Pith. sign in

REVIEW 3 major objections 4 minor 29 references

On Two Families of Funk-Type Transforms

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For an exterior center, the shifted Funk transform on the sphere is conjugate to the parallel slice transform; the paper derives from this an explicit inversion and an exact description of the kernel.

desk verdict Solid paper that closes the previously open exterior-center case of the shifted Funk transform; the main caveat is that the core identity's proof imports two analytic facts from the authors' own arXiv preprint, but spot-checks suggest the imported identities are correct. read the letter →

arxiv 1908.06794 v1 pith:OFMNBEUC submitted 2019-08-19 math.FA

Mark Agranovsky , Boris Rubin This is my paper · ORCID
classification math.FA MSC 44A1237E30
keywords shiftedFunktransformparallelsliceRadon-Johnd-planesphericaltomographyintegralgeometryinversionformulaMöbiusautomorphisminjectivity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper treats two families of integral operators on the unit sphere $S^n$: the shifted Funk transform $F_a$, which integrates a function over $k$-dimensional plane sections passing through a point $a$ outside the sphere, and the parallel slice transform $\Pi_a$, which integrates over $k$-planes parallel to a fixed direction. Its aim is to decide when these transforms are injective and to reconstruct the function from its integrals. The paper's central claim is that for $|a|>1$ the exterior-center transform is equivalent, through a Möbius change of variables together with a weight factor, to the parallel slice transform. From that equivalence it obtains an explicit inversion formula on the subspace of functions satisfying $f=W_a f$ and proves that the null space consists exactly of the functions satisfying $f=-W_a f$. This matters because spherical section integrals arise in spherical tomography, where knowing exactly which component of the function is invisible can matter as much as having a reconstruction formula.

What carries the argument

The central machinery is the conjugation identity (5.2) itself. Its main pieces are the Möbius automorphism $\phi_{a_*}$, defined by (4.1) with $a$ replaced by $a_*$, which realizes a bijection between the exterior-plane family $T_a(n+1,k)$ and the parallel-plane family $Z_a(n+1,k)$; the weighted composition operator $M_{a_*}$, which absorbs the Jacobian of the change of variables; and the parallel slice transform $\Pi_a$, which Lemma 3.1 reduces to the Radon-John $d$-plane transform on the ball in $a^\perp$. The equality converts every statement about exterior-center Funk data into a statement about parallel-slice data, so injectivity, kernel, and inversion for $F_a$ follow from the corresponding Radon-John facts.

What would settle it

Take a concrete test function and an exterior center, e.g. $f\equiv 1$ on $S^n$ with $n=2$, $k=2$, $a=(2,0,0)$, and compute both sides of (5.2) directly by elementary integration for a choice of $\tau$; the two sides must agree exactly. Alternatively, verify the skipped ingredient (6.3) numerically for the same data: the limit of the smoothed transform on the left must equal $(1-|\xi'a|^2)^{-1/2}(F_a f)(\tau_\xi)$. A mismatch at any single $\xi$ with $|\xi'a|<1$ would break the chain.

Watch

Extended reading notes

Core claim

The core discovery is the identity (5.2): for $f\in C(S^n)$, $1<k\le n$, and $|a|>1$, $$(F_a f)(\tau)=(\Pi_a M_{a_*} f)(\phi_{a_*}\tau),$$ where $a_*=a/|a|^2$, $\phi_{a_*}$ is the involutive Möbius automorphism that sends the $k$-planes through $a$ to $k$-planes parallel to $a$, and $M_{a_*}$ is the weighted composition $(M_{a_*}f)(y)=(s_{a_*}/(1-a_*\cdot y))^{k-1}(f\circ\phi_{a_*})(y)$. The paper calls this equality 'the core of the paper'. It implies that $F_a$ is injective exactly on the subspace $f=W_a f$, that its kernel is $\{f:\,f=-W_a f\}$, where $W_a$ is an involution built from the chord-reflection map $\tau_a$ and the weight $((|a|^2-1)/|a-x|^2)^{k-1}$, and that reconstruction is given by $f=M_{a_*}^{-1}\Pi_a^{-1}((F_a f)\circ\phi_{a_*})$.

Load-bearing premise

The central equality is proven only by relying on two imported analytic facts: a limit identity whose proof is skipped as a 'verbatim copy', and a change-of-variables formula cited from the authors' earlier work. If either of those is wrong, the paper's main injectivity and inversion results collapse.

