{"id":"412425ea-ea1f-4b88-918a-1185a1c0152b","arxiv_id":"2505.24793","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"AFIRE reconstructs basis images in geometric-inconsistent multispectral CT via a simplified Newton iteration built from the exact derivative of the nonlinear forward map at zero, with a convergence theorem whose key assumptions are not verified in the experiments.","lead":"AFIRE reconstructs material basis images from dual-energy CT scans whose two energy spectra used mismatched scanning geometries. It replaces expensive nonlinear optimization with a simplified Newton iteration that uses an exact derivative formula and fast filtered back-projection, and the authors prove convergence under conditions that are not checked in the experiments.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The contraction hypotheses of Theorem 2 (η̃ bound, γ=η̃κ_F(P⊗φ)<1, and residual bound for FBP approximate inverse) are never checked for the experiments, and γ is likely >1 for the reported 294,912×32,768 systems, so the convergence proof does not cover the central numerical claim.","rationale":"I agree with the reader's weakest-assumption analysis and think it identifies the decisive issue. The derivative factorization in Eq. (26) is correct, the simplified Newton update is well motivated, and the numerical evidence supports the practical claim that AFIRE reconstructs the tested basis images accurately and quickly. However, the convergence theory as written is not connected to the experiments: Theorem 2's hypotheses include a uniform contraction bound that is never verified, and the natural estimate of κ_F(P⊗φ) suggests the required γ<1 is not met for the reported large-scale systems. The continuous theorem has a separate gap, since Lemma 1 assumes geometric consistency (T^q_n=T_n), so it cannot support the geometric-inconsistent setting; the discrete theorem is the only relevant one, and its key constant is unchecked. The paper's statement that the numerical results 'confirm the convergence theory established in theorem 2' therefore exceeds what the theory currently establishes. This warrants a conditional rather than outright rejection: the algorithm may be sound, but the theoretical guarantee and its verification need to be repaired or weakened, and reproducible artifacts would strengthen the empirical claim.","tokens_in":22013,"tokens_out":7329,"duration_ms":85026,"concrete_test":"Use the exact Section 5.2 geometry to assemble P1,P2 (384×384 parallel projections, 128×128 images), compute η̃ = max_{q,d}(max_m b_dm/(b_d^T s^q) − 1, 1 − min_m b_dm/(b_d^T s^q)) from the stated SpectrumGUI spectra and MAC tables, and estimate σ_min(P⊗φ) by randomized power iterations on the 294,912×32,768 matrix to obtain κ_F(P⊗φ)=∥P⊗φ∥_F/σ_min. Then evaluate γ=η̃κ_F(P⊗φ). If γ≥1, Theorem 2's hypotheses fail for the reported noiseless experiment. Should γ<1, additionally estimate ζ=∥diag(P_1P_1^+, P_2P_2^+)−I∥_2 for the FBP inverse and check ζ<(1−γ)/(1+γ).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing unsupported premise is that Theorem 2's contraction condition γ=η̃κ_F(P⊗φ)<1 holds for the geometries in Section 5. Lemma 2 requires a global η̃ such that |b_d^T ŵ^q − b_d^T ρ| < η̃ |b_d^T ŵ^q| for every unit-simplex ρ, and Theorem 2 additionally requires the approximate-inverse residual bound ∥δ^k∥≤ζ∥K(f^k)−g∥ with ζ<(1−γ)/(1+γ). No value of η̃, κ_F(P⊗φ), or ζ is computed or bounded for the 128×128 (384 views × 384 rays, two spectra) or 256×256 (768 views × 768 rays) setups. This is not a purely technical omission: κ_F(P⊗φ) is the Frobenius-norm-to-minimum-singular-value ratio of a 294,912×32,768 block matrix; for full-scan parallel-beam projection operators this ratio is plausibly of order 10^2 or larger. Meanwhile η̃ cannot be smaller than max_{q,d}(max_m b_dm/(b_d^T s^q) − 1), since ρ may be a vertex of the unit simplex; with the 80 kV and filtered 140 kV spectra and the water/bone MAC tables this lower bound is of order 1 or larger. Hence γ≥1 is likely, making the theorem inapplicable to the very experiments that the paper says confirm it. The continuous Theorem 1 is separately restricted by Lemma 1 to T^q_n=T_n, so it does not address geometric inconsistency; the discrete theorem is the only plausible cover for the motivating case, and it is the one whose hypotheses are unchecked. The sign