The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes

T HE P USH -F ORWARD T RANSFORM FOR C ONTINUOUS AND ROBUST C OMPARISON OF DYNAMIC S HAPES A P REPRINT Roua Rouatbi

1,2,3,4

, Juan-Esteban Suarez Cardona

Ivo F. Sbalzarini

6,7

,

1,2,3,4,5,8

1

Faculty of Computer Science, Dresden University of Technology, Dresden, Germany Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany 3 Center for Systems Biology Dresden, Dresden, Germany 4 Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI) Dresden/Leipzig 5 Cluster of Excellence Physics of Life, Dresden University of Technology, Dresden, Germany 6 Chair for Mathematical Foundations of Artificial Intelligence, Ludwig-Maximilians-Universität München, Munich, Germany 7 Munich Center for Machine Learning (MCML), Munich, Germany 8 Now at: Department of Mathematical Modeling and Machine Learning, University of Zurich, Zurich, Switzerland

arXiv:2608.02306v1 [cs.CV] 3 Aug 2026

2

A BSTRACT We introduce a mathematical framework for shape comparison based on mapping functions from the shape domain to a common reference domain. This Push-Forward Transform enables invariant and robust comparison of shapes, preserving intrinsic geometric information. Quantitatively comparing shapes and their temporal evolution is a fundamental challenge in image analysis. Meaningful shape comparison requires representations that are invariant to transformations that do not alter shape itself, such as translation, rotation, reflection, re-parametrization, and uniform scaling, while remaining sensitive to intrinsic geometric variation. Existing approaches often rely on sensitive parameterizations, landmark correspondence, or learned representations that are difficult to interpret and reproduce. We show that the Push-Forward Transform (PF-T) applied to Signed Distance Functions (SDFs) yields a continuous representation that captures both boundary and interior geometry. We derive an interpretable morphometric that quantifies shape similarity and reveals features such as skeletal topology and rotational symmetries. The push-forward transform applies consistently to two- and three-dimensional shapes, extends to time-evolving geometries, and supports the joint analysis of shape and additional scalar fields defined over shapes, such as intensity or molecular signals. We present the mathematical formulation, describe an efficient algorithm, and benchmark the approach on 2D, 3D, and temporal data sets.

1

Introduction

The comparison of shapes is a fundamental problem in computer vision, medical imaging, and quantitative biology, where shape often serves as the primary descriptor of structure and function. A central challenge is to quantify shape differences in a way that is invariant to transformations that do not alter shape itself, such as translation, rotation, reflection, re-parametrization, and uniform scaling, while remaining sensitive to intrinsic geometric variation. A natural strategy is to associate each shape with one or more functions defined over its domain, such as distance functions, curvature measures, or spatially varying signals, and to compare these functions across shapes. This approach, however, runs into a fundamental difficulty: functions defined on different shape domains are not directly comparable. Even when two shapes are topologically equivalent, their domains might differ geometrically, so that point-wise comparison is ill-defined and sensitive to arbitrary choices of parameterization or alignment. Here, we address this comparability problem by introducing the Push-Forward Transform (PF-T), which maps any function defined on a

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

shape onto a common reference domain of identical topology through a smooth, structure-preserving diffeomorphism. The common reference domain enables well-defined comparison of distributions over shapes and yields metrics that are shape-preserving by construction. The PF-T can be applied to any continuous scalar-valued function over a shape. Of particular interest for deriving geometric descriptors is the signed distance function (SDF) to the shape boundary [1]. SDFs are (i) well-defined for any closed shape, (ii) encode both boundary location and interior structure in a single scalar field, and (iii) admit a smooth, computable approximation through the viscous Eikonal equation. Transforming the SDF of a shape via the PF-T yields the Push-Forward Signed Distance Function (PF-SDF), a continuous and differentiable representation in which shapes can be compared invariantly [2]. We demonstrate the utility of this framework for a range of applications. We show how our computational implementation with the viscous Eikonal equation enables the extraction of intrinsic skeletal structures, providing meaningful geometric shape descriptors that can be further used to construct parametrized reference axes. From the PF-SDF representation, we derive a spectral morphometric, the Push-Forward Signed Distance Morphometric (PF-SDM), which is interpretable, invariant to shape-preserving transformations, and robust to noise, as we show in extensive benchmarks. Finally, we apply the PF-T to arbitrary intensity fields defined over shapes, enabling the joint modeling of morphology and spatial patterns as well as their evolution over time. This unified treatment of geometry and spatial signal significantly broadens the applicability of the proposed approach. The main contributions of this work are: • We develop the theoretical foundations of the Push-Forward Transform (PF-T) as a continuous and differentiable framework for comparing shapes and functions defined on them, yielding descriptors that are invariant by construction to translation, rotation, reflection, re-parametrization, and uniform scaling. • We developed efficient approximations of the Signed Distance Function using polynomial surrogates to solve the viscous Eikonal learning problem, yielding a smooth representation from which intrinsic skeletal descriptors and parametrized reference axes (spines) of shapes can be extracted. • We derive a spectral morphometric that provides interpretable and noise-robust shape descriptors capturing meaningful geometric features, such as rotational symmetries. We demonstrate its applicability to 2D shapes, 3D shapes, and time-evolving (2D + time) shapes. • We extend the push-forward framework beyond geometry to jointly analyze shape and distributions defined over shapes, such as intensity or molecular signals, enabling unified modeling of morphology and spatial patterns across space and time. This paper is structured as follows: In Section 2, we review related work in geometric morphometrics and shape analysis. In Section 3, we present the PF-T, discuss its mathematical properties, and describe alternative choices of deformation maps. Section 4 applies the PF-T to SDFs, introduces the viscous Eikonal formulation, and uses it to extract medial axes and parameterized spines. We present experimental benchmark results in Section 5 for various applications, including rotational-symmetry-aware morphometrics, and the fusion of geometric and intensity signal features to predict the fate of time-evolving biological structures. In Section 6, we conclude with a summary of findings and directions for future research.

2

Related Work

Existing approaches to shape analysis and comparison can be broadly categorized by how shapes are represented and compared, and by whether invariance and interpretability are built into the representation or learned from data. Traditional morphometric methods rely on explicit geometric representations and offer strong interpretability. Landmark-based approaches such as Generalized Procrustes Analysis (GPA) [3] compare shapes by aligning key points, hence factoring out translation, rotation, and scale. Boundary-based spectral descriptors, including Elliptical Fourier Analysis (EFA) [4] and spherical harmonics [5], encode shapes using truncated spectral bases defined on boundaries or surfaces, respectively. These methods are mathematically grounded and computationally efficient but depend on reliable parameterizations or landmark correspondences. In particular, spherical harmonic representations require a non-trivial spherical parameterization, which becomes challenging for shapes that are not star-convex. An alternative line of work uses global region properties and moment-based descriptors such as aspect ratio, eccentricity, and compactness to summarize shape geometry in a low-dimensional, inherently invariant form. While these features are simple and fast to compute, they capture only coarse geometric characteristics and lack the expressiveness needed to distinguish nuanced morphological differences. Moreover, the choice of which features to use is itself non-trivial, leading back to the original question of what constitutes a meaningful comparison. 2

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Learning-based approaches infer shape representations directly from data and have shown strong performance on large benchmarks. Variational autoencoder models such as ShapeEmbed [6] and O2VAE [7], contrastive learning frameworks (e.g., SimCLR [8]), and transformer-based models such as masked autoencoders (MAE) [9] are among the many methods applied to shape analysis. ShapeEmbed and O2VAE are particularly notable for their treatment of rotation invariance: O2VAE enforces it architecturally through O(2)-equivariant convolutional layers, while ShapeEmbed achieves it at the representation level by encoding shapes as distance matrices between boundary points. Most learning-based approaches, however, rely on large training data sets and offer limited interpretability of the resulting latent dimensions. Unlike learning-based approaches, our proposed method is fully deterministic and does not require training data, yielding reproducible and interpretable results. At the same time, by operating on continuous functions mapped to a common reference domain, it overcomes key limitations of classical morphometric techniques, such as reliance on landmark correspondences or fragile shape parameterizations. Moreover, our framework naturally extends to additional functions defined over shapes and to time-varying dynamic shapes. This is made possible by the PF-T, as introduced in the next section.

