On Unsupervised Partial Shape Correspondence

TL;DR

A novel method is proposed that establishes direct correspondence between partial and full shapes through feature matching through feature matching bypassing the need for functional map intermediate spaces, superior also compared to supervised methods.

Abstract

While dealing with matching shapes to their parts, we often apply a tool known as functional maps. The idea is to translate the shape matching problem into "convenient" spaces by which matching is performed algebraically by solving a least squares problem. Here, we argue that such formulations, though popular in this field, introduce errors in the estimated match when partiality is invoked. Such errors are unavoidable even for advanced feature extraction networks, and they can be shown to escalate with increasing degrees of shape partiality, adversely affecting the learning capability of such systems. To circumvent these limitations, we propose a novel approach for partial shape matching. Our study of functional maps led us to a novel method that establishes direct correspondence between partial and full shapes through feature matching bypassing the need for functional map intermediate spaces. The Gromov Distance between metric spaces leads to the construction of the first part of our loss functions. For regularization we use two options: a term based on the area preserving property of the mapping, and a relaxed version that avoids the need to resort to functional maps. The proposed approach shows superior performance on the SHREC'16 dataset, outperforming existing unsupervised methods for partial shape matching.Notably, it achieves state-of-the-art results on the SHREC'16 HOLES benchmark, superior also compared to supervised methods. We demonstrate the benefits of the proposed unsupervised method when applied to a new dataset PFAUST for part-to-full shape correspondence.

Figures (9)

• Figure 1: Overview of the proposed pipeline. Basic features computed for the full and partial shapes are refined using a Siamese neural diffusion feature extractor. Next, cosine similarity is computed and fed into a softmax layer that produces a soft correspondence. While training, the loss functions are applied to the soft correspondence. At inference, the soft correspondence matrix is binarized for sharp matching.
• Figure 2: Estimating the functional map $\hat{\boldsymbol{C}}_{yx}$ between a full $\mathcal{X}$ and its part $\mathcal{Y}$ using features independently extracted for each yields errors. Recall, $\hat{\boldsymbol{C}}_{yx} = \boldsymbol{C}_{yx} + \boldsymbol{C}_{yx}^E$, where $\boldsymbol{C}_{yx}$ is the ideal functional map given the correct matching, and $\boldsymbol{C}_{yx}^E$ is an error resulting from matching the part $\mathcal{Y}$ to its complementary part in $\mathcal{X}$. Here, we plot the magnitude of entries in $|\boldsymbol{C}_{yx}|$ and $|\boldsymbol{C}_{yx}^E|$. Indeed, $\boldsymbol{C}_{yx}$ contains an informative structure, whereas $\boldsymbol{C}_{yx}^E$ is a noise-like structure-less matrix.
• Figure 3: Qualitative results on SHREC'16 CUTS (left) and HOLES (right). On SHREC'16 CUTS, we obtain visually appealing results that outperform previous unsupervised methods. On SHREC'16 HOLES, we obtain better matching results both visually and quantitatively than even the best supervised approach (DPFM).
• Figure 4: Example of shapes existing in our new PFAUST benchmark. The left shape is from PFAUST-M, while the right one is from PFAUST-H, which are of medium and hard difficulty, respectively.
• Figure 5: PCK curves of existing unsupervised methods and ours on the test sets of SHREC'16 CUTS (top left), SHREC'16 HOLES (top right), PFAUST-M (bottom left) and PFAUST-H (bottom right). Our method is systematically superior compared to competing unsupervised approaches.

Theorems & Definitions (6)

• theorem thmcountertheorem: Continuous case
• corollary thmcountercorollary
• theorem thmcountertheorem: Discrete case
• corollary thmcountercorollary
• proof
• proof