Calculation of the fundamental matrix
|
|
- Bartholomew Holmes
- 6 years ago
- Views:
Transcription
1 D reconstruction alculation of the fundamental matrix Given a set of point correspondences x i x i we wish to reconstruct the world points X i and the cameras P, P such that x i = PX i, x i = P X i, i Without any additional information it is possible to 1 Estimate the fundamental matrix from point correspondences alculate camera pairs from the fundamental matrix Estimate world coordinates X i corresponding to each point pair x i x i The reconstructed cameras and points will be unique up to a projective homography of P Depending on which information we have about the world andor the cameras (point coordinates, orthogonal line pairs, parallel line pairs, calibrated cameras, etc) we may either perform a stratified reconstruction first projectively, then affinely, the metrically, or directly perform a metric reconstruction The defining equation for the fundamental matrix is x Fx = for each pair of corresponding points x x Given enough corresponding points we may calculated F Each point pair produces one equation linear in the elements in F If x = (x, y, 1) and x = (x, y, 1) then x xf 11 +x yf 1 +x f 1 +y xf 1 +y yf +y f +xf 1 +yf +f =, that may be written as [ x x x y x y x y y y x y 1 where f is the 9-vector with elements F row-wise ] f =, p 1 alculation of the fundamental matrixf Given n corresponding points we get a linear equation system on the form Af = x 1x 1 x 1y 1 x 1 y 1x 1 y 1y 1 y 1 x 1 y 1 1 x nx n x ny n x n y nx n y ny n y n x n y n 1 5 f = The equation is homogenous, ie f can only be detered up to scale In order for a homogenous solution to exist, the rank of A cannot be larger than 8 In that case a solution f exists in the null-space of A A solution may always be detered by solving f Af with solution f = v 9 where A = UDV st f = 1, Minimal correspondence the -point algorit If the matrix A is constructed from n = correspondences, A will have a two-dimensional null-space Let the vectors f 1 and f be basis vectors for the null-space of A Then each vector f = αf 1 + (1 α)f is a solution of Af = the corresponding F-matrices are F = αf 1 + (1 α)f The condition det(f) = leads to the equation det(αf 1 + (1 α)f ) =, that is a third order equation in α with one or three real roots, ie one or three solutions are possible p
2 The -point algorithm, solutions The 8-point algorithm The easiest way to calculate the fundamental matrix is to use n 8 points and 1 alculate F that imizes the algebraic error, ie solves f Af st f = 1 Find the closest matrix F of rank, ie solve If A = UDV is the singular value decomposition of A, then the solution of problem 1 is f = v 9 If F = UDV is the singular value decomposition of F, where then D = diag(r,s, t),r s t, F F F F st rank(f ) =, where F is the Frobenius norm F = U diag(r,s, )V is a solution to problem p 5 Non rank-deficientfmatrix If F has full rank it will have an empty null-space, ie not have any point that is on all lines, ie no epipole rank(f) = rank(f) = The normalized 8-point algorithm In order for the 8-point algorithm to work in practice, it has to be normalized The algorithm then becomes: 1 Detere a transformationtand T such that ˆx i = Tx i and ˆx i = Tx i has center of gravity at the origin and a mean squared distance of from the origin alculate a fundamental matrix ˆF corresponding to the point pairs ˆx i ˆx i : (a) alculate ˆF from the solution ˆf that solves ˆf st ˆf = 1, where  is calculated from the point pairs ˆx h ˆx i (b) alculate ˆF that solves ˆF ˆF ˆF F st rank(ˆf ) = p alculatef = T ˆF T as the fundamental matrix corresponding to the original point pairs x i x i
3 OptimalF OptimalF In order to detere an optimal F, the reprojection error has to be imized This may be achieved through eg the following problem formulation: where u m d(x i, ˆx i ) + d(x i, ˆx i) i=1 st ˆx i Fˆx i =, i = 1,,m F F = 1, det(f) =, u = [f 1,,f 9, ˆx 1x, ˆx 1y,, ˆx mx, ˆx my, ˆx 1x, ˆx 1y,, ˆx mx, ˆx my] Given an algorithm to solve we get f(u) = ˆx 1 x 1 ˆx m x m ˆx 1 x 1 ˆx m x m 1 u f(u)t Wf(u) st c(u) = and c(u) = 5 ˆx 1 Fˆx 1 ˆx mfˆx m F F 1 det(f) 5 A starting approximation of F can be calculated by the normalized 8-point algorithm The initial estimates of the line points ˆx i and ˆx i are the corresponding measured points x i and x i p 9 OptimalF, normal example OptimalF, normal example res rms=85 line rms=8e 1 Starting approximation x 1 norm constraint=e 1 x 1 det constraint= 5e 1 One iteration 1 1 Solution Iteration sequence p 11