3

Push-Forward Transform

The Push-Forward Transform (PF-T) is the central mathematical concept for our framework. It maps functions defined over different shape domains to a common reference domain of identical topology, rendering a well-defined notion of comparability. This common-domain representation provides the basis for constructing descriptors that are invariant to shape-preserving transformations by construction. We start by mathematically defining the PF-T and then describe practical constructions, including radial and harmonic extensions, as well as their approximations. We denote by Ω ⊆ Rd an open, bounded, Lipschitz domain, and by ∂S ⊆ Ω a shape boundary, assumed to be a smooth and closed co-dimension-one manifold embedded in Ω. Let fS : S → R be a scalar field. The PF-T maps fS to its push-forward fS ∗ : Sr → R, defined on a reference domain Sr ⊆ (−1, 1)d , via a diffeomorphism Ψ : Sr → S. To formally define the PF-T, we start by recalling the definition of a level-set function: Definition 1 (Level-Set Function). Let Ω ⊂ Rd and let ∂S ⊆ Ω be an oriented manifold of co-dimension one. Let LS : Ω → R be a C 1 function such that ∇LS ̸= 0 on ∂S. We call LS a level-set function associated with S if and only if ∂S = {x ∈ Ω : LS (x) = 0}. The interior and exterior domains of the shape ∂S are: S := {x ∈ Ω : LS (x) > 0}, S c := {x ∈ Ω : LS (x) < 0}. The closure of S is given by S = S ∪ ∂S. Following this, we define the PF-T. Definition 2 (Push-Forward Transform). Let ∂S ⊆ Ω be a shape with shape domain S ⊆ Ω, and let ∂Sr ⊆ Ω be a reference shape with shape domain Sr ⊆ Ω. Furthermore, let Ψ : Sr → S be a diffeomorphism and fS : S → R a scalar field defined on the shape domain S. We define the Push-Forward Transform (PF-T) fS ∗ : Sr → R as fS ∗ (x) := fS (Ψ(x)),

∀x ∈ Sr .

In general, determining a smooth and continuous deformation map between two shape domains is a nontrivial task, as it should preserve the geometric structure of the scalar fields being transformed. To address this, we first construct the boundary deformation map ψS : ∂Sr → ∂S and then choose an extension Ψ : Sr → S, such that Ψ|∂Sr = ψS . Definition 3 (Boundary Deformation Map). Let ∂S, ∂Sr ⊆ Ω be the target and reference shapes, respectively. The boundary deformation map ψS : ∂Sr → ∂S is a diffeomorphism approximating the closest-point projection Z inf ∥ψS (x) − xS (x)∥22 dH(x) , (1) 1 ψS ∈C (∂Sr ,∂S) ∂Sr

where xS (x) denotes the closest-point projection of x ∈ ∂Sr onto ∂S and H is the Hausdorff-measure. To extend the boundary deformation map over the whole shape domain, different choices of extensions are possible, as we discuss next. 3

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Figure 1: Radial mapping of a star-convex shape to the unit disk. Left: original shape. Middle: radial boundary deformation map. Right: radial extension map.

Figure 2: Harmonic mapping of a star-convex shape to the unit disk. Left: original shape. Middle: arc-length boundary map. Right: harmonic extension map. 3.1

Deformation Map Extensions

We describe two possible extensions of the boundary deformation map ψS for a given shape ∂S, namely the radial and harmonic extensions. For shapes that are topological spheres, we use the unit sphere ∂Br := {x ∈ Rd : ||x||2 = 1}, with interior Br := {x ∈ Rd : ||x||2 < 1}, as the reference shape. While other reference shapes can be used for shapes of different homology groups, this choice provides a simple domain that admits natural radial and harmonic extensions from the boundary to the interior. We start by defining the radial extension: Definition 4 (Radial Extension). Let Θd ⊂ Rd−1 denote the angular parameter domain (e.g., Θ2 = [0, 2π) for d = 2 and Θ3 = [0, π] × [0, 2π) for d = 3). Let ψS : ∂Br → ∂S be a boundary deformation map expressed in spherical coordinates as ψS (θ) = (ν(θ), θ), ∀θ ∈ Θd , for some ν ∈ C 1 (Θd ; R+ ). We define the radial extension Ψr : Br → S as ∀(r, θ) ∈ [0, 1] × Θd .

Ψr (r, θ) := (r ν(θ), θ),

Figure 1 illustrates the radial boundary deformation map and its extension applied to a star-convex shape. The radial extension map shows how the original shape is stretched radially onto the reference unit circle while preserving angular coordinates. The main limitation of the radial extension is that it fails the injectivity assumption in Definition 2 for non-star-convex shapes. This motivates the harmonic extension as a less restrictive, but more complex alternative for a broader class of smooth closed shapes: Definition 5 (Harmonic Extension). Let ψS : ∂Br → ∂S be a boundary deformation map. The Harmonic extension Ψh ∈ C 2 (Br ; Rd ) is the function solving  ∆Ψh (x) = 0, ∀x ∈ Br (2) Ψh (x) = ψS (x), ∀x ∈ ∂Br , where the Laplacian ∆Ψh (x) ∈ Rd is applied component-wise. Figure 2 illustrates the harmonic extension on the same star-convex shape as in Fig. 1 using arc-length parametrization as the boundary deformation map. Unlike the radial extension map which constrains every interior point to lie on the same ray from the origin to the boundary, the harmonic map is free to bend smoothly into the interior, making it well-defined for non-star-convex shapes. We provide the following result for the radial and harmonic extensions: 4

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Theorem 1 (Diffeomorphism properties of extensions). Let ψS : ∂Br → ∂S ⊂ Rd be a C 1 boundary deformation map. (i) (Radial extension) Assume that S ⊆ Ω is star-convex with respect to the origin and that ν ∈ C 1 (∂Br ) with ν(θ) > 0,

ψS (θ) = ν(θ) ω(θ),

where ω(θ) ∈ Rd denotes the unit vector pointing in direction θ, e.g, in 2D ω(θ) = (cos(θ), sin(θ)). Then, the radial extension Ψr (r, θ) = r ν(θ) ω(θ) defines a C -diffeomorphism from Br \ {0} onto S \ {0}, which extends continuously to the origin. 1

(ii) (Harmonic extension) Let d = 2 and Ψh ∈ C 2 (Br ; Rd ) ∩ C 1 (Br ; Rd ) the harmonic extension of ψS from Definition 5. Then Ψh is a C 1 -diffeomorphism of Br onto S if and only if det DΨh > 0

on ∂Br .

(3)

The proof is given in Appendix A. There is no proof for the diffeomorphism property of the harmonic extension in d = 3 dimensions. Therefore, we only use the harmonic extension in two dimensions and restrict 3D applications to the star-convex case. For shapes on a common scale and centered at the origin1 , the PF-T is invariant to translation and uniform scaling, as well as to reparametrization of the boundary, since the deformation map Ψ is defined intrinsically through the closestpoint projection in Eq. (1) rather than through any specific parametrization of ∂S. It is, however, equivariant under rotations and reflections of the shape. To eliminate this remaining ambiguity, we formulate rotation- and reflectioninvariant features from spectral representation of the PF-T. 3.2

Spectral PF-T

If the reference domain is rotationally symmetric, such as the unit disk, an angular spectral transformation of the PF-T can be used to guarantee invariance to rotations and reflections of the shape descriptors. In 2D, this transformation is the Fourier transform, in 3D the spherical harmonics expansion. Early foundational work established Fourier descriptors in 2D as an effective tool for analyzing closed planar curves, enabling the characterization of shape invariants and symmetries with robustness to rotation and scaling [10]. We hence define the spectral PF-T: Definition 6 (Spectral Push-Forward Transform). Let S ⊆ Ω ⊆ Rd be a shape and let fS : S → R be a scalar field defined on S. Denote by fS ∗ : Br → R its PF-T to the unit ball Br ⊂ Rd . • Fourier representation (d = 2). For a fixed radius r ∈ (0, 1), the restriction fS ∗ (r, ·) is a function on the circle of radius r centered at the origin (∂Br for r = 1) with Fourier coefficients Z 2π 1 ck (r) := fS ∗ (r, θ) e−ikθ dθ, k ∈ Z. 2π 0 Given a truncation order NF ∈ N, define
c(r) = ck (r) 0≤k<N ∈ CNF . F

The normalized Fourier descriptor at radius r is then
|ck (r)| 0≤k<N F ∈ RNF . cF S ∗ (r) := PNF −1 k=0 |ck (r)| • Spherical harmonics representation (d = 3). For a fixed radius r ∈ (0, 1), the restriction fS ∗ (r, ·) is a function on the sphere of radius r centered at the origin (∂Br for r = 1). Its spherical harmonic expansion is fS ∗ (r, θ, ϕ) ≈

L X ℓ X

cℓ,m (r) Yℓm (θ, ϕ),

ℓ=0 m=−ℓ 1

This can always be achieved w.l.o.g. by a trivial pre-processing of the data.

5

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

where L is the maximum spherical harmonic degree and cℓ,m (r) are the corresponding spherical harmonic coefficients. For each degree ℓ, we define the spectral energy Eℓ (r) :=

ℓ X

|cℓ,m (r)|2 .

m=−ℓ

The spherical harmonic descriptor at radius r is defined as
L L+1 . cH S ∗ (r) := Eℓ (r) ℓ=0 ∈ R The spectral push-forward transform fS ∗ → cd , is defined by  F cS ∗ , d = 2, cd = cH S ∗ , d = 3. The following result establishes the invariance to rotation and reflection of the spectral PF-T. Proposition 1. Let S ⊆ Ω ⊆ Rd be a shape, fS : S → R a scalar field, and fS ∗ : Br → R its PF-T representation. For any rotation or reflection K ∈ O(d), define the transformed field fSK∗ (x) := fS ∗ (Kx). Then the spectral transform fS ∗ → cd is invariant, i.e., cd (fSK∗ )(r) = cd (fS ∗ )(r),

∀r ∈ (0, 1].

