Lecture 5. Epipolar Geometry. Professor Silvio Savarese Computational Vision and Geometry Lab. 21-Jan-15. Lecture 5 - Silvio Savarese
|
|
- Jasmine Henderson
- 5 years ago
- Views:
Transcription
1 Lecture 5 Epipolar Geometry Professor Silvio Savarese Computational Vision and Geometry Lab Silvio Savarese Lecture 5-21-Jan-15
2 Lecture 5 Epipolar Geometry Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F Examples Reading: [HZ] Chapter: 4 Estimation 2D perspective transformations Chapter: 9 Epipolar Geometry and the Fundamental Matrix Transformation Chapter: 11 Computation of the Fundamental Matrix F [FP] Chapter: 7 Stereopsis Chapter: 8 Structure from Motion Silvio Savarese Lecture 5-21-Jan-15
3 Recovering structure from a single view Pinhole perspective projection P p O w Calibration rig Scene C Camera K From calibration rig location/pose of the rig, K From points and lines at infinity + orthogonal lines and planes structure of the scene, K Knowledge about scene (point correspondences, geometry of lines & planes, etc
4 Recovering structure from a single view Pinhole perspective projection P p O w Calibration rig Scene C Camera K Why is it so difficult? Intrinsic ambiguity of the mapping from 3D to image (2D)
5 Recovering structure from a single view Intrinsic ambiguity of the mapping from 3D to image (2D) Courtesy slide S. Lazebnik
6 Two eyes help!
7 Two eyes help! P = l l" [Eq. 1] P? l l' K 1 =known p p K 2 =known R, T O 1 O 2 This is called triangulation
8 Triangulation Find P that minimizes d(p, M 1 P')+ d(p', M 2 P') P P [Eq. 2] M 1 P p M 2 P p image 1 image 2 O 1 O 2
9 Stereo- view geometry Correspondence: Given a point p in one image, how can I find the corresponding point p in another one? Camera geometry: Given corresponding points in two images, find camera matrices, position and pose. Scene geometry: Find coordinates of 3D point from its projection into 2 or multiple images.
10 Epipolar geometry P p p e e Epipolar Plane Epipoles e, e Baseline Epipolar Lines O 1 O 2 = intersections of baseline with image planes = projections of the other camera center
11 Example of epipolar lines
12 Example: Parallel image planes P e p p e O 1 O 2 Baseline intersects the image plane at infinity Epipoles are at infinity Epipolar lines are parallel to x axis
13 Example: Parallel Image Planes e at infinity e at infinity
14 Example: Forward translation e P O 2 e O 1 The epipoles have same position in both images Epipole called FOE (focus of expansion)
15 Epipolar Constraint p - Two views of the same object - Suppose I know the camera positions and camera matrices - Given a point on left image, how can I find the corresponding point on right image?
16 Epipolar geometry P Epipolar line 2 p p O 1 O 2
17 Epipolar Constraint
18 !!! " # % & = 1 v u M P P O 1 O 2 p p P R, T Epipolar Constraint!!! " # % & ' ' = ' 1 v u P M P [Eq. 3] [Eq. 4] M = K I 0! " # M ' = K ' R T R T T " # % &'
19 Epipolar Constraint P! # K canonical = # # " & & & % p p O 1 R, T O 2 M = K! I 0 " # K = K are known (canonical cameras) " M ' = K ' R T R T T # % &' M = [ I 0] [Eq. 5] " M ' = R T R T T # % &' [Eq. 6]
20 The cameras are related by R, T Camera 2 p T O 2 Camera 1 O 1 T = O 2 in the camera 1 reference system R is the rotation matrix such that a vector p in the camera 2 is equal to R p in camera 1.
21 Epipolar Constraint P p p O 1 R, T O 2 p in first camera reference system is = R p! T ( R p!) is perpendicular to epipolar plane p T [ T ( R p! )] = 0 [Eq. 7]
22 b a b a ] [ = " " " # % % % & ' " " " # % % % & ' = z y x x y x z y z b b b a a a a a a Cross product as matrix multiplication
23 Epipolar Constraint P p p O 1 R, T O 2 [ T (R p )] 0 p T! = [ T ] R p 0 p T! = [Eq. 8] [Eq. 9] E = Essential matrix (Longuet-Higgins, 1981)
24 Epipolar Constraint p T E p' = 0 P [Eq. 10] p l l e e p O 1 O 2 l = E p is the epipolar line associated with p l = E T p is the epipolar line associated with p E e = 0 and E T e = 0 E is 3x3 matrix; 5 DOF E is singular (rank two)
25 Epipolar Constraint P p p R, T O 1 O 2 M = K[ I 0] " M ' = K ' R T R T # T % &' p = K 1 p p! = K ' 1 p! c c [Eq. 11] [Eq. 12]
26 Epipolar Constraint p c = K 1 p p! = K ' 1 p! c P p p [Eq.9] O 1 O 2 p T c [ T ] R # 1 T p = 0 (K p) [ T ] R K! p! = 0 c 1 p T K [ T ] R K! 1 p! = 0 p T! = T F p 0 [Eq. 13]
27 Epipolar Constraint P p p O 1 O 2 Eq. 13 p T F p! = 0 F [ ] 1 = K T T R K# F = Fundamental Matrix (Faugeras and Luong, 1992) Eq. 14
28 Epipolar Constraint p T F p' = 0 P p l e e l p O 1 O 2 l = F p is the epipolar line associated with p l = F T p is the epipolar line associated with p F e = 0 and F T e = 0 F is 3x3 matrix; 7 DOF F is singular (rank two)
29 Why F is useful? p l = F T p - Suppose F is known - No additional information about the scene and camera is given - Given a point on left image, how can I find the corresponding point on right image?
