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.

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.

Equations

$${\Delta }_{\mathcal{X}}{\varphi }_i(x)$$

$${\varphi }_i(x)$$

$${\mu }_i={\int }_{\mathcal{X}}\Vert {\nabla }_{\mathcal{X}}{\varphi }_i(x){\Vert }_2^2\mathrm{d}x$$

$$H_{\mathcal{X}}f={\Delta }_{\mathcal{X}}f+vf,$$

$$H_{\mathcal{X}}{\psi }_i(x)={\lambda }_i{\psi }_i(x),$$

$$w_{ij}$$

$$a_i$$

$$(\mathbf{W}+\mathbf{A}\mathrm{diag}(\mathbf{v}))\mathbf{\Psi }=\mathbf{A}\mathbf{\Psi }\mathrm{diag}(\mathbf{\lambda }).$$

$$\displaystyle v_{\tau}(x)=\left\{\begin{array}[]{ll}0&x\in\mathcal{R}\\ \tau&x\notin\mathcal{R}\end{array}\right.$$

$${\lambda }_i{\psi }_i(x)$$

$$={\Delta }_{\mathcal{R}}{\psi }_i(x)$$

$$={\mu }_i{\varphi }_i(x).$$

$$\underset{\mathbf{v}\ge 0}{\min }\Vert \mathbf{\lambda }({\mathbf{\Delta }}_{\mathcal{X}}+\mathrm{diag}(\mathbf{v}))-\mathbf{\mu }{\Vert }_w^2,$$

$$\Vert \mathbf{\lambda }-\mathbf{\mu }{\Vert }_w^2=\sum _{i=1}^k\frac{1}{{\mu }_i^2}({\lambda }_i-{\mu }_i{)}^2,$$

$${\mathbf{v}}^{(t)}$$

$$={\mathbf{v}}^{(t-1)}-\Pi \left(2\alpha ({\mathbf{\Phi }}^{(t-1)}\circ {\mathbf{\Phi }}^{(t-1)})(({\mathbf{\lambda }}^{(t-1)}-\mathbf{\mu })\oslash {\mathbf{\mu }}^2)\right)$$

$$\underset{\mathbf{v}\in {\mathbb{R}}^n}{\min }\Vert \mathbf{\lambda }({\mathbf{\Delta }}_{\mathcal{X}}+\mathrm{diag}(\sigma (\mathbf{v})))-\mathbf{\mu }{\Vert }_w^2,$$

Full text

1 Introduction

Assessing similarity between non-rigid shapes is an active research topic in computer vision, pattern recognition and graphics [5]. At the heart of such methods lies the definition of shape descriptors, characterizing the shape either locally or globally (e.g., via the Bag-of-Words [37] paradigm or via deep learning [26]). Deformation-invariant shape descriptors often require careful tuning and hand-crafting, or sufficient training examples to enable learned-based methods. While similarity by itself is usually expressed by a numerical score, the problem as a whole is also strictly related to (and often solved in tandem with) the complementary problem of shape correspondence. In this setting, local shape descriptors are used as ‘probe’ quantities to employ in more sophisticated pipelines to infer a functional [32] or point-to-point [7] correspondence.