On Computing Elastic Shape Distances between Curves in d-dimensional Space

TL;DR

The paper advances elastic shape distance computation for curves in arbitrary dimension by (i) introducing a linear-time dynamic programming approach to compute diffeomorphisms for elastic registration, (ii) giving a purely algebraic justification of the KU rotation algorithm for rigid alignment, and (iii) redefining the elastic distance to allow simultaneous diffeomorphic reparametrization and rotation across multiple starting points, with FFT-based acceleration for closed curves. The methods are integrated into an iterative framework that alternates between optimizing diffeomorphisms and rotation matrices, enabling efficient elastic registration of open or closed curves in 3D and higher dimensions. Empirical results on 3D helices and ellipsoids show linear DP efficiency and substantial FFT-driven speedups, achieving near-zero elastic distances for congruent shapes and demonstrating practical applicability to complex curve alignment tasks.

Abstract

The computation of the elastic registration of two simple curves in higher dimensions and therefore of the elastic shape distance between them has been investigated by Srivastava et al. Assuming the first curve has one or more starting points, and the second curve has only one, they accomplish the computation, one starting point of the first curve at a time, by minimizing an L2 type distance between them based on alternating computations of optimal diffeomorphisms of the unit interval and optimal rotation matrices that reparametrize and rotate, respectively, one of the curves. We recreate the work by Srivastava et al., but in contrast to it, again for curves in any dimension, we present a Dynamic Programming algorithm for computing optimal diffeomorphisms that is linear, and justify in a purely algebraic manner the usual algorithm for computing optimal rotation matrices, the Kabsch-Umeyama algorithm, which is based on the computation of the singular value decomposition of a matrix. In addition, we minimize the L2 type distance with a procedure that alternates computations of optimal diffeomorphisms with successive computations of optimal rotation matrices for all starting points of the first curve. Carrying out computations this way is not only more efficient all by itself, but, if both curves are closed, allows applications of the Fast Fourier Transform for computing successively in an even more efficient manner, optimal rotation matrices for all starting points of the first curve.

Figures (6)

• Figure 1: Two views of the same two helices, curves in $3-$d space. The positive $z-$axis is the axis of rotation of one helix, while the positive $x-$axis is the axis of rotation of the other one. Their shapes are essentially identical thus the elastic shape distance between them should be essentially zero.
• Figure 2: On left is $\gamma^*$ from $2^{nd}$ iteration, $NI=NJ=2^2+1=5$. In center, during $3^{rd}$ iteration, $NI=NJ=2^3+1=9$; shaded bins are bins the interior of which $\gamma^*$ intersects. On right, shaded bins form adapting strip in which next $\gamma^*$ is computed. Each shaded bin is within 2 bins ($lstrp=2$) of a bin which is above, to the right of, or equal to it, and whose interior has nonempty intersection with the current $\gamma^*$.
• Figure 3: Three plots of helices. The elastic registration of the two helices in each plot and the elastic shape distance between them were computed.
• Figure 4: Optimal diffeomorphisms for pairs of helices.
• Figure 5: Views of first helix of 3 loops and second helix of 5 loops before computation of elastic shape distance and registration (left), of rotated first helix (middle) and reparametrized second helix (right) after computations.