In words: • for d = 2, the normalized Fourier descriptor cF S ∗ (r) is invariant under planar rotations and reflections; • for d = 3, the spherical harmonic descriptor cH S ∗ (r) is invariant under rotations and reflections. The proof is given in Appendix B. 3.3

Finite-Dimensional PF-T

In practice, we compute a finite-dimensional approximation of the PF-T when implementing it algorithmically. We construct this finite-dimensional approximation in a polynomial spectral space to preserve the continuity and differentiability of the PF-T. Let Ω := (−1, 1)d denote the d-dimensional hypercube. The polynomial spaces Πn (Ω) ⊆ C ∞ (Ω) of l∞ -degree n ∈ N are then defined as: d Q i Definition 7 (Polynomial Space). Let {xα }α∈An,d be the canonical basis xα := xα i for all α ∈ An,d , where An,d i=1

is the multi-index set An,d := {α ∈ Nd : ∥α∥∞ ≤ n} with |An,d | = (n + 1)d . A polynomial space of degree n ∈ N is defined as the span of the canonical basis Πn := span{xα }α∈An,d . We denote by Πn (U ) := {Q|U : Q ∈ Πn } the restriction of the polynomial space Πn to a domain U ⊆ Rd . In curved domains, we consider finite-dimensional spectral spaces generated by either Fourier modes (d = 2) or spherical harmonics (d = 3): Definition 8 (Finite-Dimensional Spectral Space). For d = 2, let An := {k ∈ Z : |k| ≤ n}, and define the space of trigonometric polynomials of order n ∈ N by  Tn2 := span eikθ k∈An . For d = 3, let Λn := {(ℓ, m) : 0 ≤ ℓ ≤ n, −ℓ ≤ m ≤ ℓ} , 6

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

and define the spherical harmonic space of degree n by Tn3 := span {Yℓm (θ, ϕ)}(ℓ,m)∈Λn , where Yℓm denotes the spherical harmonic of degree ℓ and order m. We then define the finite-dimensional spectral space  Tn := For a domain U ⊆ Rd , we denote by

Tn2 , Tn3 ,

d = 2, d = 3.

Tn (U ) := {f |U : f ∈ Tn }

its restriction to U . These continuous objects are computationally represented as polynomial surrogates : Definition 9 (Polynomial Surrogates). A polynomial surrogate of a function f : U → Rk is an approximation fξ ∈ Πn , Tn , obtained by minimizing the discrete residual of the equation satisfied by f over a set of collocation points, typically the Legendre grid [11, 12]. Since fξ ∈ C ∞ (Ω), its derivatives are computed exactly through polynomial differentiation matrices, yielding a smooth and differentiable approximation. We further define the finite-dimensional boundary deformation map : Definition 10 (Finite-Dimensional Boundary Deformation Map). Let f : S → R be a scalar field defined over a shape S ⊆ Ω := (−1, 1)d . We define the finite-dimensional boundary deformation map ψξ := Id + νξ , where νξ ∈ Tn (∂Br )d is the solution of NS 1 X 2 inf |xi + νξ (xi ) − xS (xi )| . νξ ∈Tn (∂Br )d NS i=1 S Here, {xi }N i=1 ⊆ ∂Br are sampled boundary points, xS (xi ) ∈ ∂S denotes the corresponding target boundary point.

Using Definition 5 and the non-degeneracy condition from Eq. (3), we define the finite-dimensional radial and harmonic extensions of ψξ : Definition 11 (Finite-Dimensional Extension). Let ∂S ⊆ Ω be a shape and ψξ ∈ Tn (∂Br )d a finite-dimensional boundary deformation map. Then, we define the finite-dimensional extension Ψξ as either one of: • Finite-Dimensional Radial Extension in spherical coordinates, as Ψξ (r, θ) := rψξ (θ) .

(4)

• Finite-Dimensional Harmonic Extension Ψξ ∈ (Πn (Br ))d as the polynomial surrogate solving inf Ln∆ [ξ] + Lnb [ξ] + Lnr [ξ],

(5)

Ψξ ∈(Πn (Br ))d

with X

d X

α∈An,d

i=1

Ln∆ [ξ] := Lnb [ξ] :=

! (∆Ψξxi (Pn ))2α

ωα

NS X d 1 X (Ψξxj (xi ) − (ψξ (xi ))xj )2 , NS i=1 j=1

Lnr [ξ] := −

NS

λ X log det DΨξ (xi ) , NS i=1

(6)

(7)

(8)

where Pn ⊆ Ω is the Legendre grid and λ ∈ R+ a regularization parameter. The terms Ln∆ [ξ] and Lnb [ξ] ensure that the result is a valid harmonic extension according to Def. 5, and the term Lnr [ξ] enforces a diffeomorphism according to Theorem 1. The finite-dimensional PF-T can be applied to any scalar-valued function defined over the domain of a shape. A particularly useful choice for describing the geometry of the shape is the signed distance function (SDF) of that shape. SDFs provide a continuous implicit representation of geometry, encode both boundary and interior structure, and expose singular structures related to the medial axis [13]. 7

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Figure 3: Example of the signed distance function of a star-convex shape, showing the zero level-set ∂S indicated by the cutting plane, the superlevel-set S (interior points, positive sign) in lighter colors, and the sublevel-set S c (exterior points, negative sign) in darker colors.

4

Signed Distance Functions (SDF)

We describe in detail the application of the PF-T to SDFs for shape characterization. This provides continuous shape features that are invariant under the Euclidean group. We first recall the mathematical formulation of SDFs, their connection to the Eikonal equation, and their role in encoding topological structures such as the medial axis of a shape. We start by defining the signed-distance function of a shape ∂S ⊆ Ω as the distance of any point in Ω to the closest point on ∂S with the sign distinguishing the inside and outside of the closed shape: Definition 12 (Signed-Distance Function). The Signed Distance Function (SDF) of the shape ∂S ⊆ Ω, denoted ϕS : Ω → R, is  +1, x ∈ S, ϕS (x) := σ(x) min ∥x − y∥2 , σ(x) = (9) y∈∂S −1, x ∈ S c . The SDF ϕS is a level-set function according to Definition 1. Figure 3 illustrates the zero, superlevel, and sublevel sets of the signed distance function for a star-convex shape. Following Refs. [14, 15], the SDF ϕS of a shape ∂S can be determined as the viscosity solution [16] of the Eikonal equation: Definition 13 (Viscous Eikonal Equation). Let ∂S ⊆ Ω be a given shape. The viscous Eikonal equation seeks a function ϕµS ∈ C 2 (Ω) satisfying ( |∇ϕµS |2 − µ∆ϕµS = 1 in Ω, (10) ϕµS = 0 on ∂S. The variational viscous Eikonal problem is to find ϕ ∈ H 2 (Ω) minimizing Z
2 inf |∇ϕ|2 − µ∆ϕ − 1 dx ϕ∈H 2 (Ω) Ω Z + (ϕ|∂S )2 dH(x),

(11)

∂S

with µ ∈ R+ the viscosity parameter. Figure 4 shows an example of a binary mask of a two-dimensional shape, the SDF obtained by solving the viscous Eikonal equation, and a 3D visualization of the SDF with zero-level contours. In practice we solve the variation form in Eq. (11) to obtain a globally smooth and differentiable approximation of the SDF. We use polynomial surrogates [11] and Sobolev Cubatures [12, 17] to compute the finite-dimensional polynomial SDF: 8

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Input Mask

SDF Approximation

SDF Surface Projection 0.5 0.4 0.3 0.2 0.1 0.0

Figure 4: From left to right: Input binary mask of a star-convex shape, signed distance function (SDF) approximation obtained by solving the viscous Eikonal equation, and 3D visualization of the SDF with zero-level contour highlighted in black, representing the reconstructed shape boundary. Definition 14 (Polynomial Signed-Distance Function). Let ∂S ⊆ Ω := (−1, 1)d be a shape. We define the polynomial SDF as the polynomial surrogate ϕξ ∈ Πn (Ω) solving X
2 inf ∥D∇ ϕξ (Pn )∥22 − 1 − µD∆ ϕξ (Pn ) ωα ϕξ ∈Πn (Ω)

α∈An,d N

s 1 X + ϕξ (xi )2 , Ns i=1

(12)

where D∇ , D∆ ∈ R|An,d | are the polynomial differentiation matrices for the ∇ (Nabla) and ∆ (Laplace) operators, respectively. From this, we can define the finite-dimensional PF-SDF: Definition 15 (Finite-Dimensional Push-Forward Signed Distance Function (PF-SDF)). The finite-dimensional PFSDF ϕξ∗ : Br → R is obtained by composing a polynomial SDF ϕξ with the harmonic or radial extension Ψξ (see Definition 11). More precisely, we define ϕξ∗ (x) := ϕξ (Ψξ (x)), ∀x ∈ Br , (13) where Ψξ : Br → S is either the radial Fourier or the harmonic polynomial extension map. 4.1

