REVIEW 3 major objections 3 minor 1 cited by
Complexity of Radon transforms
T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The universal inverse Radon formula forces direct Radon images to be complex-valued, and hybrid Wigner-like functions supply the missing imaginary part.
desk verdict The advertised non-trivial holonomy in Section 3 is an artifact—it compares Radon transforms of two different functions—and the rest of the complexity argument rests on the author's own prior representation, so the central claim is not established. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the hybrid (Wigner-like) function $F(x_1,x_2;k)=\int dx_3\, e^{-ikx_3} f(x_1,x_2,x_3)$, a phase-space object that is complex even when $f$ is real, together with its discrete-slice analogue $\tilde F(x_1,x_2;k)=\sum_n e^{-ikx_{3n}} f(x_1,x_2,x_{3n})$. These functions carry the direct Radon transform into the complex plane, and the inverse formula splits into a principal-value part $F_S$ and an imaginary boundary term $F_A$; the imaginary part of the complex Radon transform supplies $F_A$ naturally. A second mechanism is the non-trivial holonomy of the Radon transform developed in Section 3, where shifting the angle by $\pi$ twice and comparing two rotation paths produces a mismatch for outset functions supported in different quadrants, which the paper interprets as another source of complexity.
What would settle it
For a real compactly supported function $f$, compute the hybrid function $F$ from (5.1), its complex Radon image $R[F]$, and invert using only the real principal-value part $F_S$; if the reconstruction reproduces $f$ to the same accuracy as the full inverse $F_S+F_A$, then the imaginary contribution is not needed and the central claim fails.
Extended reading notes
Core claim
The paper claims to establish that the universal representation of the inverse Radon transform implies that the direct Radon transform must be complex-valued. The derivation starts from the Fourier slice theorem with an $ε$-regularized radial $λ$-integration, which produces an imaginary contribution beyond the standard principal-value term; in the unified inversion, the direct Radon image enters under both real and imaginary measures. Since the outset function $f$ is real, the paper introduces hybrid functions $F(x_1,x_2;k)$ (the Fourier transform of $f$ in $x_3$) and its slice-sum version $\tilde F(x_1,x_2;k)$, whose Radon transforms $R[F;\tilde F](\tau,\phi;k)$ are complex and carry the needed imaginary part. The reconstruction scheme $R[F;\tilde F] \to \{F,\tilde F\} \to f$ is the central mechanism: the momentum $k$ acts as an external parameter that aggregates all two-dimensional slices, and the imaginary part of the complex Radon transform feeds the additional term $F_A$ in the inverse formula.
Load-bearing premise
The whole need for complex direct Radon transforms rests on the assumption that the $ε$-regularized $λ$-integration in the Fourier slice theorem produces a genuine imaginary term in the universal inverse formula that cannot be absorbed by another choice of regularization.
Editorial extensions
If this is right
- In the universal inverse scheme, complex-valued direct Radon data are required, so reconstruction pipelines that assume real sinograms are missing a contribution.
- The discrete slice method can be replaced by a single complex object $\tilde F(x_1,x_2;k)$ whose Fourier parameter $k$ serves as a continuous external parameter for optimization.
- The additional imaginary contribution $F_A$ changes the inverse formula and therefore the regularization procedure for ill-posed reconstruction problems.
- Objects containing localized defects can be separated from a common background by exploiting the holonomy condition on the two rotation paths.
Reading between the lines
- A direct numerical test of the central claim is to reconstruct a phantom from complex sinograms with and without $F_A$; if the imaginary term changes the result measurably, the paper's mechanism is confirmed as practically relevant.
- The hybrid-function construction effectively performs a continuous interpolation of discrete CT slices, so it may also reduce slice-mismatch artifacts; the paper does not explore this.
- The holonomy argument compares $R[g_2](\tau,\phi+2\pi)$ with $R[g_1](\tau,\phi)$ for two different outset functions, so its force depends on whether the mismatch persists when a single function is carried around $2\pi$; a direct check would clarify how much weight this route alone can carry.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims that universal inverse Radon transforms require direct Radon transforms to be complex-valued, and it proposes two mechanisms to generate this complexity: a 'non-trivial holonomy' of Radon transforms for functions with support defects, and the use of 'hybrid (Wigner-like) functions' obtained by Fourier transforming one spatial coordinate. Section 2 re-derives the Fourier slice theorem with a particular ε-regularization of the radial Fourier integral, Section 3 attempts to exhibit a holonomy by rotating a Radon transform by 2π along two paths, Section 4 briefly discusses two-dimensional slices of three-dimensional objects, and Section 5 defines hybrid functions and states a universal inverse formula with real and imaginary parts. The paper concludes that these methods justify treating direct Radon images as complex in reconstruction and optimization.