30 Why F is useful? F captures information about the epipolar geometry of 2 views + camera parameters MORE IMPORTANTLY: F gives constraints on how the scene changes under view point transformation (without reconstructing the scene!) Powerful tool in: 3D reconstruction Multi- view object/scene matching
31 O 1 O 2 p p P Estimating F The Eight-Point Algorithm!!! " # % & ' ' = ' 1 v u p P!!! " # % & = 1 v u p P 0 F p p T =! (Hartley, 1995) (Longuet-Higgins, 1981)
32 Estimating F [Eq. 13] p T F p! = 0! F11 F12 F13 "! u '" # # u, v,1 F F F v ' = 0 ( ) # # F31 F32 F # 33 1 % &% & ( uu uv u vu vv v u v ) Let s take 8 corresponding points! F11 " # # F12 # F 13 # F # 21 ', ',, ', ',, ', ',1 # F = 0 22 # # F23 # # F31 # F 32 # F % 33 & [Eq. 14]
33 Estimating F
34 W Homogeneous system Estimating F! F11 "! u u ' u v ' u v u ' v v ' v u ' v ' 1"# F # u2u ' 2 u2v ' 2 u2 v2u ' 2 v2v ' 2 v2 u ' 2 v ' 2 1 # 12 # # F 13 # u3u ' 3 u3v ' 3 u3 v3u ' 3 v3v ' 3 v3 u ' 3 v ' 3 1# # # F21 u4u ' 4 u4v ' 4 u4 v4u ' 4 v4v ' 4 v4 u ' 4 v ' 4 1 F # 22 u5u ' 5 u5v ' 5 u5 v5u ' 5 v5v ' 5 v5 u ' 5 v ' 5 1# # # F23 # u6u ' 6 u6v ' 6 u6 v6u ' 6 v6v ' 6 v6 u ' 6 v ' 6 1# # F31 u7u ' 7 u7v ' 7 u7 v7u ' 7 v7v ' 7 v7 u ' 7 v ' 7 1# # # F32 u8u ' 8 u8v ' 8 u8 v8u ' 8 v8v ' 8 v8 u ' 8 v ' 8 1 % &# # F % 33 & # # = Wf = 0 0 f [Eqs. 15] Rank 8 If N>8 A non-zero solution exists (unique) Lsq. solution by SVD! f =1 Fˆ
35 Fˆ satisfies: p T Fˆ p! = 0 ^ and estimated F may have full rank (det(f) 0) But remember: fundamental matrix is Rank2 ^ Find F that minimizes F Fˆ = 0 Frobenius norm (*) Subject to det(f)=0 SVD (again!) can be used to solve this problem (*) Sq. root of the sum of squares of all entries
36 Find F that minimizes F Fˆ = 0 Frobenius norm (*) Subject to det(f)=0! # F =U# # "# s s & & & %& V T [HZ] pag 281, chapter 11, Computation of F Where:! # U# # "# s s s 3 & & V T = SVD( ˆF) & %&
37 Data courtesy of R. Mohr and B. Boufama.
38 Mean errors: 10.0pixel 9.1pixel
39 Problems with the 8-Point Algorithm W f = 0, Lsq solution by SVD f =1 F - Recall the structure of W: - do we see any potential (numerical) issue?