Medial axis of a shape

The solution of the inviscid Eikonal equation is Lipschitz continuous but not differentiable. In particular, there exists a set of measure zero where the derivative of the solution ϕS is discontinuous. This singular set is closely related to the medial axis of the shape ∂S [18, 19], which has been shown to be a powerful geometric descriptor of shape similarity [20]. The viscous Eikonal equation (Def. 13), which we solve here, has smooth solutions ϕµS ∈ C 2 (Ω) throughout the domain. Nevertheless, for small viscosity µ, the solution retains a diffuse signature of the singular medial set, allowing the medial axis and related geometric features to be approximated. This is illustrated in Fig. 5, where the gradients and divergence of the viscous SDF highlight the skeletal structure of the given shape, as also previously demonstrated [13]. For comparison, we show the scikit-image skeletonization result [21] for the same example shape. The information about the medial axis of a shape is contained in its diffuse signature in the polynomial SDF. We start by defining the medial axis of a shape as the set of points inside the shape that are equidistant to at least two points on the shape [19]: Definition 16 (Medial Axis). Let ∂S ⊆ Ω be a shape and ϕS : S → R its SDF. The medial axis (or topological skeleton) is defined as MS := {x ∈ S : ∃ xl , xr ∈ ∂S : xl ̸= xr , ∥x − xl ∥2 = ∥x − xr ∥2 = ϕS (x)}. Lemma 1. Let S ⊆ Ω and ϕS its SDF. Then, ϕS is not differentiable at any point c ∈ MS . The proof is given in Appendix C. The divergence ∇ · η of the normalized SDF gradient η(x) =

∇ϕξ (x) , ∥∇ϕξ (x)∥ 9

(14)

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Figure 5: Signature of a shape skeleton in the smooth polynomial SDF. Top row: solution from the viscous Eikonal equation with viscosity 0.6; bottom row: viscosity 0.001. Left to right in each row: normalized gradients (Eq. (14)), divergence of the normalized gradient field, and a reference skeleton obtained using the scikit-image skeletonization algorithm on the original binary shape mask (identical in both rows). is low along the medial axis of the shape. In all experiments in Section 5, we use ∇ · η directly as the skeleton channel of a PF-SDM. Beyond feature extraction, the same divergence field can be post-processed into a discrete skeleton point cloud and further into a smooth parametric medial-axis curve, providing an intrinsic, landmark-free longitudinal coordinate for elongated shapes. The full extraction procedure is described in Appendix D and illustrated on a real biological 3D shape.

5

Experiments and Results

Since the polynomial SDF encodes both the boundary geometry of a shape and its interior medial-axis structure, it is a natural choice to combine with the PF-T. The resulting representation can compare shapes while retaining volumetric and skeletal information. We empirically evaluate the performance and robustness (to noise and shape perturbations) of the resulting representation before analyzing its discriminative power in real-world data sets. Finally, we show an application of PF-SDM to a biological problem. 5.1

Invariance and interpretability

We first evaluate whether the PF-SDM satisfies the invariance properties required for meaningful shape comparison and whether the resulting descriptors exhibit an interpretable structure in a controlled setting. We consider a synthetic 2D data set comprising four regular polygons (equilateral triangle, square, regular pentagon, and regular hexagon) and an approximately symmetric five-petaled flower [22]. All shapes are centered at the origin, scaled to be inscribed in the unit circle, and embedded in the domain Ω := (−1, 1)2 . For each shape, we generate ten variants covering rotations, reflections, translations, uniform scaling, and additive Gaussian noise (see Appendix E for details), yielding a total of 50 contours. Since every shape in this data set is star-convex, we use radial extension (Definition 4); harmonic extension yields comparable results, but radial extension is computationally more efficient for this shape class. We compute full pairwise Euclidean distance matrices for all 50 shapes represented using three morphometrics: • PF-SDM: normalized FFT coefficients of the PF-SDF with NF = 15 truncation order. • EFA: elliptical Fourier descriptors with the first 15 harmonics, computed using the pyefd and normalized to remove rotation, translation, and scale. • GPA: generalized Procrustes analysis applied to equidistant boundary landmarks sampled at equal arc-length intervals from each contour, with pairwise Procrustes distances computed between aligned shapes. Figure 6 shows the data classes considered as well as the first two multi-dimensional scaling (MDS) components of the resulting pairwise distance matrices for each method. For PF-SDM, all augmented variants of a given shape collapse to a tight cluster, confirming invariance to the applied shape-preserving transformations and robustness to noise. In contrast, both GPA and EFA exhibit noticeably larger intra-class variance, reflecting their sensitivity to noise and residual alignment errors. Beyond invariance, the PF-SDM embedding reveals an interpretable organization of 10

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

PF-SDM

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Figure 6: Top: 2D synthetic data set of regular polygons and the approximately symmetric five-petaled flower with their legend markers. Bottom: MDS embeddings of Euclidean pairwise distance matrices for PF-SDM (left), EFA (center), and GPA (right) on the synthetic 2D data set. Each of the 50 points represents one of the five shape classes. Colors match the inset legend.

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Figure 7: MDS embeddings of PF-SDM distance matrices computed using the full normlized Fourier descriptors cF S∗ with Euclidean distance (left), and using the FFT spectrum excluding c0 with cosine distance (right). Excluding c0 emphasizes rotational symmetry over global shape similarity. shape classes. The irregular flower is clearly separated from the regular polygons, while the polygons themselves are arranged according to their geometric similarity: shapes that more closely approximate a circle (e.g., pentagon and hexagon) cluster closer together, whereas shapes that deviate more from circularity (e.g., triangle) are embedded farther away. The PF-SDM embedding thus reflects meaningful geometric relationships between shapes. Shape hierarchy and rotational symmetry While the full PF-SDM FFT spectrum emphasizes global shape similarity, the spectral representation allows explicit control over which geometric properties are emphasized. In particular, removing the Direct Current (DC) component suppresses information related to overall circularity, while using cosine emphasizes relative angular variation  distance  rather than absolute magnitude. For readability, we denote by ck := cF S ∗ k the k-th entry of the normalized Fourier descriptor from Section 3.2. If c0 is omitted, the induced distances become primarily sensitive to rotational symmetry. This is illustrated in Fig. 7. When the full FFT spectrum is used with Euclidean distance, the embedding reflects global geometric similarity. In contrast, when c0 is excluded, and cosine distance is used, the embedding reorganizes according to rotational symmetry: the five-petaled flower clusters near the pentagon, reflecting their shared five-fold symmetry. A similar effect is observed for other shapes with related symmetry structure, such as the triangle and hexagon, which exhibit three- and six-fold rotational symmetry, respectively. This embedding structure becomes even more evident in the hierarchical clustering shown in Fig. 8. Using the full descriptor, the dendrogram mainly reflects global geometry. Excluding c0 reorganizes the hierarchy around symmetry order, with shapes sharing a similar rotational structure grouped closer together. This reorganization can be explained by the mean normalized Fourier descriptors shown in Fig. 9. For regular polygons, the k-th order in ck is dominant for shapes with k-fold symmetry (e.g., c3 for the triangle, c4 for the square) 11

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Shape Hierarchy (FFT full spectrum)

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Figure 9: Mean normalized Fourier descriptors per shape class, illustrating the relationship between dominant frequency components and rotational symmetry. with higher harmonics showing up attenuated. The flower exhibits a high c5 component, consistent with its fivepetaled geometry. Because it is only approximately five-fold symmetric, the first harmonic c1 also carries substantial energy. Yet, the fifth harmonic remains the first non-trivial local maximum in the spectrum. Similarly, the square exhibits its dominant harmonic at c4 , but also carries substantial energy at c12 a harmonic it shares with the triangle (c3 , c6 , c9 , c12 , . . .) and the hexagon (c6 , c12 , . . .), since 12 is a common multiple of 3, 4, and 6. Under cosine distance on the DC-removed spectrum, this shared high-order harmonic pulls the square closer to the triangle and hexagon in the symmetry-based embedding. These properties remain true for shapes in a three-dimensional domain Ω. There, we use the unit ball Br ⊂ R3 as reference domain and compare two complementary morphometrics: • PF-SDM: Spherical harmonic descriptors cH S ∗ of the PF-SDF up to expansion order L = 20 (see Section 3.2). • Spherical parameterization: The PF-T itself induces a spherical parameterization of the shape. As in classical spherical-harmonic shape analysis we directly expand the PF-T restricted to the surface in spherical harmonics up to order L = 20. The resulting degree-wise power spectrum, corresponding to the spectral energy Eℓ defined in Section 3.2, is used as a shape descriptor. These two morphometrics emphasize different aspects of shape: PF-SDM captures the interior structure in S, while the spherical parameterization focuses on the surface ∂S and is possible because all benchmark shapes are topological spheres. Concretely, we consider six star-convex benchmark shapes in 3D: sphere, ellipsoid, cube, cone, pyramid, and a cylindric capsule shape, each augmented by five shape-preserving transformations (translations, rotations, reflections, and uniform scaling) as described in Appendix E. As in the 2D case, we use radial extension for the PF-T. Figure 10 shows the first two MDS components of the resulting pairwise distance matrices. When using the full spectrum with Euclidean distance (left column), both morphometrics organize shapes according to global geometric similarity. In this, PF-SDM is more robust to shape-preserving transformations and noise than direct spherical harmonics expansion. When the DC component (energy E0 ) is omitted, and cosine distance is used (right column), the embeddings again reorganize to emphasize rotational symmetry. Then, for both morphometrics, the cone and pyramid are close, as well as the ellipsoid and capsule. The spherical parameterization produces a nearly perfect symmetry-based organization: the sphere is isolated, and the cone–pyramid and ellipsoid–capsule pairs collapse almost exactly onto one another. This tighter clustering is expected since the spherical parameterization only evaluates the surface, and all shapes are 12

