Image Formation and Camera Calibration


 Elinor Parsons
 1 years ago
 Views:
Transcription
1 EECS 432Advanced Computer Vision Notes Series 2 Image Formation and Camera Calibration Ying Wu Electrical Engineering & Computer Science Northwestern University Evanston, IL Contents inhole Camera Model 2 erspective rojection 2 2 Weakerspective rojection 2 3 Orthographic rojection 2 2 Homogeneous Coordinates 3 3 Coordinate System Changes and Rigid Transformations 3 3 Translation 3 32 Rotation 4 33 Rigid Transformation 4 34 Summary 4 4 Image Formation (Geometrical) 5 5 Camera Calibration Inferring Camera arameters 6 5 The Setting of the roblem 6 52 Computing the rojection Matrix 7 53 Computing Intrinsic and Extrinsic arameters 7 54 Questions to Think Over 8
2 inhole Camera Model erspective rojection O Π f k O j pinhole =(x y ) p =(x y ) i Figure : Illustration of perspective projection of a pinhole camera From Figure, we can easily figure out: { x = f x/ y = f y/ () Note: Thin lenses cameras has the same geometry as pinhole cameras Thin lenses camera geometry: = f 2 Weakerspective rojection { x = (f / 0 )x y = (f / 0 )y When the scene depth is small comparing to the average distance from the camera, ie, the depth of the scene is flat, weakperspective projection is a good approximation of perspective projection It is also called scaledorthographic projection 3 Orthographic rojection (2) { x = x y = y (3) 2
3 When camera always remains at a roughly constant distance from the scene, and scene centers the optic axis, orthographic projection could be used as an approximation Obviously, orthographic projection is not real 2 Homogeneous Coordinates A 3D point is = and a plane can be written by ax + by + c d = 0 We use homogeneous coordinates to unify the representation for points and lines by adding one dimensionality, ie, we use x = y for points and Π = x y a b c d for planes, where the plane Π is defined up to a scale We have Π = 0 3 Coordinate System Changes and Rigid Transformations For convenience, We use the Craig notation frame F F means the coordinates of point in 3 Translation B = A + B O A (4) where B O A is the coordinate of the origin O A of frame A in the new coordinate system B 3
4 32 Rotation B AR = ( B i A B j A B k A ) = A i T B A j T B A k T B (5) where B i A is the coordinate of the axis i A of frame A in the new coordinate system B Then we have, B = B A R A 33 Rigid Transformation B = B A R A + B O A (6) Let s see here an advantage of the homogeneous coordinates If we make two consecutive rigid transformation, ie, from A B C, the coordinate of point in frame C will be written by: C = C B R( B AR A + B O A ) + C O B = C B R B AR A + ( C BR B O A + C O B ) It looks very awkward If we present it by homogeneous coordinates, it looks very concise [ B [ BA [ R = B O A A 0 T and So, [ C = [ C = [ CB R C O B 0 T [ CB R C O B 0 T [ B [ BA R B O A 0 T [ A 34 Summary We could write a transformation by: m m 2 m 3 m 4 T = m 2 m 22 m 23 m 24 m 3 m 32 m 33 m 34 m 4 m 42 m 43 m 44 We call it a projective transformation If we can write: [ A t T = 0 T 4
5 It becomes an affine transformation If A = R, ie, a rotation matrix (R T R = I), [ R t T = 0 T it becomes a Euclidean transformation or a rigid transformation Obviously, Euclidean transformation preserves both parallel lines and angles, but affine preserves parallel lines but not angles 4 Image Formation (Geometrical) In this section, we discuss the process of image formation in terms of geometry For a 3D point p w = [x w, y w, w T in the world coordinate system, its will be mapped to a camera coordinate system (C) from the world coordinate system (F), then map to the physical retina, ie, the physical image plane, and get image coordinates [u, v T We shall ask how this 3D point is mapped to its image coordinate y world coordin v v p p camera coordin x u image coordin u normalied image plane physical retina Figure 2: Illustration of the geometry of the image formation process under perspective projection of a pinhole camera For convenience, we introduce a normalied image plane located at the focal length f = In such a normalied image plane, the pinhole (c) is mapped to the origin of the image plane 5