40 Problems with the 8-Point Algorithm Wf = 0! F11 "! u u ' u v ' u v u ' v v ' v u ' v ' 1"# F # u2u ' 2 u2v ' 2 u2 v2u ' 2 v2v ' 2 v2 u ' 2 v ' 2 1 # 12 # # F 13 # u3u ' 3 u3v ' 3 u3 v3u ' 3 v3v ' 3 v3 u ' 3 v ' 3 1# # # F21 u4u ' 4 u4v ' 4 u4 v4u ' 4 v4v ' 4 v4 u ' 4 v ' 4 1 F # 22 u5u ' 5 u5v ' 5 u5 v5u ' 5 v5v ' 5 v5 u ' 5 v ' 5 1# # # F23 # u6u ' 6 u6v ' 6 u6 v6u ' 6 v6v ' 6 v6 u ' 6 v ' 6 1# # F31 u7u ' 7 u7v ' 7 u7 v7u ' 7 v7v ' 7 v7 u ' 7 v ' 7 1# # # F32 u8u ' 8 u8v ' 8 u8 v8u ' 8 v8v ' 8 v8 u ' 8 v ' 8 1 % &# # F % 33 & # # = Highly un- balanced (not well conditioned) Values of W must have similar magnitude This creates problems during the SVD decomposition 0 HZ pag 108
41 Normalization IDEA: Transform image coordinates such that the matrix W becomes better conditioned (pre-conditioning) For each image, apply a following transformation T (translation and scaling) acting on image coordinates such that: Origin = centroid of image points Mean square distance of the image points from origin is ~2 pixels q = T p i i q = T! p!! (normalization) i i
42 Example of normalization T 2 pixels Coordinate system of the image before applying T Coordinate system of the image after applying T Origin = centroid of image points Mean square distance of the image points from origin is ~2 pixels
43 The Normalized Eight-Point Algorithm 0. Compute T and T for image 1 and 2, respectively 1. Normalize coordinates in images 1 and 2: 2. Use the eight-point algorithm to compute from the corresponding points q and q. i q = T p i i i q! = T! p! i i ˆF q 1. Enforce the rank-2 constraint. F q such that: q T F q q! = 0 2. De-normalize F q : F T ' = T F T det( Fq ) = 0 q
44 With transformation Without transformation Mean errors: 10.0pixel 9.1pixel Mean errors: 1.0pixel 0.9pixel
45 Lecture 5 Epipolar Geometry Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F Examples Reading: [AZ] Chapter: 4 Estimation 2D perspective transformations Chapter: 9 Epipolar Geometry and the Fundamental Matrix Transformation Chapter: 11 Computation of the Fundamental Matrix F [FP] Chapter: 7 Stereopsis Chapter: 8 Structure from Motion Silvio Savarese Lecture 5-21-Jan-15
46 Example: Parallel image planes P e p p e y z K K x O 1 O 2 Hint : E=? K 1 =K 2 = known R = I T = (T, 0, 0) x parallel to O 1 O 2
47 !!! " # % & =!!! " # % & = T T T T T T T T x y x z y z I E Essential matrix for parallel images T = [ T 0 0 ] R = I [ ] R T E = [Eq. 20]
48 Example: Parallel image planes P e l p p l e y z K K O 1 What are the directions of epipolar lines? x " l = E p' = # T 0 T 0 %" ' ' ' &# u v 1 % " ' ' = ' & # 0 T T v % ' ' ' & horizontal!
49 Example: Parallel image planes P e p p e y z K K O 1 x How are p and p related? p T E p' = 0
50 Example: Parallel image planes P e p p e y z K K O 1 x How are p and p related? " # u! % 0 %! ( u v ( ) T ) & ' & ' = v! ( u v ) T Tv Tv! ( ) = = = ( 0 T 0 ) & 1 ' & Tv!', -. /. / v = v" T p E p
51 Example: Parallel image planes P e p p e y z K K O 1 x Rectification: making two images parallel Why it is useful? Epipolar constraint v = v New views can be synthesized by linear interpolation
