Hybrid data fidelity.

We employ an algebraic approach for hybrid, spectral CT reconstruction. The log-transformed, system-calibrated, and vectorized projection data is represented by for the PCD chain and by w for the EID chain, where and . n_y and n_w equal the number of detector elements times the number of projections for each chain. n_e is the number of energy bins sampled with the PCD. The projection data is used to reconstruct , the low-column-rank EID data, and , the sparse, high-resolution spectral contrast (). For simplicity, a single spectral data set from the EID chain is assumed (rank(X_L) ≈ 1); however, we note that this is not generally required for reconstruction problems based on low rank and sparse decomposition.

Hybrid reconstruction is then performed via weighted least-squares optimization: (1) The two least-squares terms enforce data fidelity relative to w and y_e, respectively, for each energy bin sampled with the PCD detector. The fidelity terms are balanced by the energy-dependent scaling parameter, λ_C,e (vectorized form: λ_C). Expanding the first data fidelity term, the ‖⋅‖_Q notation denotes least-squares weighting: (2) R is the system projection matrix for the EID data (). As in [39], the least-squares weights are calibrated based on the observed projection data, as an estimate of the expected projection data, and the system-calibrated scaling parameters g and η: (3) (4) The diagonal elements of Q represent variance estimates for each line integral of the projection data. An analogous, energy-dependent weighting matrix, Z_e, is constructed for each energy bin of the PCD chain. Notably, the use of diagonal weighting matrices approximates the detector noise as uncorrelated between neighboring detector pixels.

Returning to Eq 1, the following signal model is implied for the PCD data: (5) (6) The high-resolution PCD data is related to the observed, low-resolution PCD data through the blurring and resampling operator, B (), the PCD projection matrix, A (), and additive, zero-mean Gaussian noise, ε_e (). n_x′ denotes the number of voxels in the low-resolution PCD data. Notably, the inclusion of Gaussian noise in the signal model approximates the Poisson photon statistics as Gaussian under log transformation and under the assumption of adequate photon flux in each energy bin of the PCD data. Consistent with the use of diagonal weighting matrices, the noise is assumed uncorrelated between detector elements at the native, low resolution.

Under the (somewhat liberal) assumption that the point-spread function for each imaging chain is well approximated with a spatially-invariant Gaussian kernel (G), the B operator resamples the high-spatial-resolution EID data (G_EID) to match low-resolution PCD data (G_PCD) with an appropriate resampling kernel (G_r): (7) The Cartesian spatial vectors i and j denote discrete sampling locations (voxel centers) within the high- (x_e(i)) and low- () spatial-resolution data, respectively. Note that we define the x_e(i) notation to imply the treatment of vectors (x_e) as volumes indexed by 3D coordinates (i). The appropriate standard deviation for the resampling kernel, σ_r, is analytically computed from the measured full-width-at-half-maximum (FWHM) of the EID and PCD point-spread functions: (8) Resampling is then performed with a discrete, normalized convolution operation evaluated at each sampled spatial location within the low-resolution data, j: (9) B(j) denotes the blur operator evaluated to produce a low-resolution sample at position j. As denoted elsewhere in the paper, Bx_e denotes this resampling operation evaluated at all spatial positions, j. Assuming isotropic voxels, the scalar multiplier, c, scales attenuation measurements based on the voxel length ratio between the reconstructions of the PCD and EID data. The logical transpose of this resampling operation, which will be required in the following derivation, estimates a high-resolution sample at position i from the low-resolution samples: (10)

Penalized algebraic reconstruction with the split Bregman method.

Given that algebraic deblurring is an ill-conditioned inverse problem, particularly at noise levels encountered in small animal micro-CT, robust regularization is required for successful hybrid reconstruction. Redefining the data fidelity terms (Eq 1) as a function of x_L,e and x_S,e, f(x_L,e,x_S,e), we introduce rank and sparsity constraints on the solution to the hybrid reconstruction problem: (11) ‖X_L‖_* denotes the nuclear norm, or the sum of singular values of the matrix X_L, which is a convex proxy for column rank [40]. ‖X_S‖_BTV denotes the bilateral total variation (BTV, [21, 41]) of X_S and is a weighted L¹ norm which jointly enforces intensity gradient sparsity between the columns of X_S. BTV is discussed further in a subsequent section.

Following several previous works [27, 37], we solve Eq 11 using the split Bregman method and the add-residual-back strategy. Specifically, we split Eq 11 into a series of equivalent sub-problems, each of which is more tractable than the original problem. The nuclear norm of X_L is first reduced: (12) where D_L (d_L,e) and V_L (v_L,e) are auxiliary variables introduced during the splitting which track the regularized estimate of X_L and the regularization residuals, respectively. Eq 12 can be solved via soft thresholding of the column- (energy-) wise singular values of X_L + V_L [40]. Second, the BTV of X_S is reduced with additional auxiliary variables D_S and V_S: (13) The cost in Eq 13 is reduced by the application of joint bilateral filtration (BF) between the columns of X_S + V_S [36].

Following from the add-residual-back strategy, the residual variables associated with each regularization term are then updated: (14) (15) Notably, the use of equality signs in Eqs 14 and 15 represents overwriting of the values in the V_L and V_S variables with the computed quantities. Finally, a lower-cost solution for the original cost function (Eq 11) is found by solving the following convex optimization problem for each energy: (16) Notably, in Eq 16 we have introduced energy dependence for the regularization parameters μ_L,e and μ_S,e. These parameters are scaled with energy to account for differences in attenuation magnitude and noise level (see below). The λ_L and λ_S parameters are also, technically, energy dependent (Eqs 12 and 13); however, their effective values are adapted based on the input data (X_L + V_L, X_S + V_S; see the Spectral regularization sub-section), and they are not explicitly computed or assigned.

Because Eq 16 is convex and evaluated separately for each energy, it can be solved for each energy following differentiation with respect to x_L,e and x_S,e: (17) (18) Due to the large size of the projection operators in these linear equations (R, A; corresponding backprojection operators, R^T, A^T), we iteratively solve Eqs 17 and 18 for x_L,e and x_S,e using the biconjugate gradient stabilized method (BiCGSTAB, [42]). We have successfully employed BiCGSTAB for regularized algebraic reconstruction in previous work [36]. Multiple Bregman iterations (evaluations of Eqs 12 and 16) are typically required to achieve convergence with respect to the original cost function (Eq 11). The algorithm is terminated when a maximum number of iterations is reached or when the change in the magnitude of the residual variables between iterations falls below some tolerance.

