图像空间中的贝塞尔曲线
Bézier curves in the space of images
AI总结：
本文将贝塞尔曲线推广到无限维图像空间，通过黎曼度量和变分离散化方法计算离散贝塞尔曲线，并展示其在面部插值和形状动画中的应用。
摘要：
贝塞尔曲线是欧几里得空间中曲线设计的常用工具。本文将其概念推广到无限维图像空间。为此，图像空间配备了测量图像传输和强度变化成本的黎曼度量，该度量基于Miller和Younes的变形模型。然后通过黎曼版本的de Casteljau算法计算贝塞尔曲线，该算法基于沿测地线的层次凸组合方案。测地线通过黎曼路径能量的变分离散化进行近似。这导致了计算图像空间中合适离散贝塞尔曲线的通用de Casteljau方法。所选测试用例展示了该方法的定性性质。此外，还提出了一种通过图像草图调节面部插值和形状动画的贝塞尔方法。
英文摘要：
Bézier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of Bézier curves to the infinite-dimensional space of images. To this end the space of images is equipped with a Riemannian metric which measures the cost of image transport and intensity variation in the sense of the metamorphosis model by Miller and Younes. Bézier curves are then computed via the Riemannian version of de Casteljau's algorithm, which is based on a hierarchical scheme of convex combination along geodesic curves. Geodesics are approximated using a variational discretization of the Riemannian path energy. This leads to a generalized de Casteljau method to compute suitable discrete Bézier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a Bézier approach for the modulation of face interpolation and shape animation via image sketches is presented.
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