Computer Vision: Lecture 3
|
|
- Lizbeth McLaughlin
- 5 years ago
- Views:
Transcription
1 Computer Vision: Lecture 3 Carl Olsson Carl Olsson Computer Vision: Lecture / 28
2 Todays Lecture Camera Calibration The inner parameters - K. Projective vs. Euclidean Reconstruction. Finding the camera matrix. DLT - Direct Linear Transformation. Normalization of uncalibrated cameras. Radial distortion Carl Olsson Computer Vision: Lecture / 28
3 The Inner Parameters -K The matrix K is the upper triangular matrix: γf sf x 0 K = 0 f y f - focal length γ - aspect ratio s - skew (x 0, y 0 ) - principal points Carl Olsson Computer Vision: Lecture / 28
4 The Inner Parameters -K The focal length f fx fy 1 = f f Re scales the images (e.g. meters pixels). x y 1 Carl Olsson Computer Vision: Lecture / 28
5 The Inner Parameters -K The principal point (x 0, y 0 ) fx + x 0 fy + y 0 = 1 f 0 x 0 0 f y Re centers the image. Typically transforms the point (0, 0, 1) to the middle of the image. x y Carl Olsson Computer Vision: Lecture / 28
6 The Inner Parameters -K Aspect ratio γfx + x 0 fy + y 0 1 = γf 0 x 0 0 f y Pixels are not always squares but can be rectangular. In such cases the scaling in the x-direction should be different from the y-direction. x y 1 Skew γf sf x 0 0 f y Corrects for tilted pixels. Typically zero. Carl Olsson Computer Vision: Lecture / 28
7 Projective vs. Euclidean Reconstruction Calibrated Cameras A camera P = K [R t], where the inner parameters K are known is called calibrated. If we change coordinates in the image using x = K 1 x, we get a so called normalized (calibrated) camera x = K 1 K [R t] X = [R t] X. Carl Olsson Computer Vision: Lecture / 28
8 Projective vs. Euclidean Reconstruction Projective The reconstruction is determined up to a projective transformation. If λx = PX, then for any projective transformation X = H 1 X we have λx = PHH 1 X = PH X. PH is also a valid camera. Carl Olsson Computer Vision: Lecture / 28
9 Projective vs. Euclidean Reconstruction Euclidean The reconstruction is determined up to a similarity transformation. If λx = [R t]x, then for any similarity transformation we have X = [ λ Q v x = [R t] 1 s 0 s [ sq v 0 1 ] 1 X ] X = [RQ Since RQ is a rotation this is a normalized camera. Rv + t s ] X. Carl Olsson Computer Vision: Lecture / 28
10 Projective vs. Euclidean Reconstruction Projective Euclidean Arch of triumph, Paris. The reconstructions have exactly the same reprojection error. But the projective coordinate system makes things look strange. Carl Olsson Computer Vision: Lecture / 28
11 Projective vs. Euclidean Reconstruction Demo. Carl Olsson Computer Vision: Lecture / 28
12 Finding K See lecture notes. Carl Olsson Computer Vision: Lecture / 28
13 RQ-factorization Theorem If A is an n n matrix then there is an orthogonal matrix Q and a right triangular matrix R such that A = RQ. (If A is invertible and the diagonal elements are chosen the be positive, then the factorization is unique.) Note: In our case we will use K for the triangular matrix and R for the rotation. Carl Olsson Computer Vision: Lecture / 28
14 Finding K See lecture notes. Carl Olsson Computer Vision: Lecture / 28
15 Direct Linear Transformation - DLT Finding the camera matrix Use images of a known object to eliminate the projective ambiguity. If X i are 3d-points of a known object, and x i corresponding projections we have λ 1 x 1 = PX 1 λ 2 x 2 = PX 2. λ N x N = PX N. There are 3N equations and 11 + N unknowns. We need 3N 11 + N N 6 points to solve the problem. Carl Olsson Computer Vision: Lecture / 28
16 Direct Linear Transformation - DLT Matrix Formulation P = p T 1 p T 2 p T 3 where p i are the rows of P The first equality is X T 1 p 1 λ 1 x 1 = 0 X T 1 p 2 λ 1 y 1 = 0 X T 1 p 3 λ 1 = 0, where x 1 = (x 1, y 1, 1). In matrix form X T x 1 0 X T 1 0 y X T 1 1 λ 1 Carl Olsson Computer Vision: Lecture / 28 p 1 p 2 p 3 = 0 0 0
17 Direct Linear Transformation - DLT Matrix Formulation More equations: X T x X T 1 0 y X T X T x X T y X T X T x X T y X T } {{ } =M p 1 p 2 p 3 λ 1 λ 2 λ 3 }. {{ } =v = 0 Carl Olsson Computer Vision: Lecture / 28
18 Direct Linear Transformation - DLT Homogeneous Least Squares See lecture notes... Carl Olsson Computer Vision: Lecture / 28
19 Singular values decomposition Theorem Each m n matrix M (with real coefficients) can be factorized into M = USV T, where U and V are orthogonal (m m and n n respectively), S = [ diag(σ1, σ 2,..., σ r ) σ 1 σ 2... σ r > 0 and r is the rank of the matrix. ], Carl Olsson Computer Vision: Lecture / 28
20 Direct Linear Transformation - DLT Homogeneous Least Squares See lecture notes... Carl Olsson Computer Vision: Lecture / 28
21 Direct Linear Transformation - DLT Improving the Numerics (Normalization of uncalibrated cameras) The matrix contains entries x i, y j and ones. Since x i and y i can be about a thousand, the numerics are often greatly improved by translating the coordinates such that their center of mass is zero and then rescaling the coordinates to be roughly 1. Change coordinates according to s 0 s x x = 0 s sȳ x Solve the homogeneous linear least squares system and transform back to the original coordinate system. Similar transformations for the 3D-points X i may also improve the results. Carl Olsson Computer Vision: Lecture / 28