One primary difficulty in finding meaningful solutions to Eq 11 is the selection of the regularization parameters λ_C,e, μ_L,e, and μ_S,e (Eqs 12–16), all of which depend on energy. The λ_C,e parameter for each energy balances the EID data fidelity term with the corresponding PCD data fidelity term. For the experiments presented in this work, the field-of-view coverage and total attenuation were similar enough that λ_C,e = 1 made a reasonable choice for all energies. Selection of the μ_L,e and μ_S,e parameters for each energy, which balance the data fidelity and regularization terms of the cost function following splitting (Eq 16), was more complicated due to significant differences in noise variance between the energy bins of the PCD data. Assuming approximate spatial uniformity of the noise level within each reconstructed energy bin, the image domain noise standard deviation can be robustly estimated as the median absolute deviation (MAD) of the component of the redundant Haar wavelet transform high-pass filtered along each dimension (e.g. in 3D: HHH) [43]: (19) Practically, we note that the accuracy of this noise estimation scheme need only be relative to other energies, given that the noise estimates are only used as ratios (see below) or after multiplication with a user-specified regularization parameter (see Spectral regularization). Furthermore, the requirement of a spatially uniform noise level is relaxed when using weighted least-squares (Eq 2), since low data-fidelity weights are assigned to the nosiest (largest magnitude) line integrals. This increases the effective regularization strength in highly attenuating features.

Given MAD-based noise estimates for each energy, μ_L,e and μ_S,e are then calibrated with standard deviation ratios, δ_e, and with an empirically determined scaling factor, α: (20) (21) The notation min_e denotes that each noise level estimate is normalized relative to the smallest estimate measured over all energies. Normalization of σ_e by the expected attenuation of water in each energy bin, μ_water,e, promotes consistent noise levels when the final reconstructed results are converted to the Hounsfield scale or are used for material decomposition. The convenience of achieving good reconstruction results with a predetermined α parameter (Eq 21) and the rapid convergence of our overall algorithm rely on further internal adaptation of the regularization strength (see Spectral regularization). Taking inspiration from the right-hand sides of Eqs 17 and 18 and scaling with respect to these parameters (α, δ_e), meaningful values of μ_L,e and μ_S,e are analytically estimated with the following formulas: (22) (23) Following modifications described in the Spectral regularization sub-section, Fig 1 summarizes our application of the split Bregman method to the problem of hybrid CT reconstruction.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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, m_EID) and user-specified regularization parameters (λ_C, 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).

https://doi.org/10.1371/journal.pone.0180324.g001

Component substitution and material decomposition.

As shown in Fig 1, the first steps (1–4) of hybrid reconstruction initialize X_L (X_L,0) and X_S (X_S,0). These component substitution steps are, themselves, initialized with filtered backprojection (FBP) reconstructions. Specifically, for the experiments in this work we used the FDK algorithm [44] with a ramp filter. FBP reconstruction (denoted ) of each column (energy) of Y followed by upsampling (B^T) yields an estimate of the high-resolution PCD data at each energy: (24) Nominally, X_S,0 can then be estimated as the difference between X₀ and X_L,0. In practice, however, due to initial resolution differences between X₀ and X_L,0, this strategy was found to introduce high frequency artifacts which were not well addressed by subsequent regularization. Therefore, following from our previous work [7, 14], we instead used material decomposition to estimate X_L,0 from X₀ and then estimated X_S,0 as the difference between these resolution-matched estimates: (25) (26) The material sensitivity matrix for the PCD data, M (), relates the attenuation at each energy to material specific measurements (e.g. m_{I,e = 1} is the attenuation of 1 mg/mL of iodine at energy 1; n_m, total number of basis materials). Inversion (n_e = n_m) or, more generally, pseudo-inversion (n_e > n_m) of the material sensitivity matrix (M⁺) is used to estimate the material concentrations (fractions) C (; Eq 25). Multiplication of these concentrations by the material sensitivities measured in the EID data, m_EID (), yields the required low-resolution estimate for X_L,0. Following from the Introduction, we note this is a form of component substitution, a common strategy for pan-sharpening, given that X_S will be added to the high-resolution estimate for X_L,0 produced from the EID projection data (step 4). This estimation and substitution procedure is used to condition and accelerate the convergence of the initial algebraic reconstruction (initialization; Fig 1, step 5) prior to regularized reconstruction with the split Bregman method.

Following hybrid reconstruction with the split Bregman method, the final resolution-enhanced material maps, C, are obtained via material decomposition subject to a non-negativity constraint (Fig 1, step 19): (27) As before, this problem is readily solved by pseudo-inversion of the material sensitivity matrix, M (columns: m_e). Non-negativity is enforced on the least-squares decomposition of each voxel by orthogonal projection onto the subspace of positive material concentrations [45]. Building on the approach of Alvarez and Macovski [46], we choose the photoelectric effect (PE), Compton scattering (CS), iodine (I), and barium (Ba) as material basis functions in this work. Notably, the K-edges of iodine (33.2 keV) and barium (37.4 keV), the primary features which discriminate their attenuation from the PE and CS basis functions, are less than 5 keV apart, providing a rigorous test for the regularization strategy we propose in the next sub-section.

Joint bilateral filtration and bilateral total variation.

RSKR is based on the application of joint bilateral filtration (BF) to spectral CT data. Under the assumption of structural redundancy between energies, joint BF performs non-linear, edge-preserving smoothing to enforce matching intensity gradient sparsity patterns between energies [7]. Mathematically, we apply joint BF as follows: (28) (29) (30) The Cartesian spatial vectors o and p denote the voxel being filtered and the set of all discrete spatial offsets within the filtration neighborhood, respectively. BF then takes the form of a 3D convolution operation (Eq 28) which adapts to the local filtration domain, D(p) (b, domain radius; Eq 30), with jointly computed range weights, R_joint(o,p) (Eq 29). The contribution of each energy to the joint range kernel is scaled by the noise level measured immediately prior to filtration (σ_e, Eq 29; measured as the MAD, Eq 19) and a scalar parameter which controls regularization strength, h_e. The jointly Gaussian range weights are then computed from the intensity differences measured within the filtration domain: (31) K(t) is a spatial resampling kernel (domain indexed by t) which improves robustness in band-limited data and ensures consistent regularization results at edges. In most of our previous work, we have used a second-order resampling kernel for K(t) [7]. Here, however, we use a delta function: (32) where 0 denotes the Cartesian spatial position of the kernel’s origin. The motivation for this change is for compatibility with a tiling operation we introduce (see below) and to avoid unnecessary resolution losses prior to the application of sparse coding (see Dictionary learning).

