDD method. In the energy range ≤ the iodine K-edge energy, the nRMSE and PCC results of the latter are considerably worse than that of the former. The observation is consistent with the visual inspection results above.

Fig. 6:

Metrics nRMSE (left) and PCC (right) of VMIs in energy range 25 keV∼120 keV obtained, by using the NCPD algorithm with BRS 1 (blue solid curve, 3MDE-BRS1) and BRS 2 (orange dashed curve, 3MDE-BRS2) and by using the DDD method (green dotted curve, 2MDE-DDD).

We also compute LACs within ROIs containing water, bone, and iodine, respectively, by averaging the VMI values over their respective ROIs and plot their biases in Fig. 7. It can be observed that the NCPD algorithm with BRS 1 yields accurate LACs in energy range 25 keV∼120 keV, whereas the LACs of the DDD method have appreciable levels of bias and standard deviation (SD) for energies ≤ the iodine K-edge energy.

Fig. 7:

LAC differences between truth and VMIs in energy range 25 keV∼120 keV, along with the SDs, averaged over each of three ROIs containing water (left), bone (middle), and iodine (right) materials. The VMIs are obtained by using the NCPD algorithm with BRS 1 (blue solid curve, 3MDE-BRS1) and BRS 2 (orange dashed curve, 3MDE-BRS2) and by using the DDD method (green dotted curve, 2MDE-DDD).

4). Estimation of the IC concentration:

Using an affine relationship between the concentration and the pixel values in a basis image reconstructed [19], we estimate the IC concentrations within ROIs containing water, bone, and iodine materials, respectively, in the iodine basis image reconstructed by using the NCPD algorithm and also in the bone basis image by using the DDD method. The IC means and SDs over their ROIs are shown in Fig. 8, along with the respective truth values (dotted horizontal lines). The NCPD algorithm with BRS 1 yields IC concentrations with minimum biases and SDs, whereas that IC concentrations estimated with DDD method may be of significant biases and/or SDs.

Fig. 8:

IC concentrations, along with the SDs, averaged over each of the three ROIs selected containing water (left), bone (middle), and iodine (right) materials in the iodine basis images obtained by using the NCPD algorithm with BRS 1 (blue dot, 3MDE-BRS1) and BRS 2 (orange triangle, 3MDE-BRS2), and in the bone basis image obtained by using the DDD method (green square, 2MDE-DDD). The horizontal dotted lines indicate respective truth values.

5). Impact of the partition selection:

The study above reveals that the NCPD algorithm with BRS 1 (i.e., the OBRS) accurately yields $K=3$ basis images and VMIs from DE data. We investigate the robustness of the NCPD algorithm with respect to the BRS choice by applying the NCPD algorithm with BRS 2, which differs significantly from BRS 1 to reconstructing basis-region images from the same DE data above.

In column 2 of Figs. 3 and 4, we display the basis images and VMIs at 33 keV and 100 keV obtained with BRS 2, which are visually similar to their counterparts obtained with BRS 1, except for minor differences in the iodine region within the VMI at 33 keV shown in Fig. 5 (i.e., row 1, column 2). Conversely, the ROI images of the DDD method in Fig. 5 (i.e., row 1, columns 3) reveal a considerable level of inaccuracy, as expected. Also, as shown in Fig. 6, while the VMIs obtained with BRS 2 have nRMSE slightly inferior to that of the corresponding VMIs obtained with BRS 1, their PCC results appear close to each other.

Furthermore, as shown in Figs. 7 and 8, the NCPD algorithm with BRS 1 and BRS 2 appear to yield largely identical LACs and IC concentration. This observation is further corroborated by the results of additional studies that we have carried out with BRSs of shape and size different than that of BRS 1 and BRS 2. The robustness of the NCPD algorithm with respect to the BRS choice allows one to select, with a considerable degree of flexibility, a BRS readily by basing upon, e.g., reconstructions obtained with the existing algorithms such as the FBP algorithm from either one or both of the low- and high-kV data sets in DECT.

B. Results of six-basis-image reconstruction

1). Study conditions:

In this study, the digital torso phantom two of 200 × 256 square pixels of size 0.14 cm includes six truth basis images, shown in Fig. 9. The water and bone basis images are similar to those in Fig. 1; the iodine and gadolinium basis images split the vessels; and the titanium and stainless-steel basis images simulate the metal screws implanted into the vertebrae and the stainless-steel biopsy needle inserted into the lung region. Using the basis images in Eq. (1), we generate DE data with the scanning parameters identical to those in Sec. III-A, which are used also in image reconstruction below.

As shown in Fig. 10, we choose two partitions of BRS 1 (row 1) and BRS 2 (row 2) each of which includes five spatially complementary basis regions that contain, respectively, two basis materials: water & bone (column 1), water & iodine (column 2), water & gadolinium (column 3), water & titanium (column 4), and water & iodine (column 5), leading to 2 × $I$ unknowns in basis-region images to be reconstructed.

Fig. 10:

BRS 1 (row 1) and BRS 2 (row 2) partitions in which the basis regions highlighted in white contain, respectively, bone & water (column 1), iodine & water (column 2), gadolinium (Gd) & water (column 3), titanium (Ti) & water (column 4), stainless-steel & water (column 5) basis materials.

BRS 1 is chosen to be the OBRS partition used in Eq. (4) (or equivalently, Eq. (1)) to generate DE data from the six truth basis images with scanning parameters described above, which is used also in the NCPD algorithm for image reconstruction below. Conversely, BRS 2, which is significantly different than BRS 1, is designed for evaluating the robustness of the NCPD algorithm with BRS 2 when applied to data generated by using Eq. (4) with BRS 1 (or equivalently Eq. (1).) We use the TV values of the truth VMIs at 50 keV and 90 keV as the values of TV-constraint parameters ${\gamma }_1$ and ${\gamma }_2$ in optimization program Eq. (6) in the study.

