Camera models and calibration
|
|
- Anastasia Hines
- 5 years ago
- Views:
Transcription
1 Camera models and calibration Read tutorial chapter 2 and 3. Szeliski s book pp.29-73
2 Schedule (tentative) 2 # date topic Sep.8 Introduction and geometry 2 Sep.25 Camera models and calibration 3 Oct.2 Multiple-view geometry 4 Oct.9 Invariant features 5 Oct.6 Model fitting (RANSAC, EM, ) 6 Oct.23 Stereo Matching 7 Oct.3 Segmentation 8 Nov.6 Shape from X (silhouettes, ) 9 Nov.3 Structure from motion Nov.2 Optical flow Nov.27 Tracking (Kalman, particle filter) 2 Dec.4 Object category recognition 3 Dec. Specific object recognition 4 Dec.8 Research overview
3 Brief geometry reminder x = l l' l T x = 2D line-point coincidence relation: Point from lines: Line from points: π T X = 3D plane-point coincidence relation: T π X Point from planes: T π X X 2 = Plane from points: T π X 3 l = x x' T T 2 T 3 π = 3D line representation: [ ] (as two planes or two points) T 2x2 2D Ideal points ( x, x, ) T 2 l,, 2D line at infinity ( ) T = P Q T A B = T 3D Ideal points ( X, X2, X3,) 3D plane at infinity Π = (,,,) T 3
4 Conics and quadrics Conics l x T Cx = l T C * l = * - C = C x C l=cx Quadrics X T T * QX = π Q π = 4
5 Hierarchy of 2D transformations Projective 8dof Affine 6dof Similarity 4dof Euclidean 3dof h h h a a sr sr r r h h h a a r r sr sr h3 h 23 h 33 tx t y tx t y tx t y transformed squares invariants Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids). The line at infinity l Ratios of lengths, angles. The circular points I,J lengths, areas.
6 Hierarchy of 3D transformations Projective 5dof A T v t v Intersection and tangency Affine 2dof A T t Parallellism of planes, Volume ratios, centroids, The plane at infinity π Similarity 7dof s R T t Angles, ratios of length The absolute conic Ω Euclidean 6dof R T t Volume
7 Camera model Relation between pixels and rays in space?
8 Pinhole camera Gemma Frisius, 544
9 23 Distant objects appear smaller
10 vanishing point Parallel lines meet 24
11 Vanishing points H VPL VPR VP VP 2 25 To different directions correspond different vanishing points VP 3
12 Geometric properties of projection Points go to points Lines go to lines Planes go to whole image or half-plane Polygons go to polygons Degenerate cases: line through focal point yields point plane through focal point yields line 26
13 Pinhole camera model T T Z fy Z fx Z Y X ) /, / ( ),, ( = Z Y X f f Z fy fx Z Y X linear projection in homogeneous coordinates!
14 Pinhole camera model = Z Y X f f Z fy fx = Z Y X f f Z fy fx x = PX [ ] I,), diag( P f f =
15 Principal point offset T y x T p Z fy p Z fx Z Y X ) + /, + / ( ),, ( principal point T x p y p ), ( = + + Z Y X p f p f Z Zp fy Zp fx Z Y X y x x x
16 Principal point offset = + + Z Y X p f p f Z Zp fy Zp fx y x x x [ ] cam X x = K I = y x p f p f K calibration matrix
17 Camera rotation and translation = R( X ~ -C ~ ) X ~ cam - RC ~ X ~ R Y R - RC X cam = = X Z [ ] x = KRK I [ I -C X ~ cam ]X x = PX P = K[ R t] t = -RC ~
18 CCD camera = y y x x p p K α α = y x y x p f p f m m K
19 General projective camera = y x x x p p s K α α = y x x x p p K α α [ ] C ~ P = KR I non-singular dof (5+3+3) [ ] t P = K R intrinsic camera parameters extrinsic camera parameters
20 Radial distortion Due to spherical lenses (cheap) Model: R R ( xy, ) ( K( x y) K2( x y)...) x = y straight lines are not straight anymore