4 Automatic calculation off Example By using RANSA we may estimate F automatically Given n preliary point matches, a probability p and a distance threshold t: Draw point pairs randomly alculate a fundamental matrix from the point pairs We have 1 or solutions For each of the solutions alculate the distance d i between each point pair and the corresponding epipolar lines alculate the number of point pairs k i that are inliers, ie with distance d i < t If k i > k best or k i = k best and σ(d i ) < σ best then best = i, ɛ = k best (1 n), N max = i = i + 1 Repeat until i N max log 1 p log(1 (1 ɛ) ) Then refine the best solution by imizing the reprojection error Add correspondences that now satisfies d i < t Optionally repeat the previous step Suggested matches After calculation of F with RANSA After manual cleaning p 1 lculation of D coordinates, homogenous solution alculation of D coordinates, homogenous solutio Given a corresponding point pair x x and two camerasp,p, calculate the corresponding D point X such that x = PX and x = P X As previously with homography calculations we may dehomogenize the image points in eg image 1 and get x (PX) = or The equation systemax = is overdetered in the sense that X has degrees of freedom, but we have independent equations If the points x and x do not correspond exactly, X = will be the only solution x(p X) (p 1 X) = y(p X) (p X) = x(p X) y(p 1 X) = x Of these, only two are linearly independent If we remove the third and combine with the point in image we get AX =, where A = xp p 1 yp p x p p 1 y p p 5 l = F x l = F x e e image 1 image We thus have to solve AX = approximately, either via SVD A = UDV = A, X = v or by specifying X = (X, Y, Z, 1) p 15
5 alculation of D coordinates, Euclidian solution Minimization of the reprojection error We may also choose to imize the Euclidian distance in R If camera 1 has center in and we know another point Ũ on the line U = P+ x, then a point X on the line satisfies The optimal solution is obtained if we imize the reprojection error, ie solves X Similarly in image X = + α(ũ ) X = + α (Ũ ), where U = P + x d(x, ˆx,ˆx,X ˆx) + d(x, ˆx ) st ˆx = PX ˆx = P X x d d e e If the lines do not intersect there is no point X that satisfies both equations However, then we may imize the distance to the lines α,α, X # "Ũ α " I Ũ α 5 I X This method is called forward intersection in the photogrammetric community # which exactly corresponds to d(x, ˆx,ˆx ˆx) + d(x, ˆx ) st ˆx Fˆx = If we have detered F by imizing the reprojection error, the corresponding points have already been detered and the SVD solution of AX = gives the nullspace solution p 1 Example Starting approximate from the normalized 8-point algorithm, via the essential matrix and forward intersection One iteration Solution p 19
Multiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Structure Computation Lecture 18 March 22, 2005 2 3D Reconstruction The goal of 3D reconstruction
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 informationCS231A Course Notes 4: Stereo Systems and Structure from Motion
CS231A Course Notes 4: Stereo Systems and Structure from Motion Kenji Hata and Silvio Savarese 1 Introduction In the previous notes, we covered how adding additional viewpoints of a scene can greatly enhance
More informationTwo-View Geometry (Course 23, Lecture D)
Two-View Geometry (Course 23, Lecture D) Jana Kosecka Department of Computer Science George Mason University http://www.cs.gmu.edu/~kosecka General Formulation Given two views of the scene recover the
More informationCS 231A: Computer Vision (Winter 2018) Problem Set 2
CS 231A: Computer Vision (Winter 2018) Problem Set 2 Due Date: Feb 09 2018, 11:59pm Note: In this PS, using python2 is recommended, as the data files are dumped with python2. Using python3 might cause
More information55:148 Digital Image Processing Chapter 11 3D Vision, Geometry
55:148 Digital Image Processing Chapter 11 3D Vision, Geometry Topics: Basics of projective geometry Points and hyperplanes in projective space Homography Estimating homography from point correspondence
More information55:148 Digital Image Processing Chapter 11 3D Vision, Geometry
55:148 Digital Image Processing Chapter 11 3D Vision, Geometry Topics: Basics of projective geometry Points and hyperplanes in projective space Homography Estimating homography from point correspondence
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 informationFundamental Matrix & Structure from Motion
Fundamental Matrix & Structure from Motion Instructor - Simon Lucey 16-423 - Designing Computer Vision Apps Today Transformations between images Structure from Motion The Essential Matrix The Fundamental
More informationCSE 252B: Computer Vision II