2). Visual inspection:

We apply the NCPD algorithm with BRS 1 to reconstructing basis-region images from DE data generated and then obtain $K=6$ basis images over the entire image array. As shown in column 1 of Fig. 11, the basis images obtained are visually identical to their truth counterparts in Fig. 9, verifying again that the NCPD algorithm with BRS 1 can accurately yield $K=6$ basis images from DE data. The basis images obtained by use of the DDD method are shown also in column 3 of Fig. 11 in which the bone basis image considerably differ from its truth in Fig. 9; and the DDD method cannot yield basis images of iodine, gadolinium, titanium, or stainless steel.

Fig. 11:

Basis images of water (row 1), bone (row 2), 20-mg/ml iodine (row 3), 8-mg/ml gadolinium (row 4), titanium (row 5), and stainless steel (row 6) obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3). Display windows: [0, 1.2].

Using the basis images obtained, we subsequently create VMIs at 33 keV and 100 keV, and display them in Fig. 12 with tissue (rows 1 & 3) and lung (rows 2 & 4) display windows. Furthermore, we select three ROIs enclosed by red boxes in the truth VMI at 33 keV in Fig. 12 containing the iodine, titanium, and stainless steel basis materials, and then compute differences between the truth and obtained VMIs within the ROIs, which are displayed in Fig. 13. It can be observed that the VMIs obtained with the NCPD algorithm are visually identical to their truth, whereas the DDD method yields VMIs at 33 keV containing iodine and metal ROIs considerably different from the corresponding truth.

Fig. 12:

VMIs at 33 keV (rows 1 & 2) and 100 keV (rows 3 & 4) obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3), along with truth VMIs (column 4). VMIs in rows 1& 3 and 2 & 4 are with display windows [−1000, 1000] HU and [−1000, −500] HU for revealing tissue and lung details.

Fig. 13:

Zoomed-in views of differences of the truth and VMIs at 33 keV (rows 1–3) and 100 keV (rows 4–6) in three rectangular ROIs in Fig. 12, containing iodine (rows 1 & 4), titanium (rows 2 & 5), and stainless steel (rows 3 & 6). The VMIs are obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3). Display window: [−200, 200] HU.

3). Quantitative analysis:

We compute metrics nRMSE and PCC to evaluate quantitatively the VMIs reconstructed and display them in Fig. 14. The metrics nRMSE and PCC results reveal that the NCPD algorithm with BRS 1 yields VMIs with the lower nRMSEs and highest PCCs (∼ 1) in energy range 25 keV∼120 keV than those obtained with the DDD method. In particular, in the energy range ≤ the iodine K-edge energy, the nRMSE and PCC results of the latter are considerably worse than that of the former, because only two basis materials, i.e., water and bone, are considered in the latter. These observations are consistent with the visual inspection results above.

Fig. 14:

Metrics nRMSE (left) and PCC (right) computed from VMIs in energy range 25 keV∼120 keV obtained by using the NCPD algorithm with BRS 1 (blue solid curve, 6MDE-BRS1) and BRS 2 (orange dashed curve, 6MDE-BRS2) and the DDD method (green dotted curve, 2MDE-DDD).

We also compute the LACs within ROIs containing water, bone, iodine, gadolinium, titanium, and stainless steel materials, respectively, by averaging the VMI values over the respective ROIs selected, and plot them in Fig. 15. It can be observed that the NCPD algorithm with BRS 1 can yield accurate LACs in energy range 25 keV∼120 keV, whereas, again, the DDD method yields LACs with appreciable levels of bias and SD at energies ≤ the iodine K-edge energy.

Fig. 15:

LAC differences between truth and obtained VMIs in energy range 25 keV∼120 keV, along with the SDs, averaged over each of the six ROIs containing one of the six basis materials. Row 1: water (left), bone (middle), iodine (right); and row 2: gadolinium (left), titanium (middle), and stainless steel (right). The VMIs are obtained by using the NCPD algorithm with BRS 1 (blue solid curve, 6MDE-BRS1) and BRS 2 (orange dashed curve, 6MDE-BRS2) and by using the DDD method (green dotted curve, 2MDE-DDD).

4). Estimation of the IC concentration:

We estimate the IC concentration within ROIs corresponding to the water, bone, and iodine material regions in the iodine basis image reconstructed by use of the NCPD algorithm and in the bone basis image reconstructed by use of the DDD method, and estimate its mean and SD over the ROI. In Fig. 16, we display the means and SDs of the IC concentration estimated, along with the respective truth concentration values (dotted horizontal lines). It can be observed that the IC concentrations estimated by use of the NCPD algorithm with BRS 1 agree well with the truth, whereas some of the IC concentrations estimated by use of the DDD method appear to be with a considerable degree of bias and/or SD.

Fig. 16:

IC concentrations, along with the SDs, averaged over each of the three ROIs containing water (left), bone (middle), and iodine (right) in the iodine basis images obtained by using the NCPD algorithm with BRS 1 (blue dot, 6MDE-BRS1) and BRS 2 (orange triangle, 6MDE-BRS2), and in the bone basis image reconstructed by using the DDD method (green square, 2MDE-DDD). The horizontal dotted lines indicate the respective truth values.

5). Impact of BRS selection on basis image reconstruction:

The study above reveals that the NCPD algorithm with BRS 1 (i.e., the OBRS) can accurately yield six basis images and VMIs from DE data. We also investigate the robustness of the NCPD algorithm with respect to the BRS choice by using BRS 2 in row 2 of Fig. 10, which differs considerably from BRS 1. We apply the NCPD algorithm with BRS 2 to reconstructing basis-region images from the same DE data as that used for yielding $K=6$ basis images with BRS 1 above.