Comparing this implementation of joint BF with more common, direct sparsity constraints (e.g. TV), joint BF improves robustness because it operates on multiple scales of image derivatives. Furthermore, the abstraction of gradient information to intensity-independent probabilities enables contrast-independent spectral smoothing, while the adjustment of regularization strength and range kernel contribution based on the noise level measured at each energy adapts to the specific problem. These properties of joint BF are encapsulated in the BTV metric. Specifically, within the context of gradient-sparsity constrained, iterative reconstruction and the split Bregman method, BTV is measured as the application of the joint BF weights to the intensity gradients measured for each energy [21, 41]: (33) The application of BF to each column of X_S + V_S using the jointly computed range weights reduces BTV, yielding a lower-cost solution for D_S in the regularization update step (Eq 13).

Given this formalism for joint BF and BTV minimization, we note that BF performs sub-optimally in removing low-frequency, correlated noise which is introduced when upsampling noisy, low-resolution PCD data (B^T operator). Low frequency denoising performance of BF could be addressed by increasing the diameter of the filtration domain (i.e. b > 6, our usual value; Eq 30); however, computation time scales cubically with kernel diameter. Instead, as previously mentioned in the Introduction, we take inspiration from the problem of super-resolution, constructing several aliased volumes from each input volume (energy) prior to noise estimation with the MAD and to the application of joint BF. Given the appearance of these aliased volumes when they are constructed within the original volume (Fig 2), we call this operation tiling. Practically, tiling works well for several reasons. First, when a delta function is used for resampling (Eq 32), BF operates on the intensity of individual voxels. Combining this property with the high degree of redundancy of x-ray CT attenuation values, BF generally performs well when applied to tiled volumes. Low-frequency noise in the original volume appears as high-frequency noise in the tiled volume, more accurately reflecting the amount of noise that must be removed to match the spatial fidelity of the EID data and more accurately adapting the regularization strength between iterations of the reconstruction algorithm. The tiling operation itself negligibly increases computation time, since it involves a simple remapping of voxel spatial locations (see below), followed by BF of the tiled volume with a usual domain size (here, b = 6). The end effect is that tiled BF tends to quantize local intensity values to match global intensity modes, readily addressing much lower frequency noise than standard BF [38]. As illustrated in the Simulations sub-section of the Results, the potential downside to tiled BF is that intensity bias must be carefully managed to avoid compromising material decomposition fidelity.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 (B^T 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.

https://doi.org/10.1371/journal.pone.0180324.g002

The tiling operation is implemented as a function of the stride parameter, s, which controls the step size between voxels to be included within the same tiled sub-volume. Intuitively, s should be larger than the FWHM of the resampling kernel used to upsample the low-resolution data (measured in high-resolution voxel widths; here, s = 3 voxels is used). Using this stride, the tiling operation maps intensity values at Euclidean voxel indices within the original volume, m (intensities x(m)), to tiled voxel indices, n (intensities t(n)), within a constant, global coordinate system: (34) The 3D index vectors are themselves indexed by o, denoting the x (o = 0), y (o = 1), and z (o = 2) axes, respectively. The number of voxels along each axis is contained within the vector g (padded to integer multiples of s): (35) Consecutive integer indices are assigned to each axis of n: (36) Given these definitions, the following coordinate transformation can then be used with Eq 34 to map the input volume intensities to the tiled volume intensities: (37) The mod(a,b) function denotes the modulo operator applied to a and b, and ⌊⋅⌋ denotes rounding down to the nearest integer. Following BF and reusing the initial m and n indices, the tiling operation is then reversed with the following inverse mapping operation (detiling): (38) The tiling and detiling operations and their relation to BF are represented graphically in Fig 2.

RSKR: Joint bilateral filtration of singular vectors.

Within the context of the split Bregman method and hybrid spectral CT reconstruction, the dual objective of RSKR is to enforce matching intensity-gradient sparsity patterns and low rank between the EID and PCD reconstructions. To develop this dual rank-sparsity constraint, we first note that the constraint terms of the objective function (Eq 11), which are based on a general model for low rank and sparse matrix decomposition (i.e. robust principal component analysis, [34]), can be further specialized for the problem of hybrid spectral CT reconstruction. Specifically, Eq 11 enforces low rank between the columns of X_L and matching intensity gradient sparsity patterns between the columns of X_S; however, it does not exploit the fact that X_L and X_S must exhibit complementary intensity gradient sparsity patterns (i.e. they must add to the PCD data). Also, it does not exploit that the rank of X_L + X_S should ideally be lower than the number of columns when the number of sampled energies exceeds the number of unique materials (n_m < n_e). To incorporate these additional constraints, we replace Eqs 12 and 13 with the following combined update equation: (39) (40) The bracket notation ([· ·]) denotes concatenation of the columns of each component matrix (Eq 40). As in Eqs 12 and 13, the new regularization parameters, λ_BTV and λ_*, have some energy dependence; however, they are never explicitly computed or set (see below).

Comparing Eq 39 with the original objective function (Eq 11), the dual constraint terms appear counter-productive given the apparent difficulty in simultaneously reducing BTV and the nuclear norm subject to a data fidelity constraint. To alleviate this difficulty and to enforce matching intensity gradient sparsity patterns and low rank between the columns of X_L,S + V_L,S, we first perform a weighted, reduced singular value decomposition: (41) (42) The diagonal weighting matrix, P, (Eq 41; scalar diagonal elements p_e,e) is a function of the standard deviation ratios computed during the initialization of the algorithm, δ_e, (Eq 20) and the expected attenuation of water at each energy, μ_water,e. We call this weighting scheme prioritization because it encourages the most significant singular vectors, U₀, (i.e. the singular vectors with the largest associated singular values, E₀) to correspond with significant modes of spectral contrast rather than noise. Specifically, the water normalization roughly equalizes the attenuation magnitude at each energy, up to significant spectral differences (e.g. changes in attenuation over K-edges). The δ_e weighting factors, which are based on water-normalized noise measurements (Eq 20), further bias the most significant singular vectors to best fit the energies with the highest signal-to-noise ratios.

