OperatorNet: Recovering 3D shapes From Difference Operators
By Ruqi Huang, Marie-Julie Rakotosaona, Panos Achlioptas, Leonidas Guibas and Maks Ovsjanikov.
Abstract: This paper proposes a learning-based framework for reconstructing 3D shapes from functional operators, compactly encoded as small-sized matrices. To this end we introduce a novel neural architecture, called OperatorNet, which takes as input a set of linear operators representing a shape and produces its 3D embedding. We demonstrate that this approach significantly outperforms previous purely geometric methods for the same problem. Furthermore, we introduce a novel functional operator, which encodes the extrinsic or pose-dependent shape information, and thus complements purely intrinsic pose-oblivious operators, such as the classical Laplacian...
Latent Space Representation for Shape Analysis and Learning
By Ruqi Huang, Panos Achlioptas, Leonidas Guibas and Maks Ovsjanikov.
Abstract: We propose a novel shape representation useful for analyzing and processing shape collections, as well for a variety of learning and inference tasks. Unlike most approaches that capture variability in a collection by using a template model or a base shape, we show that it is possible to construct a full shape representation by using the latent space induced by a functional map network, allowing us to represent shapes in the context of a collection without the bias induced by selecting a template shape. Key to our construction is a novel analysis of latent functional spaces, which shows that after proper regularization they can be endowed with a natural geometric structure, giving rise to a well-defined, stable and fully informative shape representation...
Adjoint Map Representation for Shape Analysis and Matching
By Ruqi Huang, Maks Ovsjanikov. To appear in SGP 2017.[code]
Abstract: In this paper, we propose to consider the adjoint operators of functional maps, and demonstrate their utility in several tasks in geometry processing. Unlike a functional map, which represents a correspondence simply using the pull-back of function values, the adjoint operator reflects both the map and its distortion with respect to given inner products. We argue that this property of adjoint operators and especially their relation to the map inverse under the choice of different inner products, can be useful in applications including bi-directional shape matching, shape exploration, and pointwise map recovery among others...
On the Stability of Functional Maps and Shape Difference Operators
By Ruqi Huang, Frédéric Chazal, Maks Ovsjanikov. To appear in 'Computer Graphics Forum'.
Abstract: In this paper, we provide stability guarantees for two frameworks that are based on the notion of functional maps – the shape difference operators introduced in [ROA*13] and the framework of [OBCCG13] which is used to analyze and visualize the deformations between shapes induced by a functional map. We consider two types of perturbations in our analysis: one is on the input shapes and the other is on the change in scale. In theory, we formulate and justify the robustness that has been observed in practical implementations of those frameworks...
Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph
By Frédéric Chazal, Ruqi Huang, Jian Sun. 'Discrete and Computational Geometry', 2015.
Abstract: In many real-world applications data appear to be sampled around 1-dimensional filamentary structures that can be seen as topological metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and provide an efficient and easy to implement algorithm to compute such approximations in almost linear time. We illustrate the performances of our algorithm on a few data sets.
Two contributions to geometric data analysis: filamentary structures approximations, and stability properties of functional approaches for shape comparison.
Thesis by Ruqi Huang. Defended in Dec 14, 2016.
Abstract: Massive amounts of data are being generated, collected and processed all the time. A considerable portion of them are sampled from objects with geometric structures. Such objects can be tangible and ubiquitous in our daily life. Inferring the geometric information from the data, however, is not always an obvious task. Moreover, it’s not a rare case that the underlying objects are abstract and of high dimension, where the data inference is more challenging...