@article{21954,
  abstract     = {We investigate a framework for train-free MRI segmentation based on Topological Data Analysis. The pipeline proceeds in three steps, first identifying the whole object to segment via automatic thresholding, then detecting a distinctive subset whose topology is known in advance, and finally deducing the various components of the segmentation. A key ingredient is the extraction of approximate representative cycles from persistence diagrams, which provides an interpretable link between persistent features and anatomical components. To clarify the method’s scope, we make the underlying topological and intensity assumptions explicit, quantify when they hold on real data, and analyze typical failure modes. We evaluate the approach on glioblastoma and on fetal cortical plate segmentation, with comparisons to unsupervised and deep-learning references. By operating without large annotated datasets, the method is well suited to scarce-data settings and provides an interpretable baseline and practical initialization for expert refinement or learning-based pipelines.},
  author       = {François, Anton and Tinarrage, Raphaël},
  issn         = {1573-7683},
  journal      = {Journal of Mathematical Imaging and Vision},
  number       = {3},
  publisher    = {Springer Nature},
  title        = {{Train-free segmentation in MRI with cubical persistent homology}},
  doi          = {10.1007/s10851-026-01300-1},
  volume       = {68},
  year         = {2026},
}

@article{18353,
  abstract     = {Traditional models of bendable surfaces are based on the exact or approximate invariance to deformations that do not tear or stretch the shape, leaving intact an intrinsic geometry associated with it. These geometries are typically defined using either the shortest path length (geodesic distance), or properties of heat diffusion (diffusion distance) on the surface. Both measures are implicitly derived from the metric induced by the ambient Euclidean space. In this paper, we depart from this restrictive assumption by observing that a different choice of the metric results in a richer set of geometric invariants. We apply equi-affine geometry for analyzing arbitrary shapes with positive Gaussian curvature. The potential of the proposed framework is explored in a range of applications such as shape matching and retrieval, symmetry detection, and computation of Voroni tessellation. We show that in some shape analysis tasks, equi-affine-invariant intrinsic geometries often outperform their Euclidean-based counterparts. We further explore the potential of this metric in facial anthropometry of newborns. We show that intrinsic properties of this homogeneous group are better captured using the equi-affine metric.},
  author       = {Raviv, Dan and Bronstein, Alexander and Bronstein, Michael M. and Waisman, Dan and Sochen, Nir and Kimmel, Ron},
  issn         = {1573-7683},
  journal      = {Journal of Mathematical Imaging and Vision},
  pages        = {144--163},
  publisher    = {Springer Nature},
  title        = {{Equi-affine invariant geometry for shape analysis}},
  doi          = {10.1007/s10851-013-0467-y},
  volume       = {50},
  year         = {2014},
}

@article{2255,
  abstract     = {Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma-Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm.},
  author       = {Edelsbrunner, Herbert and Pausinger, Florian},
  issn         = {0924-9907},
  journal      = {Journal of Mathematical Imaging and Vision},
  number       = {1},
  pages        = {164 -- 177},
  publisher    = {Springer},
  title        = {{Stable length estimates of tube-like shapes}},
  doi          = {10.1007/s10851-013-0468-x},
  volume       = {50},
  year         = {2014},
}