The Hamilton-Jacobi skeleton, also known as the medial axis, is a powerful shape descriptor that represents binary objects in terms of the centres of maximal inscribed discs. Despite its broad applicability, the medial axis suffers from sensitivity to noise: Minor boundary variations can lead to disproportionately large and undesirable expansions of the skeleton. Classical pruning methods mitigate this shortcoming by systematically removing extraneous skeletal branches. This sequential simplification of skeletons resembles the principle of sparsification scale-spaces that embed images into a family of reconstructions from increasingly sparse pixel representations. We combine both worlds by introducing skeletonisation scale-spaces: They leverage sparsification of the medial axis to achieve hierarchical simplification of shapes. Unlike conventional pruning, our framework inherently satisfies key scale-space properties such as hierarchical architecture, controllable simplification, and equivariance to geometric transformations. We provide a rigorous theoretical foundation in both continuous and discrete formulations and extend the concept further with densification. By growing the skeleton successively instead of shrinking it, we allow inverse progression from coarse to fine scales. Densification scale-spaces can even reach beyond the original skeleton to produce overcomplete shape representations with relevancy for practical applications. Through proof-of-concept experiments, we demonstrate the effectiveness of our framework for practical tasks including robust skeletonisation, shape compression, and stiffness enhancement for additive manufacturing.

翻译：Hamilton-Jacobi骨架（亦称中轴）是一种强大的形状描述符，其通过最大内切圆盘的中心来表示二值对象。尽管应用广泛，但中轴对噪声敏感：微小的边界变化可能导致骨架产生不成比例且不期望的过度扩张。经典剪枝方法通过系统性移除多余骨架分支来弥补这一缺陷。这种骨架的序贯简化类似于稀疏化尺度空间的原理——该尺度空间将图像嵌入到由日益稀疏的像素表示重建而成的图像族中。我们通过引入骨架化尺度空间融合这两个领域：该方法利用中轴的稀疏化实现形状的分层简化。与传统剪枝不同，我们的框架天然满足关键尺度空间性质，例如层次结构、可控简化以及对几何变换的等变性。我们在连续和离散两种公式化表述中提供严格的理论基础，并通过稠密化进一步扩展该概念。通过逐步扩展而非收缩骨架，我们实现了从粗尺度到细尺度的逆推演化。稠密化尺度空间甚至能超越原始骨架，生成具有实际应用相关性的过完备形状表示。通过概念验证实验，我们证明了该框架在鲁棒骨架化、形状压缩以及增材制造刚度增强等实际任务中的有效性。