# Correspondence-Free Region Localization for Partial Shape Similarity via Hamiltonian Spectrum Alignment

## TL;DR

This paper introduces a novel, correspondence-free spectral alignment method for localizing relevant regions in partial 3D shapes, improving accuracy and efficiency in shape similarity and retrieval tasks.

## Contribution

It proposes a descriptor-free spectral alignment approach that avoids explicit correspondence solving, simplifying optimization and enhancing performance in partial shape analysis.

## Key findings

- Outperforms state-of-the-art in accuracy and computational cost
- Provides a simple alternative to shape-from-spectrum reconstruction
- Effective for partial shape similarity and retrieval tasks

## Abstract

We consider the problem of localizing relevant subsets of non-rigid geometric shapes given only a partial 3D query as the input. Such problems arise in several challenging tasks in 3D vision and graphics, including partial shape similarity, retrieval, and non-rigid correspondence. We phrase the problem as one of alignment between short sequences of eigenvalues of basic differential operators, which are constructed upon a scalar function defined on the 3D surfaces. Our method therefore seeks for a scalar function that entails this alignment. Differently from existing approaches, we do not require solving for a correspondence between the query and the target, therefore greatly simplifying the optimization process; our core technique is also descriptor-free, as it is driven by the geometry of the two objects as encoded in their operator spectra. We further show that our spectral alignment algorithm provides a remarkably simple alternative to the recent shape-from-spectrum reconstruction approaches. For both applications, we demonstrate improvement over the state-of-the-art either in terms of accuracy or computational cost.