Map-based exploration of intrinsic shape differences and variability

Abstract

Figure 1: The notion of shape difference defined in this paper provides a way to compare deformations between shape pairs. This allows us to recognize similar expressions of shape A (top row) to those of shape B (bottom row), without correspondences between A and B and without any prior learning process. AbstractWe develop a novel formulation for the notion of shape differences, aimed at providing detailed information about the location and nature of the differences or distortions between the two shapes being compared. Our difference operator, derived from a shape map, is much more informative than just a scalar global shape similarity score, rendering it useful in a variety of applications where more refined shape comparisons are necessary. The approach is intrinsic and is based on a linear algebraic framework, allowing the use of many common linear algebra tools (e.g, SVD, PCA) for studying a matrix representation of the operator. Remarkably, the formulation allows us not only to localize shape differences on the shapes involved, but also to compare shape differences across pairs of shapes, and to analyze the variability in entire shape collections based on the differences between the shapes. Moreover, while we use a map or correspondence to define each shape difference, consistent correspondences between the shapes are not necessary for comparing shape differences, although they can be exploited if available. We give a number of applications of shape differences, including parameterizing the intrinsic variability in a shape collection, exploring shape collections using local variability at different scales, performing shape analogies, and aligning shape collections.

Functional map networks for analyzing and exploring large shape collections

“…In this paper, we choose the inverse of X i j to be the Moore-Penrose pseudo-inverse X + i j := V Σ −1 U T , where U, Σ, V are given by the singular value decomposition of X i j := UΣV T . This choice inherits several nice properties from the full similarity case, as we show in Section 5 when using the shape difference operator [Rustamov et al 2013] for shape exploration.…”

“…Another application of constructing a consistent functional map network is to use the shape difference operator [Rustamov et al 2013] for shape exploration. In [Rustamov et al 2013], the authors demonstrated a few browsing operations on fully similar organic shapes.…”

“…The key feature of this method is that the segmentations are completely driven by the input shape collection -we do not need to perform initial segmentations or to specify the number of desired segments. In the second application, we show that the quality of the resulting functional maps enables us to apply the shape difference operator [Rustamov et al 2013] for visually exploring shape collections.…”

Supervised Descriptor Learning for Non-Rigid Shape Matching

“…Interestingly, it has recently been observed [27] that even articulated motion of humans can induce noticeable isometric distortion, which could explain some of the difficulties encountered by previously proposed techniques.…”

“…Although these techniques can produce good results when the deformation satisfies the a priori model, they can fail badly as soon as even moderate deviations from the model are introduced. This is especially critical since many natural deformations, such as articulated motion of humans or animals are known to induce potentially significant geodesic distortion [27]. Incorporating the possibility for such distortion into a deformation model is challenging especially using a purely axiomatic (theoretical) approach.…”