typo in Algorithm 2 line 5 and the absence of released code/data are secondary; the contraction verification is the load-bearing issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes AFIRE, a simplified Newton method for multispectral CT image reconstruction that is designed for scanning geometries that are inconsistent across different X-ray energy spectra. The core observation is that at special points such as the zero image the Jacobian of the nonlinear forward map factorizes as a block product of a diagonal operator of X-ray transforms and a small material/spectrum matrix, namely -P⊗φ in the discrete setting. This structure yields an iteration that requires only a small 2×2 matrix inversion and an approximate inversion of each projection matrix. The paper proves convergence for the continuous and discrete settings under uniform relative Lipschitz-type assumptions, and reports numerical experiments on simulated geometric-inconsistent dual-energy CT data showing fast convergence and improved accuracy over NCPD, NKM, IFBP, and INTRPL. Section 6 further demonstrates flexibility in the choice of the special point and of the approximate inverse, and extends the method to the geometric-consistent case.","tokens_in":22560,"tokens_out":6445,"duration_ms":79894,"significance":"If the convergence hypotheses were verified for the tested configurations, this would be a valuable contribution: the derivative factorization at zero is clean and correct, and the resulting algorithm is both conceptually simple and computationally attractive. The numerical results are consistent and show a clear advantage in speed and accuracy over existing methods. The paper also usefully identifies the algorithm as a framework with two adjustable modules, the special point and the approximate inverse. The main weakness is that the convergence theory is not connected to the experiments: the constants that control the discrete convergence theorem are never computed or bounded, and the continuous theorem explicitly assumes geometric consistency, so it does not cover the motivating inconsistent case. Closing this gap is necessary before the paper's central claim is fully supported.","major_comments":[{"comment":"The convergence theorem for the discrete algorithm rests on an unquantified constant η̃ and on the approximate-inverse residual bound (34). The paper never computes or bounds η̃, κ_F(P⊗φ), γ, or ζ for the geometries in Sections 5.2 and 5.3. In particular, since ρ may be a vertex of the unit simplex, η̃ is at least max_{q,d,m} |b_dm/(b_d^T ŵ^q) − 1|, and κ_F(P⊗φ) is the ratio of the Frobenius norm to the minimum singular value of a 294,912×32,768 block matrix in the 128×128 experiment. Both quantities are plausibly large enough that the condition γ = η̃κ_F(P⊗φ) < 1 is not assured. Consequently, Theorem 2 does not, as written, certify the convergence reported in Figures 2 and 6, and the statement in Section 5.2 that the results confirm the convergence theory of Theorem 2 is unsupported. Please report numerical estimates of these constants for the actual spectra, MAC tables, and projection geometries, or replace the theorem with a condition that can be checked without computing spectral gaps of a huge ill-conditioned matrix.","section":"§4.2, Lemma 2 and Theorem 2, Eqs. (25)–(26), (33)–(34)"},{"comment":"Lemma 1 explicitly assumes T_n^q = T_n for all q, i.e., geometric consistency. Therefore Theorem 1's global convergence proof applies only to the geometric-consistent case, not to the geometric-inconsistent MSCT that motivates the paper. The continuous analysis cannot be used to justify Algorithm 1 for the stated problem; the discrete Theorem 2 is the only candidate covering the motivating case, and its hypotheses are unchecked as noted above. The paper should either extend the continuous proof to genuinely inconsistent sets T_n^q or revise the claims in Sections 4 and 7 to state clearly that the continuous convergence result requires geometric consistency.","section":"§4.1, Lemma 1 and Theorem 1"},{"comment":"In the numerical experiments, P+ is implemented by FBP with a one-bandwidth Ram–Lak filter, and in Section 6 also by CG and L-BFGS. Theorem 2 requires the residual δ^k of the approximate inverse to satisfy ∥δ^k∥ ≤ ζ∥K(f^k)−g∥ with ζ < (1−γ)/(1+γ). No estimate of δ^k is given for FBP, or for CG and L-BFGS with their