Landmark shape spaces with induced metrics

Peter W. Michor; Sarang Joshi; Stefan Sommer

REVIEW 3 major objections 5 minor 34 references

Landmark shape spaces with induced metrics

T0 review · 3 major / 5 minor · reviewed 2026-07-31 · grok-4.5

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

Pith's one-line read A screened elasticity operator unifies collision-free landmark metrics with rigid-motion shape invariance.

desk verdict Real operator-level unification of Kendall quotients and LDDMM landmarks; completeness is solid for free rigid-quotient flow, thinner once scale is constrained. read the letter →

arxiv 2607.28064 v1 pith:ZTKO7BU3 submitted 2026-07-30 cs.CV math.DG

Sarang Joshi , Peter W. Michor , Stefan Sommer This is my paper

classification cs.CVmath.DG MSC 58D0553A5558E1068T45

keywords landmarkshapespacesscreenedelasticityoperatorrigidmotionsdiffeomorphismmetricsconditionallypositivedefinitekernelsquotientRiemanniangeodesicmatchingscalenormalization

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

Landmark shapes are usually handled in two incompatible ways: either by stripping rigid motions and scale from Euclidean point clouds, or by inducing smooth metrics from diffeomorphism groups so points cannot collide. This paper builds one geometry that does both. It introduces a screened elasticity operator whose null-space is exactly the infinitesimal rigid motions, so the metric descends to the rigid-motion quotient, stays defined in ambient space independent of how many landmarks are used, preserves local rigid deformations, and—with enough smoothing—keeps distinct landmarks from colliding. Scale is then fixed by restricting to a unit-radius submanifold. The construction is meant so that applied matching and geodesic computation can use regular, diffeomorphism-compatible landmark metrics without giving up the classical shape invariances.

What carries the argument

The screened elasticity operator L_s = −(Id−σ²Δ)^{s−1} ∘ div ∘ sym∇ (and its Green kernel k_s): it sits between ordinary Sobolev operators and the classical elasticity/Killing operator, keeps null-space exactly se(d), and yields a conditionally positive definite cometric on the annihilator of rigid-motion orbits that descends to the quotient.

What would settle it

Match configurations that differ only by local rigid motions inside disjoint regions: the new kernel must transport those interiors rigidly while standard identity-containing kernels do not; if landmarks still collide for s ≥ 1 + d/2, or local rigid motion is still penalized on the quotient, the unification fails.

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Extended reading notes

Core claim

The screened elasticity operator has null-space exactly the rigid motions, is equivariant under rigid motions, and for sufficient order induces a stratified Riemannian metric on landmark configurations modulo rigid motions (and on a fixed-scale submanifold) that prevents collisions, is independent of landmark count, preserves local rigid motions, and removes global ones—unifying the two classical landmark geometries.

Load-bearing premise

That smoothing the classical elasticity operator while keeping only rigid motions in its null space is enough for complete, usable geodesics once the order is high enough, even though scale is not built into the operator and must be forced by a separate constraint.

Editorial extensions

If this is right

Landmark matching can use one metric that both removes global rigid motion and preserves local rigidity, without a separate preregistration step.
Metrics on landmark shapes become compatible across different numbers of landmarks because they descend from an ambient diffeomorphism metric.
With enough screening, geodesic completeness and non-collision hold on the rigid-motion quotient for the stated dimension and order range.
Scale-normalized shape comparison becomes a constrained Hamiltonian system on the unit-radius submanifold of the quotient.
The same operator idea extends, in principle, from landmarks to surfaces and images so rigid and non-rigid registration sit in one metric framework.