21 Camera model Relation between pixels and rays in space?
22 Projector model Relation between pixels and rays in space (dual of camera)? (main geometric difference is vertical principal point offset to reduce keystone effect)
23 Meydenbauer camera vertical lens shift to allow direct ortho-photographs
24 Affine cameras
25 Action of projective camera on points and lines projection of point x = PX forward projection of line ( µ ) = P(A+ µb) = PA + µpb= a µb X + back-projection of line Π = P T l Π T X = l T PX ( l T x = ; x = PX)
26 Action of projective camera on conics and quadrics back-projection to cone Q = T T T T co P CP x Cx = X P CPX = ( x = PX) projection of quadric C * = * T T * T * T Π Q Π = l PQ P l = PQ P ( Π = P T l)
27 Resectioning Xi x i P?
28 Direct Linear Transform (DLT) x = [ i ] i i PX i x PX rank-2 matrix
29 Direct Linear Transform (DLT) Minimal solution P has dof, 2 independent eq./points 5½ correspondences needed (say 6) Over-determined solution n 6 points minimize p = Ap subject to constraint use SVD
30 Degenerate configurations (i) Points lie on plane or single line passing through projection center (ii) Camera and points on a twisted cubic
31 Data normalization Scale data to values of order. move center of mass to origin 2. scale to yield order values σ 2 D σ 3 D
32 Line correspondences Extend DLT to lines Π = P T l i (back-project line) l T i PXi l T i PX2i (2 independent eq.)
33 Geometric error
34 Gold Standard algorithm Objective Given n 6 2D to 3D point correspondences {X i x i }, determine the Maximum Likelyhood Estimation of P Algorithm (i) Linear solution: (a) Normalization: (b) DLT X ~ i = UX i x~ = Tx (ii) Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error: ~ ~ ~ i i (iii) Denormalization: P = T - ~ PU
35 Calibration example (i) Canny edge detection (ii) Straight line fitting to the detected edges (iii) Intersecting the lines to obtain the images corners typically precision </ (H&Z rule of thumb: 5n constraints for n unknowns)
36 Errors in the image (standard case) xˆ i = PX i Errors in the world i X i Errors in the image and in the world x = P X i
37 Restricted camera estimation Find best fit that satisfies skew s is zero pixels are square principal point is known complete camera matrix K is known Minimize geometric error impose constraint through parametrization Minimize algebraic error assume map from param q P=K[R -RC], i.e. p=g(q) minimize Ag(q)
38 Restricted camera estimation Initialization Use general DLT Clamp values to desired values, e.g. s=, α x = α y Note: can sometimes cause big jump in error Alternative initialization Use general DLT Impose soft constraints gradually increase weights
39
40 Image of absolute conic
41 A simple calibration device (i) compute H for each square (corners (,),(,),(,),(,)) (ii) compute the imaged circular points H(,±i,) T (iii) fit a conic to 6 circular points (iv) compute K from ω through cholesky factorization ( Zhang s calibration method)
42 Some typical calibration algorithms Tsai calibration Zhangs calibration Z. Zhang. A flexible new technique for camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(): , 2. Z. Zhang. Flexible Camera Calibration By Viewing a Plane From Unknown Orientations. International Conference on Computer Vision (ICCV'99), Corfu, Greece, pages , September
43 8 from Szeliski s book
44 Next week: Multiple-view geometry 82
3D Photography. Marc Pollefeys, Torsten Sattler. Spring 2015
3D Photography Marc Pollefeys, Torsten Sattler Spring 2015 Schedule (tentative) Feb 16 Feb 23 Mar 2 Mar 9 Mar 16 Mar 23 Mar 30 Apr 6 Apr 13 Apr 20 Apr 27 May 4 May 11 May 18 Apr 6 Introduction Geometry,
More informationHomogeneous Coordinates. Lecture18: Camera Models. Representation of Line and Point in 2D. Cross Product. Overall scaling is NOT important.
Homogeneous Coordinates Overall scaling is NOT important. CSED44:Introduction to Computer Vision (207F) Lecture8: Camera Models Bohyung Han CSE, POSTECH bhhan@postech.ac.kr (",, ) ()", ), )) ) 0 It is
More informationAugmented Reality II - Camera Calibration - Gudrun Klinker May 11, 2004
Augmented Reality II - Camera Calibration - Gudrun Klinker May, 24 Literature Richard Hartley and Andrew Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, 2. (Section 5,
More informationMore on single-view geometry class 10
More on single-view geometry class 10 Multiple View Geometry Comp 290-089 Marc Pollefeys Multiple View Geometry course schedule (subject to change) Jan. 7, 9 Intro & motivation Projective 2D Geometry Jan.