22 Pose estimation using DLT 3D points measured using scanning arm. Carl Olsson Computer Vision: Lecture / 28
23 Pose estimation using DLT 14 points used for computing the camera matrix. Carl Olsson Computer Vision: Lecture / 28
24 Pose estimation using DLT 14 points used for computing the camera matrix. Carl Olsson Computer Vision: Lecture / 28
25 Texturing the chair Project the rest of the points into the image. Carl Olsson Computer Vision: Lecture / 28
26 Texturing the chair Form triangles. Use the texture from the image. Carl Olsson Computer Vision: Lecture / 28
27 Textured chair Carl Olsson Computer Vision: Lecture / 28
28 Radial Distortion Not modeled by the K -matrix. Cannot be removed by a projective mapping since lines are not mapped onto lines (see Szeliski). Carl Olsson Computer Vision: Lecture / 28
Lecture 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 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 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 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 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 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 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 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 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 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 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 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 informationCamera models and calibration
Camera models and calibration Read tutorial chapter 2 and 3. http://www.cs.unc.edu/~marc/tutorial/ Szeliski s book pp.29-73 Schedule (tentative) 2 # date topic Sep.8 Introduction and geometry 2 Sep.25
More informationStructure from Motion. Introduction to Computer Vision CSE 152 Lecture 10
Structure from Motion CSE 152 Lecture 10 Announcements Homework 3 is due May 9, 11:59 PM Reading: Chapter 8: Structure from Motion Optional: Multiple View Geometry in Computer Vision, 2nd edition, Hartley
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 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 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 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 informationDD2429 Computational Photography :00-19:00
. Examination: DD2429 Computational Photography 202-0-8 4:00-9:00 Each problem gives max 5 points. In order to pass you need about 0-5 points. You are allowed to use the lecture notes and standard list
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 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 informationComputer Vision. Geometric Camera Calibration. Samer M Abdallah, PhD
Computer Vision Samer M Abdallah, PhD Faculty of Engineering and Architecture American University of Beirut Beirut, Lebanon Geometric Camera Calibration September 2, 2004 1 Computer Vision Geometric Camera
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 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 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 informationCamera Model and Calibration. Lecture-12
Camera Model and Calibration Lecture-12 Camera Calibration Determine extrinsic and intrinsic parameters of camera Extrinsic 3D location and orientation of camera Intrinsic Focal length The size of the
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 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 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 Geometry and Camera Calibration
3D Geometry and Camera Calibration 3D Coordinate Systems Right-handed vs. left-handed x x y z z y 2D Coordinate Systems 3D Geometry Basics y axis up vs. y axis down Origin at center vs. corner Will often
More informationGeometry of image formation
eometry of image formation Tomáš Svoboda, svoboda@cmp.felk.cvut.cz Czech Technical University in Prague, Center for Machine Perception http://cmp.felk.cvut.cz Last update: November 3, 2008 Talk Outline
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribe : Martin Stiaszny and Dana Qu LECTURE 0 Camera Calibration 0.. Introduction Just like the mythical frictionless plane, in real life we will
More informationCS6670: Computer Vision
CS6670: Computer Vision Noah Snavely Lecture 7: Image Alignment and Panoramas What s inside your fridge? http://www.cs.washington.edu/education/courses/cse590ss/01wi/ Projection matrix intrinsics projection
More informationCamera Models and Image Formation. Srikumar Ramalingam School of Computing University of Utah
Camera Models and Image Formation Srikumar Ramalingam School of Computing University of Utah srikumar@cs.utah.edu VisualFunHouse.com 3D Street Art Image courtesy: Julian Beaver (VisualFunHouse.com) 3D
More informationLecture'9'&'10:'' Stereo'Vision'
Lecture'9'&'10:'' Stereo'Vision' Dr.'Juan'Carlos'Niebles' Stanford'AI'Lab' ' Professor'FeiAFei'Li' Stanford'Vision'Lab' 1' Dimensionality'ReducIon'Machine'(3D'to'2D)' 3D world 2D image Point of observation
More informationStructure from Motion
Structure from Motion Lecture-13 Moving Light Display 1 Shape from Motion Problem Given optical flow or point correspondences, compute 3-D motion (translation and rotation) and shape (depth). 2 S. Ullman
