2,467 research outputs found

    Sign rank versus VC dimension

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    This work studies the maximum possible sign rank of N×NN \times N sign matrices with a given VC dimension dd. For d=1d=1, this maximum is {three}. For d=2d=2, this maximum is Θ~(N1/2)\tilde{\Theta}(N^{1/2}). For d>2d >2, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an O(N/log(N))O(N/\log(N)) multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the N×NN \times N adjacency matrix of a Δ\Delta regular graph with a second eigenvalue of absolute value λ\lambda and ΔN/2\Delta \leq N/2. We show that the sign rank of the signed version of this matrix is at least Δ/λ\Delta/\lambda. We use this connection to prove the existence of a maximum class C{±1}NC\subseteq\{\pm 1\}^N with VC dimension 22 and sign rank Θ~(N1/2)\tilde{\Theta}(N^{1/2}). This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC dimension". Additional results in this version: (i) Estimates on the number of maximum VC classes (answering a question of Frankl from '89). (ii) Estimates on the sign rank of large VC classes (answering a question of Ben-David et al. from '03). (iii) A discussion on the computational complexity of computing the sign-ran

    3D Reconstruction through Segmentation of Multi-View Image Sequences

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    We propose what we believe is a new approach to 3D reconstruction through the design of a 3D voxel volume, such that all the image information and camera geometry are embedded into one feature space. By customising the volume to be suitable for segmentation, the key idea that we propose is the recovery of a 3D scene through the use of globally optimal geodesic active contours. We also present an extension to this idea by proposing the novel design of a 4D voxel volume to analyse the stereo motion problem in multi-view image sequences

    Real-time 3D reconstruction of non-rigid shapes with a single moving camera

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    © . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper describes a real-time sequential method to simultaneously recover the camera motion and the 3D shape of deformable objects from a calibrated monocular video. For this purpose, we consider the Navier-Cauchy equations used in 3D linear elasticity and solved by finite elements, to model the time-varying shape per frame. These equations are embedded in an extended Kalman filter, resulting in sequential Bayesian estimation approach. We represent the shape, with unknown material properties, as a combination of elastic elements whose nodal points correspond to salient points in the image. The global rigidity of the shape is encoded by a stiffness matrix, computed after assembling each of these elements. With this piecewise model, we can linearly relate the 3D displacements with the 3D acting forces that cause the object deformation, assumed to be normally distributed. While standard finite-element-method techniques require imposing boundary conditions to solve the resulting linear system, in this work we eliminate this requirement by modeling the compliance matrix with a generalized pseudoinverse that enforces a pre-fixed rank. Our framework also ensures surface continuity without the need for a post-processing step to stitch all the piecewise reconstructions into a global smooth shape. We present experimental results using both synthetic and real videos for different scenarios ranging from isometric to elastic deformations. We also show the consistency of the estimation with respect to 3D ground truth data, include several experiments assessing robustness against artifacts and finally, provide an experimental validation of our performance in real time at frame rate for small mapsPeer ReviewedPostprint (author's final draft