We display in column 2 of Figs. 11 and 12, respectively, the six basis images and VMIs at 33 keV (rows 1 & 2) and 100 keV (rows 3 & 4) obtained from the basis-region images reconstructed by using the NCPD algorithm with BRS 2. It can be observed that the basis images and VMIs appear visually similar to their counterparts obtained with BRS 1, except for minor differences from the truth that are only visually observed in the iodine and metal regions within the VMI at energy 33 keV in rows 1–3 of Fig. 13 when the differences are displayed in a narrow display window. Also, as shown in Fig. 14, while the VMIs obtained with BRS 2 have nRMSE slightly higher than that obtained with BRS 1, the PCC results with both BRS 1 and BRS 2 appear identical.

Furthermore, for estimation of LACs and IC concentration, the results of the NCPD algorithm with both BRS 1 and BRS 2 appear to remain quantitatively virtually identical, as shown in Figs. 15 and 16. This observation is corroborated further by the results of additional studies carried out with BRSs including basis regions of shape and size different than that of BRS 1 and BRS 2 presented here.

We have also applied the NCPD algorithm to yielding basis images from DE data of phantoms containing 4, 5, or > 6 basis materials. These studies, which are not included, reveal that the NCPD algorithm can effectively yield accurately the basis images and VMIs under the conditions considered.

IV. Results of Real-Data Studies

We perform numerical studies using real DE data collected to assess the robustness of the NCPD algorithm as the real DE data contain noise and additional physical factors inconsistent with the data model in Eq. (4) that the NCPD algorithm is developed to invert. Moreover, in a real-data study, because “truth basis images” are unknown or may not even be defined meaningfully, it is unlikely that an OBRS partition can be devised that is consistent precisely with the real data. Conversely, a real-data study can be exploited for evaluating and demonstrating the robustness of the NCPD algorithm with respect to the BRS selection that is highly likely inconsistent with real data.

Real DE data were collected from the DE472 phantom and a clinical abdominal phantom by use of a DECT scanner in axial mode with 80- and 135-kV spectra [13], [20]. Data from the central row of the detector are extracted, forming an equivalent fan-beam geometry with SOR and SOD distances of 60 cm and 107 cm and a 49^◦ curved detector consisting of 896 detector elements [13], [20]. The 80- and 135-kV data were collected at identical 1200 angular views evenly distributed over 360^◦. In the studies, the physical phantoms are assumed to contain three basis materials, including water, bone, and 20-mg/ml iodine.

In the real-data studies below, using 80- and 135-kV spectra and scanning parameters described above, along with ${\mu }_{km}$ obtained with the NIST database and XCOM tool for the three basis materials, in the NCPD algorithm with a BRS selected, we reconstruct basis-region images from real DE data collected and then obtain basis images and VMIs. We consider two partitions, BRS 1 and BRS 2, containing basis regions different in shape, size, and number. Each of the basis regions in BRS 1 and BRS 2 contains ≤ 2 basis materials. The DDD method is used also for reconstructing basis images of water and bone from the real DE data.

A. Results of DE472-phantom-data study

1). Study conditions:

The DE472 phantom includes solid water, calcium inserts, and iodine inserts, and we thus reconstruct basis images of water, bone (i.e., calcium), and 20-mg/ml iodine and VMIs from its DE data collected. The basis images and VMIs are reconstructed on image arrays of 512 × 512 identical square pixels of 0.08 cm in size.

As discussed above, we select two partitions, BRS 1 and BRS 2, as shown in rows 1 and 2 of Fig. 17, by applying appropriate threshold values to the image reconstructed from the 80-kV data with the FBP algorithm. In BRS 1, the three spatially complementary basis regions, highlighted in white, are selected to contain bone & water (column 1), iodine & water (column 2), and water-only (column 3) basis materials, whereas in BRS 2, the two spatially complementary basis regions, highlighted in white, are selected to contain bone & water (column 1) and iodine & water (column 2) basis materials. Therefore, each basis region in BRS 1 or BRS 2 contains ≤ 2 basis materials, leading to $N\le 2\times I$ unknowns to be reconstructed from DE data.

Fig. 17:

BRS 1 (row 1) and BRS 2 (row 2) partitions in which the basis regions are highlighted in white. In BRS 1 (row 1), the three basis regions contain water & bone (column 1), water & 20-mg/ml iodine (column 2), and water-only (column 3) basis materials, whereas in BRS 2 (row 2), the two basis regions contain water & bone (column 1) and water & 20-mg/ml iodine (column 2).

It can be observed in Fig. 17 that the first two basis regions in BRS 1, respectively, enclose tightly the physical supports of the calcium and iodine inserts in the DE472 phantom, whereas the basis regions in BRS 2 are of shape and size considerably larger than that of the physical supports of the calcium and iodine inserts. Therefore, we use BRS 1 and BRS 2 in the NCPD algorithm to demonstrate and evaluate its robustness with respect to the BRS selection.

The TV constraints are enforced on VMIs at 50 keV and 100 keV in optimization program Eq. (6). In a real-data study, no truth VMI-TV values are available. Therefore, in an attempt to determine appropriate TV-constraint values ${\gamma }_{m_c}$ in Eq. (6), we conduct multiple reconstructions with a slew of TV-constraint values and then choose ${\gamma }_1$ and ${\gamma }_2$ that yield visually basis images and VMIs with minimal visual artifacts as appropriate TV-constraint values in Eq. (6). Additionally, we use the DDD method to reconstruct basis images and VMIs.