52 Application: view morphing S. M. Seitz and C. R. Dyer, Proc. SIGGRAPH 96, 1996, 21-30
53 Rectification P H 1 I 1 I s î 1 v u I 2 î S v u C 1 v u î 2 H 2 C s C 2
54
55
56 Morphing without rectifying
57
58
59 From its reflection!
60 The Fundamental Matrix Song
Lecture 6 Stereo Systems Multi-view geometry
Lectre 6 Stereo Systems Mlti-view geometry Professor Silvio Savarese Comptational Vision and Geometry Lab Silvio Savarese Lectre 6-29-Jan-18 Lectre 6 Stereo Systems Mlti-view geometry Stereo systems Rectification
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Scene Planes & Homographies Lecture 19 March 24, 2005 2 In our last lecture, we examined various
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribe: Tasha Vanesian LECTURE 3 Calibrated 3D Reconstruction 3.1. Geometric View of Epipolar Constraint We are trying to solve the following problem:
More informationCS4495/6495 Introduction to Computer Vision. 3D-L3 Fundamental matrix
CS4495/6495 Introduction to Computer Vision 3D-L3 Fundamental matrix Weak calibration Main idea: Estimate epipolar geometry from a (redundant) set of point correspondences between two uncalibrated cameras
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribe: Hamed Masnadi Shirazi, Solmaz Alipour LECTURE 5 Relationships between the Homography and the Essential Matrix 5.1. Introduction In practice,
More informationA Practical Method for Decomposition of the Essential Matrix
Applied Mathematical Sciences, Vol. 8, 2014, no. 176, 8755-8770 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410877 A Practical Method for Decomposition of the Essential Matrix Georgi
More informationVision 3D articielle Session 2: Essential and fundamental matrices, their computation, RANSAC algorithm
Vision 3D articielle Session 2: Essential and fundamental matrices, their computation, RANSAC algorithm Pascal Monasse monasse@imagine.enpc.fr IMAGINE, École des Ponts ParisTech Contents Some useful rules
More informationA Study of Kruppa s Equation for Camera Self-calibration
Proceedings of the International Conference of Machine Vision and Machine Learning Prague, Czech Republic, August 14-15, 2014 Paper No. 57 A Study of Kruppa s Equation for Camera Self-calibration Luh Prapitasari,
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Trifocal Tensor Lecture 21 March 31, 2005 2 Lord Shiva is depicted as having three eyes. The
More informationEPIPOLAR GEOMETRY WITH MANY DETAILS
EPIPOLAR GEOMERY WIH MANY DEAILS hank ou for the slides. he come mostl from the following source. Marc Pollefes U. of North Carolina hree questions: (i) Correspondence geometr: Given an image point in
More informationPAijpam.eu EPIPOLAR GEOMETRY WITH A FUNDAMENTAL MATRIX IN CANONICAL FORM Georgi Hristov Georgiev 1, Vencislav Dakov Radulov 2
International Journal of Pure and Applied Mathematics Volume 105 No. 4 2015, 669-683 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v105i4.8
More informationOutline. Linear Algebra for Computer Vision
Outline Linear Algebra for Computer Vision Introduction CMSC 88 D Notation and Basics Motivation Linear systems of equations Gauss Elimination, LU decomposition Linear Spaces and Operators Addition, scalar
More informationParameterizing the Trifocal Tensor
Parameterizing the Trifocal Tensor May 11, 2017 Based on: Klas Nordberg. A Minimal Parameterization of the Trifocal Tensor. In Computer society conference on computer vision and pattern recognition (CVPR).
More informationInduced Planar Homologies in Epipolar Geometry
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 4 (2016), pp. 3759 3773 Research India Publications http://www.ripublication.com/gjpam.htm Induced Planar Homologies in
More informationTrinocular Geometry Revisited
Trinocular Geometry Revisited Jean Pounce and Martin Hebert 报告人 : 王浩人 2014-06-24 Contents 1. Introduction 2. Converging Triplets of Lines 3. Converging Triplets of Visual Rays 4. Discussion 1. Introduction
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationRecovering Unknown Focal Lengths in Self-Calibration: G.N. Newsam D.Q. Huynh M.J. Brooks H.-P. Pan.
Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Congurations G.N. Newsam D.Q. Huynh M.J. Brooks H.-P. Pan fgnewsam,du,mjb,hepingg@cssip.edu.au Centre
More informationVision par ordinateur
Vision par ordinateur Géométrie épipolaire Frédéric Devernay Avec des transparents de Marc Pollefeys Epipolar geometry π Underlying structure in set of matches for rigid scenes C1 m1 l1 M L2 L1 l T 1 l
More informationDetermining the Translational Speed of a Camera from Time-Varying Optical Flow
Determining the Translational Speed of a Camera from Time-Varying Optical Flow Anton van den Hengel, Wojciech Chojnacki, and Michael J. Brooks School of Computer Science, Adelaide University, SA 5005,
More informationSingle view metrology
EECS 44 Computer vision Single view metrology Review calibration Lines and planes at infinity Absolute conic Estimating geometry from a single image Etensions Reading: [HZ] Chapters,3,8 Calibration Problem
More informationM3: Multiple View Geometry
M3: Multiple View Geometry L18: Projective Structure from Motion: Iterative Algorithm based on Factorization Based on Sections 13.4 C. V. Jawahar jawahar-at-iiit.net Mar 2005: 1 Review: Reconstruction
More informationMath Review: parameter estimation. Emma
