The multiview variety of an arrangement of cameras is the Zariski closure of the images of world points in the cameras. The prime vanishing ideal of this complex projective variety is called the multiview ideal. We show that the bifocal and trifocal polynomials from the cameras generate the multiview ideal when the foci are distinct. In the computer vision literature, many sets of (determinantal) polynomials have been proposed to describe the multiview variety. We establish precise algebraic relationships between the multiview ideal and these various ideals. When the camera foci are noncoplanar, we prove that the ideal of bifocal polynomials saturate to give the multiview ideal. Finally, we prove that all the ideals we consider coincide when dehomogenized, to cut out the space of finite images.

We present Jump, a practical system for capturing high resolution, omnidirectional stereo (ODS) video suitable for wide scale consumption in currently available virtual reality (VR) headsets. Our system consists of a video camera built using off-the-shelf components and a fully automatic stitching pipeline capable of capturing video content in the ODS format. We have discovered and analyzed the distortions inherent to ODS when used for VR display as well as those introduced by our capture method and show that they are small enough to make this approach suitable for capturing a wide variety of scenes. Our stitching algorithm produces robust results by reducing the problem to one of pairwise image interpolation followed by compositing. We introduce novel optical flow and compositing methods designed specifically for this task. Our algorithm is temporally coherent and efficient, is currently running at scale on a distributed computing platform, and is capable of processing hours of footage each day.

We present Visibility Based Preconditioning (VBP) a new technique for efficiently solving the linear least squares problems that arise in bundle adjustment (Triggs et al., 1999). Using the camera-point visibility structure of the scene, we describe the construction of two preconditioners. These preconditioners when combined with an inexact step Levenberg-Marquardt algorithm (Wright and Holt, 1985) offer state of the art performance on the BAL data set (Agarwal et al., 2010), with 3-5× reduction in execution time over currently available methods while delivering comparable or better solution quality.

We propose a novel framework for reconstructing homogenous, transparent, refractive height-fields from a single viewpoint. The height-field is imaged against a known planar background, or sequence of backgrounds. Unlike existing approaches that do a point-by-point reconstruction - which is known to have intractable ambiguities - our method estimates and optimizes for the entire height-field at the same time. The formulation supports shape recovery from measured distortions (deflections) or directly from the images themselves, including from a single image. We report results for a variety of refractive height-fields showing significant improvement over prior art.

This paper introduces a schematic representation for architectural scenes together with robust algorithms for reconstruction from sparse 3D point cloud data. The schematic models architecture as a network of transport curves, approximating a floorplan, with associated profile curves, together comprising an interconnected set of swept surfaces. The representation is extremely concise, composed of a handful of planar curves, and easily interpretable by humans. The approach also provides a principled mechanism for interpolating a dense surface, and enables filling in holes in the data, by means of a pipeline that employs a global optimization over all parameters. By incorporating a displacement map on top of the schematic surface, it is possible to recover fine details. Experiments show the ability to reconstruct extremely clean and simple models from sparse structure-from-motion point clouds of complex architectural scenes.

We present the design and implementation of new inexact Newton type Bundle Adjustment algorithms that exploit hardware parallelism for efficiently solving large scale 3D scene reconstruction problems. We explore the use of multicore CPU as well as multicore GPUs for this purpose. We show that overcoming the severe memory and bandwidth limitations of current generation GPUs not only leads to more space efficient algorithms, but also to surprising savings in runtime. Our CPU based system is up to ten times and our GPU based system is up to thirty times faster than the current state of the art methods, while maintaining comparable convergence behavior. The code and additional results are available at http://grail.cs.washington.edu/projects/mcba.

We present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality. Given a projective reconstruction, we first upgrade it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step. For each stage, we construct and minimize tight convex relaxations of the highly non-convex objective functions in a branch and bound optimization framework. We exploit the inherent problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. Chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum. Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data.

We present a system that can match and reconstruct 3D scenes from extremely large collections of photographs such as those found by searching for a given city (e.g., Rome) on Internet photo sharing sites. Our system uses a collection of novel parallel distributed matching and reconstruction algorithms, designed to maximize parallelism at each stage in the pipeline and minimize serialization bottlenecks. It is designed to scale gracefully with both the size of the problem and the amount of available computation. We have experimented with a variety of alternative algorithms at each stage of the pipeline and report on which ones work best in a parallel computing environment. Our experimental results demonstrate that it is now possible to reconstruct cities consisting of 150 K images in less than a day on a cluster with 500 compute cores.

