12 research outputs found
Functional maps representation on product manifolds
We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices
Shape Registration in the Time of Transformers
In this paper, we propose a transformer-based procedure for the efficient registration of non-rigid 3D point clouds. The proposed approach is data-driven and adopts for the first time the transformers architecture in the registration task. Our method is general and applies to different settings. Given a fixed template with some desired properties (e.g. skinning weights or other animation cues), we can register raw acquired data to it, thereby transferring all the template properties to the input geometry. Alternatively, given a pair of shapes, our method can register the first onto the second (or vice-versa), obtaining a high-quality dense correspondence between the two. In both contexts, the quality of our results enables us to target real applications such as texture transfer and shape interpolation. Furthermore, we also show that including an estimation of the underlying density of the surface eases the learning process. By exploiting the potential of this architecture, we can train our model requiring only a sparse set of ground truth correspondences (10∼20% of the total points). The proposed model and the analysis that we perform pave the way for future exploration of transformer-based architectures for registration and matching applications. Qualitative and quantitative evaluations demonstrate that our pipeline outperforms state-of-the-art methods for deformable and unordered 3D data registration on different datasets and scenarios
Intrinsic shape analysis in archaeology: A case study on ancient sundials
This paper explores a novel mathematical approach to extract archaeological
insights from ensembles of similar artifact shapes. We show that by considering
all the shape information in a find collection, it is possible to identify
shape patterns that would be difficult to discern by considering the artifacts
individually or by classifying shapes into predefined archaeological types and
analyzing the associated distinguishing characteristics. Recently, series of
high-resolution digital representations of artifacts have become available, and
we explore their potential on a set of 3D models of ancient Greek and Roman
sundials, with the aim of providing alternatives to the traditional
archaeological method of ``trend extraction by ordination'' (typology). In the
proposed approach, each 3D shape is represented as a point in a shape space --
a high-dimensional, curved, non-Euclidean space. By performing regression in
shape space, we find that for Roman sundials, the bend of the sundials'
shadow-receiving surface changes with the location's latitude. This suggests
that, apart from the inscribed hour lines, also a sundial's shape was adjusted
to the place of installation. As an example of more advanced inference, we use
the identified trend to infer the latitude at which a sundial, whose
installation location is unknown, was placed. We also derive a novel method for
differentiated morphological trend assertion, building upon and extending the
theory of geometric statistics and shape analysis. Specifically, we present a
regression-based method for statistical normalization of shapes that serves as
a means of disentangling parameter-dependent effects (trends) and unexplained
variability.Comment: accepted for publication from the ACM Journal on Computing and
Cultural Heritag
Continuous and Orientation-preserving Correspondences via Functional Maps
We propose a method for efficiently computing orientation-preserving and
approximately continuous correspondences between non-rigid shapes, using the
functional maps framework. We first show how orientation preservation can be
formulated directly in the functional (spectral) domain without using landmark
or region correspondences and without relying on external symmetry information.
This allows us to obtain functional maps that promote orientation preservation,
even when using descriptors, that are invariant to orientation changes. We then
show how higher quality, approximately continuous and bijective pointwise
correspondences can be obtained from initial functional maps by introducing a
novel refinement technique that aims to simultaneously improve the maps both in
the spectral and spatial domains. This leads to a general pipeline for
computing correspondences between shapes that results in high-quality maps,
while admitting an efficient optimization scheme. We show through extensive
evaluation that our approach improves upon state-of-the-art results on
challenging isometric and non-isometric correspondence benchmarks according to
both measures of continuity and coverage as well as producing semantically
meaningful correspondences as measured by the distance to ground truth maps.Comment: 16 pages, 22 figure
Local Geometry Processing for Deformations of Non-Rigid 3D Shapes
Geometry processing and in particular spectral geometry processing deal with many different deformations that complicate shape analysis problems for non-rigid 3D objects. Furthermore, pointwise description of surfaces has increased relevance for several applications such as shape correspondences and matching, shape representation, shape modelling and many others. In this thesis we propose four local approaches to face the problems generated by the deformations of real objects and improving the pointwise characterization of surfaces. Differently from global approaches that work simultaneously on the entire shape we focus on the properties of each point and its local neighborhood. Global analysis of shapes is not negative in itself. However, having to deal with local variations, distortions and deformations, it is often challenging to relate two real objects globally. For this reason, in the last decades, several instruments have been introduced for the local analysis of images, graphs, shapes and surfaces. Starting from this idea of localized analysis, we propose both theoretical insights and application tools within the local geometry processing domain. In more detail, we extend the windowed Fourier transform from the standard Euclidean signal processing to different versions specifically designed for spectral geometry processing. Moreover, from the spectral geometry processing perspective, we define a new family of localized basis for the functional space defined on surfaces that improve the spatial localization for standard applications in this field. Finally, we introduce the discrete time evolution process as a framework that characterizes a point through its pairwise relationship with the other points on the surface in an increasing scale of locality. The main contribute of this thesis is a set of tools for local geometry processing and local spectral geometry processing that could be used in standard useful applications. The overall observation of our analysis is that localization around points could factually improve the geometry processing in many different applications