Reading between the lines

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

Flow-based stochastic shape models that previously ignored rigid invariance could adopt this operator so evolutionary or growth trajectories quotient global pose automatically.
Because the kernel does not vanish at infinity in 2D, numerical matching on large domains may need careful truncation or boundary handling not fully settled by the present experiments.
The stratified orbit-type structure implies matching algorithms should monitor rank drops of landmark spans; crossing singular strata could silently change the effective geometry.
If the same null-space design can be ported to discrete mesh or image operators, unified rigid-plus-deformable registration might replace two-stage pipelines in imaging practice.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The useful punchline is simple: they built an SE(d)-equivariant operator whose null-space is exactly se(d), screened enough to be C¹ for s ≥ 1 + d/2, and then descended it to a stratified metric on Land_n/SE(d) that keeps local rigid motions free while killing global ones. That is the missing middle between Sobolev (penalizes local rigid), Beppo-Levi/Laplacian (too big a null-space, degenerate on the quotient), and classical elasticity (right null-space, too rough). Prior LDDMM work removed rigid/affine motion by boundary conditions or post-alignment; this does it in the metric itself.

What is actually new is the screened elasticity family L_s, the CPD story specialized to se(d) rather than full low-degree polynomials, the induced stratified cometric plus optional fixed-scale slice, and the constrained Hamiltonian numerics. The derivations are standard geometric mechanics once the operator is chosen—Fourier form, Kelvin convolution, Noether on the rigid momentum maps, horizontal lifts—but they are done carefully and the diagnosis of the three failed alternatives is clean. Experiments are qualitative demos (local rigid preservation vs Gaussian/Matérn, shooting, RATTLE constraint hold), not a bake-off; that is fine for a construction paper.

The soft spot the stress-test flags is real and proportionate: Prop. 4.5 gives all-time existence and no-collision for free CPD Hamiltonian flow with α in the annihilator. Scale is not operator-invariant; it is a holonomic constraint whose Poisson bracket with H does not vanish, so the actual Kendall-type object is a constrained system. Global existence, non-collision, and stratum-crossing on Land_n¹ are not proved—only set up and illustrated numerically. That does not sink the free rigid-quotient geometry; it means the full “Kendall + LDDMM completeness” claim is one step short. Minor: no code, free parameters (σ, s, Lamé) left to the user, applications claims ahead of the evidence.

This is for people who build landmark/LDDMM metrics or care about rigid invariance inside diffeomorphic flows. Math and citation pattern look solid. I would send it to referees; I would cite the operator and the free quotient construction. Bring it to reading group if the group does shape metrics.

Referee Report

3 major / 5 minor

Summary. The paper constructs landmark shape spaces that combine Kendall-type invariances (quotient by SE(d) and fixed scale) with LDDMM-type regularity by inducing metrics from a screened elasticity operator L_s = −(Id−σ²Δ)^{s−1} ∘ div ∘ sym∇. This operator is SE(d)-equivariant with null-space exactly se(d); its Green’s kernel k_s is conditionally positive definite w.r.t. se(d) and, for s ≥ 1 + d/2, sufficiently regular that free Hamiltonian trajectories with initial momentum in the annihilator exist for all time without landmark collision. The induced stratified cometric on Land_n/SE(d) preserves local rigid motions while removing global ones; scale is imposed by restricting to the holonomic slice R̄ = 1 and integrating a constrained Hamiltonian system (RATTLE). Matching and geodesic evolution are demonstrated numerically with k_2.

Significance. If the construction is as claimed, it fills a genuine gap between Kendall shape spaces (invariances, but Euclidean and n-dependent, no collision control) and LDDMM landmark metrics (regular, ambient, collision-free, but typically not rigid-motion free and penalizing local rigid motion). The screened elasticity operator is a clean, well-motivated choice sitting between Sobolev operators (too small null-space) and pure elasticity/Beppo-Levi (too singular or too large null-space). The derivations of null-space, equivariance, CPD structure, Noether conservation of rigid-momentum maps, and horizontal lifts are standard geometric-mechanics arguments carried out carefully. Explicit kernels for s = 2, a constrained integrator, and illustrative matching experiments (including local-rigid preservation vs Gaussian/Matérn) make the proposal usable. The main applied payoff—unified rigid-plus-deformable registration and n-independent metrics with Kendall invariances—is real if the fixed-scale geometry is placed on the same footing as the free case.

