The Persistent Homology Transform (PHT) summarizes a shape in $\mathbb{R}^m$ by collecting persistence diagrams obtained from linear height filtrations in all directions on $\mathbb{S}^{m-1}$. It enjoys strong theoretical guarantees, including continuity, stability, and injectivity on broad classes of shapes. A natural way to compare two PHTs is to use the bottleneck distance between their diagrams as the direction varies. Prior work has either compared PHTs by sampling directions or, in 2D, computed the exact \textit{integral} of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to $\tilde O(n^5)$ in place of earlier $\tilde O(n^6)$ bound. For the \textit{max} objective, we give a $\tilde O(n^3)$ algorithm in $\mathbb{R}^2$ and a $\tilde O(n^5)$ algorithm in $\mathbb{R}^3$.

翻译：持久同调变换通过收集在所有方向 $\mathbb{S}^{m-1}$ 上由线性高度过滤获得的持久图，来概括 $\mathbb{R}^m$ 中的形状。该变换具有强大的理论保证，包括连续性、稳定性以及对广泛形状类别的单射性。比较两个持久同调变换的一种自然方法是，在方向变化时使用其图之间的瓶颈距离。先前的工作要么通过采样方向来比较持久同调变换，要么在二维情况下通过动力学数据结构精确计算所有角度上瓶颈距离的\textit{积分}。我们将积分目标从先前 $\tilde O(n^6)$ 的界改进为 $\tilde O(n^5)$。对于\textit{最大值}目标，我们在 $\mathbb{R}^2$ 中给出了一个 $\tilde O(n^3)$ 算法，在 $\mathbb{R}^3$ 中给出了一个 $\tilde O(n^5)$ 算法。