arXiv subjects

Yaron Lipman

Publications and source records attributed to Yaron Lipman.

At least 55 records · Page 3Linked to original sources

arXiv2020

Multiview Neural Surface Reconstruction by Disentangling Geometry and Appearance

In this work we address the challenging problem of multiview 3D surface reconstruction. We introduce a neural network architecture that simultaneously learns the unknown geometry, camera parameters, and a neural renderer that approximates the light reflected from the surface towards the camera. The geometry is represented as a zero level-set of a neural network, while the neural renderer, derived from the rendering equation, is capable of (implicitly) modeling a wide set of lighting conditions and materials. We trained our network on real world 2D images of objects with different material properties, lighting conditions, and noisy camera initializations from the DTU MVS dataset. We found our model to produce state of the art 3D surface reconstructions with high fidelity, resolution and detail.

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arXiv2020

SALD: Sign Agnostic Learning with Derivatives

Learning 3D geometry directly from raw data, such as point clouds, triangle soups, or unoriented meshes is still a challenging task that feeds many downstream computer vision and graphics applications. In this paper, we introduce SALD: a method for learning implicit neural representations of shapes directly from raw data. We generalize sign agnostic learning (SAL) to include derivatives: given an unsigned distance function to the input raw data, we advocate a novel sign agnostic regression loss, incorporating both pointwise values and gradients of the unsigned distance function. Optimizing this loss leads to a signed implicit function solution, the zero level set of which is a high quality and valid manifold approximation to the input 3D data. The motivation behind SALD is that incorporating derivatives in a regression loss leads to a lower sample complexity, and consequently better fitting. In addition, we prove that SAL enjoys a minimal length property in 2D, favoring minimal length solutions. More importantly, we are able to show that this property still holds for SALD, i.e., with derivatives included. We demonstrate the efficacy of SALD for shape space learning on two challenging datasets: ShapeNet that contains inconsistent orientation and non-manifold meshes, and D-Faust that contains raw 3D scans (triangle soups). On both these datasets, we present state-of-the-art results.

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arXiv2020

Isometric Autoencoders

High dimensional data is often assumed to be concentrated on or near a low-dimensional manifold. Autoencoders (AE) is a popular technique to learn representations of such data by pushing it through a neural network with a low dimension bottleneck while minimizing a reconstruction error. Using high capacity AE often leads to a large collection of minimizers, many of which represent a low dimensional manifold that fits the data well but generalizes poorly. Two sources of bad generalization are: extrinsic, where the learned manifold possesses extraneous parts that are far from the data; and intrinsic, where the encoder and decoder introduce arbitrary distortion in the low dimensional parameterization. An approach taken to alleviate these issues is to add a regularizer that favors a particular solution; common regularizers promote sparsity, small derivatives, or robustness to noise. In this paper, we advocate an isometry (i.e., local distance preserving) regularizer. Specifically, our regularizer encourages: (i) the decoder to be an isometry; and (ii) the encoder to be the decoder's pseudo-inverse, that is, the encoder extends the inverse of the decoder to the ambient space by orthogonal projection. In a nutshell, (i) and (ii) fix both intrinsic and extrinsic degrees of freedom and provide a non-linear generalization to principal component analysis (PCA). Experimenting with the isometry regularizer on dimensionality reduction tasks produces useful low-dimensional data representations.

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arXiv2020

Implicit Geometric Regularization for Learning Shapes

Representing shapes as level sets of neural networks has been recently proved to be useful for different shape analysis and reconstruction tasks. So far, such representations were computed using either: (i) pre-computed implicit shape representations; or (ii) loss functions explicitly defined over the neural level sets. In this paper we offer a new paradigm for computing high fidelity implicit neural representations directly from raw data (i.e., point clouds, with or without normal information). We observe that a rather simple loss function, encouraging the neural network to vanish on the input point cloud and to have a unit norm gradient, possesses an implicit geometric regularization property that favors smooth and natural zero level set surfaces, avoiding bad zero-loss solutions. We provide a theoretical analysis of this property for the linear case, and show that, in practice, our method leads to state of the art implicit neural representations with higher level-of-details and fidelity compared to previous methods.