major comments (3)

[§4.5, §5, App. A.11, Abstract] Abstract and §1 claim the new spaces achieve both collision prevention and fixed scale as defining properties of the unification. Proposition §4.5 proves all-time existence and no-collision only for the free Hamiltonian on Land_n with α(0) ∈ se(d)(q)^ann and s ≥ 1 + d/2 (via C¹ regularity of k_s). Scale is not L_s-invariant (§5); it is imposed by the holonomic constraint R̄ = 1 whose momentum map does not Poisson-commute with H (nonzero bracket computed in App. A.11). Geodesics on the actual Kendall-type object Land_n^1 are therefore only a constrained system with Lagrange multiplier. Global existence, constraint regularity, and non-collision for that constrained flow are not proved—only locally set up and illustrated with RATTLE (§6–7). Either extend the completeness argument to the constrained system, or qualify the abstract/intro claims so that collision-free completeness is stated on
[§3.2, §4.5, App. A.10] Section 4.5 states that free Hamiltonian trajectories project to geodesics on Land_n “as long as they do not cross more singular strata,” and the orbit-space stratification (§3.2) is otherwise left open for geodesic behavior. For the unification with Kendall (where singular strata and orbit types are part of the classical picture), the paper should either (i) cite/adapt known convexity or non-crossing results for proper isometric actions under the extended positive-definite cometric of Corollary A.10 and verify the hypotheses, or (ii) explicitly restrict the main theorems and numerical claims to the regular stratum S_k (open dense) and state that crossing into lower-rank configurations is not controlled. As written, the stratified Riemannian structure is defined, but geodesic completeness of the quotient including strata is not established.
[§6–7, Figures 2–8] The numerical section supports local-rigid preservation and constraint stability but does not stress-test the load-bearing geometric claims. Experiments use only k_2, modest n, and qualitative grid/trajectory plots; there is no systematic check of long-time constraint drift vs step size, no comparison of geodesic distance or matching residual against a rigid-pre-registered Sobolev/LDDMM baseline on the same pairs, and no probe of near-collision or near-singular (low-rank) configurations where the completeness gap would matter. Adding at least one quantitative baseline comparison and a near-degeneracy experiment would substantially strengthen the applied half of the contribution without changing the theory.

minor comments (5)

[§4.2, App. A.5] Equation (4)–(5) and App. A.5 give k_2 explicitly; for s > 2 the text defers to root perturbation / repeated-root partial fractions. A short statement of the leading regularity (C^k with k in terms of s,d) already appears, but a pointer to which formula is used in code for general s would help reproducibility.
[Figure 2] Figure 2 caption compares k_2 to Gaussian and Matérn-3/2 with “rather small” σ; reporting the actual σ (and μ, λ) used in each panel would make the local-rigid claim easier to reproduce.
[§3–5] Notation switches between Land_n, Land_n bar, Land_n^1, and g_{L_s} / g_1; a small notation table early in §3–4 would reduce friction.
[throughout] Typos/style: “Mat´ ern” spacing, “Lam´ e”, “att=0”, “a to fixed-scale sphere” (§5), and occasional missing spaces before citations. Standard copy-edit pass.
[§1.1, §2.2] Related work on affine/rigid handling in LDDMM (Glaunès 2005, Younes 2006/2010) is cited; a one-sentence contrast with asymptotically affine kernels vs exact se(d) null-space would clarify novelty for readers from that line.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: honest operator construction whose claimed properties follow from null-space, equivariance, and kernel regularity by standard geometric mechanics, not by fitting or self-definitional prediction.