Math Review: parameter estimation Emma McNally@flickr Fitting lines to dots: We will cover how Slides provided by HyunSoo Park 1809, Carl Friedrich Gauss What about fitting line on a curved surface? Least
More informationAugmented Reality VU Camera Registration. Prof. Vincent Lepetit
Augmented Reality VU Camera Registration Prof. Vincent Lepetit Different Approaches to Vision-based 3D Tracking [From D. Wagner] [From Drummond PAMI02] [From Davison ICCV01] Consider natural features Consider
More informationTwo-View Segmentation of Dynamic Scenes from the Multibody Fundamental Matrix
Two-View Segmentation of Dynamic Scenes from the Multibody Fundamental Matrix René Vidal Stefano Soatto Shankar Sastry Department of EECS, UC Berkeley Department of Computer Sciences, UCLA 30 Cory Hall,
More informationIntroduction to pinhole cameras
Introduction to pinhole cameras Lesson given by Sébastien Piérard in the course Introduction aux techniques audio et vidéo (ULg, Pr JJ Embrechts) INTELSIG, Montefiore Institute, University of Liège, Belgium
More informationCamera calibration Triangulation
Triangulation Perspective projection in homogenous coordinates ~x img I 0 apple R t 0 T 1 ~x w ~x img R t ~x w Matrix transformations in 2D ~x img K R t ~x w K = 2 3 1 0 t u 40 1 t v 5 0 0 1 Translation
More informationFinal Exam Due on Sunday 05/06
Final Exam Due on Sunday 05/06 The exam should be completed individually without collaboration. However, you are permitted to consult with the textbooks, notes, slides and even internet resources. If you
More informationMulti-Frame Factorization Techniques
Multi-Frame Factorization Techniques Suppose { x j,n } J,N j=1,n=1 is a set of corresponding image coordinates, where the index n = 1,...,N refers to the n th scene point and j = 1,..., J refers to the
More informationCamera Models and Affine Multiple Views Geometry
Camera Models and Affine Multiple Views Geometry Subhashis Banerjee Dept. Computer Science and Engineering IIT Delhi email: suban@cse.iitd.ac.in May 29, 2001 1 1 Camera Models A Camera transforms a 3D
More informationAlgorithms for Computing a Planar Homography from Conics in Correspondence
Algorithms for Computing a Planar Homography from Conics in Correspondence Juho Kannala, Mikko Salo and Janne Heikkilä Machine Vision Group University of Oulu, Finland {jkannala, msa, jth@ee.oulu.fi} Abstract
More informationA Factorization Method for 3D Multi-body Motion Estimation and Segmentation
1 A Factorization Method for 3D Multi-body Motion Estimation and Segmentation René Vidal Department of EECS University of California Berkeley CA 94710 rvidal@eecs.berkeley.edu Stefano Soatto Dept. of Computer
More informationDetermining the Epipolar Geometry and its Uncertainty: A Review
International Journal of Computer Vision, 27(2), 161 198 (1998) c 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Determining the Epipolar Geometry and its Uncertainty: A Review
More informationCamera: optical system
Camera: optical system Y dαα α ρ ρ curvature radius Z thin lens small angles: α Y ρ α Y ρ lens refraction inde: n Y incident light beam deviated beam θ '' α θ ' α sin( θ '' α ) sin( θ ' α ) n θ α θ α deviation
More informationComputation of the Quadrifocal Tensor
Computation of the Quadrifocal Tensor Richard I. Hartley G.E. Corporate Research and Development Research Circle, Niskayuna, NY 2309, USA Abstract. This paper gives a practical and accurate algorithm for
More informationVisual Object Recognition
Visual Object Recognition Lecture 2: Image Formation Per-Erik Forssén, docent Computer Vision Laboratory Department of Electrical Engineering Linköping University Lecture 2: Image Formation Pin-hole, and
More informationCamera Projection Model
amera Projection Model 3D Point Projection (Pixel Space) O ( u, v ) ccd ccd f Projection plane (, Y, Z ) ( u, v ) img Z img (0,0) w img ( u, v ) img img uimg f Z p Y v img f Z p x y Zu f p Z img img x
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Basic Information Instructor: Professor Ron Sahoo Office: NS 218 Tel: (502) 852-2731 Fax: (502)
More informationCS 231A Computer Vision (Fall 2011) Problem Set 2
CS 231A Computer Vision (Fall 2011) Problem Set 2 Solution Set Due: Oct. 28 th, 2011 (5pm) 1 Some Projective Geometry Problems (15 points) Suppose there are two parallel lines that extend to infinity in
More informationPose estimation from point and line correspondences
Pose estimation from point and line correspondences Giorgio Panin October 17, 008 1 Problem formulation Estimate (in a LSE sense) the pose of an object from N correspondences between known object points
More informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab 1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed in the
More informationMirrors in motion: Epipolar geometry and motion estimation
Mirrors in motion: Epipolar geometry and motion estimation Christopher Geyer Kostas Daniilidis University of California, Berkeley University of Pennsylvania Berkeley, CA 970 Philadelphia, PA 1910 cgeyer@eecs.berkeley.edu
More informationTracking for VR and AR
Tracking for VR and AR Hakan Bilen November 17, 2017 Computer Graphics University of Edinburgh Slide credits: Gordon Wetzstein and Steven M. La Valle 1 Overview VR and AR Inertial Sensors Gyroscopes Accelerometers
More informationEE Camera & Image Formation