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 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 informationGeometric Computing. in Image Analysis and Visualization. Lecture Notes March 2007
. Geometric Computing in Image Analysis and Visualization Lecture Notes March 2007 Stefan Carlsson Numerical Analysis and Computing Science KTH, Stockholm, Sweden stefanc@bion.kth.se Contents Cameras and
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 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 informationC18 Computer Vision. Lecture 1 Introduction: imaging geometry, camera calibration. Victor Adrian Prisacariu.
C8 Computer Vision Lecture Introduction: imaging geometry, camera calibration Victor Adrian Prisacariu http://www.robots.ox.ac.uk/~victor InfiniTAM Demo Course Content VP: Intro, basic image features,
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 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 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 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 informationCamera Models and Image Formation. Srikumar Ramalingam School of Computing University of Utah
Camera Models and Image Formation Srikumar Ramalingam School of Computing University of Utah srikumar@cs.utah.edu Reference Most slides are adapted from the following notes: Some lecture notes on geometric
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 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 informationDescriptive Geometry Meets Computer Vision The Geometry of Two Images (# 82)
Descriptive Geometry Meets Computer Vision The Geometry of Two Images (# 8) Hellmuth Stachel stachel@dmg.tuwien.ac.at http://www.geometrie.tuwien.ac.at/stachel th International Conference on Geometry and
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 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 informationThe Geometry Behind the Numerical Reconstruction of Two Photos
The Geometry Behind the Numerical Reconstruction of Two Photos Hellmuth Stachel stachel@dmg.tuwien.ac.at http://www.geometrie.tuwien.ac.at/stachel ICEGD 2007, The 2 nd Internat. Conf. on Eng g Graphics
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 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 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 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 informationField of View (Zoom)
Image Projection Field of View (Zoom) Large Focal Length compresses depth 400 mm 200 mm 100 mm 50 mm 28 mm 17 mm 1995-2005 Michael Reichmann FOV depends of Focal Length f f Smaller FOV = larger Focal
More informationCALIBRATION BETWEEN DEPTH AND COLOR SENSORS FOR COMMODITY DEPTH CAMERAS. Cha Zhang and Zhengyou Zhang
CALIBRATION BETWEEN DEPTH AND COLOR SENSORS FOR COMMODITY DEPTH CAMERAS Cha Zhang and Zhengyou Zhang Communication and Collaboration Systems Group, Microsoft Research {chazhang, zhang}@microsoft.com ABSTRACT
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 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 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 informationLecture 5.3 Camera calibration. Thomas Opsahl
Lecture 5.3 Camera calibration Thomas Opsahl Introduction The image u u x X W v C z C World frame x C y C z C = 1 For finite projective cameras, the correspondence between points in the world and points
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 informationOn Plane-Based Camera Calibration: A General Algorithm, Singularities, Applications
ACCEPTED FOR CVPR 99. VERSION OF NOVEMBER 18, 2015. On Plane-Based Camera Calibration: A General Algorithm, Singularities, Applications Peter F. Sturm and Stephen J. Maybank Computational Vision Group,
More informationA Method for Interactive 3D Reconstruction of Piecewise Planar Objects from Single Images
A Method for Interactive 3D Reconstruction of Piecewise Planar Objects from Single Images Peter F Sturm and Stephen J Maybank Computational Vision Group, Department of Computer Science The University of
More informationMei Han Takeo Kanade. January Carnegie Mellon University. Pittsburgh, PA Abstract
Scene Reconstruction from Multiple Uncalibrated Views Mei Han Takeo Kanade January 000 CMU-RI-TR-00-09 The Robotics Institute Carnegie Mellon University Pittsburgh, PA 1513 Abstract We describe a factorization-based
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 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 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 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 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 informationDense 3D Reconstruction. Christiano Gava
Dense 3D Reconstruction Christiano Gava christiano.gava@dfki.de Outline Previous lecture: structure and motion II Structure and motion loop Triangulation Today: dense 3D reconstruction The matching problem
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 informationCalculation of the fundamental matrix
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
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 informationTake Home Exam # 2 Machine Vision
1 Take Home Exam # 2 Machine Vision Date: 04/26/2018 Due : 05/03/2018 Work with one awesome/breathtaking/amazing partner. The name of the partner should be clearly stated at the beginning of your report.