Editorial extensions

If this is right

  • For every exterior center $a$, every $f\in C(S^n)$ satisfying $f=W_a f$ can be reconstructed from its shifted Funk data by the explicit chain $f=M_{a_*}^{-1}\Pi_a^{-1}((F_a f)\circ\phi_{a_*})$.
  • The kernel of the exterior-center transform is exactly the set of solutions of $f=-W_a f$, so $F_a$ is never injective on all continuous functions; only the $W_a$-even part of a function is recoverable.
  • Because $\Pi_a$ is invertible on $C_a^+(S^n)$ via the Radon-John inversion formula, the reconstruction of $F_a$ inherits an explicit algorithmic structure rather than an abstract existence proof.
  • The dimension-link theorem (7.5) lets one reduce inversion for lower-dimensional sections to inversion for $k=n$, at the price of one extra integration step and a higher-order differential operator in the Radon-John inversion.
  • In the exterior case the same conjugation pattern that for $|a|<1$ related $F_a$ to the geodesic Funk transform now relates $F_a$ to parallel slices, showing that the qualitative behavior of the transform changes sharply at the sphere boundary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if (5.2) holds, it suggests a direct numerical inversion algorithm for exterior-center spherical tomography: invert the parallel-slice data on the ball using any Radon-John solver, then pull back through the Möbius map; no new quadrature over exterior planes is needed.
  • Beyond the paper: the same conjugacy idea could apply to other Möbius-invariant integral transforms on $S^n$; any transform whose sections are the images of parallel planes under a spherical automorphism should admit a parallel-slice representation with a computable Jacobian.
  • Beyond the paper: the paired-transform idea mentioned in the introduction, with one center inside and one outside the sphere, would likely give injectivity on all of $C(S^n)$, since the outside transform recovers the $W_a$-even component and an inside transform can recover what the exterior one loses.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies two families of Funk-type transforms on the unit sphere: the shifted Funk transform F_a with exterior center |a|>1, integrating over k-dimensional plane sections through a fixed point a, and the parallel slice transform Π_a, integrating over k-planes parallel to a fixed vector. The central result, Theorem 5.1, establishes the identity F_a f = (Π_a M_{a*} f)∘φ_{a*} with an explicit weight M_{a*} and the Möbius automorphism φ_{a*}. From this identity the authors derive injectivity of F_a on functions satisfying f = W_a f, the exact kernel {f = -W_a f}, and an explicit inversion formula. Section 3 reduces Π_a to the Radon-John d-plane transform on the Euclidean ball and gives inversion formulas, and Section 7 establishes a relation between shifted Funk transforms of different dimensions. The proof of the core identity is deferred to Section 6 and relies on two analytic facts imported from the authors' previous arXiv preprint [2].

Significance. If the results are correct, the paper completes the structural understanding of exterior-center Funk transforms: it gives not only injectivity but an exact description of the kernel and an explicit inversion procedure on the injective subspace. The reduction of the parallel slice transform to the Radon-John transform is self-contained, parameter-free, and connects the topic to classical integral geometry. The main weakness is completeness: the core identity (5.2) rests on two imported identities whose proofs are not included in the manuscript. I independently checked the less obvious of these, equation (6.3), in a model case and found it correct; the concern is therefore about the manuscript's self-containedness rather than about mathematical error.

