REVIEW 4 major objections 3 minor 6 references
Exact and fast inversion of the approximate discrete Radon transform from partial data
T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Exact inversion of the approximate discrete Radon transform is possible from a single quadrant of data, in $O(N \log N)$ operations.
desk verdict The paper's advertised exact partial-data inverse for the ADRT rests on an ill-posed recursion, and the key identity in Lemma 6 does not survive a natural repair; the idea is promising but the proof needs real work. 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 digital line $D_m(h,s)$, a subset of $\mathbb{Z}^2$ defined recursively in Definition 3 by splitting into two translated and reflected copies of $D_{m-1}$; it is what makes the ADRT fast and what makes exact inversion possible. The load-bearing identity is Lemma 6, which states that given the transform $R^\ell_{n,m}$ of the $\ell$-th section, the transforms $R^{2\ell-1}_{n,m-1}$ and $R^{2\ell}_{n,m-1}$ of its two child sections can be computed in $O(M)$ operations, where $M$ is the size of the parent data. The proof expresses the difference $\Delta^\ell_{n,m-1}(h,s) = R^\ell_{n,m-1}(h+1,s) - R^\ell_{n,m-1}(h,s)$ as a difference of two parent transform values; because the two digital lines in the recursive union share a child line, subtracting cancels the shared contribution. A single prefix-sum sweep over $h$, using the bounded support from Lemma 5, then turns the differences back into the child transforms. This mechanism carries the entire inversion.
What would settle it
Take a small random image, say $16\times16$ pixels, compute its single-quadrant ADRT, and run the inversion recurrence of Lemma 6 level by level; if the recovered pixels differ from the original by more than floating-point rounding, the claimed exactness is false. A more targeted check is to verify the two difference identities (2.6) and (2.7) on all $(h,s)$ pairs at one level and to enumerate $D_m(h,s)$ to confirm the Lemma 5 bound.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is Theorem 1: a $2^n \times 2^n$ image with $N = 2^{2n}$ pixels is determined exactly by its single-quadrant ADRT, and the recovery can be done in $O(N \log N)$ operations. The transform's digital lines $D_m(h,s)$ are defined recursively by splitting a line at scale $m$ into two child lines at scale $m-1$, one in each half of the image. The paper shows that this recursion is reversible in the transform domain: the differences of the child-section transform values satisfy simple identities, equations (2.6) and (2.7), that depend only on values of the parent transform at nearby parameters $(h,s)$. Once those differences are known, prefix sums reconstruct the child transforms, and repeating from $m=n$ down to $m=0$ recovers the image exactly. Only one quadrant, corresponding to digital lines with slopes in the continuous angular range $[0,\pi/4]$, is needed.
Load-bearing premise
The proof assumes that the recursive definition of the digital line $D_m(h,s)$ in Definition 3 is exactly as written, particularly the placement of the index $s$ in equation (2.2), and that the unproved bounding property in Lemma 5, $D_m(h,s) \subseteq \{h \le i \le h+s,\ 1 \le j \le 2^m\}$, holds; if either fails, the difference identities (2.6)-(2.7) no longer hold and the inversion formula collapses.
Editorial extensions
If this is right
- Exact inversion from one quadrant costs the same as the forward transform, so any pipeline that applies ADRT and then needs the image back can do so without iterative solvers or multigrid.
- Only a single quadrant of ADRT data is required for exact recovery, so storage and acquisition can be reduced by a factor of four in applications that currently compute the full transform.
- The level-by-level reversal is naturally parallel: at each step, each section can be inverted independently, which the paper notes is useful for domain decomposition and parallelization.
- Because the inversion is exact, reconstruction error is governed only by how well digital lines approximate continuous straight lines, not by inversion error, supporting the use of ADRT as a surrogate for the continuous Radon transform.
- The formula provides a direct alternative to earlier multigrid and conjugate-gradient approaches, replacing numerical approximation with a finite exact recurrence.
Reading between the lines
- The paper does not state this, but if the single-quadrant ADRT is exactly invertible, the full four-quadrant ADRT is information-theoretically redundant by a factor of four; that redundancy could be exploited for consistency checks or error correction in measured data.