CSE 5B: Computer Vision II Lecturer: Ben Ochoa Scribes: San Nguyen, Ben Reineman, Arturo Flores LECTURE Guest Lecture.. Brief bio Ben Ochoa received the B.S., M.S., and Ph.D. degrees in electrical engineering
More information1 Projective Geometry
CIS8, Machine Perception Review Problem - SPRING 26 Instructions. All coordinate systems are right handed. Projective Geometry Figure : Facade rectification. I took an image of a rectangular object, and
More informationReminder: Lecture 20: The Eight-Point Algorithm. Essential/Fundamental Matrix. E/F Matrix Summary. Computing F. Computing F from Point Matches
Reminder: Lecture 20: The Eight-Point Algorithm F = -0.00310695-0.0025646 2.96584-0.028094-0.00771621 56.3813 13.1905-29.2007-9999.79 Readings T&V 7.3 and 7.4 Essential/Fundamental Matrix E/F Matrix Summary
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 informationComputer Vision I - Robust Geometry Estimation from two Cameras
Computer Vision I - Robust Geometry Estimation from two Cameras Carsten Rother 16/01/2015 Computer Vision I: Image Formation Process FYI Computer Vision I: Image Formation Process 16/01/2015 2 Microsoft
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 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 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 informationMulti-View Geometry Part II (Ch7 New book. Ch 10/11 old book)
Multi-View Geometry Part II (Ch7 New book. Ch 10/11 old book) Guido Gerig CS-GY 6643, Spring 2016 gerig@nyu.edu Credits: M. Shah, UCF CAP5415, lecture 23 http://www.cs.ucf.edu/courses/cap6411/cap5415/,
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 informationLecture 5 Epipolar Geometry
Lecture 5 Epipolar Geometry Professor Silvio Savarese Computational Vision and Geometry Lab Silvio Savarese Lecture 5-24-Jan-18 Lecture 5 Epipolar Geometry Why is stereo useful? Epipolar constraints Essential
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 informationProject 2: Structure from Motion
Project 2: Structure from Motion CIS 580, Machine Perception, Spring 2015 Preliminary report due: 2015.04.27. 11:59AM Final Due: 2015.05.06. 11:59AM This project aims to reconstruct a 3D point cloud and
More informationProject: Camera Rectification and Structure from Motion
Project: Camera Rectification and Structure from Motion CIS 580, Machine Perception, Spring 2018 April 18, 2018 In this project, you will learn how to estimate the relative poses of two cameras and compute
More informationEpipolar Geometry and Stereo Vision
Epipolar Geometry and Stereo Vision Computer Vision Jia-Bin Huang, Virginia Tech Many slides from S. Seitz and D. Hoiem Last class: Image Stitching Two images with rotation/zoom but no translation. X x
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 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 informationMultiple Views Geometry
Multiple Views Geometry Subhashis Banerjee Dept. Computer Science and Engineering IIT Delhi email: suban@cse.iitd.ac.in January 2, 28 Epipolar geometry Fundamental geometric relationship between two perspective
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 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 - 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 information3D Reconstruction from Two Views
3D Reconstruction from Two Views Huy Bui UIUC huybui1@illinois.edu Yiyi Huang UIUC huang85@illinois.edu Abstract In this project, we study a method to reconstruct a 3D scene from two views. First, we extract
More information3D Photography: Epipolar geometry
3D Photograph: Epipolar geometr Kalin Kolev, Marc Pollefes Spring 203 http://cvg.ethz.ch/teaching/203spring/3dphoto/ Schedule (tentative) Feb 8 Feb 25 Mar 4 Mar Mar 8 Mar 25 Apr Apr 8 Apr 5 Apr 22 Apr
More informationEpipolar Geometry Prof. D. Stricker. With slides from A. Zisserman, S. Lazebnik, Seitz
Epipolar Geometry Prof. D. Stricker With slides from A. Zisserman, S. Lazebnik, Seitz 1 Outline 1. Short introduction: points and lines 2. Two views geometry: Epipolar geometry Relation point/line in two
More informationMAPI Computer Vision. Multiple View Geometry
MAPI Computer Vision Multiple View Geometry Geometry o Multiple Views 2- and 3- view geometry p p Kpˆ [ K R t]p Geometry o Multiple Views 2- and 3- view geometry Epipolar Geometry The epipolar geometry
More informationFundamental Matrix & Structure from Motion
Fundamental Matrix & Structure from Motion Instructor - Simon Lucey 16-423 - Designing Computer Vision Apps Today Review of Assignment 0. Transformations between images Structure from Motion The Essential
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 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 informationLinear Multi View Reconstruction and Camera Recovery Using a Reference Plane