chosen inner-iteration counts, in the geometrically inconsistent full-scan setting. Without such an estimate, the convergence theorem does not cover the actual algorithm whose performance is reported. At minimum, the authors should state this as an explicit assumption and discuss how it could be validated numerically, or prove a version of the theorem that does not require a global ζ.","section":"§5.1 and Theorem 2, Eq. (34)"}],"minor_comments":[{"comment":"The displayed update rules contain a spurious minus sign before the f^k term; the correct update from Eqs. (17) and (27) is f^{k+1} = f^k − (inverse Jacobian)(g − K(f^k)). Please fix the sign in both algorithms.","section":"§3.1.2, Algorithm 1 line 5 and §3.2.2, Algorithm 2 line 5, Eq. (28)"},{"comment":"The symbol ⊗ is used for a nonstandard block multiplication between P and φ and should be defined explicitly to avoid confusion with the usual Kronecker product, which has different block layout conventions.","section":"§3.2.2, Eq. (26)"},{"comment":"Lemma 2 assumes that P has full column rank, but this rank condition is not verified for the rectangular projection matrices arising in the inconsistent geometries of Section 5. Please state how it is checked or argue it from the full-scan view coverage.","section":"§4.2, Lemma 2"},{"comment":"The description of the noisy data is insufficient: it is not clear whether Gaussian noise is added to the line integrals before or after the logarithm, what noise level corresponds to 34.3 dB, and how many noise realizations are used. Please specify the exact generation procedure.","section":"§5.3"},{"comment":"The paper would benefit from a data and code availability statement, since the spectra, MAC tables, phantoms, and implementations of the compared methods are not released, making the numerical results difficult to reproduce.","section":"§5 and §6"},{"comment":"The term 'mildly full scan' is used in the abstract and Section 5 but is not defined until the experimental setup; please define it formally at first use.","section":"§2.2 and §5"},{"comment":"The sentence claiming this is the first literature to use derivative-structure-based algorithms for geometric-inconsistent MSCT is hard to verify and should be softened or supported by a more thorough literature search.","section":"§7, Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of the journal and the core algorithmic idea is promising. The main obstacle is not circularity but verifiability: the contraction constants of Theorem 2 are never checked, and the continuous theorem assumes geometric consistency. If the authors can provide numerical bounds for η̃, κ_F(P⊗φ), and ζ for the reported geometries, or replace the theorem with a weaker but checkable condition, the paper would be much stronger. The sign typo in the displayed algorithms should also be corrected before any final version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing you should know: this paper has a genuinely nice observation, but its convergence theorem is not actually connected to the numerical experiments.\n\nWhat's new and good: the authors show that the Jacobian of the discrete MSCT forward operator at f=0 factors as a block product of a diagonal projection matrix and a small 2×2 material matrix. That's correct, and it leads to a simplified Newton iteration (AFIRE) where each step is: apply FBP to the residual, then mix the result with the inverse material matrix. The numerical experiments on 128×128 and 256×256 DECT with inconsistent geometries show fast convergence — tens of iterations — and compare favorably against NCPD, NKM, IFBP, and an interpolated two-step DDD method. The flexibility tests with different special points and different approximate inverses (FBP vs CG vs L-BFGS) are sensible and add practical value.\n\nThe soft spot: Theorem 2, the only convergence result that covers the motivating geometric-inconsistent case, requires γ = η̃ κ_F(P⊗φ) < 1, where η̃ is a uniform bound on the sensitivity matrix variation over all unit-simplex densities and κ_F is the scaled condition number of a 294,912×32,768 block projection matrix. The paper never computes or bounds these quantities for the spectra, MAC tables, or geometries used in Section 5. The authors simply state that the numerical results \"confirm the convergence theory.