Estimation of smooth growth trajectories with controlled acceleration from time series shape data

Abstract

Longitudinal shape analysis often relies on the estimation of a realistic continuous growth scenario from data sparsely distributed in time. In this paper, we propose a new type of growth model parameterized by acceleration, whereas standard methods typically control the velocity. This mimics the behavior of biological tissue as a mechanical system driven by external forces. The growth trajectories are estimated as smooth flows of deformations, which are twice differentiable. This differs from piecewise geodesic regression, for which the velocity may be discontinuous. We evaluate our approach on a set of anatomical structures of the same subject, scanned 16 times between 4 and 8 years of age. We show our acceleration based method estimates smooth growth, demonstrating improved regularity compared to piecewise geodesic regression. Leave-several-out experiments show that our method is robust to missing observations, as well as being less sensitive to noise, and is therefore more likely to capture the underlying biological growth.

Fig. 1

a) and b) Shape evolution from baseline (solid) to final configuration (transparent) using a model based on piecewise geodesics (a) and our method (b) with point trajectories for selected particles displayed as black lines. c) The path of a point on the forebrain is decomposed into coordinates. Growth is estimated using 15 target shapes, highlighting the speed discontinuities present in the piecewise geodesic evolution.

Fig. 2

Volume measurements derived from our growth model are consistent with a kernel regression (σ = 0.5) performed on the sparse volume measurements. Our model describes the continuous evolution of shape and volume is measured after regression.

Fig. 3

Left: Snapshots from a continuous shape evolution of lateral ventricles estimated by our regression model. Acceleration vectors are displayed on the surface, with color denoting magnitude. Right: The impact of the number of target shapes on R².