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A Continuous and Interpretable Morphometric for Robust Quantification of Dynamic Biological Shapes

Roua Rouatbi, Juan-Esteban Suarez Cardona, Alba Villaronga-Luque, Jesse V. Veenvliet, Ivo F. Sbalzarini

TL;DR

The paper addresses robust quantification of dynamic biological shapes by introducing the Push-Forward Signed Distance Morphometric (PF-SDM), a continuous, shape-preserving morphometric invariant to shape-preserving transformations. It combines a SDF computation on a domain $Ω$, a push-forward deformation $Ψ_{ζ}$ to a reference disk $Ω_r$, and a Fourier-based morphometric derived from the PF-SDF, optionally fused with intensity fields, yielding differentiable temporal shape representations. Synthetic benchmarks show PF-SDM outperforms Elliptical Fourier Analysis and Generalized Procrustes Analysis in robustness and interpretability, while a mouse gastruloid application demonstrates that shape features plus intensity fusion achieve higher predictive accuracy with lower computational cost than a CNN baseline. The approach enables a geometry-aware, tractable framework for spatiotemporal morphometrics with potential extensions to 3D+time and Sobolev-norm variants, facilitating integration into learning pipelines.

Abstract

We introduce the Push-Forward Signed Distance Morphometric (PF-SDM) for shape quantification in biomedical imaging. The PF-SDM compactly encodes geometric and topological properties of closed shapes, including their skeleton and symmetries. This provides robust and interpretable features for shape comparison and machine learning. The PF-SDM is mathematically smooth, providing access to gradients and differential-geometric quantities. It also extends to temporal dynamics and allows fusing spatial intensity distributions, such as genetic markers, with shape dynamics. We present the PF-SDM theory, benchmark it on synthetic data, and apply it to predicting body-axis formation in mouse gastruloids, outperforming a CNN baseline in both accuracy and speed.

A Continuous and Interpretable Morphometric for Robust Quantification of Dynamic Biological Shapes

TL;DR

The paper addresses robust quantification of dynamic biological shapes by introducing the Push-Forward Signed Distance Morphometric (PF-SDM), a continuous, shape-preserving morphometric invariant to shape-preserving transformations. It combines a SDF computation on a domain , a push-forward deformation to a reference disk , and a Fourier-based morphometric derived from the PF-SDF, optionally fused with intensity fields, yielding differentiable temporal shape representations. Synthetic benchmarks show PF-SDM outperforms Elliptical Fourier Analysis and Generalized Procrustes Analysis in robustness and interpretability, while a mouse gastruloid application demonstrates that shape features plus intensity fusion achieve higher predictive accuracy with lower computational cost than a CNN baseline. The approach enables a geometry-aware, tractable framework for spatiotemporal morphometrics with potential extensions to 3D+time and Sobolev-norm variants, facilitating integration into learning pipelines.

Abstract

We introduce the Push-Forward Signed Distance Morphometric (PF-SDM) for shape quantification in biomedical imaging. The PF-SDM compactly encodes geometric and topological properties of closed shapes, including their skeleton and symmetries. This provides robust and interpretable features for shape comparison and machine learning. The PF-SDM is mathematically smooth, providing access to gradients and differential-geometric quantities. It also extends to temporal dynamics and allows fusing spatial intensity distributions, such as genetic markers, with shape dynamics. We present the PF-SDM theory, benchmark it on synthetic data, and apply it to predicting body-axis formation in mouse gastruloids, outperforming a CNN baseline in both accuracy and speed.

Paper Structure

This paper contains 10 sections, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Signed Distance Function computation
  4. Push-Forward Signed Distance Function
  5. Morphometric evaluation
  6. Results
  7. Synthetic Validation Benchmark
  8. Application to Mouse Gastruloids
  9. Conclusion
  10. Acknowledgments

Figures (3)

  • Figure 1: First two MDS components of the shape distance matrices computed with PF-SDM (left), EFA (center), and GPA (right) over ten shape-preserving transformations of each of the five test shapes shown in the inset legend. Point colors match shape-legend colors.
  • Figure 2: Shape analysis of developing mouse gastruloids. (A) Representative gastruloid images from the single-axis (blue) and multi-axes (orange) classes at early (circle), intermediate (triangle), and late (square) times, along with the posterior reference at 94 h. Brachyury::mCherry intensity is shown in red. (B) PCA trajectories of $N_F=25$ PF-SDF Fourier coefficients $\mathfrak{c}$ for Brachyury intensity (top) and shape (bottom) for the examples from (A) across 25 time points.
  • Figure 3: Comparison of classification performance between logistic regression over PF-SDM features and the CNN baseline villarongaluque_2025. Accuracy (plain) and Balanced Accuracy (shaded) are shown for shape, intensity, and fusion (shape+intensity) features. Error bars: empirical standard deviation across five stratified test splits.