major comments (3)
  1. [Section 6, Eq. (6.3)] The limit identity lim_{ε→0}(F_{a,ε}f)(ξ) = (1-|ξ'a|^2)^(-1/2)(F_a f)(τ_ξ) is the normalization step that makes the comparison in the proof of Theorem 5.1 work. Its proof is explicitly omitted: the text states it is 'a verbatim copy of Step I in [2, Section 6], and we skip it'. Since [2] is an arXiv preprint rather than a published source, this identity should either be proved in the present paper or supplied with a citable published reference. As it stands, the central equality (5.2), and with it Theorems 5.3 and 5.4, rests on an unverified imported fact.
  2. [Section 6, Eq. (6.4)] The change-of-variables formula ∫_{S^n} f(x)dx = s_{a*}^n ∫_{S^n} (f∘φ_{a*})(y)(1-a*·y)^(-n)dy, cited from [2, Lemma 2.1], is equally load-bearing: it converts the defining integral of F_{a,ε} into the y-integral on which all subsequent steps act. The identity follows from the Jacobian of φ_{a*} given in (4.3), but the manuscript should either prove it or cite a published source; a citation to an unpublished companion preprint is not sufficient for a core step of the main theorem.
  3. [Section 3, Theorem 3.2] The step 'the corresponding function φ in (3.7) is zero' uses injectivity of the Radon-John transform, but the hypotheses needed to apply Theorem 2.1 are not stated. For continuous f, the function φ(y)=2(1-|y|^2)^(-1/2)f(y+√(1-|y|^2)̃a) is in L^1 on the ball, and because k-1<n, the condition 1≤p<n/(k-1) holds with p=1. This is an easily repairable gap, but it should be made explicit so that the kernel characterization (3.8) is fully justified.
minor comments (4)
  1. [Section 3 and Section 5] Theorem 3.3 states 'a ∈ B^{n+1} \ {0}', while Section 3 opens with the assumption |a|>1 and Theorem 5.4 needs the inversion formula (3.10) for an exterior center. The intended domain appears to be a ≠ 0, and the statement should be corrected to remove this ambiguity.
  2. [Section 6, displayed formula after h_φ(z)] The displayed equation '(Fa,εf )(ξ) =' is missing a left parenthesis; it should read '(F_{a,ε}f)(ξ) ='.
  3. [Abstract and Introduction] There are several typographical errors: 'latt er' should be 'latter', 'inegrate' should be 'integrate', 'non-injectv ity' should be 'non-injectivity', and 'arbirary' should be 'arbitrary'.
  4. [References] Reference [2] is an arXiv preprint. If the authors prefer not to include full proofs of (6.3) and (6.4), they should at least update the reference to its published version once available, or add a precise statement of the cited lemma in the present paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the core identity (5.2) rests on two deferred but independent analytic facts from the authors' prior work, and the remaining derivation is self-contained.

full rationale

The paper's central result, Theorem 5.1, is an equality (5.2) linking the exterior-center shifted Funk transform F_a to the parallel slice transform Π_a. Its proof in Section 6 imports two analytic identities from the authors' previous work [2]: the limit relation (6.3), whose proof is described as 'a verbatim copy of Step I in the similar proof in [2, Section 6], and we skip it', and the change-of-variables formula (6.4), 'which was proved in [2, Lemma 2.1]'. These are real self-citations and the proof of the paper's core identity does depend on them, so a strict referee could ask for more detail. But they are not circular: both are parameter-free technical lemmas about integration under Möbius automorphisms and approximate delta-functions on sphere sections; neither states, assumes, or is equivalent to the target equality (5.2), nor do they encode injectivity or inversion. The rest of the derivation is worked out in the manuscript: Lemma 3.1 reduces Π_a to the Radon–John d-plane transform, Theorem 3.2 proves injectivity on C_a^+(S^n), Lemma 4.1 describes how φ_{a*} maps planes through a to planes parallel to a, and Lemmas 4.2 and 5.2 plus the weight computation (5.8) are algebraic and verified in the text. The formulas (5.9)–(5.11) are consequences of (5.2), not restatements of its input. There are no fitted parameters, no data-dependent constants, and no definition of a target quantity in terms of itself. The deferred items are a completeness gap, not a circular reduction. Under Rule 4, the cited identities are independent support because they are parameter-free with stated assumptions that do not include the target result. Therefore the correct circularity verdict is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard integral geometry: the Radon-John inversion theorem (quoted from Rubin [21]), the calculus of spherical Möbius automorphisms (Rudin [25], Stoll [28]), polar decomposition and determinant identities (Muirhead [14]), and two facts imported from the same authors' preprint [2]: the change of variables (6.4) and the mollifier limit (6.3). No free parameters and no invented entities: all new objects (M_{a*}, W_a, ρ_{a*}) are defined constructively from existing ones.