    Angular variation as a monocular cue for spatial percepcion

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    Monocular cues are spatial sensory inputs which are picked up exclusively from one eye. They are in majority static features that provide depth information and are extensively used in graphic art to create realistic representations of a scene. Since the spatial information contained in these cues is picked up from the retinal image, the existence of a link between it and the theory of direct perception can be conveniently assumed. According to this theory, spatial information of an environment is directly contained in the optic array. Thus, this assumption makes possible the modeling of visual perception processes through computational approaches. In this thesis, angular variation is considered as a monocular cue, and the concept of direct perception is adopted by a computer vision approach that considers it as a suitable principle from which innovative techniques to calculate spatial information can be developed. The expected spatial information to be obtained from this monocular cue is the position and orientation of an object with respect to the observer, which in computer vision is a well known field of research called 2D-3D pose estimation. In this thesis, the attempt to establish the angular variation as a monocular cue and thus the achievement of a computational approach to direct perception is carried out by the development of a set of pose estimation methods. Parting from conventional strategies to solve the pose estimation problem, a first approach imposes constraint equations to relate object and image features. In this sense, two algorithms based on a simple line rotation motion analysis were developed. These algorithms successfully provide pose information; however, they depend strongly on scene data conditions. To overcome this limitation, a second approach inspired in the biological processes performed by the human visual system was developed. It is based in the proper content of the image and defines a computational approach to direct perception. The set of developed algorithms analyzes the visual properties provided by angular variations. The aim is to gather valuable data from which spatial information can be obtained and used to emulate a visual perception process by establishing a 2D-3D metric relation. Since it is considered fundamental in the visual-motor coordination and consequently essential to interact with the environment, a significant cognitive effect is produced by the application of the developed computational approach in environments mediated by technology. In this work, this cognitive effect is demonstrated by an experimental study where a number of participants were asked to complete an action-perception task. The main purpose of the study was to analyze the visual guided behavior in teleoperation and the cognitive effect caused by the addition of 3D information. The results presented a significant influence of the 3D aid in the skill improvement, which showed an enhancement of the sense of presence.Las señales monoculares son entradas sensoriales capturadas exclusivamente por un solo ojo que ayudan a la percepción de distancia o espacio. Son en su mayoría características estáticas que proveen información de profundidad y son muy utilizadas en arte gráfico para crear apariencias reales de una escena. Dado que la información espacial contenida en dichas señales son extraídas de la retina, la existencia de una relación entre esta extracción de información y la teoría de percepción directa puede ser convenientemente asumida. De acuerdo a esta teoría, la información espacial de todo le que vemos está directamente contenido en el arreglo óptico. Por lo tanto, esta suposición hace posible el modelado de procesos de percepción visual a través de enfoques computacionales. En esta tesis doctoral, la variación angular es considerada como una señal monocular, y el concepto de percepción directa adoptado por un enfoque basado en algoritmos de visión por computador que lo consideran un principio apropiado para el desarrollo de nuevas técnicas de cálculo de información espacial. La información espacial esperada a obtener de esta señal monocular es la posición y orientación de un objeto con respecto al observador, lo cual en visión por computador es un conocido campo de investigación llamado estimación de la pose 2D-3D. En esta tesis doctoral, establecer la variación angular como señal monocular y conseguir un modelo matemático que describa la percepción directa, se lleva a cabo mediante el desarrollo de un grupo de métodos de estimación de la pose. Partiendo de estrategias convencionales, un primer enfoque implanta restricciones geométricas en ecuaciones para relacionar características del objeto y la imagen. En este caso, dos algoritmos basados en el análisis de movimientos de rotación de una línea recta fueron desarrollados. Estos algoritmos exitosamente proveen información de la pose. Sin embargo, dependen fuertemente de condiciones de la escena. Para superar esta limitación, un segundo enfoque inspirado en los procesos biológicos ejecutados por el sistema visual humano fue desarrollado. Está basado en el propio contenido de la imagen y define un enfoque computacional a la percepción directa. El grupo de algoritmos desarrollados analiza las propiedades visuales suministradas por variaciones angulares. El propósito principal es el de reunir datos de importancia con los cuales la información espacial pueda ser obtenida y utilizada para emular procesos de percepción visual mediante el establecimiento de relaciones métricas 2D- 3D. Debido a que dicha relación es considerada fundamental en la coordinación visuomotora y consecuentemente esencial para interactuar con lo que nos rodea, un efecto cognitivo significativo puede ser producido por la aplicación de métodos de L estimación de pose en entornos mediados tecnológicamente. En esta tesis doctoral, este efecto cognitivo ha sido demostrado por un estudio experimental en el cual un número de participantes fueron invitados a ejecutar una tarea de acción-percepción. El propósito principal de este estudio fue el análisis de la conducta guiada visualmente en teleoperación y el efecto cognitivo causado por la inclusión de información 3D. Los resultados han presentado una influencia notable de la ayuda 3D en la mejora de la habilidad, así como un aumento de la sensación de presencia

    3D Dynamic Scene Reconstruction from Multi-View Image Sequences

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    A confirmation report outlining my PhD research plan is presented. The PhD research topic is 3D dynamic scene reconstruction from multiple view image sequences. Chapter 1 describes the motivation and research aims. An overview of the progress in the past year is included. Chapter 2 is a review of volumetric scene reconstruction techniques and Chapter 3 is an in-depth description of my proposed reconstruction method. The theory behind the proposed volumetric scene reconstruction method is also presented, including topics in projective geometry, camera calibration and energy minimization. Chapter 4 presents the research plan and outlines the future work planned for the next two years

    Lunar Crater Identification in Digital Images

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    It is often necessary to identify a pattern of observed craters in a single image of the lunar surface and without any prior knowledge of the camera's location. This so-called "lost-in-space" crater identification problem is common in both crater-based terrain relative navigation (TRN) and in automatic registration of scientific imagery. Past work on crater identification has largely been based on heuristic schemes, with poor performance outside of a narrowly defined operating regime (e.g., nadir pointing images, small search areas). This work provides the first mathematically rigorous treatment of the general crater identification problem. It is shown when it is (and when it is not) possible to recognize a pattern of elliptical crater rims in an image formed by perspective projection. For the cases when it is possible to recognize a pattern, descriptors are developed using invariant theory that provably capture all of the viewpoint invariant information. These descriptors may be pre-computed for known crater patterns and placed in a searchable index for fast recognition. New techniques are also developed for computing pose from crater rim observations and for evaluating crater rim correspondences. These techniques are demonstrated on both synthetic and real images

