InShaDe: Invariant Shape Descriptors for visual 2D and 3D cellular and nuclear shape analysis and classification

Khaled Al-Thelaya, Marco Agus, Nauman Ullah Gilal, Yin Yang, Giovanni Pintore, Enrico Gobbetti, Corrado Calí, Pierre J. Magistretti, William Mifsud, Jens Schneider

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We present a shape processing framework for visual exploration of cellular nuclear envelopes extracted from microscopic images arising in histology and neuroscience. The framework is based on a novel shape descriptor of closed contours in 2D and 3D. In 2D, it relies on a geodesically uniform resampling of discrete curves to compute unsigned curvatures at vertices and edges based on discrete differential geometry. Our descriptor is, by design, invariant under translation, rotation, and parameterization. We achieve the latter invariance under parameterization shifts by using elliptic Fourier analysis on the resulting curvature vectors. Uniform scale-invariance is optional and is a result of scaling curvature features to z-scores. We further augment the proposed descriptor with feature coefficients obtained through sparse coding of the extracted cellular structures using K-sparse autoencoders. For the analysis of 3D shapes, we compute mean curvatures based on the Laplace-Beltrami operator on triangular meshes, followed by computing a spherical parameterization through mean curvature flow. Finally, we compute the Spherical Harmonics decomposition to obtain invariant energy coefficients. Our invariant descriptors provide an embedding into a fixed-dimensional feature space that can be used for various applications, e.g., as input features for deep and shallow learning techniques or as input for dimension reduction schemes to provide a visual reference for clustering shape collections. We demonstrate the capabilities of our framework in the context of visual analysis and unsupervised classification of 2D histology images and 3D nuclear envelopes extracted from serial section electron microscopy stacks.
Original languageEnglish (US)
Pages (from-to)105-125
Number of pages21
JournalComputers and Graphics (Pergamon)
StatePublished - May 7 2021

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • General Engineering
  • Human-Computer Interaction


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