- The difference identities (2.6)-(2.7) are purely combinatorial, so the same inversion scheme should transplant to other recursive digital-line definitions, including non-power-of-two grids or higher-dimensional pixelated lines, provided an analogue of Lemma 5 holds.
- One can turn the proof into a built-in self-test: running the inversion on a random image and comparing with the original would catch any implementation error, because exactness is a discrete identity rather than a limiting statement.
- The exactness result suggests that the gap between the continuous Radon transform and its discrete pixelated approximation is narrower than the word 'approximate' implies; quantifying how inversion error behaves under line refinement is a natural next step.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that the approximate discrete Radon transform (ADRT) of Brady can be inverted exactly from a single quadrant of transform data in O(N log N) operations for an N-pixel square image. The proposed inversion recursively recovers the ADRT of smaller sections of the image from the ADRT of larger sections, using difference identities (2.6) and (2.7). The main theorem is stated as Theorem 1 and is derived directly from Lemma 6, which asserts that each level-m ADRT can be reduced to the two level-(m-1) section transforms in linear time.
Significance. If the central claim were correct, it would be a notable result: an exact, fast inverse of the ADRT with the same asymptotic cost as the forward transform, and requiring only partial data. The paper is self-contained in its derivation and does not rely on fitted parameters or circular normalization choices, which is a strength. However, the validity of the main theorem hinges entirely on the recursive definition of digital lines and the difference identities of Lemma 6. As detailed below, the recursion as stated is ill-posed and the identities are false under a natural correction, so the claimed inversion formula is not established. The paper therefore does not, in its current form, support the advertised conclusion.
major comments (4)
- [Definition 3, Eq. (2.2)] Definition 3 is ill-posed. In Eq. (2.2), D_m(h,2t+1) and D_m(h,2t) are defined in terms of D^2_{m-1}(h+t+1,s) and D^2_{m-1}(h+t,s), respectively, where s is the level-m argument (2t+1 or 2t). But D_{m-1} is only defined for second arguments in {1,...,2^{m-1}}. For s>2^{m-1}, the expressions D^2_{m-1}(...,s) are undefined, so the recursion does not define D_m for most of its domain. This renders the subsequent use of D_m in the ADRT definition and in Lemma 6 invalid as written.
- [Lemma 6, proof display after Eq. (2.6)] The proof of Lemma 6 relies on an expansion that is inconsistent with Definition 3. For example, the term R_{\ell;n,m}(h+1,2s) is expanded using D_m(h+1,2s) = D^1_{m-1}(h+1,s) \cup D^2_{m-1}(h+s+1,s), but according to Definition 3 the second part should be D^2_{m-1}(h+s+1,2s) (or, with the natural correction of the recursion, D^2_{m-1}(h+s+1,t) with t related to s), not D^2_{m-1}(h+s+1,s). The cancellation that produces Eq. (2.6) depends on this incorrect substitution, so the identity is not proven.
- [Lemma 6, Eq. (2.6)] The claimed identity (2.6) is false under the standard corrected digital-line recursion. For the 8x8 image A(i,j)=i with n=m=3 and s=1, direct computation gives R_{3,2}(h+1,1)-R_{3,2}(h,1)=4, whereas the right-hand side of (2.6), R_{3,3}(h+1,2)-R_{3,3}(h,3), equals 3 for the same h. Thus the identity fails in a concrete case. Since Lemma 6 is the sole inversion step used to prove Theorem 1, the main theorem is not established.
- [Lemma 5] Lemma 5 is stated without proof, but its properties are load-bearing: property (iii) is used in Eq. (2.8) to truncate the prefix sum that reconstructs R_{\ell;n,m-1} from the differences, and property (v) is used to justify the lower limit of summation. As written, the lemma is an unproved input to the central derivation. This would be a gap even if the recursion in Definition 3 were corrected and the identities (2.6)-(2.7) were valid.
minor comments (3)
- [Title and Abstract] The title and abstract contain spacing artifacts (e.g., "F AST") that should be corrected.