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arXiv2020

Provably Powerful Graph Networks

Recently, the Weisfeiler-Lehman (WL) graph isomorphism test was used to measure the expressive power of graph neural networks (GNN). It was shown that the popular message passing GNN cannot distinguish between graphs that are indistinguishable by the 1-WL test (Morris et al. 2018; Xu et al. 2019). Unfortunately, many simple instances of graphs are indistinguishable by the 1-WL test. In search for more expressive graph learning models we build upon the recent k-order invariant and equivariant graph neural networks (Maron et al. 2019a,b) and present two results: First, we show that such k-order networks can distinguish between non-isomorphic graphs as good as the k-WL tests, which are provably stronger than the 1-WL test for k>2. This makes these models strictly stronger than message passing models. Unfortunately, the higher expressiveness of these models comes with a computational cost of processing high order tensors. Second, setting our goal at building a provably stronger, simple and scalable model we show that a reduced 2-order network containing just scaled identity operator, augmented with a single quadratic operation (matrix multiplication) has a provable 3-WL expressive power. Differently put, we suggest a simple model that interleaves applications of standard Multilayer-Perceptron (MLP) applied to the feature dimension and matrix multiplication. We validate this model by presenting state of the art results on popular graph classification and regression tasks. To the best of our knowledge, this is the first practical invariant/equivariant model with guaranteed 3-WL expressiveness, strictly stronger than message passing models.

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arXiv2020

SAL: Sign Agnostic Learning of Shapes from Raw Data

Recently, neural networks have been used as implicit representations for surface reconstruction, modelling, learning, and generation. So far, training neural networks to be implicit representations of surfaces required training data sampled from a ground-truth signed implicit functions such as signed distance or occupancy functions, which are notoriously hard to compute. In this paper we introduce Sign Agnostic Learning (SAL), a deep learning approach for learning implicit shape representations directly from raw, unsigned geometric data, such as point clouds and triangle soups. We have tested SAL on the challenging problem of surface reconstruction from an un-oriented point cloud, as well as end-to-end human shape space learning directly from raw scans dataset, and achieved state of the art reconstructions compared to current approaches. We believe SAL opens the door to many geometric deep learning applications with real-world data, alleviating the usual painstaking, often manual pre-process.

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arXiv2020

On Universal Equivariant Set Networks

Using deep neural networks that are either invariant or equivariant to permutations in order to learn functions on unordered sets has become prevalent. The most popular, basic models are DeepSets [Zaheer et al. 2017] and PointNet [Qi et al. 2017]. While known to be universal for approximating invariant functions, DeepSets and PointNet are not known to be universal when approximating \emph{equivariant} set functions. On the other hand, several recent equivariant set architectures have been proven equivariant universal [Sannai et al. 2019], [Keriven et al. 2019], however these models either use layers that are not permutation equivariant (in the standard sense) and/or use higher order tensor variables which are less practical. There is, therefore, a gap in understanding the universality of popular equivariant set models versus theoretical ones. In this paper we close this gap by proving that: (i) PointNet is not equivariant universal; and (ii) adding a single linear transmission layer makes PointNet universal. We call this architecture PointNetST and argue it is the simplest permutation equivariant universal model known to date. Another consequence is that DeepSets is universal, and also PointNetSeg, a popular point cloud segmentation network (used eg, in [Qi et al. 2017]) is universal. The key theoretical tool used to prove the above results is an explicit characterization of all permutation equivariant polynomial layers. Lastly, we provide numerical experiments validating the theoretical results and comparing different permutation equivariant models.

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arXiv2019

Controlling Neural Level Sets

The level sets of neural networks represent fundamental properties such as decision boundaries of classifiers and are used to model non-linear manifold data such as curves and surfaces. Thus, methods for controlling the neural level sets could find many applications in machine learning. In this paper we present a simple and scalable approach to directly control level sets of a deep neural network. Our method consists of two parts: (i) sampling of the neural level sets, and (ii) relating the samples' positions to the network parameters. The latter is achieved by a sample network that is constructed by adding a single fixed linear layer to the original network. In turn, the sample network can be used to incorporate the level set samples into a loss function of interest. We have tested our method on three different learning tasks: improving generalization to unseen data, training networks robust to adversarial attacks, and curve and surface reconstruction from point clouds. For surface reconstruction, we produce high fidelity surfaces directly from raw 3D point clouds. When training small to medium networks to be robust to adversarial attacks we obtain robust accuracy comparable to state-of-the-art methods.