6 (ĉ), and p is mapped to ˆp = [û, ˆv T û ˆp = ˆv = [ I c [ p c 0 = [ I 0 c x c y c c And we also have u v = kf 0 u 0 x c 0 lf v c 0 y c = kf 0 u 0 0 lf v 0 0 c c 0 [ I Let α = kf and β = lf We call these parameters α,β,u 0 and v 0 intrinsic parameters, which present the inner camera imaging parameters We can write x u v = α 0 u 0 0 c 0 β v c 0 0 y c c = α 0 u 0 0 [ x w 0 β v c 0 0 R t y w T w We call R and t extrinsic parameters, which represent the coordinate transformation between the camera coordinate system and the world coordinate system So, we can write, u v = M M c 2 p w = c Mpw (7) We call M the projection matrix x c y c c 5 Camera Calibration Inferring Camera arameters 5 The Setting of the roblem We are given () a calibration rig, ie, a reference object, to provide the world coordinate system, and (2) an image of the reference object The problem is to solve (a) the projection matrix, and (b) the intrinsic and extrinsic parameters Mathematically, given [x w i, yi w, i w T, i =,, n, and [u i, v i t, i =,, n, we want to solve M and M 2, st, x u w i v i = i x w M i c M 2 yi w i w = i M y w i i c i w, i 6
7 52 Computing the rojection Matrix We can write c i u i v i x m m 2 m 3 m w i 4 = m 2 m 22 m 23 m 24 y w i i m 3 m 32 m 33 m w 34 i c u i = m x w i + m 2 yi w + m 3 i w + m 4 i c v i = m 2 x w i + m 22 yi w + m 23 i w + m 24 i c = m 3 x w i + m 32 yi w + m 33 i w + m 34 Then x w i m + y w i m 2 + w i m 3 + m 4 u i x w i m 3 u i y w i m 32 u i w i m 33 = u i m 34 x w i m 2 + y w i m 22 + w i m 23 + m 24 v i x w i m 3 v i y w i m 32 v i w i m 33 = v i m 34 Then, x w y w w u x w u y w u w x w y w w v x w v y w v w x w n yn w n w u n x w n u n yn w u n n w x w n yn w n w v n x w n v n yn w v n n w m m 2 m 3 m 4 m 32 m 33 u m 34 v m 34 u 2 m 34 = v 2 m 34 u n m 34 v n m 34 (8) Obviously, we can let m 34 =, ie, the projection matrix is scaled by m 34 We have: Km = U (9) Where, K is a 2n matrix, m is a D vector, and U is a 2nD vector The least squares solution of Equation 9 is obtainted by: m = K U = (K T K) K T U (0) where K is the pseudoinverse of K Here, m and m 34 = constitute the projection matrix M This is a linear solution to the project matrix 53 Computing Intrinsic and Extrinsic arameters After computing the projection matrix M, we can compute the intrinsic and extrinsic parameters from M lease notice that the projection matrix M obtained from Equation 9 is a scaled version of the true M, since we have let m 34 = To decompose the projection 7
8 matrix in order to solve M and M 2, we need to take into account of m 34, ie, the true m 34 needs to be figured out Apparently, we have: m T r m 4 α 0 u 0 0 T t x m 34 M = m 34 m T 2 m 24 = 0 β v 0 0 r T 2 t y αr T m T r T 3 t = + u 0 r T 3 αt x + u 0 t x βr T 2 + v 0 r T 3 βt y + v 0 t x () 0 T r T 3 t x where m T i = [m i, m i2, m i3, M = {m ij } is computed from Equation 9, and r T i = [r i, r i2, r i3, R = {r ij } is the rotation matrix To see it clearly, we have: αr T + u 0 r T 3 βr T 2 + v 0 r T 3 r T 3 αt x + u 0 t x m 34 m T m 34 m 4 βt y + v 0 t x = m 34 m T 2 m 34 m 24 (2) m 34 m T 3 m 34 t x Obviously, by comparing these two matrices, it is easy to see m 34 m 3 = r 3 In addition, since R is a rotation matrix, we have r i = Then, we have m 34 = m 3 Then, it is easy to figure out all the other parameters: After that, it is also easy to get: r 3 = m 34 m 3 (3) u 0 = (αr T + u 0 r T 3 )r 3 = m 2 34m T m 3 (4) v 0 = (βr T 2 + v 0 r T 3 )r 3 = m 2 34m T 2 m 3 (5) α = m 2 34 m m 3 (6) β = m 2 34 m 2 m 3 (7) r = m 34 α (m u 0 m 3 ) (8) r 2 = m 34 β (m 2 v 0 m 3 ) (9) t = m 34 (20) t x = m 34 α (m 4 u 0 ) (2) t y = m 34 β (m 24 v 0 ) (22) Now, we have obtained all the intrinsic and extrinsic parameters For the analysis of skewed camera models (ie, θ 90 o ), please read chapter 6 of Forsyth&once 54 Questions to Think Over We v introduced a linear approach for camera calibration Let s think some questions over: 8
9 The matrix M, representing the intrinsic characteristics of cameras, has 4 independent variables; and the matrix M 2, representing the extrinsic characteristics, has 6 independent variables So the projection matrix M has 0 independent variables When letting m 34 =, we still have parameters to determine However, please note, these parameters are not independent! But our solution by Equation 9 did not address such dependency or constraints among these parameters Consequently, computation errors are not avoidable The question is that, if we can find an approach to take into account of that to enhance the accuracy? The method described above assumes that we know the 2D image coordinates How can we get these 2D image points? If the detection of these 2D image points contains noise, how does the noise affect the accuracy of the result? If we are not using pointcorespondences, can we use lines? 9
Lecture 19 Camera Matrices and Calibration