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

PF-SDM

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Figure 10: Top: the six 3D shape classes. Bottom: MDS embeddings of pairwise distance matrices computed from PF-SDM descriptors (left) and spherical parameterization descriptors (right). Top row: full spectrum with Euclidean distance. Bottom row: spectrum excluding the DC component with cosine distance. Colors match the inset legend.

topological spheres. In contrast, PF-SDM also captures interior structure, which better separates different shapes while still embedding them based on symmetry, as confirmed in Sections 5.2 and 5.3. Overall, these observations confirm the conclusions from the 2D benchmarks. Together, these results demonstrate that the PF-SDM yields invariant, noise-robust, and interpretable shape descriptors. It also provides explicit control over the geometric features emphasized in shape comparison through simple, principled adaptation of the spectral representation. 5.2

Discriminative power

We assess the discriminative power of the PF-SDM on more complex shapes from real-world computer vision data sets with annotated ground truth. Since these shapes are not generally star-convex, we use harmonic extension of the PF-T in the following experiments. Specifically, we consider the data sets MPEG-7 [23], a multi-class collection of 1 400 complex 2D shapes across 70 classes, and BBBC010 [24] from the Broad Bioimage Benchmark Collection, a binary classification task distinguishing live from dead C. elegans roundworms over 1 406 individual per-worm masks. Both data sets directly provide binary segmentation masks with examples shown in Fig. 11. On these data sets, we compare the classification performance of PF-SDM embeddings with classic morphometric descriptors and with deep-learning shape representations taking into account different levels of information: shape, skeleton, intensity. For this, we assume that the input data, e.g. an image, contain different channels. A channel is any scalar field fS : S → R defined over the domain of a shape ∂S. The PF-T (Definition 2) maps each channel to the reference domain Br , and the spectral PF-T (Section 3.2) produces a rotation- and reflection-invariant feature vector. In the experiments below, we consider three channels: • Shape channel (SDF): the polynomial SDF ϕξ , encoding boundary location and signed distance to the shape ∂S. 13

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Figure 11: Example binary masks from the benchmark data sets. (a) BBBC010 showing per-well masks of microscopy images of roundworms [24]: top row shows mostly alive worms and bottom row mostly dead worms. There are multiple worms per image; classification is done on a per-worm basis. (b) Ten example classes from the MPEG-7 shape-classification data set [23]. There is one shape per image. • Skeleton channel (Skel): the divergence of the normalized SDF gradient, ∇·η from Eq. (14), which smoothly encodes the medial-axis structure of the shape ∂S (see Section 4.1). • Intensity channel (Int): any additional scalar signal IS : S → R defined over the shape domain, such as a pixel intensity field. We refer to descriptors obtained from individual or combined channels as: • PF-SDM: shape channel only. • PF-SDM + Skel: shape and skeleton channels. • PF-SDM + Int: shape and intensity channels. • PF-SDM + Skel + Int: all three channels. For each descriptor variant, we train independent logistic-regression classifiers on the spectral features of each active channel, obtaining a per-channel class-probability vector Pc (y | x) for c ∈ C ⊆ {SDF, Skel, Int}. The fused prediction is the convex combination X X Pfused (y | x) := αc Pc (y | x) , αc = 1 , αc ≥ 0, (15) c∈C

c∈C

with weights αc selected by grid search in the validation split to maximize the target metric (F1-score in Section 5.2, balanced accuracy in Section 5.3). Ties are resolved by selecting the most uniform weighting, ensuring a deterministic and permutation-invariant fusion. For single-channel descriptors (here, PF-SDM), Eq. (15) reduces to the underlying classifier. We adopt a late fusion scheme rather than feature concatenation because it (i) decouples the regularization of channels with different amplitude scales and dimensionalities, (ii) yields interpretable channel-wise contribution weights αc , and (iii) makes single-channel ablations directly comparable across descriptor variants. All models are evaluated under an identical 70/15/15 train/validation/test split. Deep-learning methods use the training and validation sets for model fitting and regularization; for PF-SDM and the other learning-free descriptors, features are extracted from the combined training and validation sets. In all cases, the features are used to train a logistic regression classifier, and all methods are evaluated on the held-out test set. We compare two PF-SDM descriptor variants (defined above) with classification based on 19 geometric region properties from OpenCV [25] (listed in Appendix F), Elliptical Fourier Descriptors (EFA) [4] as well as four deep-learning approaches: masked autoencoders (MAE) [9] in three vision transformer (ViT) configurations, both pretrained on ImageNet-1K and trained from scratch on the target masks, SimCLR [8], O2VAE [7], and ShapeEmbed [6] with and without an appended size feature. Implementation details for all baselines are given in Appendix G. Classification performance (F1-score) is reported in Table 1, and end-to-end runtimes are compared in Table 2. Runtimes were measured on an Apple M1 Max (macOS 15.6.1). Models ran on CPU only, using their default settings, which may have included internal Pytorch optimization. We report end-to-end runtimes, which for PF-SDM include image pre-processing, solving the viscous Eikonal equation, computing the divergence for the skeleton information, computing the PF-T, and extracting the shape features. We also include the time for training, grid parameter search on the validation set, and inference of the logistic regression classifier. The higher runtimes on the MPEG-7 data set are due to the higher geometric complexity of the shapes, which leads to longer Eikonal solve times. The two data sets highlight two different regimes: MPEG-7 offers a multi-class set of shapes with high diversity and rich medial-axis structure (e.g., tools, animals), whereas BBBC010 is a binary classification of C. elegans roundworms 14

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Table 1: Shape classification performance (F1-score, higher is better) on two computer-vision benchmark data sets: MPEG-7 and BBBC010. The most accurate model is highlighted in bold. Method Region properties EFA MAE (ViT-B, pretrained) MAE (ViT-L, pretrained) MAE (ViT-H, pretrained) MAE (ViT-B) MAE (ViT-L) MAE (ViT-H) SimCLR O2VAE ShapeEmbed ShapeEmbed + Size PF-SDM PF-SDM + Skel

MPEG-7 0.45 0.56 0.39 0.40 0.37 0.87 0.63 0.56 0.81 0.31 0.90 0.77 0.85 0.91

BBBC010 0.83 0.52 0.62 0.66 0.69 0.64 0.35 0.35 0.81 0.68 0.80 0.80 0.85 0.85

Table 2: End-to-end runtimes for the different models on the benchmarks from Table 1; h = hours, m = minutes, s = seconds. The fastest model is highlighted in bold. Method Region properties EFA MAE (ViT-B, pretrained) MAE (ViT-L, pretrained) MAE (ViT-H, pretrained) MAE (ViT-B) MAE (ViT-L) MAE (ViT-H) SimCLR O2VAE ShapeEmbed ShapeEmbed + Size PF-SDM PF-SDM + Skel

MPEG-7 1m 5s 4s 4m 10m 20m 20s 10h 30m 17h 45m 39h 41m 6h 30m 7h 3h 30m 3h 30m 6m36s 10m49s

BBBC010 5.5s 3.5s 4m 11m 30s 21m 10h 26m 17h 17m 37h 6m 6h 33m 4h 30m 3h 30m 3h 30m 3m17s 5m22s

all sharing the same topology and mainly differing in boundary curvature. On MPEG-7, PF-SDM + Skel achieves the highest F1-score (0.91) of all tested methods, closely followed by ShapeEmbed (0.90) and the from-scratch MAE ViT-B (0.87). Classic approaches like region properties and EFA clearly underperform. The pretrained MAE models also perform worse than their from-scratch counterparts. On the simpler and more homogeneous shapes of BBBC010, simple region properties outperform all deep-learning baselines with large from-scratch models particularly affected. PF-SDM still performs best among all tested methods, but because all worms share the same skeletal topology, the skeleton channel adds no additional discriminative signal. Looking at runtimes, both PF-SDM variants are orders of magnitude faster than from-scratch deep-learning baselines and about as fast as forward inference using a pretrained model. Simple geometric features like EFA and region properties are expectedly the fastest. These results show that PF-SDM is competitive with state-of-the-art traditional and deep-learning methods for shape analysis both in terms of accuracy and runtime. They also show that adding the skeleton channel can further improve performance on topologically diverse shape sets. 5.3