Given the prioritized singular value decomposition of the input data, we then solve for denoised singular vectors, U (n_u columns), under a BTV constraint: (43) In terms of the pseudocode for our algorithm, we refer to the process of performing singular value decomposition and then solving Eq 43 as RSKR (Fig 3). Within a single step of a global Bregman iteration (Fig 1, step 13), we solve Eq 43 using several internal Bregman iterations and an independent set of variables which are reset between global iterations (Fig 3): (44) (45) (46)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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).

https://doi.org/10.1371/journal.pone.0180324.g003

As before (Eq 13), the cost associated with Eq 44 is reduced through the application of joint BF to the columns of U + F, including tiling and noise estimation with the MAD prior to filtration, yielding the regularized output, L (analogous to D in the global Bregman iterations). Eq 45 updates the regularization residuals stored in F (analogous to V in the global Bregman iterations). The data fidelity update equation (Eq 46) is convex and can be solved analytically for each singular vector following differentiation with respect to u_i: (47) In the simulation and in vivo experiments detailed later in this work, 3–6 Bregman iterations were required for convergence of RSKR (<1% change in the regularization residual magnitude between iterations). After convergence, the regularized estimate of the input data, D_L,S, is recovered using the regularized singular vectors, U: (48) The relative magnitudes of the columns of D_L and D_S are then recovered using the priority weights (Fig 1, step 14): (49)

From the description of RSKR, it is not immediately clear how we achieve rank reduction to enforce structural redundancy between the columns of X_L and X_S. Perhaps the most direct method would be to apply soft thresholding to the singular values (E₀, [40]) in addition to denoising the singular vectors. This idea is particularly attractive since it, ostensibly, addresses the difficulty in reducing the cost associated with the dual rank-sparsity constraint directly (Eq 39). Practically, however, we have found that thresholding singular values based on a static threshold strikes a difficult balance between performing many iterations with a small threshold, which exacerbates computation for realistically large problems, and accepting a certain level of spectral bias with a larger threshold, which exacerbates errors in bias-sensitive material decompositions. An alternative approach for rank reduction that we used in previous work [35] replaces Eq 48 with the following reprojection step to yield the regularized results, [D_L D_S]: (50) Under the assumption that orthogonality is approximately maintained between the columns of U [47], this approach intrinsically reduces the nuclear norm as a function of the amount of noise removed from the columns of U, since U will necessarily deviate from the exact orthonormal subspace for the columns of [(X_L + V_L)P (X_S + V_S)P], U₀. We believe this reprojection method will be highly effective in certain applications where some level of bias can be tolerated. Specifically, within the context of hybrid reconstruction problems, this reprojection step appears to provide some level of tolerance to missing data, cone-beam artifacts due to differences in magnification, and to more dramatic resolution differences than we deal with in this work (see [38]). In this work, however, we strictly use the more conservative method to produce the regularized results (Eq 48) because the spectral separation of iodine and barium is very sensitive to spectral bias (see Contrast and resolution phantom).

Regardless of which equation is used to produce the final, regularized results, apparent rank reduction is also achieved through manipulation of the scalar noise multiplier parameters used to scale the contribution of each singular vector to the jointly constructed range weights, h_i (h_e, Eq 29). Rather than manually choosing values for each h_i, however, we choose a single value, h₀, which works well for most problems (h₀ ≈ 1–2), and then we scale h₀ based on ratios of the singular values and on a scaling parameter, γ (γ ≈ 0.5; Fig 3, step 2): (51) The double subscript notation, e_0;i,i, refers to the scalar diagonal elements of E₀ starting from 1,1 for the first singular value. Notably, this scaling function is closely related to threshold scaling which has been proposed for non-convex soft thresholding of singular values [32]. Using the scaled h_i parameters, joint BF of the singular vectors (jBF(·); Fig 3, steps 5 and 7) places greater emphasis on preserving edge features in the most significant singular vectors which have been prioritized to contain comparatively lower levels of noise. When these same multiplier parameters are also used to control the balance between regularization and data fidelity in the data fidelity update step (Eq 46; Fig 3, step 10), RSKR has the effect of smoothing less significant singular vectors to match the image structure in more significant singular vectors over the course of multiple internal Bregman iterations.

Within the context of hybrid spectral CT reconstruction, where copies of the EID data make up one half of the inputs into RSKR (Fig 1, step 13), the most significant singular vectors are dominated by the EID data. This produces a unique regularization paradigm which balances between two extremes. In the one extreme (large γ), the joint BF kernel is entirely determined by the first singular vector, smoothing all other singular vectors to closely match the structure in the EID data. This has the benefit of copying high-frequency edge information from the EID data to the PCD data, potentially enhancing resolution, but may over-smooth spectral features which have low contrast in the EID data. This approach may also introduce significant bias between the regularized PCD reconstructions and the PCD projection data. At the other extreme (small γ), all singular vectors contribute equally to range kernel construction, preventing efficient regularization due to substantial noise is the less significant singular vectors. Choosing an intermediate value for γ (here, 0.5), provides a compromise between copying high frequencies from the EID data to the PCD data, while managing bias. During later applications of RSKR (later global Bregman iterations), when the noise level has equalized between the EID and PCD data, over-smoothing of high contrast features in the PCD data is prevented due to the non-trivial contribution of less significant singular vectors to the joint range kernel.

Dictionary learning and sparse coding.

