Topological Simplification of Nested Shapes

Abstract

AbstractWe present a method for removing unwanted topological features (e.g., islands, handles, cavities) from a sequence of shapes where each shape is nested in the next. Such sequences can be found in nature, such as a multi‐layered material or a growing plant root. Existing topology simplification methods are designed for single shapes, and applying them independently to shapes in a sequence may lose the nesting property. We formulate the nesting‐constrained simplification task as an optimal labelling problem on a set of candidate shape deletions (“cuts”) and additions (“fills”). We explored several optimization strategies, including a greedy heuristic that sequentially propagates labels, a state‐space search algorithm that is provably optimal, and a beam‐search variant with controllable complexity. Evaluation on synthetic and real‐world data shows that our method is as effective as single‐shape simplification methods in reducing topological complexity and minimizing geometric changes, and it additionally ensures nesting. Also, the beam‐search strategy is found to strike the best balance between optimality and efficiency.