We design and implement a comprehensive study of the perception of gloss. This is the largest study of its kind to date, and the first to use real material measurements. In addition, we develop a novel multi-dimensional scaling (MDS) algorithm for analyzing pairwise comparisons. The data from the psychophysics study and the MDS algorithm is used to construct a low dimensional perceptual embedding of these bidirectional reflectance distribution functions (BRDFs). The embedding is validated by correlating it with nine gloss dimensions, fitted parameters of seven analytical BRDF models, and a perceptual parameterization of Ward's model. We also introduce a novel perceptual interpolation scheme that uses the embedding to provide the user with an intuitive interface for navigating the space of gloss and constructing new materials.

Many problems in multi-view geometry, when posed as minimization of the maximum reprojection error across observations, can be solved optimally in polynomial time. We show that these problems are instances of a convex-concave generalized fractional program. We survey the major solution methods for solving problems of this form and present them in a unified framework centered around a single parametric optimization problem. We propose two new algorithms and show that the algorithm proposed by Olsson et al. [21] is a special case of a classical algorithm for generalized fractional programming. The performance of all the algorithms is compared on a variety of datasets, and the algorithm proposed by Gugat [12] stands out as a clear winner. An open source MATLAB toolbox that implements all the algorithms presented here is made available.

We present an algorithm for performing Lambertian photometric stereo in the presence of shadows. The algorithm has three novel features. First, a fast graph cuts based method is used to estimate per pixel light source visibility. Second, it allows images to be acquired with multiple illuminants, and there can be fewer images than light sources. This leads to better surface coverage and improves the reconstruction accuracy by enhancing the signal to noise ratio and the condition number of the light source matrix. The ability to use fewer images than light sources means that the imaging effort grows sublinearly with the number of light sources. Finally, the recovered shadow maps are combined with shading information to perform constrained surface normal integration. This reduces the low frequency bias inherent to the normal integration process and ensures that the recovered surface is consistent with the shadowing configuration The algorithm works with as few as four light sources and four images. We report results for light source visibility detection and high quality surface reconstructions for synthetic and real datasets.

We present an autocalibration algorithm for upgrading a projective reconstruction to a metric reconstruction by estimating the absolute dual quadric. The algorithm enforces the rank degeneracy and the positive semidefiniteness of the dual quadric as part of the estimation procedure, rather than as a post-processing step. Furthermore, the method allows the user, if he or she so desires, to enforce conditions on the plane at infinity so that the reconstruction satisfies the chirality constraints. The algorithm works by constructing low degree polynomial optimization problems, which are solved to their global optimum using a series of convex linear matrix inequality relaxations. The algorithm is fast, stable, robust and has time complexity independent of the number of views. We show extensive results on synthetic as well as real datasets to validate our algorithm.

We present a practical, stratified autocalibration algorithm with theoretical guarantees of global optimality. Given a projective reconstruction, the first stage of the algorithm upgrades it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, the algorithm upgrades this affine reconstruction to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step. For each stage, we construct and minimize tight convex relaxations of the highly non-convex objective functions in a branch and bound optimization framework. We exploit the problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data. MATLAB code for the implementation is made available to the community.

Single particle reconstruction using Cryo-Electron Microscopy (cryo-EM) is an emerging technique in structural biology for estimating the 3-D structure (density) of protein macromolecules. Unlike tomography where a large number of images of a specimen can be acquired, the number of images of an individual particle is limited because of radiation damage. Instead, the specimen consists of identical copies of the same protein macro-molecule embedded in vitreous ice at random and unknown 3-D orientations. Because the images are extremely noisy, thousands to hundreds-of-thousands of projections are needed to achieve the desired resolution of 5A. Along with differences of the imaging modality compared to photographs, single particle reconstruction provides a unique set of challenges to existing computer vision algorithms. Here, we introduce the challenge and opportunity of reconstruction from transmission electron micrographs, and briefly describe our contributions in areas of particle detection, contrast transfer function (CTF) estimation, and initial 3-D model construction. Reconstructing the Structure of Protein Macromolecules One of the most exciting challenges for biology today is understanding the molecular machinery of the cell as a working, dynamic system. Critical to this understanding is determining the 3-D structure of protein macromolecules, a task that is often accomplished using x-ray crystallography. The technique of cryo electron microscopy (cryo-EM) has a unique role to play in addressing this challenge as it can provide structural information of large macromolecular complexes in a variety of conformational and compositional states while preserved under close to physiological conditions. Traditionally the methods for cryo-EM have been time consuming and labor intensive, involving data acquisition, analysis and averaging of thousands to hundreds of thousands of images (views) of the individual macro-molecular complexes. Thus, over the last few years there has been considerable interest and substantial effort devoted to developing automated methods to improve the accuracy, robustness, ease of use, and throughput of cryo-EM [1, 2, 3, 12, 15, 17], and this abstract considers three aspects originally presented in [9, 10, 11]. 1