As we will show, the algebraic model we have proposed for hybrid spectral CT reconstruction successfully synergizes the reconstruction of EID and PCD data, yielding improved denoising performance and evidence of resolution enhancement. However, there is a subtle limitation to enforcing agreement between the EID and PCD reconstructions under L² constraints (Eq 1): some amount of correlated noise and blurring will always bleed back into the EID reconstructions. This trade-off can be managed through the λ_C regularization parameters; however, relaxing the data fidelity constraint on the PCD data will generally lead to spectral bias. Supplementing the objective function with a rank-sparsity constraint enforced with RSKR provides some robustness to this blurring and correlated noise through joint filtration, which favors the EID data, and through tiling, which improves the removal of low frequency noise. However, as previously discussed, the PCD data cannot be ignored during the construction of the joint range kernel because of the risk of over-smoothing features with low spectral contrast in the EID data. A related issue is that the piece-wise constant image structure enforced by joint BF imperfectly matches the band-limited (smooth) edge features in both the EID and PCD data, bounding the performance of resolution enhancement.

To address these limitations, we adapt regularization with dictionary learning and sparse coding to the problem of hybrid CT reconstruction. Specifically, for dictionary learning we employ the K-SVD algorithm ([48], KSVD-Box v13), and for 3D sparse coding we employ our own GPU-based implementation of batch orthogonal matching pursuit (OMP) using progressive Cholesky factorization [49]. In hybrid reconstruction, we apply dictionary learning to the FBP reconstruction of the EID data (x_{L,0;e = 1}) during the initialization phase (Fig 1, step 11; KSVD(·)) to minimize the following objective function [48]: (52) (53) X_L,P is a matrix of mean-subtracted, radial patches extracted from x_{L,0;e = 1} (n_v: number of voxels per radial patch; n_p: number of patches). The K-SVD algorithm approximates these patches as sparse, linear combinations of atoms (coefficient matrix: Φ; ‖⋅‖₀: number of non-zero entries) drawn from an overcomplete dictionary, K (unit norm, zero-mean, column vectors; n_a: total number of atoms). In more detail, the K-SVD algorithm alternates between two phases: (1) sparse coding (with OMP), to choose and update the non-zero coefficients of each column of Φ (n₀: maximum allowed number of non-zero coefficients), and (2) dictionary updating, to improve the fidelity with which the dictionary atoms represent the training patches which use them. We leave most of the details of dictionary learning to the referenced work; however, we make note of several parameter choices. First, to learn the dictionary, we extracted overlapping, 3D, radial patches (domain defined as in Eq 30; b = 4) from the input volume and stored them as columns of the matrix X_L,P. Specifically, 500,000 training patches with a variance higher than the noise level were extracted from all possible overlapping patches. The total number of dictionary atoms was chosen to be n_a = 1024 based on the heuristic criterion that the number of atoms should be four times larger than the number of voxels per patch [50]. The total number of K-SVD iterations used to learn the dictionary was 25 (one K-SVD iteration: one sparse coding phase, one dictionary updating phase). A maximum of n₀ = 48 atoms with non-zero coefficients were allowed per patch during the sparse coding phase of the K-SVD algorithm; however, at convergence, far fewer atoms were required to meet the residual error criterion for most patches (indexed by i): (54) (55) For the experiments in this work, the tile(·,s) function performed tiling as discussed in the Joint bilateral filtration section (stride, s = 3) for the purpose of robust estimation in the presence of low-frequency, correlated noise. With the exception of noise estimation, the dictionary learning and sparse coding operations were performed exclusively on untiled data. Additional details, such as the criterion for replacing under-used dictionary atoms and similar dictionary atoms, were handled with the default parameters in the referenced code.

Following convergence of the K-SVD algorithm (Fig 1, step 11), the resultant dictionary, K, is held fixed. Under the assumption that K provides a robust basis for representing new patches in related data sets, only sparse coding with OMP is performed for regularization with respect to the objective function (Eq 52). More specifically, OMP for regularization (OMP(·,K) function; Fig 1, step 15) involves several sub-steps: (1) noise estimation to calibrate the residual convergence criterion (Eq 55); (2) extraction of all overlapping radial patches from the input volume; (3) subtraction of the mean intensity from each patch; (4) sparse coding subject to the objective function (Eq 52) and the residual convergence criterion (Eq 55); (5) addition of the original patch mean back to each coded patch; and (6) recovery of the regularized volume through averaging of the overlapping patches. Our hybrid reconstruction algorithm preforms sparse coding with OMP on each column of [D_L D_S] independently following RSKR (Fig 1, step 15). For regularization, the number of non-zero coefficients (# of atoms used to code each patch) was determined by the minimum number of atoms required to satisfy the residual error criterion, up to a maximum of n₀ = 5 atoms with non-zero coefficients.

Returning to our motivations for incorporating dictionary learning and sparse coding into the hybrid reconstruction algorithm, a dictionary learned from a FBP reconstruction of the EID data encodes contrast-independent prior information about the signal characteristics which are expected in the final hybrid reconstruction results. Interestingly, there is little risk of overfitting the EID data during dictionary learning since we expect to recover nearly identical image structures in the results of hybrid reconstruction. Furthermore, because the dictionary is trained using the EID data only, any bias introduced during sparse coding with OMP will ideally make the regularized results more similar to the expected results. Finally, because the mean of each patch is strictly preserved and because the input to sparse coding should have a relatively low level of noise following RSKR, the risk of introducing spectral bias is extremely low.

Rank-sparsity constrained hybrid spectral CT reconstruction.

Now that we have detailed the individual components of our proposed hybrid spectral CT reconstruction algorithm, we briefly summarize how its components fit together into a coherent algorithm. Following the pseudocode in Fig 1, the EID (w) and PCD (Y) projection data as well their expected material sensitivities (m_EID and M, respectively; also, μ_water in the PCD data) are provided as input. The user must specify several parameters (typical values) that influence the trade-off between EID and PCD data fidelity (λ_C = 1), the amount of smoothing performed with joint BF (h₀ ≈ 1–2), the overall trade-off between data fidelity and regularization (α ∈ [0.001,0.01]), the rank reduction effect of RSKR (γ ≈ 0.5), and the bias-variance trade-off in addressing low-frequency noise (s = 3). Initial estimates for X_L (X_L,0), the redundant reconstructions of the EID data, and X_S (X_S,0), the spectral contrast contributed by the PCD data, are produced via material decomposition and component substitution (steps 1–4). Prior to regularized iterative reconstruction, several quantities must be initialized (steps 5–12). Specifically, algebraic reconstruction with weighted least-squares is performed to satisfy the hybrid data fidelity constraints and to pre-condition the regularized portion of the algorithm for faster convergence (step 5). The reconstructed noise level of the data at each energy, σ_e, is then estimated using tiling operations and MAD computations (step 6). Water-normalized ratios of these noise estimates, δ_e, (step 7) are used to calibrate the regularization parameters, μ_L,e and μ_S,e, (steps 8–9) as well as the SVD prioritization weights, p_e,e, (step 10) for each energy, e. Independently, a dictionary of characteristic atoms, K, is learned using a FBP reconstruction of the EID data, x_{L,0;e = 1}, to serve as prior information for regularization (step 11). Finally, the residual variables V_L and V_S are initialized to zero (step 12).

