Information geometry

Résumé

Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions. Introduction Historically, information geometry can be traced back to the work of C. R. Rao, who was the first to treat the Fisher matrix as a Riemannian metric. The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field. Classically, information geometry considered a parametrized statistical model as a Riemannian manifold. For such models, there is a natural choice of Riemannian metric, known as the Fisher information metric. In the special case that the statistical model is an exponential family, it is possible to induce the statistical manifold with a Hessian metric (i.e a Riemannian metric given by the potential of a convex function).

Source officielle

https://en.wikipedia.org/wiki/Information_geometry

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What is the actual information contained in light rays filling the 3-D world? Leonardo da Vinci saw the world as an infinite number of radiant pyramids caused by the objects located in it. Nowadays, the radiant pyramid is usually described as a set of light rays with various directions passing through a given point. By recording light rays at every point in space, all the information in a scene can be fully acquired. This work focuses on the analysis of the sampling models of a light field camera, a device dedicated to recording the amount of light traveling through any point along any direction in the 3-D world. In contrast to the conventional photography which only records a 2-D projection of the scene, such camera captures both the geometry information and material properties of a scene by recording 2-D angular data for each point in a 2-D spatial domain. This 4-D data is referred to as the light field. The main goal of this thesis is to utilize this 4-D data from one or multiple light field cameras based on the proposed sampling models for recovering the given scene. We first propose a novel algorithm to recover the depth information from the light field. Based on the analysis of the sampling model, we map the high dimensional light field data to a low dimensional texture signal in the continuous domain modulated by the geometric structure of the scene. We formulate the depth estimation problem as a signal recovery problem with samples at unknown locations. A practical framework is proposed to recover alternately the texture signal and the depth map. We thus acquire not only the depth map with high accuracy but also a compact representation of the light field in the continuous domain. The proposed algorithm performs especially well for scenes with fine geometric structure while also achieving state-of-the-art performance on public data-sets. Secondly, we consider multiple light fields to increase the amount of information captured from the 3-D world. We derive a motion model of the light field camera from the proposed sampling model. Given this motion model, we can extend the field of view to create light field panoramas and perform light-field super-resolution. This can help overcome the shortcoming of limited sensor resolution in current light field cameras. Finally, we propose a novel image based rendering framework to represent light rays in the 3-D space: the circular light field. The circular light field is acquired by taking photos from a circular camera array facing outwards from the center of the rig. We propose a practical framework to capture, register and stitch multiple circular light fields. The information presented in multiple circular light fields allows the creation of any virtual camera view at any chosen location with a 360-degree field of view. The new representation of the light rays can be used to generate high quality contents for virtual reality and augmented reality.

EPFL2016

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3D Face Recognition using Sparse Spherical Representations

Pascal Frossard, Effrosyni Kokiopoulou, Roser Sala Llonch, Ivana Tosic

This paper addresses the problem of 3D face recognition using spherical sparse representations. We first propose a fully automated registration process that permits to align the 3D face point clouds. These point clouds are then represented as signals on the 2D sphere, in order to take benefit of the geometry information. Simultaneous sparse approximations implement a dimensionality reduction process by subspace projection. Each face is typically represented by a few spherical basis functions that are able to capture the salient facial characteristics. The dimensionality reduction step preserves the discriminant facial information and eventually permits an effective matching in the reduced space, where it can further be combined with LDA for improved recognition performance. We evaluate the 3D face recognition algorithm on the FRGC v.1.0 data set, where it outperforms classical state-of-the-art solutions based on PCA or LDA on depth face images.

2008

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This paper addresses the problem of 3D face recognition using simultaneous sparse approximations on the sphere. The 3D face point clouds are first aligned with a novel and fully automated registration process. They are then represented as signals on the 2D sphere in order to preserve depth and geometry information. Next, we implement a dimensionality reduction process with simultaneous sparse approximations and subspace projection. It permits to represent each 3D face by only a few spherical functions that are able to capture the salient facial characteristics, and hence to preserve the discriminant facial information. We eventually perform recognition by effective matching in the reduced space, where Linear Discriminant Analysis can be further activated for improved recognition performance. The 3D face recognition algorithm is evaluated on the FRGC v.1.0 data set, where it is shown to outperform classical state-of-the-art solutions that work with depth images.

2010

Chargement

EPFL2016