Electric Electronic Engineering Bogazici University February 21, 2018 Introduction Introduction Camera models Goal: To understand the image acquisition process. Function of the camera Similar to that of
More information6.801/866. Affine Structure from Motion. T. Darrell
6.801/866 Affine Structure from Motion T. Darrell [Read F&P Ch. 12.0, 12.2, 12.3, 12.4] Affine geometry is, roughly speaking, what is left after all ability to measure lengths, areas, angles, etc. has
More informationRobert Collins CSE486, Penn State. Lecture 25: Structure from Motion
Lecture 25: Structure from Motion Structure from Motion Given a set of flow fields or displacement vectors from a moving camera over time, determine: the sequence of camera poses the 3D structure of the
More informationHigh Accuracy Fundamental Matrix Computation and Its Performance Evaluation
High Accuracy Fundamental Matrix Computation and Its Performance Evaluation Kenichi Kanatani Department of Computer Science, Okayama University, Okayama 700-8530 Japan kanatani@suri.it.okayama-u.ac.jp
More information1 Linearity and Linear Systems
Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 7-8 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)
More informationBasics of Epipolar Geometry
Basics of Epipolar Geometry Hellmuth Stachel stachel@dmg.tuwien.ac.at http://www.geometrie.tuwien.ac.at/stachel Seminar at the Institute of Mathematics and Physics Faculty of Mechanical Engineering, May
More informationOptimisation on Manifolds
Optimisation on Manifolds K. Hüper MPI Tübingen & Univ. Würzburg K. Hüper (MPI Tübingen & Univ. Würzburg) Applications in Computer Vision Grenoble 18/9/08 1 / 29 Contents 2 Examples Essential matrix estimation
More informationLecture 4.3 Estimating homographies from feature correspondences. Thomas Opsahl
Lecture 4.3 Estimating homographies from feature correspondences Thomas Opsahl Homographies induced by central projection 1 H 2 1 H 2 u uu 2 3 1 Homography Hu = u H = h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h
More informationCamera Calibration The purpose of camera calibration is to determine the intrinsic camera parameters (c 0,r 0 ), f, s x, s y, skew parameter (s =
Camera Calibration The purpose of camera calibration is to determine the intrinsic camera parameters (c 0,r 0 ), f, s x, s y, skew parameter (s = cotα), and the lens distortion (radial distortion coefficient
More informationAUDIO INTERPOLATION RICHARD RADKE 1 AND SCOTT RICKARD 2
AUDIO INTERPOLATION RICHARD RADKE AND SCOTT RICKARD 2 Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 28, USA rjradke@ecse.rpi.edu 2 Program in Applied
More informationBasic Math for
Basic Math for 16-720 August 23, 2002 1 Linear Algebra 1.1 Vectors and Matrices First, a reminder of a few basic notations, definitions, and terminology: Unless indicated otherwise, vectors are always
More informationOn the Absolute Quadratic Complex and its Application to Autocalibration
On the Absolute Quadratic Complex and its Application to Autocalibration J. Ponce and K. McHenry Beckman Institute University of Illinois Urbana, IL 61801, USA T. Papadopoulo and M. Teillaud INRIA Sophia-Antipolis
More informationGI07/COMPM012: Mathematical Programming and Research Methods (Part 2) 2. Least Squares and Principal Components Analysis. Massimiliano Pontil
GI07/COMPM012: Mathematical Programming and Research Methods (Part 2) 2. Least Squares and Principal Components Analysis Massimiliano Pontil 1 Today s plan SVD and principal component analysis (PCA) Connection
More informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab Lecture 11-1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed
More informationSelf-Calibration of a Moving Camera by Pre-Calibration
Self-Calibration of a Moving Camera by Pre-Calibration Peter Sturm To cite this version: Peter Sturm. Self-Calibration of a Moving Camera by Pre-Calibration. Robert B. Fisher and Emanuele Trucco. 7th British
More informationLines and points. Lines and points
omogeneous coordinates in the plane Homogeneous coordinates in the plane A line in the plane a + by + c is represented as (a, b, c). A line is a subset of points in the plane. All vectors (ka, kb, kc)
More informationIntroduction to Machine Learning. PCA and Spectral Clustering. Introduction to Machine Learning, Slides: Eran Halperin
1 Introduction to Machine Learning PCA and Spectral Clustering Introduction to Machine Learning, 2013-14 Slides: Eran Halperin Singular Value Decomposition (SVD) The singular value decomposition (SVD)
More informationx 2 + y 2 = 1 (1) ), ( 2 , 3
AM205: Assignment 2 solutions Problem 1 norms and Newton root finding Part (a) Write b = (x, y). Then b 2 = 1 implies and Ab 2 = 1 implies that Subtracting Eq. 1 from Eq. 2 gives x 2 + y 2 = 1 (1) (3x
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More information3D from Photographs: Camera Calibration. Dr Francesco Banterle
3D from Photographs: Camera Calibration Dr Francesco Banterle francesco.banterle@isti.cnr.it 3D from Photographs Automatic Matching of Images Camera Calibration Photographs Surface Reconstruction Dense
More informationLecture 02 Linear Algebra Basics
Introduction to Computational Data Analysis CX4240, 2019 Spring Lecture 02 Linear Algebra Basics Chao Zhang College of Computing Georgia Tech These slides are based on slides from Le Song and Andres Mendez-Vazquez.