Spatiotemporal shape and patterning analysis

Having established the performance and invariance properties of PF-SDM, we apply it to a biological problem: characterizing the developmental dynamics of mouse gastruloids. 15

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Figure 12: Shape analysis of developing mouse gastruloids. (A) Representative gastruloid images from the singleaxis (blue frame) and multi-axes (orange frame) classes at early (circle), intermediate (triangle), and late (square) times, along with a posterior reference at 94 h. Brachyury::mCherry intensity is shown in red. (B) PCA trajectories of NF = 25 PF-SDM Fourier coefficients c for Brachyury intensity (top), Skeleton information (middle) and shape (bottom) for the examples from (A) across 25 time points. Mouse gastruloids are three-dimensional in vitro models of early embryonic development that self-organize from aggregates of embryonic stem cells and recapitulate key features of body-axis formation [26]. A non-negligible fraction of gastruloids, however, fails to elongate along a single axis and instead develops multiple protrusions. The Brachyury::mCherry reporter molecule marks the developing posterior (i.e., tail-end) pole, providing a ground-truth patterning readout. We consider the problem of predicting whether a given gastruloid will develop a single axis or multiple axes before the posterior marker becomes clearly visible. Solving this task requires combining morphological information (shape and skeleton) with molecular patterning (Brachyury intensity) evolving over time. Figure 12(A) shows example microscopy images of gastruloids from the two classes at three developmental stages (74h, 87.5h, 92h) together with the a-posteriori reference at 94h. Panel (B) projects the PF-T FFT coefficients of each channel onto their first two principal components, yielding a low-dimensional trajectory for each gastruloid over the 25 imaging time points. The three channels carry visibly complementary information: the intensity trajectory captures the spatial reorganization of the Brachyury reporter as the posterior pole emerges, the skeleton trajectory reflects the 16

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

evolution of the underlying medial-axis structure, and the shape trajectory tracks the elongation and protrusion patterns of the tissue boundary. The two phenotypes separate earlier in the intensity channel than in the shape channel, where they only diverge once morphological differences become visually apparent. This qualitative observation motivates the joint use of all three channels. We therefore evaluate the combined PF-SDM + Skel + Int descriptor variant from Section 5.2 with the intensity channel given by the Brachyury::mCherry fluorescence signal on a data set of 78 gastruloids. We follow the same evaluation protocol as in our previous work [2] to enable direct comparison with the two-channel PF-SDM + Int baseline reported there. For each gastruloid, the spectral PF-T coefficients of every active channel are concatenated across the 25 time points to yield a single sample-level descriptor. We train independent logisticregression classifiers with balanced class weights for the shape, skeleton, and intensity channels, obtaining per-channel class probabilities PSDF , PSkel , PInt that are combined by the late fusion rule of Eq. (15). Performance is estimated under repeated stratified shuffle splits (k = 5, test size 20%), with the fusion weights αc tuned per split to maximize balanced accuracy. We report mean ± standard deviation across splits. The fused model in its present formulation achieves an accuracy of 0.90 ± 0.050 and a balanced accuracy of 0.91 ± 0.080, improving over the previously reported two-channel baseline (accuracy 0.875 ± 0.039, balanced accuracy 0.667 ± 0.000) [2]. The shape channel receives the highest weight (αSDF = 0.41), while the second highest weight comes from the skeleton channel with (αSkel = 0.36), indicating that medial-axis information provides a strong discriminative signal for this task. The slightly increased variance in the balanced accuracy reflects the limited number of minority-class samples. Together with the results from subsection 5.2, this application shows that PF-SDM provides strong discriminative power across shape classes of varying complexity and extends beyond pure geometry to include additional intensity signals through interpretable per-channel fusion. Discriminative power, as measured by the F1-score on the two benchmarking datasets, and by the accuracy and balanced accuracy on the gastruloid dataset, is on par with or better than state-of-the-art deep-learning models while remaining fully deterministic, training-free, and interpretable.

6

Conclusion

We introduced the Push-Forward Transform (PF-T), a continuous, smooth, and deterministic map for comparing scalar functions defined over different shape domains by mapping them to a common reference domain through a structurepreserving diffeomorphism. The framework yields descriptors that are invariant to translation, rotation, reflection, reparametrization and uniform scaling by construction, while preserving the intrinsic information encoded in the original functions. Applied to signed distance functions (SDF), the PF-T induces the PF-SDF and an associated morphometric, the PF-SDM, in either Fourier (in 2D) or spherical-harmonic (in 3D) bases and provides explicit control over which geometric properties dominate the comparison. Exploiting the viscous formulation of the Eikonal equation, we derived an algorithm that extracts a parameterized medial axis (spine) directly from the smooth polynomial SDF approximation, providing an intrinsic, landmark-free coordinate for elongated shapes. Across 2D, 3D, and time-resolved data sets, PF-SDM matches or exceeds the classification accuracy of state-of-the-art traditional and deep-learning methods while requiring no training data and running orders of magnitude faster than deep-learning models. In dynamic biological gastruloid data, fusing PF-SDM shape descriptors with PF-T intensity and skeleton features further improved predictive accuracy over the previous state of the art, illustrating the value of the framework for joint geometric and signal analysis. The proposed framework also has limitations. The PF-T, as was defined here, requires a smooth diffeomorphism between the reference and target domains. It is therefore restricted to shapes that are topologically equivalent to the reference domain. Here, we only considered topological spheres. Extending the framework to other topologies would require choosing a topology-matched reference domain and defining a boundary deformation map and extension that remain diffeomorphic. If the reference domain lacks rotational symmetry, rotation invariance is no longer obtained from the spectral representation and may require an additional transformation. Furthermore, the parameterized spine algorithm relies on PCA-based ordering of the skeletal point cloud, which restricts its applicability to elongated shapes with a single dominant axis. Extending it to branched and near-isotropic shapes would require replacing it with graph-based traversal. Finally, the presented PF-T is restricted to scalar-valued functions. Generalizing it to vector- and tensor-valued fields would require extending the notion to the push-forward of differential forms using exterior calculus [27]. Several possibilities for future applications of the PF-T follow naturally. First, the entire PF-T computational pipeline—from solving the viscous Eikonal equation to extending the boundary deformation map—is differentiable 17

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

in its parameters, offering the possibility of end-to-end gradient-based optimization. Second, the framework provides a principled deterministic feature extractor that can be integrated with deep-learning models to combine geometric inductive biases with data-driven flexibility. More broadly, because the PF-T applies to any continuous scalar function defined over a shape domain, the framework provides a unified mathematical formulation for the joint analysis of morphology and spatial signals in any setting where shapes and shape-correlated distributions must be compared invariantly and at scale.

7

Software and Data Availability

The 2D implementation of PF-SDM is publicly available at https://git.mpi-cbg.de/mosaic/software/ machine-learning/pf-sdm, together with a napari plugin at https://git.mpi-cbg.de/mosaic/software/ machine-learning/pf-sdm-napari-plugin, as released with our previous work [2]. In this work, we additionally release the 3D version of PF-SDM, including parameterized medial-axis (spline) extraction and the synthetic demo shapes and mesh data used in Section 5 and in Appendix D, at https://git.mpi-cbg.de/mosaic/software/ machine-learning/3d.

Acknowledgments This work was supported by the German Ministry of Research, Technology and Space (BMFTR, Bundesministerium für Forschung, Technologie und Raumfahrt) as part of the Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI) Dresden/Leipzig. Large language models were used for proof-reading and improving the quality of the text, as well as for code generation in the publicly available implementation.

A

Proof of Theorem 1

Proof. Let x = (x1 , . . . , xd ) ∈ Rd be the Cartesian coordinate vector and θ ∈ Θd ⊂ Rd−1 the collection of all angular coordinates. We start by proving (i). By construction, the radial extension Ψr : Br \ {0} → S \ {0} is of class C 1 . Denoting by Dx Ψr its Jacobian in Cartesian coordinates and by DΨr its Jacobian in spherical coordinates (r, θ). The change of variables (r, θ) to x with Jacobian factor rd−1 Jω (θ) gives det Dx Ψr (x) =

det DΨr (r, θ) ν(θ)d rd−1 Jω (θ) = rd−1 Jω (θ) rd−1 Jω (θ) = ν(θ)d > 0,

where Jω (θ) := det(ω(θ), ∂θ1 ω(θ), . . . , ∂θd −1 ω(θ)) . depends only on the angular parametrization. Thus, det Dx Ψr (r, θ) > 0 for all r > 0. For instance, in dimension d = 2, a direct computation yields det DΨr (r, θ) = r ν(θ)2 (cos2 (θ) + sin2 (θ)) = r ν(θ)2 , and since the deformation map ν(θ) ̸= 0 by definition, det Dx Ψr (r, θ) = ν(θ)2 > 0. Injectivity follows from the star-convexity of ∂S, which ensures that each point x ∈ S \ {0} admits a unique representation of the form x = ρ ω(θ(x)), for some ρ ∈ (0, ν(θ(x))). This yields a well-defined inverse of Ψr , proving global injectivity. Since det DΨr > 0 on Br \ {0}, the inverse function theorem implies that Ψr is locally invertible. Combined with global injectivity, this yields a global C 1 inverse on S \ {0}. Hence, Ψr is a C 1 -diffeomorphism. For d = 2, the proof for the harmonic extension follows by applying the Alessandrini–Nesi theorem [28], using that the boundary deformation map satisfies the injectivity and non-degeneracy conditions, i.e., det(DΨh ) > 0 on ∂Br .