\" That is not justified — the sufficient condition is likely violated, since κ_F is probably large and η̃ is of order one or larger. So the proof does not cover the experiments, even if the algorithm works. Separately, the continuous Theorem 1 assumes geometric consistency, so the motivating inconsistent case is only handled by the discrete theorem with the unverified conditions. There is also a sign error in Algorithm 2's update (the leading minus on the previous iterate is wrong), and no code or data were released.\n\nAssessment: the core idea is sound and the algorithm appears effective from the experiments. The theoretical claims are overreaching in their current form. This is a meaningful contribution to the MSCT reconstruction subfield, not a modality-changer. A serious referee should ask for either numerical verification of the contraction assumptions (or at least computation of γ for the test cases) or a restatement of the theorem as conditional with applicability left open. The paper also needs the pseudocode fixed and reproducibility artifacts.\n\nRecommendation: worth a full peer review, but with major revision. I'd bring it to a reading group focused on nonlinear inverse problems and imaging.","headline":"Nice factorization-based simplified Newton method for geometric-inconsistent spectral CT, but the convergence theorem's key contraction condition is never checked for the experiments it claims to confirm.","tokens_in":22913,"tokens_out":3832,"would_cite":false,"duration_ms":43209,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65J22","44A12","92C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims geometric-inconsistent multispectral CT can be solved by a simplified Newton iteration whose Jacobian at zero factorizes into projection matrices and a tiny 2×2 matrix, with a convergence proof.","keywords":["multispectral CT","dual-energy CT","geometric inconsistency","simplified Newton method","nonlinear inverse problem","basis image decomposition","convergence analysis","filtered back-projection"],"falsifier":"Estimate $\\tilde\\eta$ by maximizing $\\bigl|b_d^T(w_j^{(q)}(f)-\\hat w^{(q)})\\bigr|/\\bigl|b_d^T \\hat w^{(q)}\\bigr|$ over rays, unit-simplex weights, and $d$, and evaluate $\\kappa_F(P\\otimes\\phi)$ for the actual $P$ built from the two inconsistent parallel-beam geometries. If $\\gamma = \\tilde\\eta\\,\\kappa_F(P\\otimes\\phi)\\ge 1$, or if the FBP-based residual bound (34) is violated at the reported iteration counts, then the hypotheses of Theorem 2 do not hold for the experiments that are used to demonstrate convergence.","tokens_in":21794,"feed_emoji":"🩻","tokens_out":12277,"duration_ms":130962,"temperature":0.7,"pith_summary":"Multispectral CT reconstruction becomes hard when the scanning geometries of different energy spectra do not align, because the nonlinear forward map can no longer be split into independent per-ray equations. This paper shows that at special points, in particular at zero, the Jacobian of that map reduces to a block product of the per-spectrum projection matrices and a very small matrix that encodes the energy spectra and mass-attenuation coefficients. That factorization converts the large nonlinear inverse problem into a simplified Newton iteration: invert each spectrum's projection approximately, then combine the residual sinograms with a $2\\times2$ matrix. The paper proves convergence of this iteration under a contraction condition and demonstrates numerically that in dual-energy CT it reaches high accuracy in fewer than fifty noiseless iterations and about fifteen noisy ones, outperforming NCPD, NKM, IFBP, and INTRPL in accuracy and speed. The same structure also covers geometric-consistent scans and tolerates different choices of the special point.","feed_headline":"Mismatched dual-energy CT reconstructed in tens of iterations","feed_subtitle":"One filtered back-projection per spectrum recovers basis images from mismatched geometries, with a convergence proof.","key_machinery":"The carrying object is the identity $\\nabla K(\\hat f) = -P\\otimes\\phi$, where $\\otimes$ denotes block multiplication: the $Q\\times D$ matrix $\\phi$ is applied to the $D$ basis-image updates after each per-spectrum projection matrix $P_q$ has acted on the residual. The matrix $\\phi$ is defined by $\\phi_d^{(q)} = (\\hat w^{(q)})^T b_d$, i.e., the spectrum-weighted average of the