Significance. If the central claim were established, it would imply that standard tomographic inversion, for real objects, necessarily involves complex-valued Radon data, a nonstandard and potentially impactful statement. However, the paper does not establish this claim. Section 2 correctly re-derives the standard Fourier slice theorem, but this is a textbook result. Section 3's holonomy argument contains a function-substitution error that invalidates the claimed mechanism. Section 5 does show that a partial Fourier transform of a real function is complex, but this is a trivial observation and it changes the object under consideration rather than showing that the Radon transform of the original real function is complex. The manuscript's reliance on the author's prior universal inverse representation [3,4] is also not made self-contained. No machine-checked proofs, reproducible code, or parameter-free derivations are provided; the main novelty claims are therefore not supported by the presented evidence.
major comments (3)
- [Section 3, Eqs. (3.5)-(3.9)] The holonomy argument is invalid because it compares Radon transforms of different functions. For a fixed function G, the Radon transform satisfies R[G](τ, φ + 2π) = R[G](τ, φ), since n_{φ+2π} = n_φ. Equation (3.7) computes R[g2](τ, φ + π + π), not R[G](τ, φ + 2π); the latter is R[g1](τ, φ) by (3.8). Thus inequality (3.9) merely states that R[g2](τ, φ + 2π) ≠ R[g1](τ, φ), which is a comparison of two different functions, not a holonomy of a single Radon transform. The same substitution error appears in (3.12). Since the Abstract and Conclusions advertise this holonomy as a source of complexity, the paper's central claim loses its main independent support.
- [Section 1 and Section 5, Eqs. (5.1)-(5.3)] The claimed necessity of complex direct Radon transforms is not established for the original real outset function. Section 1 states that the complex measure in the universal inverse representation 'should be compensated' by the complexity of the direct Radon image, but this is an assertion, not a proof. Section 5 then defines a hybrid function F(x1,x2;k) by a one-dimensional Fourier transform of f, notes that F is complex, and computes R[F]. This only shows that the Radon transform of a complex-valued function can be complex; it does not show that the Radon transform R[f] of the original real f must be complex. The reconstruction scheme (5.7) inverts R[F] and then recovers f, but this is a different inverse problem. The argument is therefore circular with respect to the premise from [3,4] and does not provide independent support for the abstract's claim.
- [Section 2, Eq. (2.10) and following paragraph] The assertion that the imaginary part of δ+(η) 'disappears provided the full region of angular integration is considered' is not correct as stated. With δ+(η) = 1/(η - iε), the imaginary part is π δ(η). Substituting this into (2.9) gives an imaginary contribution proportional to ∫ dφ R[f](⟨n_φ, x⟩, φ), which is not generally zero; for example, if f is a point mass at x0, this integral equals 2/|x - x0|. Thus the imaginary part does not vanish under full angular integration, and the claimed cancellation needs an explicit proof or a corrected statement.
minor comments (3)
- [Section 3, Eq. (3.5)] Equation (3.5) writes R[g1](τ, φ) where the integral actually involves g1(⃗x)ΘI(⃗x); the same shorthand appears in (3.6). This notation obscures the distinction between the restricted and unrestricted Radon transforms and contributes to the error in (3.7).
- [Throughout] There are numerous typographical and grammatical errors, including 'r.h.sof', 'semultaneously', 'opimization', 'Ratdon' in the Conclusions, and 'The Introducing' in Section 2. These should be corrected.
- [Section 1 and Eq. (2.8)] The notation 'f.r.' is used without definition; it is later stated to mean 'full regions of variations,' but this should be defined at first use. Similarly, the 'AC-regularization' mentioned in the Introduction is never defined.
Circularity Check
The claimed 'needed complexity' of direct Radon transforms is built into the author's universal inverse representation via self-citation, and the Section 5 'hybrid' method produces it by definition (a partial Fourier transform); the central claim reduces to its own construction.
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self definitional
[Section 1, paragraph 3 (page 1); Abstract]
"In [3, 4], the universal (or unified) representation of inverse Radon transform has been proposed and studied for any dimension of space. ... In particular, within the reconstruction procedure, the additional new contribution in the unified inverse Radon transforms is given by the integration with the complex measure that should be compensated, generally speaking, by the complexity of direct Radon image."
The argument is: the universal inverse representation contains a complex measure, therefore the direct Radon image must be complex to compensate. But this universal representation is the author's own prior construction, so the 'needed complexity' is the consistency condition of that chosen inverse formula, not an independent property of the Radon transform. The premise (complex measure) already contains the conclusion (complex direct image); no independent theorem shows that R[f] of a real f must be complex.
