Fig 1.
Pseudocode for hybrid CT reconstruction with the split Bregman method.
The objective of hybrid CT reconstruction is to synthesize high-resolution, spectral CT reconstructions from high-resolution, energy-integrated projection data, w, and lower-resolution and nosier photon-counted projection data, Y. In addition to the projection data, expected material sensitivity values (M, mEID) and user-specified regularization parameters (λC, h0, α, γ, s) are provided as inputs. FBP reconstruction and component substitution provide estimates of the reconstructed results (steps 1–4) which are further refined with algebraic reconstruction during initialization (steps 5–12). Following initialization, the hybrid reconstruction results are refined under low spectral rank and intensity-gradient sparsity constraints through iterative application of the split Bregman method (step 13–18). Following reconstruction, the hybrid results are used to compute high-resolution material decompositions (step 19).
Fig 2.
Bilateral filtration (BF) with tiling.
Given a volume to be filtered (1), a tiling operation is first applied to reduce correlation in the noise and to effectively extend the filtration domain size (2). BF is applied to this tiled volume (3), and then the output volume is recovered by reversing the tiling operation (detiling, 4). Note the difference in the median absolute deviation (MAD) measured in the input (1) and tiled (2) volumes (bottom), given an input volume which has been upsampled (BT operator). As with all figures in this work, the window width and level for each panel as well as the appropriate units are as indicated by the calibration bar (3, bottom; HU: Hounsfield units). The absolute scale is as shown by the scale bar (4, bottom right). All CT reconstructions and material decompositions in this work are presented as single 2D slices, without averaging between slices, unless otherwise specified.
Fig 3.
Pseudocode for regularization with rank-sparse kernel regression (RSKR).
The objective of RSKR is to enforce matching intensity gradient sparsity patterns and low column rank on a matrix of spectral CT data taken as input. To achieve this, RSKR operates on a weighted singular value decomposition of the spectral data (step 1) and calibrates the regularization strength for each singular vector based on ratios of the corresponding singular values (step 2). Intensity gradient sparsity patterns are copied between singular vectors through joint bilateral filtration (jBF; steps 5 and 7). A rank reduction effect is achieved by allowing proportionally stronger regularization for less significant singular vectors (step 2, step 10) over the course of several internal Bregman iterations. Within the context of hybrid spectral CT reconstruction, RSKR is embedded as a sub-step within our proposed algorithm (Fig 1, step 13).
Fig 4.
Preprocessing applied to the PCD projection data.
(A) Example log-transformed PCD projection prior to processing (threshold: 26 keV). The yellow, dotted line denotes a single detector row. The readout of this row is shown as a function of angle in the bottom row of this figure (sinogram). (B) Corresponding PCD projection following the three forms of correction described in the text. In the single projection (row 1), an inset (corresponding yellow boxes) and red arrows highlight overly dark pixels before (A) and after (B) ring artifact prevention. In the sinogram (row 2), similar insets and arrows denote detector pixel readouts notably affected by ring artifact correction. (C) Absolute difference computed between (A) and (B). Note bright bands where the detector gaps were interpolated (red arrows). Also note the differences in windowing between columns (A) and (B) (below (A)) and column (C) (below (C)).
Fig 5.
3D digital simulation phantom.
The phantom we constructed for assessing our proposed hybrid spectral CT reconstruction algorithm consists of three key features: line pairs to assess spatial resolution, spheres to assess detection, and vials to make volumetric spectral measurements. These features are arranged in three disks along the z-axis. The line pairs and spheres in each disk exclusively contain one of the three contrast materials: iodine (red), calcium (blue), and barium (green). The phantom is synthesized from material fraction maps which denote fractions of the maximum concentration of each contrast material (1.0: 15 mg/ml, iodine; 15 mg/ml, barium; 75 mg/ml calcium). Here, and elsewhere in this paper, material decompositions are shown as overlaid material maps, coded by basis function (color) and concentration (intensity; multiple relative to water). The window width and level for the CT data and for each material map are as shown.
Fig 6.