full rationale

The paper defines the screened elasticity operator L_s = −(Id−σ²Δ)^{s−1} ∘ div ∘ sym∇ so that its null-space is exactly se(d), inherits SE(d)-equivariance from metric operations, and obtains a stratified cometric on Land_n/SE(d) by the usual right-invariant descent / CPD construction. Local rigid-motion preservation, removal of global rigid motions, ambient n-independence, and C^1 non-collision for free Hamiltonian flow (s ≥ 1 + d/2) are consequences of that definition and of standard landmark-kernel arguments, not renamings of fitted constants or uniqueness theorems imported from the authors’ prior work. Self-citations (Michor diffeomorphism groups, LDDMM lineage, completeness for positive-definite kernels) supply background Lie-group and shape-space facts; the central operator and quotient metric are derived in-place. Scale normalization is imposed separately as a holonomic constraint and is not claimed to follow from free L_s-invariance—gaps there are completeness/correctness issues, not circular reduction of a prediction to its inputs. No fitted-input-as-prediction loop, no self-definitional X⇔Y, no load-bearing uniqueness-from-authors step.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The result rests on standard infinite-dimensional Riemannian geometry of diffeomorphism groups, classical linear elasticity, and CPD kernel theory, plus the modeling choice that SE(d) (not Aff or full similarity inside the operator) is the symmetry to kill, with scale handled by a separate sphere constraint. Free parameters are kernel/order/material constants chosen for numerics. The main invented object is the screened elasticity operator itself.

free parameters (4)

screening length σ
Sets the spatial scale of the Sobolev screening factor (Id−σ²Δ)^{s−1}; chosen by hand in experiments and controls deformation locality.
Sobolev order s = s=2 used in experiments
Integer/half-integer order of L_s; must satisfy s ≥ 1 + d/2 for the stated C¹/no-collision regime; chosen, not derived from data.
Lamé parameters μ, λ
Enter the elasticity/Kelvin structure inside L_s and k_s; free material constants in the operator family.
integration step size / step count and optimizer settings
RATTLE Δt, number of steps (e.g. 500 on [0,2]), and Adam-style shooting hyperparameters affect numerical figures only.

assumptions (5)

domain assumption Right-invariant weak Riemannian metrics on Diff_A(R^d) induce metrics on landmark orbits via Riemannian submersion when the inertia operator yields a usable kernel/cometric.
Standard LDDMM/shape-space setup recalled in §2 and App. A.6–A.8; load-bearing for ‘descending metric’ language.
standard math Classical sym∇ / linear elasticity has null-space exactly se(d) and is SE(d)-equivariant via Euclidean metric operations.
§2.3 derivation; used as the unscreened seed of L_s.
domain assumption Only SE(d) (not Aff(d) or full similarity) should be quotiented inside the operator; scale is fixed by restricting to R̄=1.
§4.1 and App. A.12 argue affine-equivariant linear vector-field operators are order zero; scale non-invariance forces §5 constraint.
domain assumption For C¹ kernels with s ≥ 1 + d/2, Hamiltonian trajectories with α(0) in se(d)(q)^{ann} exist for all time and landmarks do not collide.
Prop. §4.5, by analogy with TY05/BBM14/HPS25 completeness results for positive-definite kernels; not fully re-proved from scratch for the semi-definite CPD case.
ad hoc to paper Stratified orbit space Land_n/SE(d) carries the induced cometric on annihilators as a stratified Riemannian metric suitable for geodesic shooting away from singular strata.
§3–4 construct g_{L_s} stratumwise; behavior at stratum boundaries is only partially controlled.

invented entities (2)

screened elasticity operator L_s and kernels k_s
purpose: Provide an SE(d)-equivariant inertia operator with exact rigid-motion null-space and enough regularity for non-colliding landmark metrics.
Defined in §4.2 as −(Id−σ²Δ)^{s−1}∘div∘sym∇ with Green kernel k_{H_{s−1}}∗Φ; central new object of the paper.
quotient landmark shape space Land_n with metric g_{L_s} and scale slice Land_n_1
purpose: Realize Kendall-type rigid/scale invariances on an LDDMM-style induced metric.
§3–5; mathematical space defined by the construction rather than an external empirical entity.

