[Paper Review] Shape Complementarity Analysis for Objects of Arbitrary Shape

Morad Behandish, Horea T. Ilieş|arXiv (Cornell University)|Dec 1, 2017

3D Shape Modeling and Analysis9 references3 citations

TL;DR

This paper proposes a general framework for shape complementarity analysis using a space-continuous skeletal density function (SDF) to quantify geometric fit between arbitrary 2D and 3D objects. By modeling shape skeletons as implicit density fields and computing their cross-correlation via fast Fourier transforms, the method enables robust, gradient-friendly optimization for applications in assembly planning and protein docking.

ABSTRACT

The basic problem of shape complementarity analysis appears fundamental to applications as diverse as mechanical design, assembly automation, robot motion planning, micro- and nano-fabrication, protein-ligand binding, and rational drug design. However, the current challenge lies in the lack of a general mathematical formulation that applies to objects of arbitrary shape. We propose that a measure of shape complementarity can be obtained from the extent of approximate overlap between shape skeletons. A space-continuous implicit generalization of the skeleton, called the skeletal density function (SDF) is defined over the Euclidean space that contains the individual assembly partners. The SDF shape descriptors capture the essential features that are relevant to proper contact alignment, and are considerably more robust than the conventional explicit skeletal representations. We express the shape complementarity score as a convolution of the individual SDFs. The problem then breaks down to a global optimization of the score over the configuration space of spatial relations, which can be efficiently implemented using fast Fourier transforms (FFTs) on nonequispaced samples. We demonstrate the effectiveness of the scoring approach for several examples from 2D peg-in-hole alignment to more complex 3D examples in mechanical assembly and protein docking. We show that the proposed method is reliable, inherently robust against small perturbations, and effective in steering gradient-based optimization.

Motivation & Objective

• Address the lack of a general mathematical formulation for shape complementarity in objects of arbitrary shape.
• Develop a robust, continuous representation of shape skeletons that captures essential geometric features for alignment.
• Enable efficient global optimization of shape fit using convolution and fast Fourier transforms on nonequispaced samples.
• Provide a reliable, differentiable score function suitable for gradient-based optimization in complex geometric problems.
• Demonstrate applicability across diverse domains, from mechanical assembly to protein-ligand docking.

Proposed method

• Define the skeletal density function (SDF) as a space-continuous, implicit representation of shape skeletons, derived from distance projections to boundary elements.
• Use a complex-valued kernel function ϕ: ℂ → ℂ to encode both proximal (inverse-square) and medial (nearest-neighbor region extent) effects in the SDF.
• Formulate the shape complementarity score as the convolution of two SDFs, enabling global optimization over configuration space.
• Leverage nonequispaced fast Fourier transforms (NFFT) to efficiently compute the score across spatial configurations.
• Apply the score function to steer gradient-based optimization, with smooth variations enabling convergence in few iterations.
• Unify various affinity formulations under a common conceptual framework by treating the kernel ϕ as the key design variable.

Experimental results

Research questions

• RQ1 How can a general, mathematically rigorous measure of shape complementarity be defined for arbitrary 2D and 3D shapes?
• RQ2 Can a continuous, implicit skeleton representation (SDF) outperform traditional explicit skeletal models in robustness and geometric fidelity?
• RQ3 Does the SDF-based convolution score enable reliable and efficient optimization for complex assembly and docking problems?
• RQ4 How does the SDF formulation compare to existing methods in terms of smoothness and suitability for gradient-based search?
• RQ5 Can the proposed method generalize across diverse applications, from mechanical parts to protein-ligand complexes?

Key findings

• The SDF-based score function exhibits smooth, broad support across the configuration space, enabling stable gradient-based optimization, unlike the narrow, spike-prone scores from explicit skeleton methods.
• The method correctly identifies the native configuration in protein docking (e.g., HIV protease–Saquinavir complex) with high accuracy, even in the presence of complex surface topologies.
• For mechanical assemblies with sharp and dull features, the SDF-based score function reliably predicts the optimal fit, demonstrating robustness to geometric complexity.
• The SDF formulation produces a low-energy valley in the score landscape, providing clear optimization pathways for gradient methods, leading to fast convergence in few iterations.
• The use of nonequispaced FFTs enables efficient evaluation of the convolution score across spatial configurations, supporting real-time or interactive applications.
• The framework unifies diverse affinity formulations under a single conceptual model, with differences attributed solely to the choice of the kernel ϕ.