Spectral modeling of PCD data acquisition.
(A) 80 kVp tungsten source spectrum. (B) Quantum efficiency for 1 mm of CdTe in the PILATUS3 detector. (C) Idealized spectral binning (rect functions) applied to the element-wise product of the source spectrum and the detector sensitivity function for each of the energy thresholds used for imaging in this paper. (D) Spectral basis functions considered in this work. Note that the PE and CS basis functions are scaled to sum to the attenuation of water (each with a coefficient of one). The attenuation coefficient curves for iodine and barium are scaled to a density of 20 mg/ml for display purposes. (E) Unit-norm material sensitivity vectors for each of the basis materials, as predicted by the spectral model. (F) Unit-norm material sensitivity vectors as predicted by the spectral model, including a Gaussian energy-spread function with a standard deviation of 3.25 keV, to match the conditioning of sensitivity vectors derived for our in vivo experiment (G).
Fig 7.
Pseudocode for PCD only reconstruction with the split Bregman method.
As a control experiment for the proposed hybrid spectral CT reconstruction algorithm (Fig 1), which uses both EID (w) and PCD (Y) projection data, we performed a second iterative reconstruction using only the PCD projection data. This PCD only reconstruction algorithm is largely analogous to the hybrid reconstruction algorithm, with two main exceptions. First, the PCD data is reconstructed directly (X) rather than indirectly as a function of the EID reconstruction (XL) and the spectral contrast provided by the PCD data (XS). Second, during initialization, the dictionary is trained using a variance () weighted average of the initialized reconstruction results (avge(X), step 7) rather than using a FBP reconstruction of the EID data (Fig 1, step 11).
Fig 8.
Simulated results of component substitution and algebraic initialization in hybrid reconstruction.
(A) Initial FBP reconstruction using the EID projection data. A magnified inset (yellow box) shows the set of line pairs with a spatial frequency of 2.84 lp/mm. Note that the magnified inset (only) is intensity averaged over 21 consecutive, 2D slices for improved visibility. (B) Initial upsampled, FBP reconstructions using the PCD projection data corresponding with thresholds of 26 (lowest noise level) and 45 (highest noise level) keV by row (Fig 1, step 1). (C) The results of component substitution (XL,0 + XS,0; Fig 1, steps 3–4). (D) Results of algebraic initialization (Fig 1, step 5). The RMSE, computed over the entire reconstructed volume, is reported in HU at the bottom-left corner of each panel. Note that all simulation results in this paper are shown with a voxel size of 88 μm3.
Fig 9.
Regularization during the first iteration of hybrid reconstruction.
(A) Initialization results for the 37 keV threshold in the iodine disk. Rows show XL (EID reconstruction), XS (PCD contrast), and the absolute difference between XL + XS and the expected reconstruction result (|Residual|). In the first row, yellow boxes denote three regions of interest which are concatenated and magnified at the bottom of each panel for improved visibility. Note that these regions of interest represent a single 2D slice. (B) RSKR applied to (A) without tiling (s = 1). (C) RSKR applied to (A) with tiling (s = 3). A contrast-enhanced and 21-slice averaged inset in the second row (red box) highlights striping artifacts introduced because of tiling. (D) RSKR applied to (A) and following the steps outlined in Fig 3 steps 4–8 (“Average”). Red arrows denote intensity bias introduced when striding is used (row 2: contrast inversion in XS; row 3: localized bias in XL + XS). (E) OMP applied to (D).
Fig 10.
Quantitative analysis of the simulated regularization and reconstruction results.
These measurements were taken in each of the phantom’s calibration vials and then averaged to yield a single value per energy threshold ((A) absolute bias, Eq 63; (B) standard deviation; (C) RMSE, Eq 62). The first five sets of bars (“Iteration 1 (Hybrid Reconstruction)”) match the experimental conditions of the results summarized in Fig 9. The last two sets of bars (“Final Reconstruction”) represent the final reconstruction results after 6 iterations of the split Bregman method (“PCD Only”: Fig 7, steps 9–13; “Hybrid”: Fig 1, steps 13–18). Borrowing from the layout in Fig 9, the bar plots are accompanied by a visual comparison of the results (right column; 37 keV threshold; XL + XS).
