arXiv subjects

Meirav Galun

Publications and source records attributed to Meirav Galun.

29 records · Page 2Linked to original sources

arXiv2020

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 $κ$, convergence at a point $\x \in \Sphere^{d-1}$ occurs in time $O(κ^d/p(\x))$ 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.

cs.LG↗

arXiv2020

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.

cs.CV↗

arXiv2019

Learning to Optimize Multigrid PDE Solvers

Constructing fast numerical solvers for partial differential equations (PDEs) is crucial for many scientific disciplines. A leading technique for solving large-scale PDEs is using multigrid methods. At the core of a multigrid solver is the prolongation matrix, which relates between different scales of the problem. This matrix is strongly problem-dependent, and its optimal construction is critical to the efficiency of the solver. In practice, however, devising multigrid algorithms for new problems often poses formidable challenges. In this paper we propose a framework for learning multigrid solvers. Our method learns a (single) mapping from a family of parameterized PDEs to prolongation operators. We train a neural network once for the entire class of PDEs, using an efficient and unsupervised loss function. Experiments on a broad class of 2D diffusion problems demonstrate improved convergence rates compared to the widely used Black-Box multigrid scheme, suggesting that our method successfully learned rules for constructing prolongation matrices.

math.NA↗

arXiv2019

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.

cs.CV↗

arXiv2017

On Detection of Faint Edges in Noisy Images

A fundamental question for edge detection in noisy images is how faint can an edge be and still be detected. In this paper we offer a formalism to study this question and subsequently introduce computationally efficient multiscale edge detection algorithms designed to detect faint edges in noisy images. In our formalism we view edge detection as a search in a discrete, though potentially large, set of feasible curves. First, we derive approximate expressions for the detection threshold as a function of curve length and the complexity of the search space. We then present two edge detection algorithms, one for straight edges, and the second for curved ones. Both algorithms efficiently search for edges in a large set of candidates by hierarchically constructing difference filters that match the curves traced by the sought edges. We demonstrate the utility of our algorithms in both simulations and applications involving challenging real images. Finally, based on these principles, we develop an algorithm for fiber detection and enhancement. We exemplify its utility to reveal and enhance nerve axons in light microscopy images.

cs.CV↗

arXiv2017

A New Rank Constraint on Multi-view Fundamental Matrices, and its Application to Camera Location Recovery

Accurate estimation of camera matrices is an important step in structure from motion algorithms. In this paper we introduce a novel rank constraint on collections of fundamental matrices in multi-view settings. We show that in general, with the selection of proper scale factors, a matrix formed by stacking fundamental matrices between pairs of images has rank 6. Moreover, this matrix forms the symmetric part of a rank 3 matrix whose factors relate directly to the corresponding camera matrices. We use this new characterization to produce better estimations of fundamental matrices by optimizing an L1-cost function using Iterative Re-weighted Least Squares and Alternate Direction Method of Multiplier. We further show that this procedure can improve the recovery of camera locations, particularly in multi-view settings in which fewer images are available.

cs.CV↗

arXiv2015

Wide baseline stereo matching with convex bounded-distortion constraints

Finding correspondences in wide baseline setups is a challenging problem. Existing approaches have focused largely on developing better feature descriptors for correspondence and on accurate recovery of epipolar line constraints. This paper focuses on the challenging problem of finding correspondences once approximate epipolar constraints are given. We introduce a novel method that integrates a deformation model. Specifically, we formulate the problem as finding the largest number of corresponding points related by a bounded distortion map that obeys the given epipolar constraints. We show that, while the set of bounded distortion maps is not convex, the subset of maps that obey the epipolar line constraints is convex, allowing us to introduce an efficient algorithm for matching. We further utilize a robust cost function for matching and employ majorization-minimization for its optimization. Our experiments indicate that our method finds significantly more accurate maps than existing approaches.

cs.CV↗

arXiv2015

Fast Detection of Curved Edges at Low SNR

Detecting edges is a fundamental problem in computer vision with many applications, some involving very noisy images. While most edge detection methods are fast, they perform well only on relatively clean images. Indeed, edges in such images can be reliably detected using only local filters. Detecting faint edges under high levels of noise cannot be done locally at the individual pixel level, and requires more sophisticated global processing. Unfortunately, existing methods that achieve this goal are quite slow. In this paper we develop a novel multiscale method to detect curved edges in noisy images. While our algorithm searches for edges over a huge set of candidate curves, it does so in a practical runtime, nearly linear in the total number of image pixels. As we demonstrate experimentally, our algorithm is orders of magnitude faster than previous methods designed to deal with high noise levels. Nevertheless, it obtains comparable, if not better, edge detection quality on a variety of challenging noisy images.

cs.CV↗

arXiv2012

A Multiscale Framework for Challenging Discrete Optimization

Current state-of-the-art discrete optimization methods struggle behind when it comes to challenging contrast-enhancing discrete energies (i.e., favoring different labels for neighboring variables). This work suggests a multiscale approach for these challenging problems. Deriving an algebraic representation allows us to coarsen any pair-wise energy using any interpolation in a principled algebraic manner. Furthermore, we propose an energy-aware interpolation operator that efficiently exposes the multiscale landscape of the energy yielding an effective coarse-to-fine optimization scheme. Results on challenging contrast-enhancing energies show significant improvement over state-of-the-art methods.

cs.CV↗

arXiv2012

A Unified Multiscale Framework for Discrete Energy Minimization

Discrete energy minimization is a ubiquitous task in computer vision, yet is NP-hard in most cases. In this work we propose a multiscale framework for coping with the NP-hardness of discrete optimization. Our approach utilizes algebraic multiscale principles to efficiently explore the discrete solution space, yielding improved results on challenging, non-submodular energies for which current methods provide unsatisfactory approximations. In contrast to popular multiscale methods in computer vision, that builds an image pyramid, our framework acts directly on the energy to construct an energy pyramid. Deriving a multiscale scheme from the energy itself makes our framework application independent and widely applicable. Our framework gives rise to two complementary energy coarsening strategies: one in which coarser scales involve fewer variables, and a more revolutionary one in which the coarser scales involve fewer discrete labels. We empirically evaluated our unified framework on a variety of both non-submodular and submodular energies, including energies from Middlebury benchmark.

cs.CV↗

arXiv2011

Large Scale Correlation Clustering Optimization

Clustering is a fundamental task in unsupervised learning. The focus of this paper is the Correlation Clustering functional which combines positive and negative affinities between the data points. The contribution of this paper is two fold: (i) Provide a theoretic analysis of the functional. (ii) New optimization algorithms which can cope with large scale problems (>100K variables) that are infeasible using existing methods. Our theoretic analysis provides a probabilistic generative interpretation for the functional, and justifies its intrinsic "model-selection" capability. Furthermore, we draw an analogy between optimizing this functional and the well known Potts energy minimization. This analogy allows us to suggest several new optimization algorithms, which exploit the intrinsic "model-selection" capability of the functional to automatically recover the underlying number of clusters. We compare our algorithms to existing methods on both synthetic and real data. In addition we suggest two new applications that are made possible by our algorithms: unsupervised face identification and interactive multi-object segmentation by rough boundary delineation.

cs.CV↗