More informationChapter 7: Computation of the Camera Matrix P
Chapter 7: Computation of the Camera Matrix P Arco Nederveen Eagle Vision March 18, 2008 Arco Nederveen (Eagle Vision) The Camera Matrix P March 18, 2008 1 / 25 1 Chapter 7: Computation of the camera Matrix
More informationCamera model and calibration
and calibration AVIO tristan.moreau@univ-rennes1.fr Laboratoire de Traitement du Signal et des Images (LTSI) Université de Rennes 1. Mardi 21 janvier 1 AVIO tristan.moreau@univ-rennes1.fr and calibration
More informationStructure from motion
Structure from motion Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R 1,t 1 R 2,t R 2 3,t 3 Camera 1 Camera
More informationComputer Vision I - Algorithms and Applications: Multi-View 3D reconstruction
Computer Vision I - Algorithms and Applications: Multi-View 3D reconstruction Carsten Rother 09/12/2013 Computer Vision I: Multi-View 3D reconstruction Roadmap this lecture Computer Vision I: Multi-View
More informationStructure from motion
Structure from motion Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R 1,t 1 R 2,t 2 R 3,t 3 Camera 1 Camera
More informationProjective geometry for Computer Vision
Department of Computer Science and Engineering IIT Delhi NIT, Rourkela March 27, 2010 Overview Pin-hole camera Why projective geometry? Reconstruction Computer vision geometry: main problems Correspondence
More informationVisual Recognition: Image Formation
Visual Recognition: Image Formation Raquel Urtasun TTI Chicago Jan 5, 2012 Raquel Urtasun (TTI-C) Visual Recognition Jan 5, 2012 1 / 61 Today s lecture... Fundamentals of image formation You should know
More informationInvariance of l and the Conic Dual to Circular Points C
Invariance of l and the Conic Dual to Circular Points C [ ] A t l = (0, 0, 1) is preserved under H = v iff H is an affinity: w [ ] l H l H A l l v 0 [ t 0 v! = = w w] 0 0 v = 0 1 1 C = diag(1, 1, 0) is
More informationComputer Vision. 2. Projective Geometry in 3D. Lars Schmidt-Thieme
Computer Vision 2. Projective Geometry in 3D Lars Schmidt-Thieme Information Systems and Machine Learning Lab (ISMLL) Institute for Computer Science University of Hildesheim, Germany 1 / 26 Syllabus Mon.
More informationUnit 3 Multiple View Geometry
Unit 3 Multiple View Geometry Relations between images of a scene Recovering the cameras Recovering the scene structure http://www.robots.ox.ac.uk/~vgg/hzbook/hzbook1.html 3D structure from images Recover
More informationcalibrated coordinates Linear transformation pixel coordinates
1 calibrated coordinates Linear transformation pixel coordinates 2 Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial
More information3D Sensing and Reconstruction Readings: Ch 12: , Ch 13: ,
3D Sensing and Reconstruction Readings: Ch 12: 12.5-6, Ch 13: 13.1-3, 13.9.4 Perspective Geometry Camera Model Stereo Triangulation 3D Reconstruction by Space Carving 3D Shape from X means getting 3D coordinates
More informationCamera calibration. Robotic vision. Ville Kyrki
Camera calibration Robotic vision 19.1.2017 Where are we? Images, imaging Image enhancement Feature extraction and matching Image-based tracking Camera models and calibration Pose estimation Motion analysis
More informationMETR Robotics Tutorial 2 Week 2: Homogeneous Coordinates
METR4202 -- Robotics Tutorial 2 Week 2: Homogeneous Coordinates The objective of this tutorial is to explore homogenous transformations. The MATLAB robotics toolbox developed by Peter Corke might be a
More informationCamera model and multiple view geometry
Chapter Camera model and multiple view geometry Before discussing how D information can be obtained from images it is important to know how images are formed First the camera model is introduced and then
More informationMultiple View Geometry in computer vision
Multiple View Geometry in computer vision Chapter 8: More Single View Geometry Olaf Booij Intelligent Systems Lab Amsterdam University of Amsterdam, The Netherlands HZClub 29-02-2008 Overview clubje Part
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 More on Single View Geometry Lecture 11 2 In Chapter 5 we introduced projection matrix (which
More informationCamera Calibration. Schedule. Jesus J Caban. Note: You have until next Monday to let me know. ! Today:! Camera calibration
Camera Calibration Jesus J Caban Schedule! Today:! Camera calibration! Wednesday:! Lecture: Motion & Optical Flow! Monday:! Lecture: Medical Imaging! Final presentations:! Nov 29 th : W. Griffin! Dec 1
More information3D Sensing. 3D Shape from X. Perspective Geometry. Camera Model. Camera Calibration. General Stereo Triangulation.