2). Visual inspection:

In column 1 of Fig. 18, we display the water, bone, and iodine basis images, along with VMIs at 33 keV and 100 keV, of the DE472 phantom reconstructed by using the NCPD algorithm with BRS 1. It can be observed that few visual artifacts are observed in the basis images and VMIs obtained. Conversely, the corresponding water and bone basis images and VMIs of the DE472 phantom reconstructed by use of the DDD method are displayed in column 3 of Fig. 18, where streaks connecting high-contrast inserts and elevated noise levels appear visible in both water and bone basis images and the 33-keV VMI.

Fig. 18:

Basis images of solid water (row 1), bone (row 2), and 20-mg/ml iodine (row 3), and VMIs at 33 keV (row 4) and 100 keV (row 5) obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3). Display windows: [0, 1.2] for basis images and [−1000, 1000] HU for VMIs.

3). Quantitative analysis:

We compute the LACs in ROIs within the supports of the solid water, calcium inserts, and iodine inserts in the DE472 phantom, and calculate their biases and SDs over the respective ROIs relative to the reference values obtained by use of the NIST XCOM tool with the specs for each inserts of the phantom, provided by the manufacturer, and plot the estimated biases and SDs in Fig. 19.

Fig. 19:

LAC differences between references and VMIs in energy range 25 keV∼120 keV, along with the SD bars, averaged over pixels within, respectively, one of three ROIs selected in the supports of the solid water (left), 600-mg/ml calcium insert (middle), and 20-mg/ml iodine insert (right). The VMIs are obtained by using the NCPD algorithm with BRS 1 (blue solid curve, 3MDE-BRS1) and BRS 2 (orange dashed curve, 3MDE-BRS2) and by using the DDD method (green dotted curve, 2MDE-DDD).

It can be observed that the NCPD algorithm with BRS 1 shown in Fig. 17 can yield accurate LACs in energy range 25 keV∼120 keV and that the DDD method yields LACs with significant biases and SDs in the energy range ≤ the K-edge energy, while it may produce LACs of reduced biases and SDs within the solid-water, calcium, and iodine ROIs for an energy range considerably above the iodine K-edge,

4). Estimation of the IC concentration:

We estimate the IC concentrations within ROIs of the inserts in the DE472 phantom. For each ROI, we compute the mean and SD of the IC concentrations and show them in Fig. 20, respectively, for ROIs within the inserts of solid water, 600-mg/ml calcium, and 20-mg/ml iodine. It can be observed that the NCPD algorithm with BRS 1 can yield estimates of IC concentrations comparable to the reference IC concentrations (i.e., the horizontal dotted lines) in the inserts, whereas the DDD method yields biased IC concentrations with significant SDs.

Fig. 20:

IC concentrations, along with their SDs, estimated within three ROIs in each of the basis images of solid water (left), 600-mg/ml calcium insert (middle), and 20-mg/ml iodine insert (right) obtained by using the NCPD algorithm with BRS 1 (blue dot, 3MDE-BRS1) and BRS 2 (orange triangle, 3MDE-BRS2) and by using the DDD method (green square, 2MDE-DDD). The horizontal dotted lines indicate the respective reference values.

5). Impact of basis-region selection on image reconstruction:

We also investigate the robustness of the NCPD algorithm with BRS 2 from real DE data. As displayed in column 2 of Fig. 18, the water, bone (i.e., calcium,) and iodine basis images, along with VMIs at 33 and 100 keV, are obtained from basis-region images reconstructed by use of the NCPD algorithm with BRS 2 from the same DE data as that used in the study with BRS 1. It can be observed that the basis images and VMIs obtained appear visually comparable to their counterparts in column 1 of Fig. 18 obtained with BRS 1. The observation is consistent with the quantitative results of LACs and IC concentrations estimated, which are shown in Figs. 19 and 20.

B. Results of abdominal-phantom-data study

1). Study conditions:

In the study, DE data were collected from a clinical abdominal phantom with the DECT system described above; and we consider reconstruction of basis images of water, bone, and 20-mg/ml iodine on an image array of 432 × 656 identical square pixels of 0.08 cm in size.

We choose two partitions, BRS 1 and BRS 2, as shown in Fig. 21 by applying appropriate threshold values to the image reconstructed from the 80-kV data with the FBP algorithm. In BRS 1 or BRS 2, the basis images highlighted in white contain, respectively, bone & water (column 1), iodine & water (column 2), and water only (column 3) basis materials. Therefore, each of the basis regions in BRS 1 or BRS 2 contains exactly 2 basis materials, resulting in $N=2\times I$ unknowns. In particular, BRS 1 and BRS 2 are chosen to contain basis regions substantially different in shape and size and are used for evaluation of the robustness of the NCPD algorithm with respect to the basis-region selection.

Fig. 21:

BRS 1 (row 1) and BRS 2 (row 2) partitions in which the basis regions highlighted in white contain, respectively, water & bone (column 1), water & 20-mg/ml iodine (column 2), and water-only (column 3) basis materials.

The TV constraints are enforced on VMIs at 50 keV and 100 keV in optimization program Eq. (6). As no truth values of ${\gamma }_{m_c}$ are available in the real-data study, we conduct, as that in the DE472-phantom study, multiple reconstructions with a slew of TV-constraint values and then choose ${\gamma }_1$ and ${\gamma }_2$ that yield visually basis images and VMIs with minimal visual artifacts as appropriate TV-constraint values in Eq. (6). In addition to the NCPD algorithm, we also use the DDD method to reconstruct basis images of water and bone and VMIs.

2). Visual inspection:

In columns 1 and 3 of Fig. 22, we show the basis images of water, bone, and iodine, along with VMIs at 33 keV and 100 keV, obtained by using the NCPD algorithm with BRS 1 and by using the DDD method. No significant visual artifacts are observed in the NCPD reconstructions, whereas the DDD reconstructions appear to be with a visible level of noise and possibly overly estimated IC concentration in the 33-keV VMI below the iodine K-edge.

