Visual Complexity of Point Set Mappings
Wouter Meulemans, Arjen Simons, Kevin Verbeek
TL;DR
The paper addresses measuring the visual complexity of transitions between labeled point sets by modeling transitions as grouped translations of subsets of indices. It presents a formal framework with inputs A,B in Euclidean space, a per-point movement collection, and a solution consisting of a family of groups and translations that collectively realize the required displacements. The main contributions are a problem-classification framework for variants and results on the algorithmic complexity of these variants (polynomial-time algorithms, NP-hardness, and approximations for open problems), along with discussion of extensibility and practical implications for visual transitions. These insights enable design of animated transitions that exploit common motion to reduce cognitive load and guide future work on additional dimensions of the problem.
Abstract
We study the visual complexity of animated transitions between point sets. Although there exist many metrics for point set similarity, these metrics are not adequate for this purpose, as they typically treat each point separately. Instead, we propose to look at translations of entire subsets/groups of points to measure the visual complexity of a transition between two point sets. Specifically, given two labeled point sets A and B in R^d, the goal is to compute the cheapest transformation that maps all points in A to their corresponding point in B, where the translation of a group of points counts as a single operation in terms of complexity. In this paper we identify several problem dimensions involving group translations that may be relevant to various applications, and study the algorithmic complexity of the resulting problems. Specifically, we consider different restrictions on the groups that can be translated, and different optimization functions. For most of the resulting problem variants we are able to provide polynomial time algorithms, or establish that they are NP-hard. For the remaining open problems we either provide an approximation algorithm or establish the NP-hardness of a restricted version of the problem. Furthermore, our problem classification can easily be extended with additional problem dimensions giving rise to new problem variants that can be studied in future work.