    Efficient Structure and Motion: Path Planning, Uncertainty and Sparsity

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    This thesis explores methods for solving the structure-and-motion problem in computer vision, the recovery of three-dimensional data from a series of two-dimensional image projections. The first paper investigates an alternative state space parametrization for use with the Kalman filter approach to simultaneous localization and mapping, and shows it has superior convergence properties compared with the state-of-the-art. The second paper presents a continuous optimization method for mobile robot path planning, designed to minimize the uncertainty of the geometry reconstructed from images taken by the robot. Similar concepts are applied in the third paper to the problem of sequential 3D reconstruction from unordered image sequences, resulting in increased robustness, accuracy and a reduced need for costly bundle adjustment operations. In the final paper, a method for efficient solution of bundle adjustment problems based on a junction tree decomposition is presented, exploiting the sparseness patterns in typical structure-and-motion input data

    Deep Learning-Based 6-DoF Object Pose Estimation With Synthetic Data: A Case Study in Underwater Environments

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    In this thesis we aim to address the image based 6-DoF pose estimation problem, or 3D pose estimation problem, for Autonomous Underwater Vehicles (AUVs). The results of the object pose estimation will be used, for example, to estimate the global location of the AUV or to approach more accurately the underwater infrastructures. Actually, an autonomous robot or a team of autonomous robots need accurate location skills to safely and effectively move within an underwater environment, where communications are sparse and unreliable, and to accomplish high-level tasks such as: underwater exploration, mapping of the surrounding environment, multi-robot conveyance and many other multi-robot problems. Several state-of-the-art approaches will be analysed and tested on real datasets. Collecting underwater images and providing them with an accurate ground-truth estimate of the object's pose is an expansive and extremely time-consuming activity To this end, we addressed the problem using only synthetic datasets. In fact, it was not possible to use the standard datasets used in the analyzed papers, since they are datasets with objects and conditions very different from those in which the AUVs operate. Hence, we exploited an unpaired image-to-image translation network is employed to bridge the gap between the rendered and the real images, producing photorealistic synthetic training images. Promising preliminary results confirm the goodness of the made choices.In this thesis we aim to address the image based 6-DoF pose estimation problem, or 3D pose estimation problem, for Autonomous Underwater Vehicles (AUVs). The results of the object pose estimation will be used, for example, to estimate the global location of the AUV or to approach more accurately the underwater infrastructures. Actually, an autonomous robot or a team of autonomous robots need accurate location skills to safely and effectively move within an underwater environment, where communications are sparse and unreliable, and to accomplish high-level tasks such as: underwater exploration, mapping of the surrounding environment, multi-robot conveyance and many other multi-robot problems. Several state-of-the-art approaches will be analysed and tested on real datasets. Collecting underwater images and providing them with an accurate ground-truth estimate of the object's pose is an expansive and extremely time-consuming activity To this end, we addressed the problem using only synthetic datasets. In fact, it was not possible to use the standard datasets used in the analyzed papers, since they are datasets with objects and conditions very different from those in which the AUVs operate. Hence, we exploited an unpaired image-to-image translation network is employed to bridge the gap between the rendered and the real images, producing photorealistic synthetic training images. Promising preliminary results confirm the goodness of the made choices

    Robust surface modelling of visual hull from multiple silhouettes

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    Reconstructing depth information from images is one of the actively researched themes in computer vision and its application involves most vision research areas from object recognition to realistic visualisation. Amongst other useful vision-based reconstruction techniques, this thesis extensively investigates the visual hull (VH) concept for volume approximation and its robust surface modelling when various views of an object are available. Assuming that multiple images are captured from a circular motion, projection matrices are generally parameterised in terms of a rotation angle from a reference position in order to facilitate the multi-camera calibration. However, this assumption is often violated in practice, i.e., a pure rotation in a planar motion with accurate rotation angle is hardly realisable. To address this problem, at first, this thesis proposes a calibration method associated with the approximate circular motion. With these modified projection matrices, a resulting VH is represented by a hierarchical tree structure of voxels from which surfaces are extracted by the Marching cubes (MC) algorithm. However, the surfaces may have unexpected artefacts caused by a coarser volume reconstruction, the topological ambiguity of the MC algorithm, and imperfect image processing or calibration result. To avoid this sensitivity, this thesis proposes a robust surface construction algorithm which initially classifies local convex regions from imperfect MC vertices and then aggregates local surfaces constructed by the 3D convex hull algorithm. Furthermore, this thesis also explores the use of wide baseline images to refine a coarse VH using an affine invariant region descriptor. This improves the quality of VH when a small number of initial views is given. In conclusion, the proposed methods achieve a 3D model with enhanced accuracy. Also, robust surface modelling is retained when silhouette images are degraded by practical noise
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