Lecture 19 Camera Matrices and Calibration Project Suggestions Texture Synthesis for InPainting Section 10.5.1 in Szeliski Text Project Suggestions Image Stitching (Chapter 9) Face Recognition Chapter
More informationGeometric Camera Parameters
Geometric Camera Parameters What assumptions have we made so far? All equations we have derived for far are written in the camera reference frames. These equations are valid only when: () all distances
More informationModule 6: Pinhole camera model Lecture 30: Intrinsic camera parameters, Perspective projection using homogeneous coordinates
The Lecture Contains: Pinhole camera model 6.1 Intrinsic camera parameters A. Perspective projection using homogeneous coordinates B. Principalpoint offset C. Imagesensor characteristics file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_1.htm[12/31/2015
More information2D Geometric Transformations. COMP 770 Fall 2011
2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrixvector multiplication Matrixmatrix multiplication
More informationStructured light systems
Structured light systems Tutorial 1: 9:00 to 12:00 Monday May 16 2011 Hiroshi Kawasaki & Ryusuke Sagawa Today Structured light systems Part I (Kawasaki@Kagoshima Univ.) Calibration of Structured light
More informationAffine Transformations. University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell
Affine Transformations University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell Logistics Required reading: Watt, Section 1.1. Further reading: Foley, et al, Chapter 5.15.5. David
More informationLecture 2: Homogeneous Coordinates, Lines and Conics
Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4
More informationCamera calibration and epipolar geometry. Odilon Redon, Cyclops, 1914
Camera calibration and epipolar geometry Odilon Redon, Cyclops, 94 Review: Alignment What is the geometric relationship between pictures taken by cameras that share the same center? How many points do
More informationEpipolar Geometry. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. Right Image. Left Image. e(p ) Epipolar Lines. e(q ) q R.
Epipolar Geometry We consider two perspective images of a scene as taken from a stereo pair of cameras (or equivalently, assume the scene is rigid and imaged with a single camera from two different locations).
More informationGeometric Objects and Transformations
Geometric Objects and ransformations How to represent basic geometric types, such as points and vectors? How to convert between various represenations? (through a linear transformation!) How to establish
More informationgeometric transforms
geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition
More informationThe Geometry of Perspective Projection
The Geometry o Perspective Projection Pinhole camera and perspective projection  This is the simplest imaging device which, however, captures accurately the geometry o perspective projection. Rays o
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribes: Jia Mao, Andrew Rabinovich LECTURE 9 Affine and Euclidean Reconstruction 9.1. Stratified reconstruction Recall that in 3D reconstruction from
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
More information3D Scanner using Line Laser. 1. Introduction. 2. Theory
. Introduction 3D Scanner using Line Laser Di Lu Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute The goal of 3D reconstruction is to recover the 3D properties of a geometric
More informationUsing Spatially Distributed Patterns for Multiple View Camera Calibration
Using Spatially Distributed Patterns for Multiple View Camera Calibration Martin Grochulla, Thorsten Thormählen, and HansPeter Seidel MaxPlancInstitut Informati, Saarbrücen, Germany {mgrochul, thormae}@mpiinf.mpg.de
More informationB4 Computational Geometry
3CG 2006 / B4 Computational Geometry David Murray david.murray@eng.o.ac.uk www.robots.o.ac.uk/ dwm/courses/3cg Michaelmas 2006 3CG 2006 2 / Overview Computational geometry is concerned with the derivation
More informationWii Remote Calibration Using the Sensor Bar
Wii Remote Calibration Using the Sensor Bar Alparslan Yildiz Abdullah Akay Yusuf Sinan Akgul GIT Vision Lab  http://vision.gyte.edu.tr Gebze Institute of Technology Kocaeli, Turkey {yildiz, akay, akgul}@bilmuh.gyte.edu.tr
More information2D Geometric Transformations
2D Geometric Transformations (Chapter 5 in FVD) 2D Geometric Transformations Question: How do we represent a geometric object in the plane? Answer: For now, assume that objects consist of points and lines.