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renaming known result
[Section 5, Eqs. (5.1)-(5.3) and closing paragraph]
"We have F(x1,x2;k)=∫(dx3)e^{−ikx3}f(x1,x2,x3), where ... it is clear that f(x1,x2,x3)∈Re by definition, while F(x1,x2;k)∈C already. ... Then, we calculate the Radon image of the Fourier F-functions as ... where R[F;eF](τ,ϕ;k)∈C. ... the necessary step to obtain the complexity of direct Radon transform is given thanks for the introduction and the transition to the hybrid functions F(x1,x2;k) or eF(x1,x2;k)."
The complexity is inserted by the partial Fourier transform, a standard complexifying operation, before the Radon transform is applied; by linearity of the Radon integral, R[F] is then complex automatically. Calling F a 'hybrid (Wigner-like) function' and presenting this as the source of the 'needed complexity' is a renaming of a known operation. The original direct Radon transform of the real object f, namely R[f], is untouched and remains real; no necessity for R[f] itself to be complex follows from this construction.
1 more flagged steps
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self citation load bearing
[Section 5, first paragraph; Section 4, first sentence]
"As explained in a series of papers [2–4], the regularized inverse Radon operator involves two contributions. If one of them is related to the real integration measure, another is associated with the imaginary integration measure (that is a result of the Cauchy theorem)."
The two-contribution inverse formula with an imaginary measure is the load-bearing premise that makes complex direct Radon transforms 'needed'. Its only cited support is [2], [3], and [4], all authored or coauthored by the present author, and the formula is not re-derived or independently validated in this manuscript. The paper's central conclusion is therefore carried by the author's own prior series rather than by an externally verified result.
full rationale
The paper's advertised derivation chain has two load-bearing moves, and both reduce to constructions rather than to independent properties of the Radon transform. First, the Abstract and Introduction assert that the universal inverse representation 'implies the needed complexity' of the direct transform; but the universal representation is the author's prior construction [3,4] whose complex measure is an input, and the 'compensation' requirement is the tautological condition that this measure not produce an imaginary part for a real object. The conclusion that R[f] must be complex is thus the consistency condition of the chosen inverse formula, not an independent theorem. Second, Section 5 obtains complex direct Radon images by first replacing the real object f by its partial Fourier transform F (or the Fourier-series analogue eF); since F is complex for nonzero k and R is linear, R[F] is complex by construction. The paper then presents this definitional complexification as 'the necessary step to obtain the complexity of direct Radon transform'. This is renaming a standard Fourier transform as a 'hybrid (Wigner-like) function'. The Section 3 holonomy argument, offered as an additional independent mechanism, is not itself circular, but it is not needed for the circularity verdict; moreover, Eqs. (3.5)-(3.9) compare R[g2](φ+2π) with R[g1](φ) after switching functions, while for a single G the 2π-periodicity of R[G] rules out holonomy. The net effect is that the central claim that direct Radon transforms must be complex is forced by the author's own definitions and self-citations; the paper does not validate it against independent data or external benchmarks.
Assumptions & free parameters
assumptions (2)
- ad hoc to paper The universal inverse Radon representation from [3,4] is valid and contains a complex measure term.
- domain assumption The support restrictions of the characteristic indicators Θ_I and Θ_III, together with the 2π-periodicity of R[G] in φ, justify the comparisons in equations (3.5)-(3.8).
invented entities (1)
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Hybrid (Wigner-like) function F(x1,x2;k)
Cite this review
Pith. "Pith review of Complexity of Radon transforms." pith.science (2026). https://pith.science/paper/5PEW25FY
@misc{pith2026250618911,
author = {Pith},
title = {Pith review of: Complexity of Radon transforms},
year = {2026},
howpublished = {\url{https://pith.science/paper/5PEW25FY}},
note = {Machine review of arXiv:2506.18911}
}
read the original abstract
For the reconstruction problem, the universal representation of inverse Radon transforms implies the needed complexity of the direct Radon transforms which leads to the additional contributions. In the standard theory of generalized functions, if the outset (origin) function which generates the Radon image is a pure-real function, as a rule, the complexity of Radon transforms becomes in question. In the paper, we discuss the Fourier slice theorem analyzing the degenerated (singular) points as possible sources of the complexity. We also demonstrate the different methods to generate the needed complexity on the intermediate stage of calculations. Besides, we show that the introduction of the hybrid (Wigner-like) function ensures naturally the corresponding complexity. The discussed complexity provides not only the additional contribution to the inverse Radon transforms, but also it makes an essential impact on the reconstruction and optimization procedures within the frame of the incorrect problems. The presented methods can be effectively used for the practical tasks of reconstruction problems.
Forward citations
Cited by 1 Pith paper
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Non-integral geometry: additional term $f_A$ as a regularizing term
The additional term f_A in the universal inverse Radon transform removes complex singularities that arise for non-symmetric supports, provided certain log-branch parameters are chosen appropriately.
Reference graph
Works this paper leans on
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Reviewed August 7, 2026 · model on record in the stance chip above.
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