Gautam Pai

Ramana Sundararaman, Gautam Pai, Maks Ovsjanikov

Establishing a correspondence between two non-rigidly deforming shapes is one of the most fundamental problems in visual computing. Existing methods often show weak resilience when presented with challenges innate to real-world data such as noise, outliers, self-occlusion etc. On the other hand, auto-decoders have demonstrated strong expressive power in learning geometrically meaningful latent embeddings. However, their use in \emph{shape analysis} has been limited. In this paper, we introduce an approach based on an auto-decoder framework, that learns a continuous shape-wise deformation field over a fixed template. By supervising the deformation field for points on-surface and regularising for points off-surface through a novel \emph{Signed Distance Regularisation} (SDR), we learn an alignment between the template and shape \emph{volumes}. Trained on clean water-tight meshes, \emph{without} any data-augmentation, we demonstrate compelling performance on compromised data and real-world scans.

Gijs Bellaard, Daan L. J. Bon, Gautam Pai et al.

Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalises G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously 1) reduce network complexity, 2) increase classification performance, and 3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry.

Souhaib Attaiki, Gautam Pai, Maks Ovsjanikov

We consider the problem of computing dense correspondences between non-rigid shapes with potentially significant partiality. Existing formulations tackle this problem through heavy manifold optimization in the spectral domain, given hand-crafted shape descriptors. In this paper, we propose the first learning method aimed directly at partial non-rigid shape correspondence. Our approach uses the functional map framework, can be trained in a supervised or unsupervised manner, and learns descriptors directly from the data, thus both improving robustness and accuracy in challenging cases. Furthermore, unlike existing techniques, our method is also applicable to partial-to-partial non-rigid matching, in which the common regions on both shapes are unknown a priori. We demonstrate that the resulting method is data-efficient, and achieves state-of-the-art results on several benchmark datasets. Our code and data can be found online: https://github.com/pvnieo/DPFM

Gautam Pai, Mor Joseph-Rivlin, Ron Kimmel

A majority of shape correspondence frameworks are based on devising pointwise and pairwise constraints on the correspondence map. The functional maps framework allows for formulating these constraints in the spectral domain. In this paper, we develop a functional map framework for the shape correspondence problem by constructing pairwise constraints using point-wise descriptors. Our core observation is that, every point-wise descriptor allows for the construction a pairwise kernel operator whose low frequency eigenfunctions depict regions of similar descriptor values at various scales of frequency. By aggregating the pairwise information from the descriptor and the intrinsic geometry of the surface encoded in the heat kernel, we construct a hybrid kernel and call it the bilateral operator. Analogous to the edge preserving bilateral filter in image processing, the action of the bilateral operator on a function defined over the manifold yields a descriptor dependent local smoothing of that function. By forcing the correspondence map to commute with the Bilateral operator, we show that we can maximally exploit the information from a given set of pointwise descriptors in a functional map framework.

Moshe Lichtenstein, Gautam Pai, Ron Kimmel

A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update scheme. We replace the formulaic local numerical solver with a trained neural network to provide highly accurate estimates of local distances for a variety of different geometries and sampling conditions. Our learning approach generalizes not only to flat Euclidean domains but also to curved surfaces enabled by the incorporation of certain invariant features in the neural network architecture. We show a considerable gain in performance, validated by smaller errors and higher orders of accuracy for the numerical solutions of the Eikonal equation computed on different surfaces The proposed approach leverages the approximation power of neural networks to enhance the performance of numerical algorithms, thereby, connecting the somewhat disparate themes of numerical geometry and learning.

Gautam Pai, Ronen Talmon, Alex Bronstein et al.

This paper explores a fully unsupervised deep learning approach for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We use the Siamese configuration to train a neural network to solve the problem of least squares multidimensional scaling for generating maps that approximately preserve geodesic distances. By training with only a few landmarks, we show a significantly improved local and nonlocal generalization of the isometric mapping as compared to analogous non-parametric counterparts. Importantly, the combination of a deep-learning framework with a multidimensional scaling objective enables a numerical analysis of network architectures to aid in understanding their representation power. This provides a geometric perspective to the generalizability of deep learning.

Yoni Choukroun, Gautam Pai, Ron Kimmel

The discrete Laplace operator is ubiquitous in spectral shape analysis, since its eigenfunctions are provably optimal in representing smooth functions defined on the surface of the shape. Indeed, subspaces defined by its eigenfunctions have been utilized for shape compression, treating the coordinates as smooth functions defined on the given surface. However, surfaces of shapes in nature often contain geometric structures for which the general smoothness assumption may fail to hold. At the other end, some explicit mesh compression algorithms utilize the order by which vertices that represent the surface are traversed, a property which has been ignored in spectral approaches. Here, we incorporate the order of vertices into an operator that defines a novel spectral domain. We propose a method for representing 3D meshes using the spectral geometry of the Hamiltonian operator, integrated within a sparse approximation framework. We adapt the concept of a potential function from quantum physics and incorporate vertex ordering information into the potential, yielding a novel data-dependent operator. The potential function modifies the spectral geometry of the Laplacian to focus on regions with finer details of the given surface. By sparsely encoding the geometry of the shape using the proposed data-dependent basis, we improve compression performance compared to previous results that use the standard Laplacian basis and spectral graph wavelets.

Gautam Pai, Aaron Wetzler, Ron Kimmel

We propose a metric learning framework for the construction of invariant geometric functions of planar curves for the Eucledian and Similarity group of transformations. We leverage on the representational power of convolutional neural networks to compute these geometric quantities. In comparison with axiomatic constructions, we show that the invariants approximated by the learning architectures have better numerical qualities such as robustness to noise, resiliency to sampling, as well as the ability to adapt to occlusion and partiality. Finally, we develop a novel multi-scale representation in a similarity metric learning paradigm.