More informationAffine Structure From Motion
EECS43-Advanced Computer Vision Notes Series 9 Affine Structure From Motion Ying Wu Electrical Engineering & Computer Science Northwestern University Evanston, IL 68 yingwu@ece.northwestern.edu Contents
More informationRevisiting Hartley s Normalized Eight-Point Algorithm
Revisiting Hartley s Normalized Eight-Point Algorithm Wojciech Chojnacki, Michael J. Brooks, Anton van den Hengel, and Darren Gawley Abstract Hartley s eight-point algorithm has maintained an important
More informationThe Multibody Trifocal Tensor: Motion Segmentation from 3 Perspective Views
The Multibody Trifocal Tensor: Motion Segmentation from 3 Perspective Views Richard Hartley 1,2 and RenéVidal 2,3 1 Dept. of Systems Engineering 3 Center for Imaging Science Australian National University
More informationStructure from Motion. CS4670/CS Kevin Matzen - April 15, 2016
Structure from Motion CS4670/CS5670 - Kevin Matzen - April 15, 2016 Video credit: Agarwal, et. al. Building Rome in a Day, ICCV 2009 Roadmap What we ve seen so far Single view modeling (1 camera) Stereo
More informationCSE 554 Lecture 7: Alignment
CSE 554 Lecture 7: Alignment Fall 2012 CSE554 Alignment Slide 1 Review Fairing (smoothing) Relocating vertices to achieve a smoother appearance Method: centroid averaging Simplification Reducing vertex
More informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationLecture 19: Isometries, Positive operators, Polar and singular value decompositions; Unitary matrices and classical groups; Previews (1)
Lecture 19: Isometries, Positive operators, Polar and singular value decompositions; Unitary matrices and classical groups; Previews (1) Travis Schedler Thurs, Nov 18, 2010 (version: Wed, Nov 17, 2:15
More informationarxiv: v2 [cs.cv] 22 Mar 2019
Closed-Form Optimal Triangulation Based on Angular Errors Seong Hun Lee Javier Civera IA, University of Zaragoza, Spain {seonghunlee, jcivera}@unizar.es arxiv:190.09115v [cs.cv] Mar 019 Abstract In this
More informationCS 231A Section 1: Linear Algebra & Probability Review. Kevin Tang
CS 231A Section 1: Linear Algebra & Probability Review Kevin Tang Kevin Tang Section 1-1 9/30/2011 Topics Support Vector Machines Boosting Viola Jones face detector Linear Algebra Review Notation Operations
More informationCS 231A Section 1: Linear Algebra & Probability Review
CS 231A Section 1: Linear Algebra & Probability Review 1 Topics Support Vector Machines Boosting Viola-Jones face detector Linear Algebra Review Notation Operations & Properties Matrix Calculus Probability
More informationCamera Calibration. (Trucco, Chapter 6) -Toproduce an estimate of the extrinsic and intrinsic camera parameters.
Camera Calibration (Trucco, Chapter 6) What is the goal of camera calibration? -Toproduce an estimate of the extrinsic and intrinsic camera parameters. Procedure -Given the correspondences beteen a set
More informationCoding the Matrix Index - Version 0
0 vector, [definition]; (2.4.1): 68 2D geometry, transformations in, [lab]; (4.15.0): 196-200 A T (matrix A transpose); (4.5.4): 157 absolute value, complex number; (1.4.1): 43 abstract/abstracting, over
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)
More informationLinear Differential Algorithm for Motion Recovery: A Geometric Approach
International Journal of Computer Vision 361), 71 89 000) c 000 Kluwer Academic Publishers. Manufactured in The Netherlands. Linear Differential Algorithm for Motion Recovery: A Geometric Approach YI MA,
More informationFeature extraction: Corners and blobs
Feature extraction: Corners and blobs Review: Linear filtering and edge detection Name two different kinds of image noise Name a non-linear smoothing filter What advantages does median filtering have over
More informationSegmentation of Dynamic Scenes from the Multibody Fundamental Matrix
ECCV Workshop on Vision and Modeling of Dynamic Scenes, Copenhagen, Denmark, May 2002 Segmentation of Dynamic Scenes from the Multibody Fundamental Matrix René Vidal Dept of EECS, UC Berkeley Berkeley,
More informationRecap: edge detection. Source: D. Lowe, L. Fei-Fei
Recap: edge detection Source: D. Lowe, L. Fei-Fei Canny edge detector 1. Filter image with x, y derivatives of Gaussian 2. Find magnitude and orientation of gradient 3. Non-maximum suppression: Thin multi-pixel
More informationComputational math: Assignment 1
Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange
More informationUNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T