Following initialization, regularized iterative reconstruction of the hybrid CT data is performed using the split Bregman method (steps 13–18). Each global Bregman iteration repeats the following steps: (1) serial regularization with RSKR (steps 13–14) and dictionary-based sparse coding (OMP, step 15); (2) regularization residual updates (steps 16–17); and (3) data fidelity updates relative to the regularized estimates of X_L and X_S, D_L and D_S, at each energy (step 18). Fewer than six global Bregman iterations were required to achieve convergence (negligible change in X_L and X_S between iterations) for both the simulation and in vivo experiments. Convergence is achieved in such a small number of iterations given the initialization of X_L and X_S prior to regularized reconstruction and due to the data-adaptive nature of the regularization strategies employed. Following convergence of the hybrid reconstruction algorithm, material decomposition is performed to obtain maps of the relevant basis materials, C (step 19).

System configuration, data acquisition, and projection preprocessing.

The EID chain of our hybrid micro-CT scanner consists of an Epsilon high frequency x-ray generator (EMD Technologies, Saint-Eustache, QC), a G297 x-ray tube (Varian Medical Systems, Palo Alto, CA; fs = 0.3/0.8 mm; tungsten rotating anode; filtration: 0.7 mm Al, 3 mm PMMA), and a XDI-VHR CCD x-ray detector (Photonic Science Limited, Robertsbridge, UK; 22 μm² pixels) with a Gd₂O₂S scintillator. The 22 μm² EID pixels were binned to 88 μm² prior to image reconstruction (1002x667 88 μm² pixels / projection). 360 EID projections were acquired over a single 360° rotation (1 angular increment). The source-detector and source-object distances were 80 cm and 70 cm, respectively, resulting in a geometric magnification of 1.14 times. Each projection was acquired with an 80 kVp tungsten spectrum with a 100-mA current and a 10 ms projection integration time (resulting in ~5e4 photons / line integral). The absorbed radiation dose associated with the EID scan was ~48 mGy.

The PCD chain of the hybrid micro-CT scanner consisted of a PXS-10 65W micro-focus x-ray source (Thermo Fisher Scientific, Waltham, MA; tungsten anode; filtration: 0.25 mm beryllium) and a PILATUS3 CdTe 300K PCD with a single hardware-based energy threshold (on loan from Dectris AG, Baden-Dättwil, Switzerland; 1 mm CdTe thickness; 487x619 172 μm² pixels / projection) [51–55]. With this PCD chain, 400 projections were acquired over a single 360° rotation (0.9° angular increment) for each of five threshold settings: 26, 34, 37, 39, and 45 keV. The source-detector distance was 27 cm and source-object distance was 20 cm, resulting in a geometric magnification of 1.35 times. Each projection was acquired with an 80 kVp tungsten spectrum, a 251 μA current, and a 20 ms projection integration time (resulting in ~4.2e3 photons / line integral prior to considering the energy threshold). The absorbed radiation dose per PCD scan (per threshold) was ~14 mGy.

For both the simulation experiment and the in vivo experiment, the EID data was reconstructed with a voxel size of 88 μm³. The PCD data was reconstructed with a voxel size of 127 μm³. At these voxel sizes and magnifications, the FWHM of the point spread function, estimated from edge-spread measurements in FBP reconstructions, was approximately twice the voxel width. With respect to the Gaussian kernels used to approximate the point spread function in the algebraic forward model (Hybrid data fidelity sub-section), this yielded the following parameter values: FWHM_PCD = 0.254 mm; FWHM_EID = 0.176 mm; σ_r = 0.078 mm.

For the in vivo experiment and only for the PCD projection data, three preprocessing steps were performed, in order, prior to reconstruction and following bright-field normalization and log-transformation (Fig 4): (1) ring artifact prevention, (2) ring artifact correction, and (3) detector gap interpolation. (1) To prevent significant ring artifacts, permanently under- and over-exposed detector pixels were identified in exposure-averaged, bright-field images as detector pixels with signal values which deviated by more than 25% from the median signal recorded in air. Within the log-transformed projection data and at each angle, the new intensity values assigned to these outlier pixels were interpolated using a 2D Gaussian kernel (standard deviation: 1 pixel). (2) More subtle ring artifacts were corrected by estimating the intensity bias associated with specific detector pixels. Specifically, low-pass filtration of the log-transformed projection data was applied independently along the projection columns and along the angular dimension (1D box filter; length: 5 voxels or 5, 0.9° angular increments). For each detector pixel across all angles, the average difference between these two low-pass filtered sets of projections was used as a correction factor to correct the estimated intensity bias in the unfiltered projection data. The magnitude of the applied correction was not allowed to exceed the uncorrected magnitude recorded for each individual detector pixel at each projection angle. (3) As a final preprocessing step, gaps in the detector were filled in via 1D interpolation perpendicular to the orientation of the gaps. Specifically, these values were filled in using a 1D Gaussian kernel with a diameter of 41 pixels and a standard deviation of 8 pixels, with appropriate magnitude normalization. The primary vertically oriented gap was interpolated to prevent a ring artifact near the center of rotation. The larger horizontal gaps were interpolated to mitigate the deleterious impact of reprojecting partial rays during iterative reconstruction.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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)).

https://doi.org/10.1371/journal.pone.0180324.g004

Geometric calibration and affine registration.

