De Casteljau's Algorithm in Geometric Data Analysis: Theory and Application

TL;DR

A survey of the recent theoretical developments in de Casteljau's algorithm as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology are provided.

Abstract

For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing, and discrete geometry to name but a few. With increasing interest in nonlinear data science, its constructive approach has been shown to provide a principled way to generalize parametric smooth curves to manifolds. These curves have found remarkable new applications in the analysis of parameter-dependent, geometric data. This article provides a survey of the recent theoretical developments in this exciting area as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology.

Figures (10)

• Figure 1: Cubic Bézier curve $\beta$ on the sphere $\mathcal{S}^2$ and the construction of $\beta(1/2)$ by the de Casteljau algorithm.
• Figure 2: Examples of Bézier splines. The unlabeled green points are dependent control points.
• Figure 3: Sketch of regression with non-geodesic Bézier splines in $\mathcal{S}^2$. The orange dots are data points, while $B$ indicates the underlying spline $B$ in (\ref{['eq:regression_model_manifold']}).
• Figure 4: Normalization w.r.t. some parameter $t$ for a single data group $(q_j,t_j)$, $j=1,2,3,4$, in a Riemannian manifold $M$. The curve $B$ is the result of spline regression w.r.t. $t$. The points $B(t_j)$ are depicted in light grey, while the tangent vectors $v_j$ are the black arrows attached to them. Finally, the parallel translation $w_j$ of each $v_j$ is a tangent vector at $B(t_0)$ (also black); it yields the normalized data $\widetilde{q}_j$ shown in orange.
• Figure 6: Conceptual comparison of a hierarchical model and regression analysis on a longitudinal data set. The measurements are shown as grey dots (without time). Each subject's (correlated) measurements follow its trend (broken lines). Spline regression on the full data set yields the red curve; its direction deviates strongly from all individual trends. The mean of the individual trends (green curve) is a better estimator of the average longitudinal trend in the cohort.

Theorems & Definitions (18)

• Definition 2.1: De Casteljau's Algorithm on Manifolds
• Theorem 1
• Corollary 2.1
• Definition 2.2
• Remark 1
• Definition 3.1: Bézierfolds
• Theorem 2
• proof
• Conjecture
• Corollary 3.1