assumptions (5)
  • standard math Radon-John d-plane transform inversion (Theorem 2.1)
    Quoted from Rubin [21, Theorem 3.5]; used to invert the parallel slice transform in Theorems 3.2 and 3.3. Assumes L^p integrability with 1 ≤ p < n/d, satisfied in the application.
  • standard math Change of variables for the spherical Möbius automorphism, formula (6.4)
    ∫ f dx = s_{a*}^n ∫ (f∘φ_{a*})(y)(1 - a*·y)^{-n} dy, invoked in the proof of Theorem 5.1 and cited from the same authors' preprint [2, Lemma 2.1]. It is the Jacobian of the automorphism φ_{a*}, consistent with the standard identity (4.3).
  • standard math Möbius automorphism properties (4.1)-(4.3), including that φ_a maps the ball and sphere to themselves
    Taken from Rudin [25, Section 2.2.1] and Stoll [28]; used throughout Sections 4 to 6 to relate planes through a to planes parallel to a.
  • standard math Polar decomposition of the (n+1) x (n+1-k) matrix Q_{a*}ξ as ηρ^{1/2}
    Used in Lemma 4.1 and Section 6; cited to Muirhead [14, pp. 66, 591] and the determinant identity det(I + AB) = det(I + BA).
  • standard math Convergence of smoothed section integrals to the section integral, formula (6.3)
    The limit lim_{ε→0} (F_{a,ε} f)(ξ) = (1 - |ξ'a|^2)^{-1/2} (F_a f)(τ_ξ) for a bump function ω_ε. The paper states this is proved verbatim in [2, Section 6] and omits the proof; a standard mollifier argument, but the only genuinely deferred step.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Two Families of Funk-Type Transforms." pith.science (2026). https://pith.science/paper/OFMNBEUC

@misc{pith2026190806794,
  author       = {Pith},
  title        = {Pith review of: On Two Families of Funk-Type Transforms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFMNBEUC}},
  note         = {Machine review of arXiv:1908.06794}
}
read the original abstract

We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon-John d-plane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 22 canonical work pages

  1. [2]

    Non-geodesic Spherical Funk Transforms with One and Two Centers

    M. Agranovsky and B. Rubin, Non-geodesic spherical Funk tran sforms with one and two centers, 2019, arXiv:1904.11457 [math.F A]

    work page Pith review arXiv 2019

  2. [1]

    Abouelaz and R

    A. Abouelaz and R. Daher, Sur la transformation de Radon de la sp h` ere Sd. Bull. Soc. Math. France 121 (1993), 353-382

    work page 1993

  3. [3]

    P. G. Funk, ¨Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien”, The- sis, Georg-August-Universit¨ at G¨ ottingen, 1911

    work page 1911

  4. [4]

    P. G. Funk, ¨Uber Fl¨ achen mit lauter geschlossenen geod¨ atschen Linen.Math. Ann., 74 (1913), 278–300

    work page 1913

  5. [5]

    R. J. Gardner, Geometric Tomography (second edition). Cambridge University Press, New York, 2006. 20 M. AGRANOVSKY AND B. RUBIN

    work page 2006

  6. [6]

    I. M. Gelfand, S. G. Gindikin, and M. I. Graev. Selected Topics in Integral Ge- ometry, Translations of Mathematical Monographs, AMS, Providence, Rh ode Island, 2003

    2003

  7. [7]

    Gindikin, J

    S. Gindikin, J. Reeds, and L. Shepp, Spherical tomography and s pherical in- tegral geometry. In Tomography, impedance imaging, and integral geometry (South Hadley, MA, 1993) , 83–92, Lectures in Appl. Math., 30, Amer. Math. Soc., Providence, RI (1994)

    work page 1994

  8. [8]

    Helgason, The totally geodesic Radon transform on constant curvature spaces

    S. Helgason, The totally geodesic Radon transform on constant curvature spaces. Contemp. Math. , 113 (1990), 141–149

    work page 1990

Show all 29 references
  1. [9]

    Helgason, Integral geometry and Radon transform

    S. Helgason, Integral geometry and Radon transform . Springer, New York- Dordrecht-Heidelberg-London, 2011

    2011

  2. [10]

    Hielscher and M

    R. Hielscher and M. Quellmalz, Reconstructing a function on the s phere from its means along vertical slices. Inverse Probl. Imaging , 10(3) (2016), 711-739

    2016

  3. [11]

    S. G. Kazantsev, Funk-Minkowski transform and spherical c onvolution of Hilbert type in reconstructing functions on the sphere. Siberian Ele ctronic Mathematical Reports 15, 1630–1650

  4. [12]