mass-attenuation vector $b_d$ at the special point, making it a $Q\\times D$ matrix (size $2\\times2$ in the dual-energy tests). The actual iteration (29) applies an approximate inverse $P^{+}$ to each spectrum's residual and then mixes the results with the small inverse $\\phi^{-1}$, so a single outer step costs about one filtered back-projection per spectrum. The convergence argument is carried by a uniform bound $\\tilde\\eta$ on how far the per-ray weight vector $w_j^{(q)}(f)$ can depart from $\\hat w^{(q)}$, combined with the scaled condition number $\\kappa_F(P\\otimes\\phi)$ to control the resulting perturbation of the Jacobian.","core_discovery":"The central discovery is a factorization of the Jacobian of the discrete forward map at a point whose ray integrals are constant: $\\nabla K(\\hat f) = -P\\otimes\\phi$ (Eq. (26)), where $P = \\operatorname{diag}(P_1,\\ldots,P_Q)$ is the block-diagonal matrix of the projection matrices of the $Q$ spectra and $\\phi$ is the small $Q\\times D$ matrix with entries $\\phi_d^{(q)} = (\\hat w^{(q)})^T b_d$ (Eq. (25)). Because $\\phi$ is tiny, the inverse of the Jacobian is effectively the per-spectrum inverse of the projections followed by a $D\\times D$ matrix multiplication. This yields the simplified Newton iteration $f^{k+1} = f^k - \\phi^{-1}\\otimes P^{+}(g - K(f^k))$, where $P^{+}$ is any sufficiently accurate approximate inverse such as filtered back-projection. The paper proves global convergence to the unique solution of the discrete system when the approximate-inverse residual satisfies (34) and the contraction factor $\\gamma = \\tilde\\eta\\,\\kappa_F(P\\otimes\\phi)$ is below one, and it reports numerical convergence in the tested DECT geometries within tens of iterations.","pith_inferences":["Editorial inference: because the factorization only needs $Q=D$ and the same constant-ray condition, the scheme should carry over to photon-counting or three-material decompositions by resizing $\\phi$; for $Q>D$ one would naturally invert the small normal system in a least-squares sense.","Editorial inference: the convergence hypotheses of Lemma 2 are never verified numerically; computing $\\gamma = \\tilde\\eta\\,\\kappa_F(P\\otimes\\phi)$ for the reported 128$\\times$128 and 256$\\times$256 geometries would turn the proof into a testable prediction.","Editorial inference: the continuous theorem (Theorem 1) assumes geometric consistency ($T_q^n=T_n$), so the proof for the motivating inconsistent regime rests entirely on the discrete argument; a continuous proof allowing $T_q^n\\neq T_n$ would be needed to close that gap.","Editorial inference: replacing $P^{+}$ with a regularized or learned reconstruction operator would adapt the same outer iteration to sparse-view or low-dose MSCT, since the convergence proof only requires the residual bound (34) to hold."],"forward_implications":["In completely geometric-inconsistent dual-energy CT, AFIRE reconstructs the water and bone basis images to high accuracy in under 50 noiseless iterations and under 15 noisy iterations, while NCPD, NKM, IFBP, and INTRPL lag behind or diverge.","Each outer iteration costs essentially one approximate projection inversion per spectrum plus a $2\\times2$ mixing step, so the algorithm combines the speed of analytic FBP with the accuracy of an iterative one-step method.","Theorem 2 guarantees convergence to the unique solution when $\\gamma = \\tilde\\eta\\,\\kappa_F(P\\otimes\\phi) < 1$ and the approximate-inverse residual obeys the bound in (34); Corollary 1 covers the exact-inverse iteration (27).","The algorithm is not sensitive to the choice of the special point: three random choices of the constants $\\{C_d^{(q)}\\}$ give nearly identical convergence in the reported experiments, so no explicit knowledge of an image $\\hat f$ is needed.","The same iteration solves the geometric-consistent case and outperforms the two-step DDD benchmark, and the module $P^{+}$ can be implemented by FBP, CG, or L-BFGS with a documented accuracy and efficiency trade-off."],"supporting_citations":[{"why":"Supplies the simplified Newton method template that the AFIRE iteration is built on.","marker":"[12]"},{"why":"Provides the classic iterative solution framework for nonlinear equations that the scheme employs.","marker":"[33]"},{"why":"Provides the invertibility condition for X-ray transforms (Theorem 2.11) and the continuity estimate used in Propositions 1 and 2.","marker":"[32]"},{"why":"Introduces the energy-selective basis decomposition model that defines the nonlinear forward map.","marker":"[4]"},{"why":"Documents the fast kV-switching projection mismatch that motivates the geometric-inconsistent setting.","marker":"[38]"},{"why":"The non-convex primal-dual baseline that AFIRE is compared against in the experiments.","marker":"[10]"},{"why":"The nonlinear Kaczmarz method baseline and its convergence framework, used for comparison in the tests.","marker":"[16]"},{"why":"The iterative FBP baseline designed for mismatched dual-energy projections, compared in the experiments.","marker":"[26]"},{"why":"Supplies the mass-attenuation tables for water and bone used in the numerical experiments.","marker":"[22]"},{"why":"An open-source X-ray spectrum simulator used to generate the 80-kV and filtered 140-kV spectra in the tests.","marker":"[34]"}],"fun_headline_variants":["AFIRE tames mismatched CT with a tiny matrix","Geometric-inconsistent CT solved in tens of iterations","Simplified Newton slashes multispectral CT reconstruction time","Derivative trick speeds up dual-energy CT imaging"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof for the inconsistent case rests on the discrete theorem, which assumes an unquantified bound on how much the per-ray energy weights change and that this change times the scaled condition number of the block system stays below one; neither is checked for the test geometries.","fun_headline_variants_meta":{"raw":{"variants":["AFIRE tames mismatched CT with a tiny matrix","Geometric-inconsistent CT solved in tens of iterations","Simplified Newton slashes multispectral CT reconstruction time","Derivative trick speeds up dual-energy CT imaging"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000222,"raw_usage":{"total_tokens":1490,"prompt_tokens":1017,"completion_tokens":473,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":410}},"tokens_in":633,"tokens_out":473,"duration_ms":5561,"temperature":1.0,"reasoning_tokens":410,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:13:44.699094+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Estimate $\\tilde\\eta$ by maximizing $\\bigl|b_d^T(w_j^{(q)}(f)-\\hat w^{(q)})\\bigr|/\\bigl|b_d^T \\hat w^{(q)}\\bigr|$ over rays, unit-simplex weights, and $d$, and evaluate $\\kappa_F(P\\otimes\\phi)$ for the actual $P$ built from the two inconsistent parallel-beam geometries. If $\\gamma = \\tilde\\eta\\,\\kappa_F(P\\otimes\\phi)\\ge 1$, or if the FBP-based residual bound (34) is violated at the reported iteration counts, then the hypotheses of Theorem 2 do not hold for the experiments that are used to demonstrate convergence.","supporting_citations":[{"cited_title":"Deuflhard","cited_arxiv_id":null,"evidence_quote":"Supplies the simplified Newton method template that the AFIRE iteration is built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classic iterative solution framework for nonlinear equations that the scheme employs."},{"cited_title":"Natterer and F","cited_arxiv_id":null,"evidence_quote":"Provides the invertibility condition for X-ray transforms (Theorem 2.11) and the continuity estimate used in Propositions 1 and 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the energy-selective basis decomposition model that defines the nonlinear forward map."},{"cited_title":"Zou and M","cited_arxiv_id":null,"evidence_quote":"Documents the fast kV-switching projection mismatch that motivates the geometric-inconsistent setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The non-convex primal-dual baseline that AFIRE is compared against in the experiments."},{"cited_title":"Gao and C","cited_arxiv_id":null,"evidence_quote":"The nonlinear Kaczmarz method baseline and its convergence framework, used for comparison in the tests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The iterative FBP baseline designed for mismatched dual-energy projections, compared in the experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the mass-attenuation tables for water and bone used in the numerical experiments."},{"cited_title":"Philippe","cited_arxiv_id":null,"evidence_quote":"An open-source X-ray spectrum simulator used to generate the 80-kV and filtered 140-kV spectra in the tests."}],"review_version":1}