International Journal of Computer Vision 49(2/3), 117 141, 2002 c 2002 Kluwer Academic Publishers. Manufactured in The Netherlands. Linear Multi View Reconstruction and Camera Recovery Using a Reference
More informationTwo-view geometry Computer Vision Spring 2018, Lecture 10
Two-view geometry http://www.cs.cmu.edu/~16385/ 16-385 Computer Vision Spring 2018, Lecture 10 Course announcements Homework 2 is due on February 23 rd. - Any questions about the homework? - How many of
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 informationVision par ordinateur
Epipolar geometry π Vision par ordinateur Underlying structure in set of matches for rigid scenes l T 1 l 2 C1 m1 l1 e1 M L2 L1 e2 Géométrie épipolaire Fundamental matrix (x rank 2 matrix) m2 C2 l2 Frédéric
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 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 informationProject: Camera Rectification and Structure from Motion
Project: Camera Rectification and Structure from Motion CIS 580, Machine Perception, Spring 2018 April 26, 2018 In this project, you will learn how to estimate the relative poses of two cameras and compute
More informationMinimizing Algebraic Error
Minimizing Algebraic Richard I. Hartley, G.E. Corporate Research and Development Research Circle Niskayuna, NY 39 Abstract This paper gives a widely applicable technique for solving many of the parameter
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 informationCS231M Mobile Computer Vision Structure from motion
CS231M Mobile Computer Vision Structure from motion - Cameras - Epipolar geometry - Structure from motion Pinhole camera Pinhole perspective projection f o f = focal length o = center of the camera z y
More informationStereo and Epipolar geometry
Previously Image Primitives (feature points, lines, contours) Today: Stereo and Epipolar geometry How to match primitives between two (multiple) views) Goals: 3D reconstruction, recognition Jana Kosecka
More informationLast lecture. Passive Stereo Spacetime Stereo
Last lecture Passive Stereo Spacetime Stereo Today Structure from Motion: Given pixel correspondences, how to compute 3D structure and camera motion? Slides stolen from Prof Yungyu Chuang Epipolar geometry
More informationStructure from motion
Multi-view geometry Structure rom motion Camera 1 Camera 2 R 1,t 1 R 2,t 2 Camera 3 R 3,t 3 Figure credit: Noah Snavely Structure rom motion? Camera 1 Camera 2 R 1,t 1 R 2,t 2 Camera 3 R 3,t 3 Structure:
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 informationComputing F class 13. Multiple View Geometry. Comp Marc Pollefeys
Computing F class 3 Multiple View Geometr Comp 90-089 Marc Pollefes Multiple View Geometr course schedule (subject to change) Jan. 7, 9 Intro & motivation Projective D Geometr Jan. 4, 6 (no class) Projective
More informationStereo Vision. MAN-522 Computer Vision
Stereo Vision MAN-522 Computer Vision What is the goal of stereo vision? The recovery of the 3D structure of a scene using two or more images of the 3D scene, each acquired from a different viewpoint in
More informationUndergrad HTAs / TAs. Help me make the course better! HTA deadline today (! sorry) TA deadline March 21 st, opens March 15th
Undergrad HTAs / TAs Help me make the course better! HTA deadline today (! sorry) TA deadline March 2 st, opens March 5th Project 2 Well done. Open ended parts, lots of opportunity for mistakes. Real implementation
More informationThe end of affine cameras
The end of affine cameras Affine SFM revisited Epipolar geometry Two-view structure from motion Multi-view structure from motion Planches : http://www.di.ens.fr/~ponce/geomvis/lect3.pptx http://www.di.ens.fr/~ponce/geomvis/lect3.pdf
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 informationRectification and Distortion Correction
Rectification and Distortion Correction Hagen Spies March 12, 2003 Computer Vision Laboratory Department of Electrical Engineering Linköping University, Sweden Contents Distortion Correction Rectification
More informationUnsupervised learning in Vision
Chapter 7 Unsupervised learning in Vision The fields of Computer Vision and Machine Learning complement each other in a very natural way: the aim of the former is to extract useful information from visual
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 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 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 informationCSCI 5980/8980: Assignment #4. Fundamental Matrix
Submission CSCI 598/898: Assignment #4 Assignment due: March 23 Individual assignment. Write-up submission format: a single PDF up to 5 pages (more than 5 page assignment will be automatically returned.).
