Layered Neural Atlases for Consistent Video Editing

We present a method that decomposes, or "unwraps", an input video into a set of layered 2D atlases, each providing a unified representation of the appearance of an object (or background) over the video. For each pixel in the video, our method estimates its corresponding 2D coordinate in each of the atlases, giving us a consistent parameterization of the video, along with an associated alpha (opacity) value. Importantly, we design our atlases to be interpretable and semantic, which facilitates easy and intuitive editing in the atlas domain, with minimal manual work required. Edits applied to a single 2D atlas (or input video frame) are automatically and consistently mapped back to the original video frames, while preserving occlusions, deformation, and other complex scene effects such as shadows and reflections. Our method employs a coordinate-based Multilayer Perceptron (MLP) representation for mappings, atlases, and alphas, which are jointly optimized on a per-video basis, using a combination of video reconstruction and regularization losses. By operating purely in 2D, our method does not require any prior 3D knowledge about scene geometry or camera poses, and can handle complex dynamic real world videos. We demonstrate various video editing applications, including texture mapping, video style transfer, image-to-video texture transfer, and segmentation/labeling propagation, all automatically produced by editing a single 2D atlas image.

Volume Rendering of Neural Implicit Surfaces

Neural volume rendering became increasingly popular recently due to its success in synthesizing novel views of a scene from a sparse set of input images. So far, the geometry learned by neural volume rendering techniques was modeled using a generic density function. Furthermore, the geometry itself was extracted using an arbitrary level set of the density function leading to a noisy, often low fidelity reconstruction. The goal of this paper is to improve geometry representation and reconstruction in neural volume rendering. We achieve that by modeling the volume density as a function of the geometry. This is in contrast to previous work modeling the geometry as a function of the volume density. In more detail, we define the volume density function as Laplace's cumulative distribution function (CDF) applied to a signed distance function (SDF) representation. This simple density representation has three benefits: (i) it provides a useful inductive bias to the geometry learned in the neural volume rendering process; (ii) it facilitates a bound on the opacity approximation error, leading to an accurate sampling of the viewing ray. Accurate sampling is important to provide a precise coupling of geometry and radiance; and (iii) it allows efficient unsupervised disentanglement of shape and appearance in volume rendering. Applying this new density representation to challenging scene multiview datasets produced high quality geometry reconstructions, outperforming relevant baselines. Furthermore, switching shape and appearance between scenes is possible due to the disentanglement of the two.

Deep Permutation Equivariant Structure from Motion

Existing deep methods produce highly accurate 3D reconstructions in stereo and multiview stereo settings, i.e., when cameras are both internally and externally calibrated. Nevertheless, the challenge of simultaneous recovery of camera poses and 3D scene structure in multiview settings with deep networks is still outstanding. Inspired by projective factorization for Structure from Motion (SFM) and by deep matrix completion techniques, we propose a neural network architecture that, given a set of point tracks in multiple images of a static scene, recovers both the camera parameters and a (sparse) scene structure by minimizing an unsupervised reprojection loss. Our network architecture is designed to respect the structure of the problem: the sought output is equivariant to permutations of both cameras and scene points. Notably, our method does not require initialization of camera parameters or 3D point locations. We test our architecture in two setups: (1) single scene reconstruction and (2) learning from multiple scenes. Our experiments, conducted on a variety of datasets in both internally calibrated and uncalibrated settings, indicate that our method accurately recovers pose and structure, on par with classical state of the art methods. Additionally, we show that a pre-trained network can be used to reconstruct novel scenes using inexpensive fine-tuning with no loss of accuracy.

On the Similarity between the Laplace and Neural Tangent Kernels

Recent theoretical work has shown that massively overparameterized neural networks are equivalent to kernel regressors that use Neural Tangent Kernels(NTK). Experiments show that these kernel methods perform similarly to real neural networks. Here we show that NTK for fully connected networks is closely related to the standard Laplace kernel. We show theoretically that for normalized data on the hypersphere both kernels have the same eigenfunctions and their eigenvalues decay polynomially at the same rate, implying that their Reproducing Kernel Hilbert Spaces (RKHS) include the same sets of functions. This means that both kernels give rise to classes of functions with the same smoothness properties. The two kernels differ for data off the hypersphere, but experiments indicate that when data is properly normalized these differences are not significant. Finally, we provide experiments on real data comparing NTK and the Laplace kernel, along with a larger class of{\gamma}-exponential kernels. We show that these perform almost identically. Our results suggest that much insight about neural networks can be obtained from analysis of the well-known Laplace kernel, which has a simple closed-form.