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arXiv2019

Surface Networks via General Covers

Developing deep learning techniques for geometric data is an active and fruitful research area. This paper tackles the problem of sphere-type surface learning by developing a novel surface-to-image representation. Using this representation we are able to quickly adapt successful CNN models to the surface setting. The surface-image representation is based on a covering map from the image domain to the surface. Namely, the map wraps around the surface several times, making sure that every part of the surface is well represented in the image. Differently from previous surface-to-image representations, we provide a low distortion coverage of all surface parts in a single image. Specifically, for the use case of learning spherical signals, our representation provides a low distortion alternative to several popular spherical parameterizations used in deep learning. We have used the surface-to-image representation to apply standard CNN architectures to 3D models as well as spherical signals. We show that our method achieves state of the art or comparable results on the tasks of shape retrieval, shape classification and semantic shape segmentation.

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arXiv2019

On the Universality of Invariant Networks

Constraining linear layers in neural networks to respect symmetry transformations from a group $G$ is a common design principle for invariant networks that has found many applications in machine learning. In this paper, we consider a fundamental question that has received little attention to date: Can these networks approximate any (continuous) invariant function? We tackle the rather general case where $G\leq S_n$ (an arbitrary subgroup of the symmetric group) that acts on $\mathbb{R}^n$ by permuting coordinates. This setting includes several recent popular invariant networks. We present two main results: First, $G$-invariant networks are universal if high-order tensors are allowed. Second, there are groups $G$ for which higher-order tensors are unavoidable for obtaining universality. $G$-invariant networks consisting of only first-order tensors are of special interest due to their practical value. We conclude the paper by proving a necessary condition for the universality of $G$-invariant networks that incorporate only first-order tensors.

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arXiv2019

Invariant and Equivariant Graph Networks

Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular transformations, at-least), a full characterization of invariant and equivariant linear layers for graphs is not known. In this paper we provide a characterization of all permutation invariant and equivariant linear layers for (hyper-)graph data, and show that their dimension, in case of edge-value graph data, is 2 and 15, respectively. More generally, for graph data defined on k-tuples of nodes, the dimension is the k-th and 2k-th Bell numbers. Orthogonal bases for the layers are computed, including generalization to multi-graph data. The constant number of basis elements and their characteristics allow successfully applying the networks to different size graphs. From the theoretical point of view, our results generalize and unify recent advancement in equivariant deep learning. In particular, we show that our model is capable of approximating any message passing neural network Applying these new linear layers in a simple deep neural network framework is shown to achieve comparable results to state-of-the-art and to have better expressivity than previous invariant and equivariant bases.

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arXiv2019

Multi-chart Generative Surface Modeling

This paper introduces a 3D shape generative model based on deep neural networks. A new image-like (i.e., tensor) data representation for genus-zero 3D shapes is devised. It is based on the observation that complicated shapes can be well represented by multiple parameterizations (charts), each focusing on a different part of the shape. The new tensor data representation is used as input to Generative Adversarial Networks for the task of 3D shape generation. The 3D shape tensor representation is based on a multi-chart structure that enjoys a shape covering property and scale-translation rigidity. Scale-translation rigidity facilitates high quality 3D shape learning and guarantees unique reconstruction. The multi-chart structure uses as input a dataset of 3D shapes (with arbitrary connectivity) and a sparse correspondence between them. The output of our algorithm is a generative model that learns the shape distribution and is able to generate novel shapes, interpolate shapes, and explore the generated shape space. The effectiveness of the method is demonstrated for the task of anatomic shape generation including human body and bone (teeth) shape generation.

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arXiv2018

(Probably) Concave Graph Matching

In this paper we address the graph matching problem. Following the recent works of \cite{zaslavskiy2009path,Vestner2017} we analyze and generalize the idea of concave relaxations. We introduce the concepts of conditionally concave and probably conditionally concave energies on polytopes and show that they encapsulate many instances of the graph matching problem, including matching Euclidean graphs and graphs on surfaces. We further prove that local minima of probably conditionally concave energies on general matching polytopes (e.g., doubly stochastic) are with high probability extreme points of the matching polytope (e.g., permutations).