More informationEpipolar Geometry and the Essential Matrix
Epipolar Geometry and the Essential Matrix Carlo Tomasi The epipolar geometry of a pair of cameras expresses the fundamental relationship between any two corresponding points in the two image planes, and
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 informationEpipolar Geometry CSE P576. Dr. Matthew Brown
Epipolar Geometry CSE P576 Dr. Matthew Brown Epipolar Geometry Epipolar Lines, Plane Constraint Fundamental Matrix, Linear solution + RANSAC Applications: Structure from Motion, Stereo [ Szeliski 11] 2
More informationWeek 2: Two-View Geometry. Padua Summer 08 Frank Dellaert
Week 2: Two-View Geometry Padua Summer 08 Frank Dellaert Mosaicking Outline 2D Transformation Hierarchy RANSAC Triangulation of 3D Points Cameras Triangulation via SVD Automatic Correspondence Essential
More informationEpipolar Geometry and Stereo Vision
Epipolar Geometry and Stereo Vision Computer Vision Shiv Ram Dubey, IIIT Sri City Many slides from S. Seitz and D. Hoiem Last class: Image Stitching Two images with rotation/zoom but no translation. X
More informationElements of Computer Vision: Multiple View Geometry. 1 Introduction. 2 Elements of Geometry. Andrea Fusiello
Elements of Computer Vision: Multiple View Geometry. Andrea Fusiello http://www.sci.univr.it/~fusiello July 11, 2005 c Copyright by Andrea Fusiello. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike
More informationMultiview Stereo COSC450. Lecture 8
Multiview Stereo COSC450 Lecture 8 Stereo Vision So Far Stereo and epipolar geometry Fundamental matrix captures geometry 8-point algorithm Essential matrix with calibrated cameras 5-point algorithm Intersect
More information3D Reconstruction with two Calibrated Cameras
3D Reconstruction with two Calibrated Cameras Carlo Tomasi The standard reference frame for a camera C is a right-handed Cartesian frame with its origin at the center of projection of C, its positive Z
More informationComputer Graphics. Coordinate Systems and Change of Frames. Based on slides by Dianna Xu, Bryn Mawr College
Computer Graphics Coordinate Systems and Change of Frames Based on slides by Dianna Xu, Bryn Mawr College Linear Independence A set of vectors independent if is linearly If a set of vectors is linearly
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 informationContents. 1 Introduction Background Organization Features... 7
Contents 1 Introduction... 1 1.1 Background.... 1 1.2 Organization... 2 1.3 Features... 7 Part I Fundamental Algorithms for Computer Vision 2 Ellipse Fitting... 11 2.1 Representation of Ellipses.... 11
More informationComputer Vision Projective Geometry and Calibration
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 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 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 informationStructure from Motion. Prof. Marco Marcon
Structure from Motion Prof. Marco Marcon Summing-up 2 Stereo is the most powerful clue for determining the structure of a scene Another important clue is the relative motion between the scene and (mono)
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 informationMulti-view geometry problems
Multi-view geometry Multi-view geometry problems Structure: Given projections o the same 3D point in two or more images, compute the 3D coordinates o that point? Camera 1 Camera 2 R 1,t 1 R 2,t 2 Camera
More informationMultiple Motion Scene Reconstruction from Uncalibrated Views
Multiple Motion Scene Reconstruction from Uncalibrated Views Mei Han C & C Research Laboratories NEC USA, Inc. meihan@ccrl.sj.nec.com Takeo Kanade Robotics Institute Carnegie Mellon University tk@cs.cmu.edu
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 information3D Computer Vision. Structure from Motion. Prof. Didier Stricker
3D Computer Vision Structure from Motion Prof. Didier Stricker Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche Intelligenz http://av.dfki.de 1 Structure
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 informationInterlude: Solving systems of Equations
Interlude: Solving systems of Equations Solving Ax = b What happens to x under Ax? The singular value decomposition Rotation matrices Singular matrices Condition number Null space Solving Ax = 0 under
More informationStructure and motion in 3D and 2D from hybrid matching constraints
Structure and motion in 3D and 2D from hybrid matching constraints Anders Heyden, Fredrik Nyberg and Ola Dahl Applied Mathematics Group Malmo University, Sweden {heyden,fredrik.nyberg,ola.dahl}@ts.mah.se
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 informationThe real voyage of discovery consists not in seeking new landscapes, but in having new eyes.
