Fig. 1.
Quantitative analysis of the movies: method. (A) The frames are segmented and aligned to obtain images of Fa containing information about the shape alone, devoid of translation, rotation, or textures. A B-spline curve, given by its control polygon Pa (black circles), is fitted to the boundary of Fa and is a generating curve of the axisymmetric representation of the pellicle. Original frame image courtesy of Richard E. Triemer. (B) The segmented and aligned frames (i), representative of the shapes adopted by the cell, are embedded in low dimensions by a nonlinear dimensionality reduction technique called Isomap (15). The algorithm maps each frame to a low-dimensional point (circles in ii and iii, color labeling chronological order), so that the intrinsic distance between frames (shape dissimilarity) is preserved as much as possible by the low-dimensional representation. The embedding in the plane (τ1, τ2) (ii), shows that the stroke is a closed nonreciprocal path in shape space. Consequently, it can be most compactly described by embedding the frames in a periodic 1D segment (iii), from which we parametrize the stroke as a function of τ by interpolation (iv) with smooth basis functions wa(τ). At any given τ, the synthetic stroke is a weighted average of the curves fitting video frames whose 1D embedding is in the vicinity of τ. The parameter τ is not proportional to physical time, ignored by the manifold learning algorithm, but rather to arc length in shape space. See SI Text, section Video Processing and Parametrization of the Stroke, for technical details.