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Cite this review

Pith. "Pith review of Landmark shape spaces with induced metrics." pith.science (2026). https://pith.science/paper/ZTKO7BU3

@misc{pith2026260728064,
  author       = {Pith},
  title        = {Pith review of: Landmark shape spaces with induced metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTKO7BU3}},
  note         = {Machine review of arXiv:2607.28064}
}

read the original abstract

We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeomorphism group. The resulting new landmark shape spaces achieve the defining properties of both approaches: The regularity of the descending metric prevents landmarks from colliding, the metric is defined in the ambient space independent of the number of landmarks, local rigid transformations are preserved, global rigid motions are removed, and scale fixed. To achieve this, we define a particular Sobolev-type operator, the screened elasticity operator, whose null-space consists exactly of the rigid motions, we show how this operator descends to achieve the desired geometry, and we present approaches to solving matching problems and computing geodesics numerically. The resulting construction allows the use of landmark configuration spaces with sufficiently regular metrics in applications while retaining the shape invariances that are a hallmark of Kendall's shape spaces.

Figures

Figures reproduced from arXiv: 2607.28064 by the authors.

**Figure 2.** 1.2 Contributions We present a new landmark shape space that has the defining properties of both Kendall’s shape spaces and LDDMM shape spaces. From the Kendall shape space point of view, the model has the regularity to prevent points from colliding and the metric is defined in the ambient space thus inducing compatible metrics with varying numbers of landmarks. From the LDDMM viewpoint, the 3 [PITH_FULL_IMAGE:figu… view at source ↗

**Figure 2.** Landmark shape matching example, comparing the screened elas [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Figures from the paper (6 more)

**Figure 3.** Figure 3: Vector-field visualization of the two columns of the [PITH_FULL_IMAGE:figures/full_fig_p014_3.png]

**Figure 4.** Figure 4: Landmark geodesics with the k2 kernel solved numerically by integrating the Hamiltonian flow. Landmarks initially in a circular configuration (blue points) and random momentum projected to the rigid-motion constraint subspace. Bottom row has 4× larger initial momentum…

**Figure 5.** Figure 5: Forwards and backwards integration of the landmark geodesics [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: Shooting-based matching produced by optimizing the initial mo [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: Landmark matching for two circles rotated 90 degrees in opposite [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: Landmark matching for the butterflies in Figure 1 with and with [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

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Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

[1] [1]

Higher-order fractional Laplacians : An overview

Nicola Abatangelo. Higher-order fractional Laplacians : An overview. Bruno Pini Mathematical Analysis Seminar , 12(1):53--80, 2021

2021

[2] [2]

Adams and Lars I

David R. Adams and Lars I. Hedberg. Function Spaces and Potential Theory . Springer Science & Business Media, November 1999

1999

[3] [3]

Dmitri Alekseevsky, Andreas Kriegl, Mark Losik, and Peter W. Michor. The R iemannian geometry of orbit spaces---the metric, geodesics, and integrable systems. Publ. Math. Debrecen , 62(3-4):247--276, 2003. Dedicated to Professor Lajos Tam \'a ssy on the occasion of his 80th birthday

2003

[4] [4]

velocity

Hans C Andersen. Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations. Journal of Computational Physics , 52(1):24--34, October 1983

1983

[5] [5]

Martin Bauer, Martins Bruveris, and Peter W. Michor. Overview of the Geometries of Shape Spaces and Diffeomorphism Groups . J. Math. Imaging Vis. , 50(1-2):60--97, September 2014

2014

[6] [6]

Bookstein

Fred L. Bookstein. Morphometric Tools for Landmark Data : Geometry and Biology . Cambridge University Press, June 1997

1997

[7] [7]

Fast Large Deformation Matching with the Energy Distance Kernel , May 2025

Siwan Boufadene, Fran c ois-Xavier Vialard, and Jean Feydy. Fast Large Deformation Matching with the Energy Distance Kernel , May 2025

2025

[8] [8]

Droske and M

M. Droske and M. Rumpf. A variational approach to nonrigid morphological image registration. SIAM Journal on Applied Mathematics , 64(2):668--687, 2004

2004