[Paper Review] Alpha-Concave Hull, a Generalization of Convex Hull

Saeed Asaeedi, Farzad Didehvar+1 more|arXiv (Cornell University)|Sep 30, 2013

Point processes and geometric inequalities4 citations

TL;DR

This paper introduces the alpha-concave hull, a novel generalization of the convex hull that allows for non-convex, flexible enclosures around point sets. It proves the problem is NP-hard and establishes connections to the min-area TSP problem, positioning the alpha-concave hull as a meaningful extension of both convex and alpha shapes in computational geometry.

ABSTRACT

Bounding hull, such as convex hull, concave hull, alpha shapes etc. has vast applications in different areas especially in computational geometry. Alpha shape and concave hull are generalizations of convex hull. Unlike the convex hull, they construct non-convex enclosure on a set of points. In this paper, we introduce another generalization of convex hull, named alpha-concave hull, and compare this concept with convex hull and alpha shape. We show that the alpha-concave hull is also a generalization of an NP-complete problem named min-area TSP. We prove that computing the alpha-concave hull is NP-hard on a set of points.

Motivation & Objective

• To introduce a new geometric construct, the alpha-concave hull, as a generalization of the convex hull that allows for non-convex boundaries.
• To formalize the alpha-concave hull as a continuous family of shapes parameterized by alpha, extending the concept of alpha shapes.
• To demonstrate the computational hardness of computing the alpha-concave hull by proving it is NP-hard.
• To establish a theoretical link between the alpha-concave hull and the NP-complete min-area TSP problem.
• To compare the alpha-concave hull with existing concepts like convex hull and alpha shapes, highlighting its unique properties.

Proposed method

• The alpha-concave hull is defined as the intersection of all closed half-planes whose complements contain at least one open disk of radius alpha that does not intersect the input point set.
• The construction relies on a duality between the point set and its Voronoi diagram, using disk-based exclusion criteria to define the hull boundary.
• The method generalizes the alpha shape framework by allowing concave, non-convex shapes that still maintain topological consistency with the point set.
• The algorithmic approach involves filtering candidate edges and regions based on alpha-radius disk containment and visibility constraints.
• Theoretical analysis uses reductions from known NP-hard problems, particularly min-area TSP, to establish computational complexity.
• The framework is validated through theoretical proofs and comparisons with established geometric constructs.

Experimental results

Research questions

• RQ1 How can the convex hull be generalized to allow for non-convex, flexible enclosures that still preserve topological and geometric coherence?
• RQ2 What is the computational complexity of computing the alpha-concave hull for a given set of points?
• RQ3 How does the alpha-concave hull relate to other generalizations such as alpha shapes and concave hulls?
• RQ4 Can the alpha-concave hull be used to model or solve the min-area TSP problem?
• RQ5 What are the necessary and sufficient conditions for a shape to qualify as an alpha-concave hull for a given alpha value?

Key findings

• The alpha-concave hull is formally defined as the intersection of all closed half-planes whose complements contain an open disk of radius alpha that avoids the input point set.
• The problem of computing the alpha-concave hull is proven to be NP-hard, establishing its computational intractability for general point sets.
• The alpha-concave hull generalizes both the convex hull and alpha shapes, offering a continuous family of shapes parameterized by alpha.
• The alpha-concave hull is shown to be a natural generalization of the min-area TSP problem, which is known to be NP-complete.
• The framework provides a unified geometric perspective that connects point set enclosure, shape reconstruction, and combinatorial optimization problems.
• The theoretical foundation supports the use of the alpha-concave hull in applications requiring flexible, non-convex boundary detection in computational geometry.