Fig 11.
Comparison of hybrid and PCD only reconstruction results using identical PCD projection data.
(A) Expected reconstruction results in a single 2D slice through the center of each material disk (37 keV threshold). The magenta box denotes a set of line pairs that are magnified and contrast-enhanced for comparison (single 2D slice; 2.84 lp/mm). The red box denotes the spherical lesions used for detectability analysis in (G). (B) Final PCD only reconstruction results (X). (C) Absolute difference between (A) and (B). (D) Final hybrid reconstruction results (XL + XS). (E) Absolute difference between (A) and (D). (F) Gaussian MTFs fitted from MT measurements taken in all 3 material disks and all 5 energy thresholds (error bars: ±1 SD). (G) The increase in the detectability index associated with hybrid reconstruction over PCD only reconstruction for each of the spherical lesions. The results are organized by material disk, diameter, and material concentration (in mg/ml) and are averaged over all 5 energy thresholds.
Fig 12.
Material decomposition results for the phantom simulation experiment.
(A) Expected material decomposition shown in the central 2D slice of each material disk. (B) Reference material decomposition performed following algebraic initialization (PCD only reconstruction; Fig 7, step 2). (C) Final PCD only material decomposition results (Fig 7, step 14). (D) Final hybrid material decomposition results (Fig 1, step 19). The global RMSE computed for each expected material map is shown in the upper-left of each panel in the assigned units (see calibration bars, upper-right). Insets (yellow boxes in column 1) compare the expected decomposition results (inset, top) with the final hybrid results (inset, bottom; 2.84 lp/mm; expected material only, by column). (E) Line profiles (white, dotted line in (D)) for each expected material. Note the PCD only line profile is taken from (C).
Fig 13.
Hybrid, spectral CT reconstruction in an in vivo mouse model of soft tissue sarcoma.
(A) 2D, sagittal slice through the algebraic initialization results shown for the least noisy data set (26 keV threshold) and the most noisy data set (45 keV threshold) by column (Fig 1, step 5; XL + XS). A yellow oval denotes the location of the sarcoma tumor on the flank of the mouse. Red squares denote a region of interest (“ROI”) which is magnified, at right, for both thresholds. (B) Final PCD only reconstruction results (X; 6 iterations of regularized reconstruction). (C) Final hybrid reconstruction results for XL (6 iterations of regularized reconstruction). Red arrows within the magnified region of interest denote high-contrast features which appear to be better resolved in the hybrid reconstruction. (D) Final hybrid reconstruction results for X = XL + XS. Magenta arrows denote attenuation artifacts around bone.
Fig 14.
In vivo material decomposition.
(A) PCD only (left) and hybrid (right) decomposition results for the 2D sagittal slice shown in each panel of Fig 13. Magenta arrows in the main panels (1) denote a region of barium enhancement within the tumor where the concentration appears lower in the hybrid results than in the PCD only results. Magenta boxes and the arrows within them (2) refer to the same ROI and features as in Fig 13. (B) Axial slice through the left kidney, liver, spleen, and spine of the mouse. Yellow labels and arrows denote three material calibration vials used to calibrate the material sensitivity matrix. A magenta arrow near the spleen (3) denotes material decomposition errors which result from physiological motion between scans.
Fig 15.
In vivo material decomposition (continued).
(A) Maximum intensity projections (MIPs) computed through the segmented sarcoma tumor using the XL component of the final hybrid reconstruction results. The same tumor is shown in coronal, axial, and sagittal orientations. (B) Comparable MIPs through the material decomposition of the final PCD only reconstruction results. (C) Comparable MIPs through the material decomposition of the final hybrid reconstruction results. Magenta arrows in (B) and (C) denote features of interest: (1) potentially misclassified vessels; (2) iodine artifact resulting from localized misregistration between the EID and PCD data; and (3) apparent material decomposition errors around bone.
Table 1.
User-specified parameters.