3D Sensing 3D Shape from X Perspective Geometry Camera Model Camera Calibration General Stereo Triangulation 3D Reconstruction 3D Shape from X shading silhouette texture stereo light striping motion mainly
More informationVision Review: Image Formation. Course web page:
Vision Review: Image Formation Course web page: www.cis.udel.edu/~cer/arv September 10, 2002 Announcements Lecture on Thursday will be about Matlab; next Tuesday will be Image Processing The dates some
More informationCHAPTER 3. Single-view Geometry. 1. Consequences of Projection
CHAPTER 3 Single-view Geometry When we open an eye or take a photograph, we see only a flattened, two-dimensional projection of the physical underlying scene. The consequences are numerous and startling.
More informationProjective geometry, camera models and calibration
Projective geometry, camera models and calibration Subhashis Banerjee Dept. Computer Science and Engineering IIT Delhi email: suban@cse.iitd.ac.in January 6, 2008 The main problems in computer vision Image
More informationGeometric camera models and calibration
Geometric camera models and calibration http://graphics.cs.cmu.edu/courses/15-463 15-463, 15-663, 15-862 Computational Photography Fall 2018, Lecture 13 Course announcements Homework 3 is out. - Due October
More informationBIL Computer Vision Apr 16, 2014
BIL 719 - Computer Vision Apr 16, 2014 Binocular Stereo (cont d.), Structure from Motion Aykut Erdem Dept. of Computer Engineering Hacettepe University Slide credit: S. Lazebnik Basic stereo matching algorithm
More information(Geometric) Camera Calibration
(Geoetric) Caera Calibration CS635 Spring 217 Daniel G. Aliaga Departent of Coputer Science Purdue University Caera Calibration Caeras and CCDs Aberrations Perspective Projection Calibration Caeras First
More informationIndex. 3D reconstruction, point algorithm, point algorithm, point algorithm, point algorithm, 263
Index 3D reconstruction, 125 5+1-point algorithm, 284 5-point algorithm, 270 7-point algorithm, 265 8-point algorithm, 263 affine point, 45 affine transformation, 57 affine transformation group, 57 affine
More informationIndex. 3D reconstruction, point algorithm, point algorithm, point algorithm, point algorithm, 253
Index 3D reconstruction, 123 5+1-point algorithm, 274 5-point algorithm, 260 7-point algorithm, 255 8-point algorithm, 253 affine point, 43 affine transformation, 55 affine transformation group, 55 affine
More informationOutline. ETN-FPI Training School on Plenoptic Sensing
Outline Introduction Part I: Basics of Mathematical Optimization Linear Least Squares Nonlinear Optimization Part II: Basics of Computer Vision Camera Model Multi-Camera Model Multi-Camera Calibration
More informationComputer Vision cmput 428/615
Computer Vision cmput 428/615 Basic 2D and 3D geometry and Camera models Martin Jagersand The equation of projection Intuitively: How do we develop a consistent mathematical framework for projection calculations?