Fig. 22:

Basis images of water (row 1), bone (row 2), and iodine (row 3), and VMIs at 33 keV (row 4) and 100 keV (row 5) obtained by using the NCPD algorithm with BRS 1 (column 1) and BRS 2 (column 2) and by using the DDD method (column 3). Display windows: [0, 1.0] for basis images and [−160, 240] HU for VMIs.

3). Impact of basis-region selection on image reconstruction:

We also investigate the robustness of the NCPD algorithm with BRS 2 when applied to real DE data. As displayed in column 2 of Fig. 22, the water, bone, and iodine basis images and VMIs at 33 and 100 keV are obtained by using the NCPD algorithm with BRS 2 from the same DE data used in the study with BRS 1. It can be observed that the basis images and VMIs obtained appear visually comparable to their counterparts in column 1 of Fig. 22 of BRS 1, indicating again the robustness of the NCPD algorithm relative to the BRS selection.

V. Discussion

We carry out in the work investigation and development of algorithms to yield accurately multiple ($K>2$) basis images and VMIs directly from DE data in CT, which remain of theoretical and practical interests. An observation can be made that CT images in some applications may be partitioned into spatially complementary basis regions of sizes smaller than the image array, and that each of the basis regions may contain a number of basis materials smaller than $K$.

Exploiting this observation, we derive the data model based upon basis regions and then tailor the NCPD algorithm to reconstruct basis-region images of a reduced number of unknowns directly from DE data. From basis-region images reconstructed, multiple basis images can be obtained subsequently on the entire image array.

Following numerical studies with simulated DE data to verify the NCPD algorithm’s reconstruction accuracy, we study its robustness when it uses an image-array partition inconsistent with the OBRS partition used in the data generation, and the result reveals that under this inconsistent condition, the NCPD algorithm remains to yield accurate basis images and VMIs, in terms of visual inspection, reconstruction quantification, and estimation of LACs and IC concentration.

Furthermore, we conduct real-data studies to evaluate the impact of multi-sources of inconsistencies on the robustness of the NCPD algorithm. The first type of inconsistencies stems from the likely differences between the image-array partition used in the NCPD reconstruction and the OBRS partition, which is unknown, underlying real data, whereas the other types of inconsistencies are in the forms of noise and additional physical factors inconsistent with the data model upon which the NCPD algorithm is based. The results of visual inspection, reconstruction quantification, and estimation of LACs and IC concentration reveal that the NCPD algorithm can robustly yield accurate multiple ($K>2$) basis images and VMIs over energy range 25 keV∼120 keV of clinical interest directly from real DE data containing various inconsistencies.

In real-data studies, both BRS 1 & 2 are obtained by applying a single threshold value to yield visually reasonable boundaries of the basis regions, without any additional finetuning steps. It is a straightforawrd and relatively efficient process to manually find the appropriate threshold values. On the other hand, it is possible to develop semi- or fully-automated techniques to obtain a BRS partition based on existing methods in image segmentation [21].

The NCPD algorithm has a flexibility to reconstruct images directly from low- and high-kV data collected over non-overlapping rays, unlike the DDD method requiring that low- and high-kV data are measured over completely overlapping rays. This flexibility allows for the application of the NCPD algorithm to possibly yielding $K>2$ basis images from DE data collected over partially-overlapping or non-overlapping rays [19], [22] in scan configurations different than that used in diagnostic DECT for meeting the unique imaging-workflow needs in emerging clinical and other applications.

While the NCPD algorithm can readily be generalized to the cases in which a basis region may contain > 2 basis materials, it remains to be investigated the impact of an increased number (> 2) of basis materials within a basis region on the performance of the NCPD algorithm. Furthermore, additional constraints on, e.g., the basis volume conservation, may be incorporated into the NCPD algorithm for expanding its applicability to reconstructing $K>2$ basis images from DE data collected over LARs [23] and/or at sparse views [24] with an increased degree of ill-conditionedness. The work can also be extended to yield $K$ basis images from $S$ sets of spectral or photo-counting data [25]–[28], where $K\ge S$ and $S>2$. We point out that the studies presented in the work is limited to DECT data containing no motion artifacts. In some in vivo experiments, cardiac or respiratory motion can be a source of inconsistent BRS, and its impact on the reconstruction remains to be investigated.

VI. Conclusions

In the work, we develop the NCPD algorithm to accurately yield multiple ($K>2$) basis images and VMIs directly from DE data in CT. Results of quantitative studies involving simulated, and real, DE data verify the accuracy, and demonstrate the robustness, of the NCPD algorithm. The NCPD algorithm may be extended to reconstruct multiple ($K>2$) basis images from DE data collected over LARs and/or at sparse views of application-workflow significance; and it can readily be extended to obtain a number of basis images that is larger than the number of spectral or photo-counting data sets.

Appendix I. Derivation of the data model based upon basis regions

In Sec. II-B, the standard data model based upon the entire image array in Eq. (1) can be re-expressed as

$$g_{sj}\left(\left\{b_{ki}\right\}\right)=-\ln \sum _{m_e=1}^{M_e}{q}_{sj{m}_e}\exp \left(-\sum _{i=1}^I{a}_{sji}{f}_{mi}\left(\left\{b_{ki}\right\}\right)\right),$$ (8)

where each of the $K$ basis images is of size $I$, and

$$f_{mi}\left(\left\{b_{ki}\right\}\right)=\sum _{k=1}^K{\mu }_{km}{b}_{ki}$$ (9)

denotes element $i$ of VMI $f_m\left(\left\{{\text{b}}_k\right\}\right)$ at energy $m$ given by Eq. (2). The total number of unknowns in the standard data model is ${\sum }_{i=1}^I{\sum }_{k=1}^{K}1=K\times I$, which can, when $K>2$, be significantly larger than the total number of 2 × $I$ unknowns in two basis images, making it challenging to accurately reconstruct $K>2$ basis images from DE data

a). Image-array partition and basis regions:

Observing, however, that in medical or industrial CT imaging, basis materials in some of the basis images are confined often within regions that may be considerably smaller than the image array, we discuss a partition of the image array into $L$ spatially complementary regions, referred to as basis regions, characterized by region mask ${\mathcal{R}}_{l{I}_l}$ and set ${\text{Φ}}_{l{K}_l}$, where mask ${\mathcal{R}}_{l{I}_l}$ is a diagonal matrix of size $I\times I$, with element $r_{lii}=1$, or 0, if voxel $i$ is within, or outside of, basis region $l$, and $I_l=\text{tr}\left({\mathcal{R}}_{l{I}_l}\right)\le I$ denoting the total number of voxels within basis region $l$, and set ${\text{Φ}}_{l{K}_l}$ of size $K_l=\text{card}\left({\text{Φ}}_{l{K}_l}\right)\le K$ includes elements that are indices $k\text{'s}$ of the $K_l$ basis materials contained in basis region $l$, with $l=1,2,\dots ,L$.

It can be observed that

$$\sum _{l=1}^L{\mathcal{R}}_{l{I}_l}=\mathcal{I}\text{and}{\text{b}}_k=\sum _{l=1}^L{\mathcal{R}}_{l{I}_l}{\text{b}}_k,$$ (10)

where $\mathcal{I}$ denotes the identity matrix of size $I\times I$. Substituting Eq. (10) into Eq. (9) yields

$$f_m\left(\left\{{\text{b}}_k\right\}\right)=\sum _{l=1}^L\sum _{k=1}^K{\mu }_{km}{\mathcal{R}}_{l{I}_l}{\text{b}}_k.$$ (11)

b). Orthogonal basis-region set (OBRS):

Given basis images, the basis-region set of a partition of the image array is referred to as the orthogonal basis-region set (OBRS) if the number and types of basis materials within each basis region are known, satisfying ${\mathcal{R}}_{l{I}_l}{\text{b}}_k=0$ for $k\notin {\text{Φ}}_{l{K}_l}$.

c). Data model based upon the basis regions:

For the OBRS partition, we can rewrite Eq. (11) as

$$f_m\left(\left\{b_k\right\}\right)=\sum _{l=1}^L\sum _{k\in {\text{Φ}}_{l{K}_l}}{\mu }_{km}{\mathcal{R}}_{l{I}_l}{b}_k.$$ (12)

We also construct matrix ${\mathcal{R}}_l^{\prime }$ of size $I_l\times I$ from matrix ${\mathcal{R}}_{l{I}_l}$ by removing all rows containing all zeros in ${\mathcal{R}}_{l{I}_l}$ and use $b_{lk}^{\prime }={\mathcal{R}}_l^{\prime }{b}_k$ to denote the basis-region image of material $k$ in basis region $l$ in the OBRS partition. It can be shown that

$${\mathcal{R}}_l^{\prime \top }{b}_{lk}^{\prime }={\mathcal{R}}_l^{\prime \top }{\mathcal{R}}_l^{\prime }{b}_k={\mathcal{R}}_{l{I}_l}{b}_k,$$ (13)

where matrix ${\mathcal{R}}_l^{\prime \top }$ of size $I\times I_l$ denotes the transpose of ${\mathcal{R}}_l^{\prime }$. Replacing Eq. (13) in Eq. (12) yields

$$f_m\left(\left\{b_k\right\}\right)=f_m^{\prime }\left(\left\{b_{lk}^{\prime }\right\}\right)=\sum _{l=1}^L\sum _{k\in {\text{Φ}}_{l{K}_l}}{\mu }_{km}{\mathcal{R}}_l^{\prime \top }{b}_{lk}^{\prime },$$ (14)

where vector $f_m^{\prime }\left(\left\{b_{lk}^{\prime }\right\}\right)$ of size $I$ is the basis-region VMI, with element $i$ given by

$$f_{mi}\left(\left\{b_{ki}\right\}\right)=f_{mi}^{\prime }\left(\left\{b_{lk{i}^{\prime }}^{\prime }\right\}\right)=\sum _{l=1}^L\sum _{k\in {\text{Φ}}_{l{K}_l}}{\mu }_{km}\sum _{i^{\prime }=1}^{I_l}{r}_{li{i}^{\prime }}^{\prime }{b}_{lk{i}^{\prime }}^{\prime },$$ (15)

where $r_{li{i}^{\prime }}^{\prime }$ is element $\left(i,i^{\prime }\right)$ in matrix ${\mathcal{R}}_l^{\prime \top }$. Finally, replacing $f_{mi}\left(\left\{b_{ki}\right\}\right)$ in Eq. (8) with Eq. (15) yields

$$\begin{gathered} g_{sj}\left(\left\{b_{ki}\right\}\right)=g_{sj}^{\prime }\left(\left\{b_{lk{i}^{\prime }}^{\prime }\right\}\right) \\ =-\ln \sum _{m_e=1}^{M_e}{q}_{sj{m}_e}\exp \left(-\sum _{l=1}^L\sum _{k\in {\text{Φ}}_{l{K}_l}}{\mu }_{k{m}_e}\sum _{i^{\prime }=1}^{I_l}{a}_{sjl{i}^{\prime }}^{\prime }{b}_{lk{i}^{\prime }}^{\prime }\right), \end{gathered}$$ (16)

Where $a_{sjl{i}^{\prime }}^{\prime }={\sum }_{i=1}^I{a}_{sji}{r}_{li{i}^{\prime }}^{\prime }$. We refer to Eq. (16) as the data model based upon basis regions, which is equivalent to the standard data model in Eq. (1) or (8) under the OBRS-partition condition.