B

Proof of Proposition 1

Proof. We prove the claim of the invariance proposition from the main text separately for d = 2 and d = 3. 18

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Case d = 2. Any orthogonal transformation K ∈ O(2) acts on the angular variable as θ → σθ + α,

σ ∈ {1, −1},

where σ = 1 corresponds to a rotation and σ = −1 to a reflection. Hence fSK∗ (r, θ) = fS ∗ (r, σθ + α). The corresponding Fourier coefficients satisfy iβk cK cσk (r) k (r) = e K for some phase factor eiβk with |eiβk | = 1. Therefore, |cK k (r)| = |cσk (r)|. If σ = 1, this directly yields |ck (r)| = K |ck (r)|. If σ = −1, then, since fS ∗ is real-valued, c−k (r) = ck (r), and hence |ck (r)| = |c−k (r)| = |ck (r)|. Thus, in both cases, |cK k (r)| = |ck (r)|. Moreover, the normalization factor is preserved: NX F −1

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F Consequently, cF S ∗ (r) = cS ∗ (r), which proves invariance under rotations and reflections in dimension 2. K

Case d = 3. Let K ∈ O(3) be an orthogonal transformation that includes rotations and reflections. For each degree ℓ, the spherical harmonic coefficients of the transformed field are obtained from the original coefficients by an orthogonal, unitary change of basis within the same degree: ℓ X

cK ℓ,m (r) =

ℓ Dm,m ′ (K) cℓ,m′ (r),

m′ =−ℓ

where Dℓ (K) denotes the representation matrix associated with degree ℓ. Since Dℓ (K) is unitary, the degree-wise spectral energy is preserved: EℓK (r) =

ℓ X

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H Therefore, the spherical harmonic descriptor satisfies cH ∗ (r) = cS ∗ (r) and is invariant under rotations and reflections SK in dimension 3.

C

Proof of Lemma 1

Proof. Let c ∈ MS , and let xl , xr ∈ ∂S be two distinct closest points, i.e. ∥c − xl ∥2 = ∥c − xr ∥2 = ϕS (c) =: d . Define zlδ := c − δ

c − xl , ∥c − xl ∥2

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c − xr , ∥c − xr ∥2

for 0 < δ < d, and let xδl , xδr ∈ ∂S be the closest points to zlδ , zrδ , respectively. Then, we have that zlδ → c and zrδ → c as δ → 0. Moreover, since ∂S is a compact set, there exists a subsequence such that xδl → xl and xδr → xr , for xl , xr ∈ ∂S. We prove that xl = xl . By optimality of xδl we have ∥zlδ − xδl ∥2 ≤ ∥zlδ − xl ∥2 .

(A1)

Taking the limit δ → 0 and using the continuity of the norm, we obtain ∥c − xl ∥2 ≤ ∥c − xl ∥2 .

(A2)

∥c − xl ∥2 ≤ ∥c − xl ∥2 ,

(A3)

Since xl is a closest point to c, we also have

19

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

and therefore ∥c − xl ∥2 = ∥c − xl ∥2 . To show that the minimizer of ∥c − x∥2 is unique, we prove that ∥zlδ − xl ∥2 < ∥zlδ − x∥2 , ∀x ∈ ∂S, s.t. ∥c − x∥2 = ∥c − xl ∥2 .

(A4)

To show this, let x ∈ ∂S such that ∥c − xl ∥22 = ∥c − x∥22 , then compute ∥zlδ − xl ∥22 = ∥c − xl ∥22 − 2δ∥c − xl ∥2 + δ 2 .

(A5)

On the other side,

∥zlδ − x∥22 = ∥c − x∥22 − 2

δ ⟨c − x, c − xl ⟩2 + δ 2 . ||c − xl ||2

(A6)

Using that ∥c − xl ∥22 = ∥c − x∥22 and canceling the common terms, we have that ∥zlδ − xl ∥2 < ∥zlδ − x∥2 ⇐⇒ ∥c − xl ∥22 > ⟨c − x, c − xl ⟩2 . By the Cauchy–Schwarz inequality, ⟨c−x, c−xl ⟩2 ≤ ∥c−x∥2 ∥c−xl ∥2 = ∥c−xl ∥22 , and since x ̸= xl , the inequality is strict. This shows that xl ∈ ∂S is the unique minimizer of the constrained optimization ∥zlδ − x∥2 .

inf x∈∂S

(A7)

∥c−x∥2 =∥c−xl ∥2

Therefore, xl = xl . The proof follows analogously for xδr ∈ ∂S. Thus,

∇ϕS (zlδ ) =

zlδ − xl , ∥zlδ − xl ∥2

∇ϕS (zrδ ) =

zrδ − xr . ∥zrδ − xr ∥2

c − xl , ∥c − xl ∥2

∇ϕS (zrδ ) →

c − xr . ∥c − xr ∥2

In the limit δ → 0, we obtain

∇ϕS (zlδ ) →

Since xl ̸= xr , these limits differ, and therefore ϕS is not differentiable at c.

D

Parametric Medial-Axis Extraction

Given the polynomial SDF ϕξ and its normalized-gradient divergence ∇ · η (Eq. (14) of the main text), Algorithm A1 extracts a smooth parametric medial axis γ : [0, 1] → Rd for elongated shapes with a single dominant principal axis. 20

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

Figure A1: Extraction of a parameterized medial axis for a 3D biological gastruloid shape from Ref. [26], following Algorithm A1: (a) shape surface mesh; (b) central plane cut of the polynomial SDF values sampled inside the volume; (c) central plane cut of the divergence of the normalized gradient; (d) filtered skeleton point cloud; (e) ordered and smoothed skeleton points obtained by PCA projection (red); (f) final medial axis as a smooth parametric B-spline (blue).

Algorithm A1 Parametric medial-axis extraction Require: Polynomial SDF ϕξ ∈ Πn (S) over shape domain S, uniform grid GS ⊆ S, quantile level α ∈ (0, 1), step size h > 0, B-spline order K Ensure: Parametric medial axis γ : [0, 1] → Rd 1: Compute η(x) = ∇ϕξ (x)/∥∇ϕξ (x)∥ and (∇ · η)(x) for all x ∈ GS 2: Compute the α-quantile qα of ∇ · η over GS 3: Form the skeleton point cloud Pskel = {x ∈ GS : (∇ · η)(x) ≤ qα } 4: Compute the centroid x̄ and first principal component v1 of Pskel ; order points by scalar projection ti = (xi −

x̄)⊤ v1 and smooth to obtain a discrete centerline (x̂1 , . . . , x̂m ) −x̂m−1 x̂2 −x̂1 5: Compute endpoint tangents τ1 = ∥x̂ and τm = ∥x̂x̂m 2 −x̂1 ∥ m −x̂m−1 ∥ 6: while ϕξ (x̂1 ) > 0 do 7: x̂1 ← x̂1 − h τ1 8: end while 9: while ϕξ (x̂m ) > 0 do 10: x̂m ← x̂m + h τm 11: end while 12: Assign chord-length parameters si ∈ [0, 1] to the extended centerline and fit a B-spline of order K by solving min

P1 ,...,PK

13: return γ(s) =

E

m X

x̂i −

i=1

K X j=1

2

Bj (si ) Pj

2

PK

j=1 Bj (s) Pj

Synthetic Shape Generation

The five 2D shape classes are equilateral triangle, square, regular pentagon, regular hexagon, and the five-petaled flower from [22], each class is represented by ten variants: the original shape, rotations by 30◦ , 45◦ , 60◦ , reflection about the y-axis, scalings by 2 and 3, translation by (1, 3), and Gaussian perturbations with σ = 0.005 and σ = 0.01. For each augmented contour we rasterize a binary image, segment it and follow the same preprocessing pipeline for 21

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

all. Specifically we extract 100 contour points at equal arc-length intervals, the contours are then centered to the origin and rescaled to be inscribed in the unit circle. In our synthetic 3D benchmark, each class is represented by five transformed trimesh meshes (https://github. com/mikedh/trimesh). The base classes are a sphere, ellipsoid, cube, cone, pyramid, and capsule. For each class, we generate five variants using rotations by 60◦ , 90◦ , 120◦ , 150◦ , and 180◦ about the axis (0.5, 2, 1), translations by (1, −1, 2), (2, −2, 4), (3, −3, 6), (4, −4, 8), and (5, −5, 10), and uniform scalings by 6, 11, 16, 21, and 26, respectively. After transformation, each mesh is centered at the origin and uniformly rescaled to fit inside the unit ball. Since these synthetic objects are available directly as surface meshes, we evaluate signed-distance samples from the mesh surface and fit the same polynomial SDF surrogate used by the PF-SDM pipeline. Thus, the subsequent push-forward map, polynomial representation, and spherical-harmonic PF-SDM descriptors remain unchanged.

