eCurves: A Temporal Shape Encoding

Objective: This paper presents a framework for temporal shape analysis to capture the shape and changes of anatomical structures from three-dimensional+t(ime) medical scans. Method: We first encode the shape of a structure at each time point with the spectral signature, i.e., the eigenvalues and eigenfunctions of the Laplace operator. We then expand it to capture morphing shapes by tracking the eigenmodes across time according to the similarity of their eigenfunctions. The similarity metric is motivated by the fact that small-shaped deformations lead to minor changes in the eigenfunctions. Following each eigenmode from the beginning to end results in a set of eigenmode curves representing the shape and its changes over time. Results: We apply our encoding to a cardiac dataset consisting of series of segmentations outlining the right and left ventricles over time. We measure the accuracy of our encoding by training classifiers on discriminating healthy adults from patients that received reconstructive surgery for Tetralogy of Fallot (TOF). The classifiers based on our encoding significantly surpass deformation-based encodings of the right ventricle, the structure most impacted by TOF. Conclusion: The strength of our framework lies in its simplicity: It only assumes pose invariance within a time series but does not assume point-to-point correspondence across time series or a (statistical or physical) model. In addition, it is easy to implement and only depends on a single parameter, i.e., the number of curves.