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 informationCS 6320 Computer Vision Homework 2 (Due Date February 15 th )
CS 6320 Computer Vision Homework 2 (Due Date February 15 th ) 1. Download the Matlab calibration toolbox from the following page: http://www.vision.caltech.edu/bouguetj/calib_doc/ Download the calibration
More informationLecture 9 & 10: Stereo Vision
Lecture 9 & 10: Stereo Vision Professor Fei- Fei Li Stanford Vision Lab 1 What we will learn today? IntroducEon to stereo vision Epipolar geometry: a gentle intro Parallel images Image receficaeon Solving
More informationComputer Vision Lecture 17
Announcements Computer Vision Lecture 17 Epipolar Geometry & Stereo Basics Seminar in the summer semester Current Topics in Computer Vision and Machine Learning Block seminar, presentations in 1 st week
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 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 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 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 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 information3D FACE RECONSTRUCTION BASED ON EPIPOLAR GEOMETRY
IJDW Volume 4 Number January-June 202 pp. 45-50 3D FACE RECONSRUCION BASED ON EPIPOLAR GEOMERY aher Khadhraoui, Faouzi Benzarti 2 and Hamid Amiri 3,2,3 Signal, Image Processing and Patterns Recognition
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 informationA Method for Interactive 3D Reconstruction of Piecewise Planar Objects from Single Images
A Method for Interactive 3D Reconstruction of Piecewise Planar Objects from Single Images Peter Sturm Steve Maybank To cite this version: Peter Sturm Steve Maybank A Method for Interactive 3D Reconstruction
More informationMosaics. Today s Readings
Mosaics VR Seattle: http://www.vrseattle.com/ Full screen panoramas (cubic): http://www.panoramas.dk/ Mars: http://www.panoramas.dk/fullscreen3/f2_mars97.html Today s Readings Szeliski and Shum paper (sections
More informationMultiple 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 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 informationEuclidean Reconstruction Independent on Camera Intrinsic Parameters
Euclidean Reconstruction Independent on Camera Intrinsic Parameters Ezio MALIS I.N.R.I.A. Sophia-Antipolis, FRANCE Adrien BARTOLI INRIA Rhone-Alpes, FRANCE Abstract bundle adjustment techniques for Euclidean
More informationCalibrating a Structured Light System Dr Alan M. McIvor Robert J. Valkenburg Machine Vision Team, Industrial Research Limited P.O. Box 2225, Auckland
Calibrating a Structured Light System Dr Alan M. McIvor Robert J. Valkenburg Machine Vision Team, Industrial Research Limited P.O. Box 2225, Auckland New Zealand Tel: +64 9 3034116, Fax: +64 9 302 8106
More informationIntroduction to Computer Vision
Introduction to Computer Vision Michael J. Black Nov 2009 Perspective projection and affine motion Goals Today Perspective projection 3D motion Wed Projects Friday Regularization and robust statistics
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 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 informationA linear algorithm for Camera Self-Calibration, Motion and Structure Recovery for Multi-Planar Scenes from Two Perspective Images
A linear algorithm for Camera Self-Calibration, Motion and Structure Recovery for Multi-Planar Scenes from Two Perspective Images Gang Xu, Jun-ichi Terai and Heung-Yeung Shum Microsoft Research China 49
More informationLecture 14: Basic Multi-View Geometry
Lecture 14: Basic Multi-View Geometry Stereo If I needed to find out how far point is away from me, I could use triangulation and two views scene point image plane optical center (Graphic from Khurram
More informationN-Views (1) Homographies and Projection
CS 4495 Computer Vision N-Views (1) Homographies and Projection Aaron Bobick School of Interactive Computing Administrivia PS 2: Get SDD and Normalized Correlation working for a given windows size say
More information