More informationMultiple View Geometry in Computer Vision
http://www.cs.unc.edu/~marc/ Multiple View Geometry in Computer Vision 2011.09. 2 3 4 5 Visual 3D Modeling from Images 6 AutoStitch http://cs.bath.ac.uk/brown/autostitch/autostitch.html 7 Image Composite
More informationRobotics - Projective Geometry and Camera model. Matteo Pirotta
Robotics - Projective Geometry and Camera model Matteo Pirotta pirotta@elet.polimi.it Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano 14 March 2013 Inspired from Simone
More informationComputer Vision Projective Geometry and Calibration. Pinhole cameras
Computer Vision Projective Geometry and Calibration Professor Hager http://www.cs.jhu.edu/~hager Jason Corso http://www.cs.jhu.edu/~jcorso. Pinhole cameras Abstract camera model - box with a small hole
More informationRobot Vision: Projective Geometry
Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals Understand homogeneous coordinates Understand points, line, plane parameters and interpret them geometrically
More informationCoordinate Transformations, Tracking, Camera and Tool Calibration
Coordinate Transformations, Tracking, Camera and Tool Calibration Tassilo Klein, Hauke Heibel Computer Aided Medical Procedures (CAMP), Technische Universität München, Germany Motivation 3D Transformations
More informationComputer Vision: Lecture 3
Computer Vision: Lecture 3 Carl Olsson 2019-01-29 Carl Olsson Computer Vision: Lecture 3 2019-01-29 1 / 28 Todays Lecture Camera Calibration The inner parameters - K. Projective vs. Euclidean Reconstruction.
More informationAuto-calibration. Computer Vision II CSE 252B
Auto-calibration Computer Vision II CSE 252B 2D Affine Rectification Solve for planar projective transformation that maps line (back) to line at infinity Solve as a Householder matrix Euclidean Projective
More informationImage Transformations & Camera Calibration. Mašinska vizija, 2018.
Image Transformations & Camera Calibration Mašinska vizija, 2018. Image transformations What ve we learnt so far? Example 1 resize and rotate Open warp_affine_template.cpp Perform simple resize
More informationCS 664 Slides #9 Multi-Camera Geometry. Prof. Dan Huttenlocher Fall 2003
CS 664 Slides #9 Multi-Camera Geometry Prof. Dan Huttenlocher Fall 2003 Pinhole Camera Geometric model of camera projection Image plane I, which rays intersect Camera center C, through which all rays pass
More informationStructure from Motion
11/18/11 Structure from Motion Computer Vision CS 143, Brown James Hays Many slides adapted from Derek Hoiem, Lana Lazebnik, Silvio Saverese, Steve Seitz, and Martial Hebert This class: structure from
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Projective 3D Geometry (Back to Chapter 2) Lecture 6 2 Singular Value Decomposition Given a
More informationStructure from Motion CSC 767
Structure from Motion CSC 767 Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R,t R 2,t 2 R 3,t 3 Camera??
More informationComputer Vision I - Appearance-based Matching and Projective Geometry
Computer Vision I - Appearance-based Matching and Projective Geometry Carsten Rother 05/11/2015 Computer Vision I: Image Formation Process Roadmap for next four lectures Computer Vision I: Image Formation
More informationComputer Vision Projective Geometry and Calibration. Pinhole cameras
Computer Vision Projective Geometry and Calibration Professor Hager http://www.cs.jhu.edu/~hager Jason Corso http://www.cs.jhu.edu/~jcorso. Pinhole cameras Abstract camera model - box with a small hole
More informationPin Hole Cameras & Warp Functions
Pin Hole Cameras & Warp Functions Instructor - Simon Lucey 16-423 - Designing Computer Vision Apps Today Pinhole Camera. Homogenous Coordinates. Planar Warp Functions. Motivation Taken from: http://img.gawkerassets.com/img/18w7i1umpzoa9jpg/original.jpg
More informationCOSC579: Scene Geometry. Jeremy Bolton, PhD Assistant Teaching Professor
COSC579: Scene Geometry Jeremy Bolton, PhD Assistant Teaching Professor Overview Linear Algebra Review Homogeneous vs non-homogeneous representations Projections and Transformations Scene Geometry The
More informationCamera Geometry II. COS 429 Princeton University
Camera Geometry II COS 429 Princeton University Outline Projective geometry Vanishing points Application: camera calibration Application: single-view metrology Epipolar geometry Application: stereo correspondence
More informationBasic principles - Geometry. Marc Pollefeys
Basic principles - Geometr Marc Pollees Basic principles - Geometr Projective geometr Projective, Aine, Homograph Pinhole camera model and triangulation Epipolar geometr Essential and Fundamental Matri
