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Multi-View Geometry of the Refractive Plane
"... Transparent refractive objects are one of the main problems in geometric vision that have been largely unexplored. The imaging and multi-view geometry of scenes with transparent or translucent objects with refractive properties is relatively less well understood than for opaque objects. The main obj ..."
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Transparent refractive objects are one of the main problems in geometric vision that have been largely unexplored. The imaging and multi-view geometry of scenes with transparent or translucent objects with refractive properties is relatively less well understood than for opaque objects. The main objective of our work is to analyze the underlying multi-view relationships between cameras, when the scene being viewed contains a single refractive planar surface separating two different media. Such a situation might occur in scenarios like underwater photography. Our main result is to show the existence of geometric entities like the fundamental matrix, and the homography matrix in such instances. In addition, under special circumstances we also show how to compute the relative pose between two cameras immersed in one of the two media. 1
Analytical Forward Projection for Axial Non-Central Dioptric & Catadioptric Cameras
"... Abstract. Wepresentatechniqueformodelingnon-centralcatadioptric cameras consisting of a perspective camera and a rotationally symmetric conic reflector. While previous approaches use a central approximation and/or iterative methods for forward projection, we present an analytical solution. This allo ..."
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Abstract. Wepresentatechniqueformodelingnon-centralcatadioptric cameras consisting of a perspective camera and a rotationally symmetric conic reflector. While previous approaches use a central approximation and/or iterative methods for forward projection, we present an analytical solution. This allows computation of the optical path from a given 3D point to the given viewpoint by solving a 6 th degree forward projection equation for general conic mirrors. For a spherical mirror, the forward projection reduces to a 4 th degree equation, resulting in a closed form solution. We also derive the forward projection equation for imaging through a refractive sphere (non-central dioptric camera) and show that it is a 10 th degree equation. While central catadioptric cameras lead to conic epipolar curves, we show the existence of a quartic epipolar curve for catadioptric systems using a spherical mirror. The analytical forward projection leads to accurate and fast 3D reconstruction via bundle adjustment. Simulations and real results on single image sparse 3D reconstruction are presented. We demonstrate ∼ 100 times speed up using the analytical solution over iterative forward projection for 3D reconstruction using spherical mirrors. 1
What is a Camera?
, 2009
"... This paper addresses the problem of characterizing a general class of cameras under reasonable, “linear” assumptions. Concretely, we use the formalism and terminology of classical projective geometry to model cameras by two-parameter linear families of straight lines—that is, degenerate reguli (rank ..."
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Cited by 11 (2 self)
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This paper addresses the problem of characterizing a general class of cameras under reasonable, “linear” assumptions. Concretely, we use the formalism and terminology of classical projective geometry to model cameras by two-parameter linear families of straight lines—that is, degenerate reguli (rank-3 families) and non-degenerate linear congruences (rank-4 families). This model captures both the general linear cameras of Yu and McMillan [16] and the linear oblique cameras of Pajdla [8]. From a geometric perspective, it affords a simple classification of all possible camera configurations. From an analytical viewpoint, it also provides a simple and unified methodology for deriving general formulas for projection and inverse projection, triangulation, and binocular and trinocular geometry.
Calibration of Central Catadioptric Cameras Using a DLT-Like Approach
"... the date of receipt and acceptance should be inserted later Since their introduction to the computer vision community, catadioptric omnidirectional cameras have been utilized in many application areas such as surveillance [1], tele-presence [2], robot navigation [3] and 3D reconstruction [4]. Omnidi ..."
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the date of receipt and acceptance should be inserted later Since their introduction to the computer vision community, catadioptric omnidirectional cameras have been utilized in many application areas such as surveillance [1], tele-presence [2], robot navigation [3] and 3D reconstruction [4]. Omnidirectional cameras being singleviewpoint are searched, since it is an important property. If single-viewpoint cameras are used, directions of the light rays coming into the camera can easily be calculated and combined in a multiview geometric framework [5]. Catadioptric systems, combinations of camera lenses and mirrors were extensively studied by Baker and Nayar [6]. They showed which of these systems are able to provide the single-viewpoint property, i.e., if the mirror has a focal point which can behave like an effective pinhole. Among those systems the most useful ones are the para-catadioptric and the hyper-catadioptric models, using a mirror of parabolic/hyperbolic shape, coupled with an orthographic/perspective camera. Swaminathan et al. [7] conducted a detailed study on the geometry of non-single-viewpoint systems. There also exist studies for approximating a viewpoint in non-singleviewpoint systems as Derrien and Konolige proposed for spherical mirrors [8]. Camera calibration is essential when we want to extract metric information from images. It establishes a relationship between the 3D rays and their corresponding pixels in the image. This relationship makes possible to measure distances in a real world from their projections on the images [9]. Camera calibration is bainria-00590268,
Dlt-like calibration of central catadioptric cameras
- In: Proceedings of the Eight Workshop on Omnidirectional Vision
"... Abstract. In this study, we present a calibration technique that is valid for all single-viewpoint catadioptric cameras. We are able to represent the projection of 3D points on a catadioptric image linearly with a 6×10 projection matrix, which uses lifted coordinates for image and 3D points. This pr ..."
