The Persistent Homology Transform (PHT) summarizes a shape in ℝ^m by collecting persistence diagrams obtained from linear height filtrations in all directions on 𝕊^{m-1}. It enjoys strong theoretical guarantees, including continuity, stability, and injectivity. 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 integral of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to Õ(n⁵) in place of the earlier Õ(n⁶) bound, where n denotes the number of simplices. For the max objective, we give an Õ(n³) expected-time algorithm in ℝ² and an Õ(n⁵) expected-time algorithm in ℝ³.