    A. Markoe. Analytic Tomography. Encyclopedia of Mathematics and its Appli- cations 106, Cambridge Univ. Press, 2006

    2006

  5. [13]

    Minkowski, ¨Uber die K¨ orper konstanter Breite [in Russian]

    H. Minkowski, ¨Uber die K¨ orper konstanter Breite [in Russian]. Mat. Sbornik . 25 (1904), 505–508; German translation in Gesammelte Abhandlungen 2, Bd. (Teubner, Leipzig, (1911), 277–279

    1904

  6. [14]

    Muirhead, Aspects of multivariate statistical theory , John Wiley & Sons

    R.J. Muirhead, Aspects of multivariate statistical theory , John Wiley & Sons. Inc., New York, 1982

    1982

  7. [15]

    Palamodov

    V. Palamodov. Reconstructive Integral Geometry. Monographs in Mathematics,

  8. [16]

    Palamodov, Reconstruction from Integral Data

    V.P. Palamodov, Reconstruction from Integral Data. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016

    2016

  9. [17]

    Quellmalz, A generalization of the Funk-Radon transform

    M. Quellmalz, A generalization of the Funk-Radon transform. In verse Problems 33, no. 3, 035016, 26 pp. (2017)

    2017

  10. [18]

    Quellmalz, The Funk-Radon transform for hyperplane sectio ns through a common point, Preprint, arXiv:1810.08105 (2018)

    M. Quellmalz, The Funk-Radon transform for hyperplane sectio ns through a common point, Preprint, arXiv:1810.08105 (2018)

    2018 arXiv

  11. [19]

    Rubin, Inversion formulas for the spherical Radon transfo rm and the gener- alized cosine transform

    B. Rubin, Inversion formulas for the spherical Radon transfo rm and the gener- alized cosine transform. Advances in Appl. Math. , 29 (2002), 471–497

    2002

  12. [20]

    Rubin, Reconstruction of functions from their integrals ove r k-dimensional planes

    B. Rubin, Reconstruction of functions from their integrals ove r k-dimensional planes. Israel J. of Math. 141 (2004), 93–117

    2004

  13. [21]

    Rubin, On the Funk-Radon-Helgason inversion method in integ ral geometry

    B. Rubin, On the Funk-Radon-Helgason inversion method in integ ral geometry. Contemp. Math. , 599 (2013), 175–198

    2013

  14. [22]

    B. Rubin, Introduction to Radon transforms: With elements of fractio nal calcu- lus and harmonic analysis (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2015

    2015

  15. [23]

    B. Rubin, Reconstruction of functions on the sphere from the ir in- tegrals over hyperplane sections, Analysis and Mathematical Phys ics, https://doi.org/10.1007/s13324-019-00290-1, 2019

    2019 doi

  16. [24]

    Rubin, The vertical slice transform in spherical tomography , 2018, arXiv:1807.07689

    B. Rubin, The vertical slice transform in spherical tomography , 2018, arXiv:1807.07689

    2018 arXiv

  17. [25]

    Rudin, Function theory in the unit ball of Cn, Springer-Verlag, New York, NY, 1980

    W. Rudin, Function theory in the unit ball of Cn, Springer-Verlag, New York, NY, 1980. FUNK-TYPE TRANSFORMS 21

    1980

  18. [26]

    Salman, An inversion formula for the spherical transform in S2 for a special family of circles of integration

    Y. Salman, An inversion formula for the spherical transform in S2 for a special family of circles of integration. Anal. Math. Phys., 6, no. 1 (2016), 43–58

    2016

  19. [27]

    Salman, Recovering functions defined on the unit sphere by in tegration on a special family of sub-spheres

    Y. Salman, Recovering functions defined on the unit sphere by in tegration on a special family of sub-spheres. Anal. Math. Phys. 7, no. 2 (2017), 165–185

    2017

  20. [28]

    Stoll, Harmonic and subharmonic function theory on the hyperbolic ball

    M. Stoll, Harmonic and subharmonic function theory on the hyperbolic ball. LMS Lecture Notes in Mathematics, vol. 155, Cambridge University Press, 2016. Department of Mathematics, Bar Ilan University, Ramat-Gan, 5290002, and Holon Institute of Technology, Holon, 5810201, Israe...

    2016

  21. [98]

    Birkh¨ auser Verlag, Basel, 2004

    2004

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.