The real voyage of discovery consists not in seeking new landscapes, but in having new eyes. - Marcel Proust University of Texas at Arlington Camera Calibration (or Resectioning) CSE 4392-5369 Vision-based
More informationLecture 6 Stereo Systems Multi- view geometry Professor Silvio Savarese Computational Vision and Geometry Lab Silvio Savarese Lecture 6-24-Jan-15
Lecture 6 Stereo Systems Multi- view geometry Professor Silvio Savarese Computational Vision and Geometry Lab Silvio Savarese Lecture 6-24-Jan-15 Lecture 6 Stereo Systems Multi- view geometry Stereo systems
More informationCOMP 558 lecture 19 Nov. 17, 2010
COMP 558 lecture 9 Nov. 7, 2 Camera calibration To estimate the geometry of 3D scenes, it helps to know the camera parameters, both external and internal. The problem of finding all these parameters is
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 informationTriangulation from Two Views Revisited: Hartley-Sturm vs. Optimal Correction
Triangulation from Two Views Revisited: Hartley-Sturm vs. Optimal Correction Kenichi Kanatani 1, Yasuyuki Sugaya 2, and Hirotaka Niitsuma 1 1 Department of Computer Science, Okayama University, Okayama
More informationCEE598 - Visual Sensing for Civil Infrastructure Eng. & Mgmt.
CEE598 - Visual Sensing for Civil Infrastructure Eng. & Mgmt. Session 4 Affine Structure from Motion Mani Golparvar-Fard Department of Civil and Environmental Engineering 329D, Newmark Civil Engineering
More informationMultiple-View Structure and Motion From Line Correspondences
ICCV 03 IN PROCEEDINGS OF THE IEEE INTERNATIONAL CONFERENCE ON COMPUTER VISION, NICE, FRANCE, OCTOBER 003. Multiple-View Structure and Motion From Line Correspondences Adrien Bartoli Peter Sturm INRIA
More informationShort on camera geometry and camera calibration
Short on camera geometry and camera calibration Maria Magnusson, maria.magnusson@liu.se Computer Vision Laboratory, Department of Electrical Engineering, Linköping University, Sweden Report No: LiTH-ISY-R-3070
More informationCamera system: pinhole model, calibration and reconstruction
Camera system: pinhole model, calibration and reconstruction Francesco Castaldo, Francesco A.N. Palmieri December 22, 203 F. Castaldo (francesco.castaldo@unina2.it) and F. A. N. Palmieri are with the Dipartimento
More informationA Factorization Method for Structure from Planar Motion
A Factorization Method for Structure from Planar Motion Jian Li and Rama Chellappa Center for Automation Research (CfAR) and Department of Electrical and Computer Engineering University of Maryland, College
More informationAn Improved Evolutionary Algorithm for Fundamental Matrix Estimation
03 0th IEEE International Conference on Advanced Video and Signal Based Surveillance An Improved Evolutionary Algorithm for Fundamental Matrix Estimation Yi Li, Senem Velipasalar and M. Cenk Gursoy Department
More informationStructure from Motion and Multi- view Geometry. Last lecture
Structure from Motion and Multi- view Geometry Topics in Image-Based Modeling and Rendering CSE291 J00 Lecture 5 Last lecture S. J. Gortler, R. Grzeszczuk, R. Szeliski,M. F. Cohen The Lumigraph, SIGGRAPH,
More informationCS 231A Computer Vision (Winter 2015) Problem Set 2
CS 231A Computer Vision (Winter 2015) Problem Set 2 Due Feb 9 th 2015 11:59pm 1 Fundamental Matrix (20 points) In this question, you will explore some properties of fundamental matrix and derive a minimal
More informationProjective Rectification from the Fundamental Matrix
Projective Rectification from the Fundamental Matrix John Mallon Paul F. Whelan Vision Systems Group, Dublin City University, Dublin 9, Ireland Abstract This paper describes a direct, self-contained method
More information