F

Region Properties Used as Baseline Features

For the region-properties classification baseline, we extract 19 shape features from binary masks using OpenCV [25]: area, convex area, perimeter, major and minor axis lengths, extent, eccentricity, solidity, orientation, bounding-box width and height, maximum Feret diameter, and the seven Hu moments computed from the image moments.

G

Benchmark Implementation Details

We used the same stratified 70/15/15 split for all methods. This produced 980/210/210 train/validation/test samples for MPEG-7 and 984/211/211 for BBBC010. Contours were extracted using the same ConvertBinaryMasksToDMs procedure as also used for ShapeEmbed [6]: the longest object contour was retained and periodically spline-resampled to 64 points. G.1

Deterministic Baselines

Elliptical Fourier Descriptors We used the pyefd library to compute EFA. Descriptors were extracted with pyefd.elliptic_fourier_descriptors using order 30 (order 15 for the interpretability MDS in Fig. 6). Each harmonic contributes four coefficients (an , bn , cn , dn ), yielding 120-dimensional (respectively 60-dimensional) feature vectors per shape. The coefficients were then normalized with pyefd.normalize_efd to enforce invariance to translation, rotation, and scale. PF-SDM and PF-SDM + Skel Both variants follow the definitions in Section 5.2 of the main text. For the BBBC010 data set, we excluded the GFP channel which reports worm viability (dead worms accumulate the dye and fluoresce brightly while live worms remain dark). This was to prevent clever-Hans behavior and keep the benchmarking shapefocused. The excluded PF-SDM + Int variant has the same runtime as PF-SDM + Skel and achieves an F1 test score of 0.86. G.2

Deep-Learning Baselines

For the deep-learning baselines, we limited training to 200 epochs as was also done for ShapeEmbed [6]. ShapeEmbed and ShapeEmbed + Size We used the official ShapeEmbed [6] implementation with a 128dimensional latent space. Each input was the 64×64 Euclidean distance matrix computed from the resampled contour. For the ShapeEmbed + Size variant, we concatenated the latent mean with ShapeEmbed’s recovered original scale feature. MAE We used the official facebookresearch/mae [9] implementation. Binary masks were resized to 224 × 224. For pretrained MAE, we loaded the ImageNet-1K MAE checkpoints for the evaluated ViT configurations. For the data-trained MAE models, we initialized the MAE architecture from scratch and trained it on the training masks for 200 epochs using a batch size of 16. SimCLR We used the sthalles/SimCLR [8] implementation with a ResNet18 backbone and a 128-dimensional projection head. Binary masks were resized to 224 × 224 and positive pairs were generated with the default SimCLR augmentation pipeline: random resized crops, horizontal flips, color jitter, random grayscale conversion, and Gaussian blur. The model was trained on the training split for 200 epochs with batch size 256. 22

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

O2VAE We used the native O2VAE [7] implementation with binary masks as inputs, since the benchmark focuses on shape. Masks were resized to 64 × 64. The O2VAE used an O(2)-equivariant CNN encoder and a 128-dimensional latent space. Training used batch size 256, 200 epochs, random rotations, and vertical flips.

References [1] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences. New York, NY: Springer, 2003. [2] R. Rouatbi, J.-E. Suarez Cardona, A. Villaronga-Luque, J. V. Veenvliet, and I. F. Sbalzarini, “A continuous and interpretable morphometric for robust quantification of dynamic biological shapes,” in 2026 IEEE 23rd Int. Symp. Biomedical Imaging (ISBI), 2026, pp. 1–5. [3] J. C. Gower, “Generalized Procrustes analysis,” Psychometrika, vol. 40, no. 1, pp. 33–51, 1975. [4] F. P. Kuhl and C. R. Giardina, “Elliptic Fourier features of a closed contour,” Comput. Graph. Image Process., vol. 18, no. 3, pp. 236–258, 1982. [5] M. A. Wieczorek and M. Meschede, “SHTools: Tools for working with spherical harmonics,” Geochem. Geophys. Geosyst., vol. 19, no. 8, pp. 2574–2592, 2018. [6] A. Foix Romero, C. Russell, A. Krull, and V. Uhlmann, “ShapeEmbed: A self-supervised learning framework for 2D contour quantification,” arXiv:2507.01009, 2025. [7] J. Burgess, J. J. Nirschl, M.-C. Zanellati, A. Lozano, S. Cohen, and S. Yeung-Levy, “Orientation-invariant autoencoders learn robust representations for shape profiling of cells and organelles,” Nature Commun., vol. 15, no. 1, 2024. [8] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton, “A simple framework for contrastive learning of visual representations,” in Proc. 37th Int. Conf. Mach. Learn. (ICML), 2020. [9] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick, “Masked autoencoders are scalable vision learners,” in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), 2022, pp. 15979–15988. [10] C. T. Zahn and R. Z. Roskies, “Fourier descriptors for plane closed curves,” IEEE Trans. Comput., vol. C-21, no. 3, pp. 269–281, 1972. [11] J.-E. Suarez Cardona, P.-A. Hofmann, and M. Hecht, “Negative order Sobolev cubatures: Preconditioners of partial differential equation learning tasks circumventing numerical stiffness,” Mach. Learn.: Sci. Technol., vol. 5, no. 3, p. 035029, 2024. [12] J.-E. Suarez Cardona and M. Hecht, “Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power,” Mach. Learn.: Sci. Technol., vol. 4, no. 4, p. 045005, 2023. [13] K. Siddiqi, S. Bouix, A. Tannenbaum, and S. W. Zucker, “The Hamilton-Jacobi skeleton,” in Proc. 7th IEEE Int. Conf. Comput. Vis., 1999, pp. 828–834. [14] T. Sakai, “On Riemannian manifolds admitting a function whose gradient is of constant norm,” Kodai Math. J., vol. 19, no. 1, pp. 39–51, 1996. [15] S. Osher and R. Fedkiw, “Signed distance functions,” in Applied Mathematical Sciences. New York, NY: Springer, 2003, pp. 17–22. [16] M. G. Crandall and P.-L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Trans. Amer. Math. Soc., vol. 277, no. 1, p. 1, 1983. [17] J. E. Suarez Cardona, H. Boche, and G. Kutyniok, “A variational framework for the algorithmic complexity of PDE solutions,” arXiv:2510.21290, 2025. [18] H. Xia and P. G. Tucker, “Distance solutions for medial axis transform,” in Proc. 18th Int. Meshing Roundtable. Berlin, Heidelberg: Springer, 2009, pp. 247–265. [19] F. Chazal and A. Lieutier, “The λ-medial axis,” Graphical Models, vol. 67, no. 4, pp. 304–331, 2005. [20] A. Torsello and E. R. Hancock, “A skeletal measure of 2D shape similarity,” Comput. Vis. Image Underst., vol. 95, no. 1, pp. 1–29, 2004. [21] S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in Python,” PeerJ, vol. 2, p. e453, 2014. 23

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A P REPRINT

[22] P. Gandhi, M.-V. Ciocanel, K. Niklas, and A. T. Dawes, “Identification of approximate symmetries in biological development,” Phil. Trans. R. Soc. A, vol. 379, no. 2213, 2021. [23] “MPEG-7 Core Experiment CE-Shape-1 Test Set: Benchmarking image database for shape recognition techniques,” 2009. [Online]. Available: https://www.ehu.eus/ccwintco/index.php/MPEG-7_Core_ Experiment_CE-Shape-1_Test_Set._Benchmarking_image_database_for_shape_recognition_ techniques [24] V. Ljosa, K. L. Sokolnicki, and A. E. Carpenter, “Annotated high-throughput microscopy image sets for validation,” Nature Methods, vol. 9, no. 7, p. 637, 2012. [25] G. Bradski, “The OpenCV library,” Dr. Dobb’s J. Softw. Tools, 2000. [26] R. G. Savill, A. Villaronga-Luque, M. Trani Bustos, Y. Maroudas-Sacks, J. Batki, A. Meissner, A. Q. Ryan, C. D. Modes, O. Campàs, and J. V. Veenvliet, “SpinePy enables automated 3D spatiotemporal quantification of multicellular in vitro systems,” bioRxiv, doi:10.1101/2025.09.10.674634, 2025. [27] H. Cartan, Differential Forms, Dover Books on Mathematics. Mineola, NY: Dover Publications, 2006. [28] G. Alessandrini and V. Nesi, “Invertible harmonic mappings, beyond Kneser,” Ann. Sc. Norm. Super. Pisa Cl. Sci., pp. 451–468, 2009.

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