More informationTwo-View Multibody Structure from Motion
International Journal of Computer Vision 68(), 7 5, 006 c 006 Springer Science + Business Media, LLC Manufactured in he Netherlands DOI: 0007/s63-005-4839-7 wo-view Multibody Structure from Motion RENÉ
More informationRoutines for Relative Pose of Two Calibrated Cameras from 5 Points
Routines for Relative Pose of Two Calibrated Cameras from 5 Points Bill Triggs INRIA Rhône-Alpes, 655 avenue de l Europe, 38330 Montbonnot, France. http://www.inrialpes.fr/movi/people/ Triggs Bill.Triggs@inrialpes.fr
More informationCamera Self-Calibration Using the Singular Value Decomposition of the Fundamental Matrix
Camera Self-Calibration Using the Singular Value Decomposition of the Fundamental Matrix Manolis I.A. Lourakis and Rachid Deriche Projet Robotvis, INRIA Sophia-Antipolis 00 route des Lucioles, BP 3 00
More informationScale & Affine Invariant Interest Point Detectors
Scale & Affine Invariant Interest Point Detectors Krystian Mikolajczyk and Cordelia Schmid Presented by Hunter Brown & Gaurav Pandey, February 19, 2009 Roadmap: Motivation Scale Invariant Detector Affine
More informationMachine Learning. Principal Components Analysis. Le Song. CSE6740/CS7641/ISYE6740, Fall 2012
Machine Learning CSE6740/CS7641/ISYE6740, Fall 2012 Principal Components Analysis Le Song Lecture 22, Nov 13, 2012 Based on slides from Eric Xing, CMU Reading: Chap 12.1, CB book 1 2 Factor or Component
More informationMATH 350: Introduction to Computational Mathematics
MATH 350: Introduction to Computational Mathematics Chapter V: Least Squares Problems Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2011 fasshauer@iit.edu MATH
More informationLecture 13 Visual recognition
Lecture 13 Visual recognition Announcements Silvio Savarese Lecture 13-20-Feb-14 Lecture 13 Visual recognition Object classification bag of words models Discriminative methods Generative methods Object
More information3D Computer Vision - WT 2004
3D Computer Vision - WT 2004 Singular Value Decomposition Darko Zikic CAMP - Chair for Computer Aided Medical Procedures November 4, 2004 1 2 3 4 5 Properties For any given matrix A R m n there exists
More informationFactorization with Uncertainty
International Journal of Computer Vision 49(2/3), 101 116, 2002 c 2002 Kluwer Academic Publishers Manufactured in The Netherlands Factorization with Uncertainty P ANANDAN Microsoft Corporation, One Microsoft
More informationThe structure tensor in projective spaces
The structure tensor in projective spaces Klas Nordberg Computer Vision Laboratory Department of Electrical Engineering Linköping University Sweden Abstract The structure tensor has been used mainly for
More informationSimilarity transformation in 3D between two matched points patterns.
Similarity transformation in 3D between two matched points patterns. coordinate system 1 coordinate system 2 The first 3D coordinate system is transformed through a three-dimensional R followed by a translation
More informationCorners, Blobs & Descriptors. With slides from S. Lazebnik & S. Seitz, D. Lowe, A. Efros
Corners, Blobs & Descriptors With slides from S. Lazebnik & S. Seitz, D. Lowe, A. Efros Motivation: Build a Panorama M. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003 How do we build panorama?
More informationHomographies and Estimating Extrinsics
Homographies and Estimating Extrinsics Instructor - Simon Lucey 16-423 - Designing Computer Vision Apps Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince Review: Motivation
More informationDETERMINING THE EGO-MOTION OF AN UNCALIBRATED CAMERA FROM INSTANTANEOUS OPTICAL FLOW
DETERMINING THE EGO-MOTION OF AN UNCALIBRATED CAMERA FROM INSTANTANEOUS OPTICAL FLOW MICHAEL J. BROOKS, WOJCIECH CHOJNACKI, AND LUIS BAUMELA Abstract. The main result of this paper is a procedure for self-calibration
More informationDegeneracies, Dependencies and their Implications in Multi-body and Multi-Sequence Factorizations
Degeneracies, Dependencies and their Implications in Multi-body and Multi-Sequence Factorizations Lihi Zelnik-Manor Michal Irani Dept. of Computer Science and Applied Math The Weizmann Institute of Science
More informationEdges and Scale. Image Features. Detecting edges. Origin of Edges. Solution: smooth first. Effects of noise
Edges and Scale Image Features From Sandlot Science Slides revised from S. Seitz, R. Szeliski, S. Lazebnik, etc. Origin of Edges surface normal discontinuity depth discontinuity surface color discontinuity
More information