It can be observed that the number of unknowns $\left\{b_{lk{i}^{\prime }}^{\prime }\right\}$ in Eq. (16) is

$$N=\sum _{l=1}^L\sum _{k\in {\text{Φ}}_{l{K}_l}}\sum _{i^{\prime }=1}^{I_l}1=\sum _{l=1}^L{K}_l\times I_l.$$ (17)

If any of the $K_l$’s is smaller than $K$, we have $N<K\times I$, i.e., $N$ is smaller than the total number of unknowns in the standard data model in Eq. (1) or (8). In each of the numerical studies in Secs. III and IV, a BRS partition is assumed with $K_l\le 2$ for $l=1,2,\dots ,L$, leading to $N\le 2\times I$ that is substantially smaller than $K\times I$ for, e.g., the case of $K=3$ or $K=6$ considered, allowing for possibly accurate reconstruction of multiple basis images from DE data, as demonstrated in Secs. III and IV.

d). Construction of the basis images from basis-region images:

For the OBRS partition, we use ${\text{Ψ}}_{k{L}_k}$ to denote a set encompassing elements that are indices of the $L_k=\text{card}\left({\text{Ψ}}_{k{L}_k}\right)\le L$ basis regions containing basis material $k$, where $k=1,2,\dots ,K$. Using Eqs. (10) and (13) and noticing ${\mathcal{R}}_{l{I}_l}{b}_k=0$ for $k\notin {\text{Ψ}}_{k{L}_k}$, we obtain

$$b_k=\sum _{l=1}^L{\mathcal{R}}_{l{I}_l}{b}_k=\sum _{l\in {\text{Ψ}}_{k{L}_k}}{\mathcal{R}}_{l{I}_l}{b}_k=\sum _{l\in {\text{Ψ}}_{k{L}_k}}{\mathcal{R}}_l^{\prime \top }{b}_{lk}^{\prime }.$$ (18)

Therefore, Eq. (18) provides a recipe for constructing exactly basis images $b_k$ from basis-region image $\left\{b_{lk}^{\prime }\right\}$ reconstructed under the OBRS partition condition.

Appendix II. The pseudo-codes and convergence conditions of the NCPD algorithm

a). Derivation of the NCPD algorithm:

We derive the NCPD algorithm following the same strategy used in the previous work [12], [13] to numerically accurately solve the constrained, non-convex optimization program in Eq. (6). Recognizing that the non-convexity of the optimization program in Eq. (6) is solely due to the non-linear data model in Eq. (4) (equivalently, Eq. (5)), or specifically non-linear component $\text{Δ}{\mathbf{\text{g}}}^{\prime }\left({\mathbf{\text{b}}}^{\prime }\right)$, we first convexify the objective function by replacing non-linear component $\text{Δ}{\mathbf{\text{g}}}^{\prime }\left({\mathbf{\text{b}}}^{\prime }\right)$ with $\text{Δ}\overline{\mathbf{\text{g}}}$ that is independent of $b^{\prime }$. An instance of the general PD algorithm [29], [30] can then be derived for solving the convexified constrained optimization program involving “measured data” $\left[g^{[\mathcal{M}]}-\text{Δ}\overline{g}\right]$ and linear model $\mathcal{H}\prime b^{\prime }$. Finally, by replacing $\text{Δ}\overline{g}$ with $\text{Δ}{g}^{\prime }\left(b^{\prime (n)}\right)$, where $b^{\prime (n)}$ is the basis-region images reconstructed at iteration $n$, in the derived PD instance for the convexified optimization program, we obtain the algorithm, referred to as the NCPD algorithm, for empirically solving the non-convex optimization program in Eq. (6). Its pseudo-codes are summarized below.

Algorithm 1.

Pseudo-codes of the NCPD algorithm solving Eq. (6)

1:	INPUT: $g^{[\mathcal{M}]},\mathcal{H}\prime ,{\gamma }_1,{\gamma }_2,\rho$
2:	$\nu \leftarrow \frac{{\Vert \mathcal{H}\prime \Vert }_2}{\parallel \mathcal{I}{\parallel }_2},{\alpha }_1\leftarrow \frac{{\Vert \mathcal{H}\prime \Vert }_2}{{\Vert \nabla {\mathcal{U}}_1\Vert }_2},{\alpha }_2\leftarrow \frac{{\Vert \mathcal{H}\prime \Vert }_2}{{\Vert \nabla {\mathcal{U}}_2\Vert }_2}$
3:	$L\leftarrow \parallel \mid \mathcal{K}{\parallel }_2;\sigma \leftarrow {\rho }^{\prime }/L;\tau \leftarrow 1/\rho L$
4:	INITIALIZE $b^{\prime (0)}$, $u^{(0)}$, $v^{(0)}$, $w^{(0)}$, $p^{(0)}$ to zeros
5:	${\overline{b}}^{\prime (0)}\leftarrow b^{\prime (0)}$
6:	repeat
7:	$u^{(n+1)}=\frac{u^{(n)}-\sigma \left(\left[g^{[\mathcal{M}]}-\text{Δ}{g}^{\prime }\left(b^{\prime (n)}\right)\right]-\mathcal{H}\prime {\overline{b}}^{\prime (n)}\right)}{1+\sigma }$
8:	$v^{\prime (n)}=v^{(n)}+\sigma {\alpha }_1\nabla {\mathcal{U}}_1{\overline{b}}^{\prime (n)}$
9:	$w^{\prime (n)}=w^{(n)}+\sigma {\alpha }_2\nabla {\mathcal{U}}_2{\overline{b}}^{\prime (n)}$
10:	$v^{(n+1)}=v^{\prime (n)}-\sigma \frac{v^{\prime (n)}}{\left|v^{\prime (n)}\right|}{\text{POL}}_1{\text{B}}_{{\alpha }_1{\gamma }_1}\left(\frac{\left|v^{\prime (n)}\right|}{\sigma }\right)$
11:	$w^{(n+1)}=w^{\prime (n)}-\sigma \frac{w^{\prime (n)}}{\left|w^{\prime (n)}\right|}{\text{POL}}_1{\text{B}}_{{\alpha }_2{\gamma }_2}\left(\frac{\left|w^{\prime (n)}\right|}{\sigma }\right)$
12:	$p^{(n+1)}=neg\left(p^{(n)}+\sigma \nu {\overline{b}}^{\prime (n)}\right))$
13:	$b^{\prime (n+1)}=b^{\prime (n)}-\tau ({\mathcal{H}}^{\prime \top }{u}^{(n+1)}+{\alpha }_1{\mathcal{U}}_1^{\top }{\nabla }^{\top }{v}^{(n+1)}+{\alpha }_2{\mathcal{U}}_2^{\top }{\nabla }^{\top }{w}^{(n+1)}+\nu p^{(n+1)})$
14:	${\overline{b}}^{\prime (n+1)}=2b^{\prime (n+1)}-b^{\prime (n)}$
15:	until convergence conditions are satisfied
16:	OUTPUT: $b^{\prime \ast }=b^{\prime (n)}$