More informationProjective Geometry and Camera Models
/2/ Projective Geometry and Camera Models Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem Note about HW Out before next Tues Prob: covered today, Tues Prob2: covered next Thurs Prob3:
More informationCOS429: COMPUTER VISON CAMERAS AND PROJECTIONS (2 lectures)
COS429: COMPUTER VISON CMERS ND PROJECTIONS (2 lectures) Pinhole cameras Camera with lenses Sensing nalytical Euclidean geometry The intrinsic parameters of a camera The extrinsic parameters of a camera
More informationDD2423 Image Analysis and Computer Vision IMAGE FORMATION. Computational Vision and Active Perception School of Computer Science and Communication
DD2423 Image Analysis and Computer Vision IMAGE FORMATION Mårten Björkman Computational Vision and Active Perception School of Computer Science and Communication November 8, 2013 1 Image formation Goal:
More informationMultiple View Geometry in Computer Vision Second Edition
Multiple View Geometry in Computer Vision Second Edition Richard Hartley Australian National University, Canberra, Australia Andrew Zisserman University of Oxford, UK CAMBRIDGE UNIVERSITY PRESS Contents
More informationPinhole Camera Model 10/05/17. Computational Photography Derek Hoiem, University of Illinois
Pinhole Camera Model /5/7 Computational Photography Derek Hoiem, University of Illinois Next classes: Single-view Geometry How tall is this woman? How high is the camera? What is the camera rotation? What
More informationRigid Body Motion and Image Formation. Jana Kosecka, CS 482
Rigid Body Motion and Image Formation Jana Kosecka, CS 482 A free vector is defined by a pair of points : Coordinates of the vector : 1 3D Rotation of Points Euler angles Rotation Matrices in 3D 3 by 3
More informationCamera Model and Calibration
Camera Model and Calibration Lecture-10 Camera Calibration Determine extrinsic and intrinsic parameters of camera Extrinsic 3D location and orientation of camera Intrinsic Focal length The size of the
More informationAgenda. Rotations. Camera calibration. Homography. Ransac
Agenda Rotations Camera calibration Homography Ransac Geometric Transformations y x Transformation Matrix # DoF Preserves Icon translation rigid (Euclidean) similarity affine projective h I t h R t h sr
More informationCoplanar circles, quasi-affine invariance and calibration
Image and Vision Computing 24 (2006) 319 326 www.elsevier.com/locate/imavis Coplanar circles, quasi-affine invariance and calibration Yihong Wu *, Xinju Li, Fuchao Wu, Zhanyi Hu National Laboratory of
More informationEpipolar Geometry class 11
Epipolar Geometry class 11 Multiple View Geometry Comp 290-089 Marc Pollefeys Multiple View Geometry course schedule (subject to change) Jan. 7, 9 Intro & motivation Projective 2D Geometry Jan. 14, 16
More information3D reconstruction class 11
3D reconstruction class 11 Multiple View Geometry Comp 290-089 Marc Pollefeys Multiple View Geometry course schedule (subject to change) Jan. 7, 9 Intro & motivation Projective 2D Geometry Jan. 14, 16
More informationComputer Vision. Coordinates. Prof. Flávio Cardeal DECOM / CEFET- MG.
Computer Vision Coordinates Prof. Flávio Cardeal DECOM / CEFET- MG cardeal@decom.cefetmg.br Abstract This lecture discusses world coordinates and homogeneous coordinates, as well as provides an overview
More informationLecture 9: Epipolar Geometry
Lecture 9: Epipolar Geometry Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F (Problem Set 2
More informationCamera Calibration from the Quasi-affine Invariance of Two Parallel Circles
Camera Calibration from the Quasi-affine Invariance of Two Parallel Circles Yihong Wu, Haijiang Zhu, Zhanyi Hu, and Fuchao Wu National Laboratory of Pattern Recognition, Institute of Automation, Chinese
More informationCS201 Computer Vision Camera Geometry
CS201 Computer Vision Camera Geometry John Magee 25 November, 2014 Slides Courtesy of: Diane H. Theriault (deht@bu.edu) Question of the Day: How can we represent the relationships between cameras and the
More informationStructure from Motion
Structure from Motion Outline Bundle Adjustment Ambguities in Reconstruction Affine Factorization Extensions Structure from motion Recover both 3D scene geoemetry and camera positions SLAM: Simultaneous
More informationProjective geometry for 3D Computer Vision
Subhashis Banerjee Computer Science and Engineering IIT Delhi Dec 16, 2015 Overview Pin-hole camera Why projective geometry? Reconstruction Computer vision geometry: main problems Correspondence problem:
More informationCamera Projection Models We will introduce different camera projection models that relate the location of an image point to the coordinates of the
Camera Projection Models We will introduce different camera projection models that relate the location of an image point to the coordinates of the corresponding 3D points. The projection models include:
More informationFlexible Calibration of a Portable Structured Light System through Surface Plane
Vol. 34, No. 11 ACTA AUTOMATICA SINICA November, 2008 Flexible Calibration of a Portable Structured Light System through Surface Plane GAO Wei 1 WANG Liang 1 HU Zhan-Yi 1 Abstract For a portable structured
More informationLecture 10: Multi-view geometry
Lecture 10: Multi-view geometry Professor Stanford Vision Lab 1 What we will learn today? Review for stereo vision Correspondence problem (Problem Set 2 (Q3)) Active stereo vision systems Structure from
More informationStructure from Motion
/8/ Structure from Motion Computer Vision CS 43, Brown James Hays Many slides adapted from Derek Hoiem, Lana Lazebnik, Silvio Saverese, Steve Seitz, and Martial Hebert This class: structure from motion
More informationIntroduction to Homogeneous coordinates
Last class we considered smooth translations and rotations of the camera coordinate system and the resulting motions of points in the image projection plane. These two transformations were expressed mathematically
More informationDiscriminant Functions for the Normal Density
Announcements Project Meetings CSE Rm. 4120 HW1: Not yet posted Pattern classification & Image Formation CSE 190 Lecture 5 CSE190d, Winter 2011 CSE190d, Winter 2011 Discriminant Functions for the Normal
More informationRobotics - Projective Geometry and Camera model. Marcello Restelli
Robotics - Projective Geometr and Camera model Marcello Restelli marcello.restelli@polimi.it Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano Ma 2013 Inspired from Matteo
More informationPin Hole Cameras & Warp Functions
Pin Hole Cameras & Warp Functions Instructor - Simon Lucey 16-423 - Designing Computer Vision Apps Today Pinhole Camera. Homogenous Coordinates. Planar Warp Functions. Example of SLAM for AR Taken from:
More informationEpipolar geometry. x x
Two-view geometry Epipolar geometry X x x Baseline line connecting the two camera centers Epipolar Plane plane containing baseline (1D family) Epipoles = intersections of baseline with image planes = projections
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribe: Sameer Agarwal LECTURE 1 Image Formation 1.1. The geometry of image formation We begin by considering the process of image formation when a
More informationModule 4F12: Computer Vision and Robotics Solutions to Examples Paper 2
Engineering Tripos Part IIB FOURTH YEAR Module 4F2: Computer Vision and Robotics Solutions to Examples Paper 2. Perspective projection and vanishing points (a) Consider a line in 3D space, defined in camera-centered
More information3-D D Euclidean Space - Vectors
3-D D Euclidean Space - Vectors Rigid Body Motion and Image Formation A free vector is defined by a pair of points : Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Coordinates of the vector : 3D Rotation
More informationToday. Stereo (two view) reconstruction. Multiview geometry. Today. Multiview geometry. Computational Photography
Computational Photography Matthias Zwicker University of Bern Fall 2009 Today From 2D to 3D using multiple views Introduction Geometry of two views Stereo matching Other applications Multiview geometry
More informationMachine vision. Summary # 11: Stereo vision and epipolar geometry. u l = λx. v l = λy
1 Machine vision Summary # 11: Stereo vision and epipolar geometry STEREO VISION The goal of stereo vision is to use two cameras to capture 3D scenes. There are two important problems in stereo vision:
More informationPart I: Single and Two View Geometry Internal camera parameters
!! 43 1!???? Imaging eometry Multiple View eometry Perspective projection Richard Hartley Andrew isserman O p y VPR June 1999 where image plane This can be written as a linear mapping between homogeneous
More informationColorado School of Mines. Computer Vision. Professor William Hoff Dept of Electrical Engineering &Computer Science.
Professor Willia Hoff Dept of Electrical Engineering &Coputer Science http://inside.ines.edu/~whoff/ 1 Caera Calibration 2 Caera Calibration Needed for ost achine vision and photograetry tasks (object
More informationC / 35. C18 Computer Vision. David Murray. dwm/courses/4cv.