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arXiv2018

Sinkhorn Algorithm for Lifted Assignment Problems

Recently, Sinkhorn's algorithm was applied for approximately solving linear programs emerging from optimal transport very efficiently. This was accomplished by formulating a regularized version of the linear program as Bregman projection problem onto the polytope of doubly-stochastic matrices, and then computing the projection using the efficient Sinkhorn algorithm, which is based on alternating closed-form Bregman projections on the larger polytopes of row-stochastic and column-stochastic matrices. In this paper we suggest a generalization of this algorithm for solving a well-known lifted linear program relaxations of the Quadratic Assignment Problem (QAP), which is known as the Johnson Adams (JA) Relaxation. First, an efficient algorithm for Bregman projection onto the JA polytope by alternating closed-form Bregman projections onto one-sided local polytopes is devised. The one-sided polytopes can be seen as a high-dimensional, generalized version of the row/column-stochastic polytopes. Second, a new method for solving the original linear programs using the Bregman projections onto the JA polytope is developed and shown to be more accurate and numerically stable than the standard approach of driving the regularizer to zero. The resulting algorithm is considerably more scalable than standard linear solvers and is able to solve significantly larger linear programs.

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arXiv2018

Point Convolutional Neural Networks by Extension Operators

This paper presents Point Convolutional Neural Networks (PCNN): a novel framework for applying convolutional neural networks to point clouds. The framework consists of two operators: extension and restriction, mapping point cloud functions to volumetric functions and vise-versa. A point cloud convolution is defined by pull-back of the Euclidean volumetric convolution via an extension-restriction mechanism. The point cloud convolution is computationally efficient, invariant to the order of points in the point cloud, robust to different samplings and varying densities, and translation invariant, that is the same convolution kernel is used at all points. PCNN generalizes image CNNs and allows readily adapting their architectures to the point cloud setting. Evaluation of PCNN on three central point cloud learning benchmarks convincingly outperform competing point cloud learning methods, and the vast majority of methods working with more informative shape representations such as surfaces and/or normals.

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arXiv2018

A Linear Variational Principle for Riemann Mappings and Discrete Conformality

We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in $H^1$, even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge uniformly and are known to be bijective. As a consequence we show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the Riemann mappings between each Lipschitz domain and the triangle.

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arXiv2017

Exact Recovery with Symmetries for Procrustes Matching

The Procrustes matching (PM) problem is the problem of finding the optimal rigid motion and labeling of two point sets so that they are as close as possible. Both rigid and non-rigid shape matching problems can be formulated as PM problems. Recently [Maron et al.] presented a novel convex semi-definite programming relaxation (PM-SDP) for PM which achieves state of the art results on common shape matching benchmarks. In this paper we analyze the successfulness of PM-SDP in solving PM problems without noise (Exact PM problems). We begin by showing Exact PM to be computationally equivalent to the graph isomorphism problem. We demonstrate some natural theoretical properties of the relaxation, and use these properties together with the moment interpretation of [Lasserre] to show that for exact PM problems and for (generic) input shapes which are asymmetric or bilaterally symmetric, the relaxation returns a correct solution of PM. For symmetric shapes, PM has multiple solutions. The non-convex set of optimal solutions of PM is strictly contained in the convex set of optimal solutions of PM-SDP, so that `most' solutions of PM-SDP will not be solutions of PM. We deal with this by showing the solution set of PM to be the extreme points of the solution set of PM-SDP, and suggesting a random algorithm which returns a solution of PM with probability one, and returns all solutions of PM with equal probability. We also show these results can be extended to the almost-exact case. To the best of our knowledge, our work is the first to achieve exact recovery in the presence of multiple solutions.

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arXiv2017

Photometric Stereo by Hemispherical Metric Embedding

Photometric Stereo methods seek to reconstruct the 3d shape of an object from motionless images obtained with varying illumination. Most existing methods solve a restricted problem where the physical reflectance model, such as Lambertian reflectance, is known in advance. In contrast, we do not restrict ourselves to a specific reflectance model. Instead, we offer a method that works on a wide variety of reflectances. Our approach uses a simple yet uncommonly used property of the problem - the sought after normals are points on a unit hemisphere. We present a novel embedding method that maps pixels to normals on the unit hemisphere. Our experiments demonstrate that this approach outperforms existing manifold learning methods for the task of hemisphere embedding. We further show successful reconstructions of objects from a wide variety of reflectances including smooth, rough, diffuse and specular surfaces, even in the presence of significant attached shadows. Finally, we empirically prove that under these challenging settings we obtain more accurate shape reconstructions than existing methods.

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