While not exclusive to hybrid spectral CT, two related calibration operations were fundamental to the success of our in vivo experiment: geometric calibration and dual-chain, affine registration. The objective of geometric calibration is to determine the parameters of the system projection matrix which describe the view-dependent relationship between the x-ray source and detector positions [56]. Once determined, these parameters are used to perform projection and backprojection operations in a highly parallel fashion (e.g. on the GPU) by computing elements of the projection matrix as needed. In past work with dual chain micro-CT (i.e. with two independent sets of sources and detectors; [57]), we have applied geometric calibration to each imaging chain independently. Given well-calibrated reconstructions from each imaging chain, data from both chains can be combined in the image domain by computing an affine registration matrix. This matrix maps voxels from the secondary imaging chain into the space of the primary imaging chain. Resolution-losses due to interpolation associated with this affine transform can be mitigated by incorporating the inverse of the affine transform directly into the system projection matrix for the secondary imaging chain.

Specific to the problem of hybrid spectral CT, the geometric calibration and affine registration operations are complicated by fundamental differences between the EID and PCD data: noise level, spatial resolution, geometric magnification, field-of-view coverage, etc. Considering these differences, the precision with which the PCD data is calibrated and registered relative to the EID data will directly limit the potential for resolution enhancement through hybrid reconstruction. To achieve high-fidelity geometric calibration and registration between the EID and PCD imaging chains, we first performed geometric calibration for each chain independently at the native resolution of each imaging chain (EID: 88 μm² pixels, 88 μm³ voxels; PCD: 172 μm² pixels, 127 μm³ voxels). Then, using the previously described upsampling operator, B^T, the projection matrix for the EID data, R, and the FBP operator for the PCD data, , we refined the geometric calibration and affine registration of the PCD imaging chain using the EID projection data, w: (56) The fit between the two sets of data was evaluated with the mutual information similarity metric (MI, [58]). The geometric parameters for the PCD data as well as the affine transform parameters to register the PCD data into the space of the EID data (six free parameters describing translation and rotation) were directly incorporated into the PCD backprojection matrix, A^T (). To reduce noise, the energy dimension of the PCD projection data was reduced with a variance-weighted average (variance measured in air). This resulting energy-averaged projection data, y_avg, was used solely for calibration and registration. Similar to the approach in [59], geometric and affine parameters, which maximized the MI between the PCD and EID projections, were found using the covariance matrix adaptation evolution strategy (CMA-ES, [60]). Practically, the MI similarity metric appeared to be very robust to the interpolated gaps in the projection data and to the lower resolution of the reprojected PCD data.

Contrast and resolution phantom.

Fig 5 details the digital contrast and resolution phantom we constructed to assess the fidelity of hybrid reconstruction. The digital phantom is a variation on the ACR 464 phantom used for quality assurance in clinical CT scanners [61]. Specifically, the phantom consists of a cylinder of water with a diameter scaled to match the diameter of the cradle we use for scanning adult mice, 3.4 cm. The cylinder is divided into three segments (disks) along the z-axis. Each disk is dominated by a single contrast material we aim to separate in vivo: iodine (red), calcium (blue), and barium (green). The materials are present in realistic concentrations for small animal micro-CT—15 mg/ml of iodine, 75 mg/ml of calcium, and 15 mg/ml of barium in water (material fraction = 1.0). To assess spatial resolution, each disk contains a set of line pairs which discretely represent spatial frequencies from 0.71 to 5.68 line pairs per mm (lp/mm). To assess the trade-offs in feature detection with feature size and material concentration, a grid of spheres is included within each disk. Along the y-axis, the diameters of these spheres vary from 2.0 mm to 0.5 mm in increments of 0.5 mm, with some truncation due to discretization. Along the x-axis, the concentrations of each disk’s material take on the following fractions of the maximum concentration: 1.0, 0.66, 0.33, and 0.20. To monitor spectral fidelity during iterative reconstruction, volumetric measures are taken in cylindrical “vials” which contain 0.5 times the maximum concentration of each material in all three disks along the z-axis. An additional vial of water, also used for monitoring spectral fidelity, is positioned to the right of the vials visible in Fig 5.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pone.0180324.g005

Using this digital phantom and the previously defined scanning configurations (System configuration sub-section), projection measurements, I, were synthesized with the following spectral model: (57) (58) Specifically, given the maximum concentration, c_max(m), for each material, m (n_m = 4 basis materials), and the material fraction maps used to construct the phantom, frac(m,r) (Fig 5), the linear attenuation coefficients (Eq 58) were integrated as a function of position, r, and energy, e (Eq 57). The mass attenuation coefficients, , were derived from Spektr [62]. For the EID projection data, the detected signal in the absence of the phantom, I_D(e), was modeled as follows: (59) I_0,EID(e) denotes the 80 kVp source spectrum previously described in the System configuration sub-section. S_EID(e) denotes the normalized detector sensitivity function, which includes the response of the Gd₂O₂S scintillator and additional PMMA filtration. For the PCD projection data, the detected signal model included a single hardware-based energy threshold, t, consistent with the specifications of the PILATUS3 detector: (60) (61) Again, the 80 kVp source spectrum, I_0,PCD(e), was as described in the System configuration sub-section. The normalized PCD sensitivity function, S_PCD(e), included the expected quantum efficiency of detection with 1 mm of CdTe (Fig 6B; [63]), but did not model more complex physical phenomena such as charge sharing and pulse pile-up. Rectangular (rect) functions (Eq 61) were used to represent idealized spectral bins. The components of the PCD spectral model are summarized in Fig 6.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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).

https://doi.org/10.1371/journal.pone.0180324.g006

Given these basic spectral models for the EID and PCD data acquisition, additional steps were taken to better match the (1) image noise, (2) spatial resolution, and (3) energy resolution characteristics of the in vivo data (In vivo experiment sub-section). To reproduce the observed noise levels (1), the total number of counts associated with I_0,EID (5e4 photons / line integral) and I_0,PCD (4.2e3 photons / line integral) were empirically determined. Specifically, these values were adjusted such that when the recorded counts, I, (Eq 57) were drawn from a Poisson distribution with a mean equal to I, the resultant FBP reconstructions reproduced the expected noise standard deviations measured in water. For the EID data, the noise standard deviation in water was ~80 HU (88 μm³ voxels). For the PCD data, the noise standard deviations ranged from ~200 HU (26 keV threshold) to ~450 HU (45 keV threshold; 127 μm³ voxels). Prior to the computation of the projection data and to approximately reproduce the observed spatial resolution (2), the material density maps (c_max(m) * frac(m,r), Eq 58), discretized with 88 μm³ voxels, were resampled with the B operator. As previously reported, the Gaussian FWHM for the approximated point spread function of the EID data was 176 μm, while the FWHM for the PCD data was 254 μm. The B operator applied to the PCD data also resampled the voxel size from 88 μm³ to 127 μm³.