This paper addresses the problem of reconstructing the density of a scene from multiple projection images produced by modalities such as x-ray, electron microscopy, etc. where an image value is related to the integral of the scene density along a 3D line segment between a radiation source and a point on the image plane. While computed tomography (CT) addresses this problem when the absolute orientation of the image plane and radiation source directions are known, this paper addresses the problem when the orientations are unknown - it is akin to the structure-from-motion (SFM) problem when the extrinsic camera parameters are unknown. We study the problem within the context of reconstructing the density of protein macro-molecules in Cryogenic Electron Microscopy (cryo-EM), where images are very noisy and existing techniques use several thousands of images. In a non-degenerate configuration, the viewing planes corresponding to two projections, intersect in a line in 3D. Using the geometry of the imaging setup, it is possible to determine the projections of this 3D line on the two image planes. In turn, the problem can be formulated as a type of orthographic structure from motion from line correspondences where the line correspondences between two views are unreliable due to image noise. We formulate the task as the problem of denoising a correspondence matrix and present a Bayesian solution to it. Subsequently, the absolute orientation of each projection is determined followed by density reconstruction. We show results on cryo-EM images of proteins and compare our results to that of Electron Micrograph Analysis (EMAN) - a widely used reconstruction tool in cryo-EM.

Recently there has been considerable interest in learning with higher order relations (i.e., three-way or higher) in the unsupervised and semi-supervised settings. Hypergraphs and tensors have been proposed as the natural way of representing these relations and their corresponding algebra as the natural tools for operating on them. In this paper we argue that hypergraphs are not a natural representation for higher order relations, indeed pairwise as well as higher order relations can be handled using graphs. We show that various formulations of the semi-supervised and the unsupervised learning problem on hypergraphs result in the same graph theoretic problem and can be analyzed using existing tools.

We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms.

We present a framework for motion segmentation that combines the concepts of layer-based methods and feature-based motion estimation. We estimate the initial correspondences by comparing vectors of filter outputs at interest points, from which we compute candidate scene relations via random sampling of minimal subsets of correspondences. We achieve a dense, piecewise smooth assignment of pixels to motion layers using a fast approximate graph-cut algorithm based on a Markov random field formulation. We demonstrate our approach on image pairs containing large inter-frame motion and partial occlusion. The approach is efficient and it successfully segments scenes with inter-frame disparities previously beyond the scope of layer-based motion segmentation methods.

Accurate spectral decomposition is essential for the analysis and diagnosis of histologically stained tissue sections. In this paper we present the first automated system for performing this decompo- sition. We compare the performance of our system with ground truth data and report favorable results.

We introduce structured importance sampling , a new technique for efficiently rendering scenes illuminated by distant natural illumination given in an environment map. Our method handles occlusion, high-frequency lighting, and is significantly faster than alternative methods based on Monte Carlo sampling. We achieve this speedup as a result of several ideas. First, we present a new metric for stratifying and sampling an environment map taking into account both the illumination intensity as well as the expected variance due to occlusion within the scene. We then present a novel hierarchical stratification algorithm that uses our metric to automatically stratify the environment map into regular strata. This approach enables a number of rendering optimizations, such as pre-integrating the illumination within each stratum to eliminate noise at the cost of adding bias, and sorting the strata to reduce the number of sample rays. We have rendered several scenes illuminated by natural lighting, and our results indicate that structured importance sampling is better than the best previous Monte Carlo techniques, requiring one to two orders of magnitude fewer samples for the same image quality.

The increased interest in region based image coding has given rise to graph coloring based partition encoding methods. These methods are based on the four color theorem for planar graphs, and assume that a coloring for a graph with the minimum possible number of colors will result in the most compressible representation. We show that this assumption is wrong. We show that there exist graphs with chromatic number k that can be colored with k + 1 colors resulting in bitmaps representing image partitions which are more compressible than, the corresponding bitmaps generated using any k coloring of the same graph. We conclude with some conjectures on optimal coloring of weighted graphs.