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Abstract. In this study, we present a calibration technique that is valid for all single-viewpoint catadioptric cameras. We are able to represent the projection of 3D points on a catadioptric image linearly with a 6×10 projection matrix, which uses lifted coordinates for image and 3D points. This projection matrix can be computed with enough number of 3D-2D correspondences (minimum 20 points distributed in three different planes). We show how to decompose it to obtain intrinsic and extrinsic parameters. Moreover, we use this parameter estimation followed by a non-linear optimization to calibrate various types of cameras. Our results are based on the sphere camera model which considers that every central catadioptric system can be modeled using two projections, one from 3D points to a unitary sphere and then a perspective projection from the sphere to the image plane. We tested our method both with simulations and real images. 1
Author manuscript, published in "International Conference on Computer Vision (2009)" DOI: 10.1109/ICCV.2009.5459336 Plane-Based Calibration of Central Catadioptric Cameras
"... niques for calibrating central cameras have been suggested; an early work by Geyer and Daniilidis [9] aimed at calibrating paracatadioptric cameras from images of lines. Barreto and Araujo extended this procedure to general central catadioptric systems [4] while Ying and Hu [20] used geometric invar ..."
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niques for calibrating central cameras have been suggested; an early work by Geyer and Daniilidis [9] aimed at calibrating paracatadioptric cameras from images of lines. Barreto and Araujo extended this procedure to general central catadioptric systems [4] while Ying and Hu [20] used geometric invariants provided by lines and spheres to calibrate central catadioptric cameras. However, the calibration based on line images is usually difficult to use in practice because it requires the fitting of a conic starting from a small arc, which makes it quite inaccurate and unreliable. In general, if many lines are involved, it is difficult to solve the 2D-3D correspondence because not all the conics are images of lines. However, Barreto and Araujo determined some sufficient properties that a conic curve must satisfy to correspond to a paracatadioptric line image [3].
CHARI, STURM: MULTI-VIEW GEOMETRY OF THE REFRACTIVE PLANE 1 Multi-View Geometry of the Refractive Plane
"... Transparent refractive objects are one of the main problems in geometric vision that have been largely unexplored. The imaging and multi-view geometry of scenes with transparent or translucent objects with refractive properties is relatively less well under-stood than for opaque objects. The main ob ..."
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Transparent refractive objects are one of the main problems in geometric vision that have been largely unexplored. The imaging and multi-view geometry of scenes with transparent or translucent objects with refractive properties is relatively less well under-stood than for opaque objects. The main objective of our work is to analyze the under-lying multi-view relationships between cameras, when the scene being viewed contains a single refractive planar surface separating two different media. Such a situation might occur in scenarios like underwater photography. Our main result is to show the existence of geometric entities like the fundamental matrix, and the homography matrix in such instances. In addition, under special circumstances we also show how to compute the relative pose between two cameras immersed in one of the two media. 1
Author manuscript, published in "Dans The 8th Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras- OMNIVIS (2008)" DLT-Like Calibration of Central Catadioptric Cameras
, 2008
"... Abstract. In this study, we present a calibration technique that is valid for all single-viewpoint catadioptric cameras. We are able to represent the projection of 3D points on a catadioptric image linearly with a 6×10 projection matrix, which uses lifted coordinates for image and 3D points. This pr ..."
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Abstract. In this study, we present a calibration technique that is valid for all single-viewpoint catadioptric cameras. We are able to represent the projection of 3D points on a catadioptric image linearly with a 6×10 projection matrix, which uses lifted coordinates for image and 3D points. This projection matrix can be computed with enough number of 3D-2D correspondences (minimum 20 points distributed in three different planes). We show how to decompose it to obtain intrinsic and extrinsic parameters. Moreover, we use this parameter estimation followed by a non-linear optimization to calibrate various types of cameras. Our results are based on the sphere camera model which considers that every central catadioptric system can be modeled using two projections, one from 3D points to a unitary sphere and then a perspective projection from the sphere to the image plane. We tested our method both with simulations and real images. 1
Noname manuscript No. (will be inserted by the editor) Hybrid Homographies and Fundamental Matrices Mixing Uncalibrated Omnidirectional and Conventional Cameras.
"... the date of receipt and acceptance should be inserted later Abstract In this paper we present a deep analysis of the hybrid two-view relations combining images acquired with uncalibrated central catadioptric systems and conventional cameras. We consider both, hybrid fundamental matrices and hybrid p ..."
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the date of receipt and acceptance should be inserted later Abstract In this paper we present a deep analysis of the hybrid two-view relations combining images acquired with uncalibrated central catadioptric systems and conventional cameras. We consider both, hybrid fundamental matrices and hybrid planar homographies. These matrices contain useful geometric information. We study three different types of matrices, varying in complexity depending on their capacity to deal with a single or multiple types of central catadioptric systems. The first and simplest one is designed to deal with para-catadioptric systems, the second one and more complex, considers the combination of a perspective camera and any central catadioptric system. The last one is the complete and generic model which is able to deal with any combination of central catadioptric systems. We show that the generic and most complex model sometimes is not the best option when we deal with real images. Simpler models are not as accurate as the complete model in the ideal case, but they provide a better and more accurate behavior in presence of noise, being simpler and requiring less correspondences to be computed. Experiments with simulated data and real images are performed. To show the potential of