- [Definition 2] In Definition 2, the range of ℓ is stated as ℓ=1,...,2^{n-m}, but for m=n this range is just ℓ=1, which is consistent, yet the later use of ℓ in Lemma 6 for ℓ up to 2^{n-m+1} deserves an explicit clarification.
- [Notation for t in Definition 3] The range t=1,...,2^{m-1} in Definition 3 excludes the case s=1 when s is written as 2t+1 (there t=0), so the recursion does not cover all s=1,...,2^m; the indexing should be stated carefully.
Circularity Check
No circularity: the inversion is derived from the forward transform's recursive definition with no fitted parameters or load-bearing self-citations.
full rationale
The derivation chain in Theorem 1 is self-contained. Lemma 6 obtains the lower-level ADRT sections from the higher-level section by algebraic subtraction of sums over recursively defined digital lines, as expressed in Eqs. (2.4)-(2.8), and Theorem 1 simply iterates this reduction from m=n down to m=0. The only external input is Brady's forward ADRT definition, which is the object being inverted rather than the output of the inversion. The two numerical-observation citations ([5] Press and [6] Rim) appear only as context and are not used in the proof of Theorem 1. Lemma 5's unproved digital-line properties are assumptions about the forward transform, not conclusions forced by the inversion algorithm. There is no fitted parameter later renamed as a prediction, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in by citation. A skeptical reviewer's concern that Definition 3 is ill-posed and that Eq. (2.6) may fail in small cases would be a correctness objection, not a circularity objection: even if the recursion is misstated or the lemma is false, the paper's inversion claim is not equivalent by construction to any of its inputs.
Assumptions & free parameters
assumptions (3)
- domain assumption The digital-line recursion in Definition 3 correctly defines the ADRT as introduced by Brady [1].
- domain assumption Lemma 5 properties (i)-(v) hold, particularly the bound that D_m(h,s) is contained in {h to h+s} by {1 to 2^m}.
- standard math The identities (2.6) and (2.7) in Lemma 6 follow from direct computation using the recursion.
Cite this review
Pith. "Pith review of Exact and fast inversion of the approximate discrete Radon transform from partial data." pith.science (2026). https://pith.science/paper/YTF5B4XD
@misc{pith2026190800887,
author = {Pith},
title = {Pith review of: Exact and fast inversion of the approximate discrete Radon transform from partial data},
year = {2026},
howpublished = {\url{https://pith.science/paper/YTF5B4XD}},
note = {Machine review of arXiv:1908.00887}
}
abstract
We give an exact inversion formula for the approximate discrete Radon transform introduced in [Brady, SIAM J. Comput., 27(1), 107--119] that is of cost $O(N \log N)$ for a square 2D image with $N$ pixels and requires only partial data.
Figures
Reference graph
Works this paper leans on
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[5]
W. H. Press , Discrete Radon transform has an exact, fast inverse and zes t o operations other than sums along lines , Proceedings of the National Academy of Sciences, 103 (2006 ), pp. 19249–19254, doi:10.1073/pnas.0609228103
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[1]
M. L. Brady , A fast discrete approximation algorithm for the Radon trans form, SIAM Journal on Computing, 27 (1998), pp. 107–119, doi:10.1137/S0097539793256673
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[2]
W. G ¨otz and H. Druckm ¨uller, A fast digital radon transform: an efficient means for evaluating the hough transform , Pattern Recognition, 29 (1996), pp. 711 – 718, doi:10.1016/0031-3203(96)00015-5
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[3]
S. Helgason , Integral Geometry and Radon Transforms , Springer New York, New York, NY, 2011, doi:10.1007/978-1-4419-6055-9
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[4]
F. Natterer , The Mathematics of Computerized Tomography , Society for Industrial and Ap- plied Mathematics, 2001, doi:10.1137/1.9780898719284
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[6]
D. Rim , Dimensional splitting of hyperbolic partial differential e quations using the Radon transform, SIAM Journal on Scientific Computing, 40 (2018), pp. A4184– A4207, doi:10.1137/17M1135633. 4
Reviewed August 14, 2026 · model on record in the stance chip above.
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