b). Pseudo codes of the NCPD algorithm:

In the pseudo-codes, stacked matrix $\mathcal{K}$, of size $(J+2I+N)\times N$, and linear transforms ${\mathcal{U}}_{m_c}$, of size $I\times N$, for $m_c=1$ or 2 are defined as

$$\begin{array}{l}\mathcal{K}={\left({\mathcal{H}}^{\prime \top },{\alpha }_1{\left(\nabla {\mathcal{U}}_1\right)}^{\top },{\alpha }_2{\left(\nabla {\mathcal{U}}_2\right)}^{\top },\nu \mathcal{I}\right)}^{\top },\\ {\mathcal{U}}_{m_c}=\left({\mathcal{U}}_{1m_c}^{\prime },{\mathcal{U}}_{2m_c}^{\prime },\cdots ,{\mathcal{U}}_{L{m}_c}^{\prime }\right),\\ {\mathcal{U}}_{l{m}_c}^{\prime }=\left({\mathcal{R}}_l^{\prime \top }{\mu }_{k_1{m}_c},{\mathcal{R}}_l^{\prime \top }{\mu }_{k_2{m}_c},\cdots ,{\mathcal{R}}_l^{\prime \top }{\mu }_{k_{K_l}{m}_c}\right),\end{array}$$ (19)

where =$k_1,\cdots ,k_{K_l}\in {\text{Φ}}_{l{K}_l},\parallel \cdot {\parallel }_2$ computes the largest singular value of a matrix, neg(·) enforces the non-positivity by hard thresholding; operator POL₁B_r(·) indicates a ${\ell }_2$-projection onto the ${\ell }_1$-ball of size $r$ [31]; and $|\cdot |$ computes the magnitude image. Final basis-region image $\mathbf{\text{b}}\prime \ast$ is achieved when multiple necessary convergence conditions [13] are satisfied numerically, as summarized below.

c). Necessary convergence conditions of the NCPD algorithm:

We first introduce below the convergence metrics:

$$D_g^{(n)}=\Vert \left[g^{[\mathcal{M}]}-\text{Δ}{g}^{\prime }\left(b^{\prime (n)}\right)\right]-\mathcal{H}\prime b^{\prime (n)}{\Vert }_2/\Vert g^{[\mathcal{M}]}{\Vert }_2,$$ (20)

$$D_{f_{m_c}^{\prime }-\text{TV}}^{(n)}=\left|\left(\Vert \nabla {\mathcal{U}}_{m_c}{b}^{\prime (n)}{\Vert }_1-{\gamma }_{m_c}\right)\right|/{\gamma }_{m_c}\left(m_c=1,2\right),$$ (21)

$$D_{\text{b}-\text{NNEG}}^{(n)}={\Vert \text{neg}\left(b^{\prime (n)}\right)\Vert }_2,$$ (22)

where $b^{\prime (n)}$ denotes the congregate basis-region image reconstructed at iteration $n$ and $g^{[\mathcal{M}]}$ measured DE data in a congregate form. At iteration $n$, Eq. (20) measures the difference between measured data and model data estimated; Eq. (21) depicts the relative differences between the TV of the basis-region VMIs and the values selected for respective TV-constraint parameters at two energy levels; and Eq. (22) computes the magnitude of negative voxels within the congregate basis-region image.

In the simulated-data study, measured data $g^{[\mathcal{M}]}$ are generated by use of the data model in Eq. (4) and thus consistent with model data $g^{\prime }\left(b^{\prime \text{truth}}\right)$, i.e., $g^{[\mathcal{M}]}=g^{\prime }\left(b^{\prime \text{truth}}\right)$. Conversely, in the real-data study, measured data $g^{[\mathcal{M}]}$ contain physical factors that are inconsistent with the data model in Eq. (4). Using the convergence metrics defined in Eqs. (20)−(22), we introduce necessary convergence conditions for the NCPD algorithm in Algorithm 1 as

$$\begin{array}{l}{D}_g^{(n)}\to e,\\ D_{{\text{f}}_{m_c}^{\prime }-\text{TV}}^{(n)}\to 0\text{for}{m}_c=1\text{and}2,\\ D_{\text{b-NNEG}}^{(n)}\to 0,\end{array}$$ (23)

as $n\to \infty$, where constant $e=0$ in the consistent simulated-data study or $e>0$ in the inconsistent simulated- and real-data study.