C18 2015 1 / 35 C18 Computer Vision David Murray david.murray@eng.ox.ac.uk www.robots.ox.ac.uk/ dwm/courses/4cv Michaelmas 2015 C18 2015 2 / 35 Computer Vision: This time... 1. Introduction; imaging geometry;
More informationRobust Geometry Estimation from two Images
Robust Geometry Estimation from two Images Carsten Rother 09/12/2016 Computer Vision I: Image Formation Process Roadmap for next four lectures Computer Vision I: Image Formation Process 09/12/2016 2 Appearance-based
More informationAgenda. Rotations. Camera models. Camera calibration. Homographies
Agenda Rotations Camera models Camera calibration Homographies D Rotations R Y = Z r r r r r r r r r Y Z Think of as change of basis where ri = r(i,:) are orthonormal basis vectors r rotated coordinate
More informationProjective Geometry and Camera Models
Projective Geometry and Camera Models Computer Vision CS 43 Brown James Hays Slides from Derek Hoiem, Alexei Efros, Steve Seitz, and David Forsyth Administrative Stuff My Office hours, CIT 375 Monday and
More informationProjective 2D Geometry
Projective D Geometry Multi View Geometry (Spring '08) Projective D Geometry Prof. Kyoung Mu Lee SoEECS, Seoul National University Homogeneous representation of lines and points Projective D Geometry Line
More informationCamera Calibration from Surfaces of Revolution
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. Y, MONTH YEAR 1 Camera Calibration from Surfaces of Revolution Kwan-Yee K. Wong, Paulo R. S. Mendonça and Roberto Cipolla Kwan-Yee
More informationCS4670: Computer Vision
CS467: Computer Vision Noah Snavely Lecture 13: Projection, Part 2 Perspective study of a vase by Paolo Uccello Szeliski 2.1.3-2.1.6 Reading Announcements Project 2a due Friday, 8:59pm Project 2b out Friday
More informationCritical Motion Sequences for the Self-Calibration of Cameras and Stereo Systems with Variable Focal Length
Critical Motion Sequences for the Self-Calibration of Cameras and Stereo Systems with Variable Focal Length Peter F Sturm Computational Vision Group, Department of Computer Science The University of Reading,
More informationCS231A Midterm Review. Friday 5/6/2016
CS231A Midterm Review Friday 5/6/2016 Outline General Logistics Camera Models Non-perspective cameras Calibration Single View Metrology Epipolar Geometry Structure from Motion Active Stereo and Volumetric
More informationPart 0. The Background: Projective Geometry, Transformations and Estimation
Part 0 The Background: Projective Geometry, Transformations and Estimation La reproduction interdite (The Forbidden Reproduction), 1937, René Magritte. Courtesy of Museum Boijmans van Beuningen, Rotterdam.
More informationLecture 3: Camera Calibration, DLT, SVD
Computer Vision Lecture 3 23--28 Lecture 3: Camera Calibration, DL, SVD he Inner Parameters In this section we will introduce the inner parameters of the cameras Recall from the camera equations λx = P
More informationCamera Self-calibration Based on the Vanishing Points*
Camera Self-calibration Based on the Vanishing Points* Dongsheng Chang 1, Kuanquan Wang 2, and Lianqing Wang 1,2 1 School of Computer Science and Technology, Harbin Institute of Technology, Harbin 150001,
More informationRecovering structure from a single view Pinhole perspective projection
EPIPOLAR GEOMETRY The slides are from several sources through James Hays (Brown); Silvio Savarese (U. of Michigan); Svetlana Lazebnik (U. Illinois); Bill Freeman and Antonio Torralba (MIT), including their
More information3D Vision Real Objects, Real Cameras. Chapter 11 (parts of), 12 (parts of) Computerized Image Analysis MN2 Anders Brun,
3D Vision Real Objects, Real Cameras Chapter 11 (parts of), 12 (parts of) Computerized Image Analysis MN2 Anders Brun, anders@cb.uu.se 3D Vision! Philisophy! Image formation " The pinhole camera " Projective
More informationStructure from motion
Structure from motion Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R 1,t 1 R 2,t 2 R 3,t 3 Camera 1 Camera
More informationCamera Calibration using Vanishing Points
Camera Calibration using Vanishing Points Paul Beardsley and David Murray * Department of Engineering Science, University of Oxford, Oxford 0X1 3PJ, UK Abstract This paper describes a methodformeasuringthe
More information