Finally, to better model the limited energy resolution of the PCD (3), the rect function used for ideal spectral binning (Eq 61) was convolved with an energy-invariant, Gaussian energy spread function, G_esf (Eq 60), parameterized by g, the energy offset from e, and σ_esf, the standard deviation of the energy spread function. The energy (e) dependence of the energy spread function for CdTe-based PCDs is well documented [16]; however, in the absence of a precise model for this dependence in the PILATUS3 detector, we used an alternative approach to make the simulation more realistic. Specifically, we chose a fixed value for the σ_esf parameter such that the resultant material sensitivity matrix derived from the simulated data had a condition number closely matching that of the material sensitivity matrix derived from the in vivo data. Mathematically, the condition number is computed as the ratio of the largest and smallest singular values of a matrix, and it quantifies the potential for error amplification in solving linear inverse problems (e.g. material decomposition). Fig 6E plots the simulated material sensitivities, following magnitude normalization per material and using ideal (rectangular) spectral bins (condition number = 40). By contrast, Fig 6F plots the same simulated, normalized material sensitivities when an energy-invariant Gaussian energy spread function is included in the forward model (σ_esf = 3.25 keV; condition number = 100). While not identical to the material sensitivities derived from the real data (Fig 6G), the degradation of iodine (34 keV threshold) and barium (37, 39 keV thresholds) K-edge contrast is convincingly reproduced.

Note that the phantom is constructed from calcium, water, barium, and iodine, but the phantom is decomposed into basis functions for the PE, CS, barium, and iodine. Our PE and CS basis function were derived using the analytical, energy-dependent expressions for the PE and CS [46] and energy-effective attenuation calculations using our forward model (Eq 57). These derived PE and CS basis functions were then modified to exactly describe the derived basis functions for calcium and water across all five PCD thresholds and the EID data. Specifically, we projected the derived PE and CS basis functions onto the orthonormal subspace defined by the basis functions for calcium and water (<3% difference before and after projection). This fitting procedure was important to maximize the separation of the PE and CS basis functions from the basis functions for iodine and barium (i.e. to minimize the condition number of the material sensitivity matrix). For convenience, the PE and CS basis functions were also scaled to each decompose to a coefficient of one in pure water. Setting the lower bound of the PE display window to one then visually differentiated calcium from water in overlaid material decompositions (Fig 5).

Quantitative evaluation.

Several metrics were used to quantitatively evaluate the results of hybrid spectral CT reconstruction using the digital phantom. To globally assess the fidelity of the reconstructed results at a single energy, x_e, we computed the root-mean-square error (RMSE) relative to the expected reconstruction results, : (62) Here, i indexes all n_x voxels of the reconstructed volume at a single energy. To assess spectral fidelity independent of denoising performance, we also computed the absolute bias within volumetric regions of interest expected to have constant intensity, k: (63) The spatial vector r indexes the n_roi total voxels within the defined volumetric region of interest. To assess spatial resolution in the reconstructed results, we computed the modulation transfer (MT) at each spatial frequency sampled by the bar patterns in the digital phantom: (64) The variables a and b correspond with average attenuation measurements taken from the expected locations of a set of contrast-enhanced lines and from the corresponding, expected gaps between the lines, respectively. Prior to the MT calculation, the expected attenuation of water was subtracted from each measurement. Any negative values of MT were set to zero. Given a vector of 120 independent MT measurements, n, taken in the digital phantom (n_l = 120; 8 sets of line pairs, 3 disks, 5 thresholds; Fig 5), a Gaussian modulation transfer function (MTF) was fitted to match the observed measurements under a least-squares penalty: (65) (66) The standard deviation of the MTF, σ, is the only free parameter. The vector l contains the spatial frequencies in lp/mm for each corresponding measurement.

Finally, the detectability index was computed to assess the visibility of the spherical lesions within the digital phantom under a non-prewhitening observer model [64]: (67) Specifically, the detectability index was computed over a cubic region of interest defined around each spherical lesion (side length: 32 voxels; no overlap between regions). All voxels within each cubic region were indexed by the Cartesian offset vector, j. Specific to our implementation of the detectability index calculation, MTF_3D denotes the 1D MTF from Eq 65 extrapolated to 3D in an isotropic fashion based on the Euclidean distance of each voxel center from the origin of each region of interest. We defined the task function, W, as the discrete Fourier transform of the same cubic region of interest within the expected reconstruction following the subtraction of the attenuation of water from each voxel. The local noise power spectrum, NPS, was approximated as the power spectral density of the residuals computed between the hybrid reconstruction results and the expected reconstruction results within each cubic region of interest.

Our primary use of the detectability index is to compare the results of hybrid spectral CT reconstruction with the results of a second control reconstruction where the EID projection data is not used. We refer to the results of this control experiment as “PCD only” results. The algorithm applied for PCD only reconstruction is very similar to the algorithm applied for hybrid reconstruction, which includes the EID data (compare the pseudocode in Fig 7 with Fig 1). However, there are two significant differences. First, the PCD data is reconstructed directly (X) rather than indirectly as a function of the EID reconstruction (X_L) and the spectral contrast provided by the PCD data (X_S). Second, during initialization, the dictionary is trained using a variance () weighted average of the initialized reconstruction results (avg_e(X), step 7) rather than using a FBP reconstruction of the EID data (Fig 1, step 11). Given these changes, the purpose of the PCD only control experiment is to establish a base-line for the resolution enhancement and denoising performance improvements afforded by the incorporation of EID data into the PCD reconstruction problem.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

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 (X_L) and the spectral contrast provided by the PCD data (X_S). Second, during initialization, the dictionary is trained using a variance () weighted average of the initialized reconstruction results (avg_e(X), step 7) rather than using a FBP reconstruction of the EID data (Fig 1, step 11).

https://doi.org/10.1371/journal.pone.0180324.g007