End-To-End Convolutional Neural Network for 3D Reconstruction of Knee Bones From Bi-Planar X-Ray Images

We present an end-to-end Convolutional Neural Network (CNN) approach for 3D reconstruction of knee bones directly from two bi-planar X-ray images. Clinically, capturing the 3D models of the bones is crucial for surgical planning, implant fitting, and postoperative evaluation. X-ray imaging significantly reduces the exposure of patients to ionizing radiation compared to Computer Tomography (CT) imaging, and is much more common and inexpensive compared to Magnetic Resonance Imaging (MRI) scanners. However, retrieving 3D models from such 2D scans is extremely challenging. In contrast to the common approach of statistically modeling the shape of each bone, our deep network learns the distribution of the bones' shapes directly from the training images. We train our model with both supervised and unsupervised losses using Digitally Reconstructed Radiograph (DRR) images generated from CT scans. To apply our model to X-Ray data, we use style transfer to transform between X-Ray and DRR modalities. As a result, at test time, without further optimization, our solution directly outputs a 3D reconstruction from a pair of bi-planar X-ray images, while preserving geometric constraints. Our results indicate that our deep learning model is very efficient, generalizes well and produces high quality reconstructions.

Frequency Bias in Neural Networks for Input of Non-Uniform Density

Recent works have partly attributed the generalization ability of over-parameterized neural networks to frequency bias -- networks trained with gradient descent on data drawn from a uniform distribution find a low frequency fit before high frequency ones. As realistic training sets are not drawn from a uniform distribution, we here use the Neural Tangent Kernel (NTK) model to explore the effect of variable density on training dynamics. Our results, which combine analytic and empirical observations, show that when learning a pure harmonic function of frequency $\kappa$, convergence at a point $\x \in \Sphere^{d-1}$ occurs in time $O(\kappa^d/p(\x))$ where $p(\x)$ denotes the local density at $\x$. Specifically, for data in $\Sphere^1$ we analytically derive the eigenfunctions of the kernel associated with the NTK for two-layer networks. We further prove convergence results for deep, fully connected networks with respect to the spectral decomposition of the NTK. Our empirical study highlights similarities and differences between deep and shallow networks in this model.

Averaging Essential and Fundamental Matrices in Collinear Camera Settings

Global methods to Structure from Motion have gained popularity in recent years. A significant drawback of global methods is their sensitivity to collinear camera settings. In this paper, we introduce an analysis and algorithms for averaging bifocal tensors (essential or fundamental matrices) when either subsets or all of the camera centers are collinear. We provide a complete spectral characterization of bifocal tensors in collinear scenarios and further propose two averaging algorithms. The first algorithm uses rank constrained minimization to recover camera matrices in fully collinear settings. The second algorithm enriches the set of possibly mixed collinear and non-collinear cameras with additional, "virtual cameras," which are placed in general position, enabling the application of existing averaging methods to the enriched set of bifocal tensors. Our algorithms are shown to achieve state of the art results on various benchmarks that include autonomous car datasets and unordered image collections in both calibrated and unclibrated settings.

The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies

We study the relationship between the speed at which a neural network learns a function and the frequency of the function. We build on recent results that show that the dynamics of overparameterized neural networks trained with gradient descent can be well approximated by a linear system. When normalized training data is uniformly distributed on a hypersphere, the eigenfunctions of this linear system are spherical harmonic functions. We derive the corresponding eigenvalues for each frequency after introducing a bias term in the model. This bias term had been omitted from the linear network model without significantly affecting previous theoretical results. However, we show theoretically and experimentally that a shallow neural network without bias cannot learn simple, low frequency functions with odd frequencies, in the limit of large amounts of data. Our results enable us to make specific predictions of the time it will take a network with bias to learn functions of varying frequency. These predictions match the behavior of real shallow and deep networks.

Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings

Essential matrix averaging, i.e., the task of recovering camera locations and orientations in calibrated, multiview settings, is a first step in global approaches to Euclidean structure from motion. A common approach to essential matrix averaging is to separately solve for camera orientations and subsequently for camera positions. This paper presents a novel approach that solves simultaneously for both camera orientations and positions. We offer a complete characterization of the algebraic conditions that enable a unique Euclidean reconstruction of $n$ cameras from a collection of $(^n_2)$ essential matrices. We next use these conditions to formulate essential matrix averaging as a constrained optimization problem, allowing us to recover a consistent set of essential matrices given a (possibly partial) set of measured essential matrices computed independently for pairs of images. We finally use the recovered essential matrices to determine the global positions and orientations of the $n$ cameras. We test our method on common SfM datasets, demonstrating high accuracy while maintaining efficiency and robustness, compared to existing methods.

Resultant Based Incremental Recovery of Camera Pose from Pairwise Matches

Incremental (online) structure from motion pipelines seek to recover the camera matrix associated with an image $I_n$ given $n-1$ images, $I_1,...,I_{n-1}$, whose camera matrices have already been recovered. In this paper, we introduce a novel solution to the six-point online algorithm to recover the exterior parameters associated with $I_n$. Our algorithm uses just six corresponding pairs of 2D points, extracted each from $I_n$ and from \textit{any} of the preceding $n-1$ images, allowing the recovery of the full six degrees of freedom of the $n$'th camera, and unlike common methods, does not require tracking feature points in three or more images. Our novel solution is based on constructing a Dixon resultant, yielding a solution method that is both efficient and accurate compared to existing solutions. We further use Bernstein's theorem to prove a tight bound on the number of complex solutions. Our experiments demonstrate the utility of our approach.