Wormhole Loss for Partial Shape Matching

TL;DR

The approach proposed treats surfaces as manifolds equipped with geodesic distances, and addresses the partial shape matching challenge by introducing a novel criterion to meticulously search for consistent distances between pairs of points.

Abstract

When matching parts of a surface to its whole, a fundamental question arises: Which points should be included in the matching process? The issue is intensified when using isometry to measure similarity, as it requires the validation of whether distances measured between pairs of surface points should influence the matching process. The approach we propose treats surfaces as manifolds equipped with geodesic distances, and addresses the partial shape matching challenge by introducing a novel criterion to meticulously search for consistent distances between pairs of points. The new criterion explores the relation between intrinsic geodesic distances between the points, geodesic distances between the points and surface boundaries, and extrinsic distances between boundary points measured in the embedding space. It is shown to be less restrictive compared to previous measures and achieves state-of-the-art results when used as a loss function in training networks for partial shape matching.

Figures (13)

• Figure 1: For a distance preserving map between full surfaces $\mathcal{X}$ and $\mathcal{Y}$, also known as an isometry, the minimal geodesics in the partial version $\mathcal{Y}'$ may not correspond to those in the full surfaces $\mathcal{Y}$. That is, the geodesic distances between corresponding points may get larger.
• Figure 2: Venn diagrams showing the relation between all pairs of points, consistent and guaranteed pairs for a surface with boundaries. All guaranteed pairs are consistent. Our criterion $\mathcal{C}_{\mathcal{W}}$ is more inclusive than that of bronstein2006efficientrosman2008topologically, $\mathcal{C}_{\mathcal{T}}$. All guaranteed pairs by $\mathcal{C}_{\mathcal{T}}$ are also guaranteed by $\mathcal{C}_{\mathcal{W}}$.
• Figure 3: Example of inconsistent and consistent pairs of points. The minimal geodesic paths are colored red, while paths to the boundary points are colored blue. Different boundary points are selected in $\mathcal{C}_{\mathcal{T}}$bronstein2006efficientrosman2008topologically and $\mathcal{C}_{\mathcal{W}}$. The Euclidean lines connecting the boundary points selected by $\mathcal{C}_{\mathcal{W}}$ are dashed blue. Both criteria correctly reject inconsistent pairs (left). Since $\mathcal{C}_{\mathcal{T}}$ ignores the distance between boundary points, it discards many consistent pairs (middle). Criterion $\mathcal{C}_{\mathcal{W}}$ finds more consistent pairs by including the extrinsic Euclidean distance between boundary points (right).
• Figure 4: We plot in green the set of points that together with the blue point satisfy $\mathcal{C}_{\mathcal{T}}$bronstein2006efficientrosman2008topologically (left) and $\mathcal{C}_{\mathcal{W}}$ (right). The blue point and any of the green ones form a guaranteed pairs. The cat surface is taken from the SHREC'16 HOLES dataset cosmo2016shrec.
• Figure 5: MDS embedding of various manifold learning methods on a whole Swiss roll (top) and on two versions of it with Gaussian noise, having either a rectangular hole (middle) or a rectangular cut (bottom). Other methods are referred to in \ref{['fig: mds all less stretch refRoll single page no noise full full hole broken', 'fig: mds all less stretch refRoll single page small noise full full hole broken']} in \ref{['sec: sup mat Multi-dimensional scaling']}.

Theorems & Definitions (8)

• Definition 1: Consistent Pair of Points
• Definition 2: Guaranteed Pair of Points
• Theorem 1: Euclidean bound
• Theorem 2: $\mathcal{C}_{\mathcal{W}}$ guarantees
• proof
• proof
• Theorem 3: $\mathcal{C}_{\mathcal{W}}$ guarantees
• proof