More informationMath 497C Sep 9, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say
More informationRotation Matrices and Homogeneous Transformations
Rotation Matrices and Homogeneous Transformations A coordinate frame in an ndimensional space is defined by n mutually orthogonal unit vectors. In particular, for a twodimensional (2D) space, i.e., n
More informationw = COI EYE view direction vector u = w ( 010,, ) cross product with yaxis v = w u up vector
. w COI EYE view direction vector u w ( 00,, ) cross product with ais v w u up vector (EQ ) Computer Animation: Algorithms and Techniques 29 up vector view vector observer center of interest 30 Computer
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationRelating Vanishing Points to Catadioptric Camera Calibration
Relating Vanishing Points to Catadioptric Camera Calibration Wenting Duan* a, Hui Zhang b, Nigel M. Allinson a a Laboratory of Vision Engineering, University of Lincoln, Brayford Pool, Lincoln, U.K. LN6
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationRESEARCHES ON HAZARD AVOIDANCE CAMERAS CALIBRATION OF LUNAR ROVER
ICSO International Conference on Space Optics 48 October RESERCHES ON HZRD VOIDNCE CMERS CLIBRTION OF LUNR ROVER Li Chunyan,,Wang Li,,LU Xin,,Chen Jihua 3,Fan Shenghong 3. Beijing Institute of Control
More informationSection 1.7 22 Continued
Section 1.5 23 A homogeneous equation is always consistent. TRUE  The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE  The equation
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationIntroduction Epipolar Geometry Calibration Methods Further Readings. Stereo Camera Calibration
Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration 12.10.2004 Overview Introduction Summary / Motivation Depth Perception Ambiguity of Correspondence
More informationVPITS : VIDEO BASED PHONOMICROSURGERY INSTRUMENT TRACKING SYSTEM. Ketan Surender
VPITS : VIDEO BASED PHONOMICROSURGERY INSTRUMENT TRACKING SYSTEM by Ketan Surender A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science (Electrical Engineering)
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More information1 Scalars, Vectors and Tensors
DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationImage Formation. The two parts of the image formation process. Asimple model
Image Formation The two parts of the image formation process The geometry of image formation which determines where in the image plane the projection of a point in the scene will be located. The physics
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationGeometric Transformations and Image Warping: Mosaicing
Geometric Transformations and Image Warping: Mosaicing CS 6640 Ross Whitaker, Guido Gerig SCI Institute, School of Computing University of Utah (with slides from: Jinxiang Chai, TAMU) faculty.cs.tamu.edu/jchai/cpsc641_spring10/lectures/lecture8.ppt
More informationINTRODUCTION TO RENDERING TECHNIQUES
INTRODUCTION TO RENDERING TECHNIQUES 22 Mar. 212 Yanir Kleiman What is 3D Graphics? Why 3D? Draw one frame at a time Model only once X 24 frames per second Color / texture only once 15, frames for a feature
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More information2D Geometrical Transformations. Foley & Van Dam, Chapter 5
2D Geometrical Transformations Fole & Van Dam, Chapter 5 2D Geometrical Transformations Translation Scaling Rotation Shear Matri notation Compositions Homogeneous coordinates 2D Geometrical Transformations
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationGiven a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes:
3 3.1 2D Given a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes: 1. Change coordinate frames (world, window, viewport, device, etc). 2. Compose objects
More informationVector algebra Christian Miller CS Fall 2011
Vector algebra Christian Miller CS 354  Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationGeometric Transformations
CS3 INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and 3D CS3 INTRODUCTION TO COMPUTER GRAPHICS Grading Plan to be out Wednesdas one week after the due date CS3 INTRODUCTION TO COMPUTER
More informationEuler Angle Formulas. Contents. David Eberly Geometric Tools, LLC Copyright c All Rights Reserved.
Euler Angle Formulas David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: December 1, 1999 Last Modified: March 10, 2014 Contents 1 Introduction
More informationGeometric Image Transformations
Geometric Image Transformations Part One 2D Transformations Spatial Coordinates (x,y) are mapped to new coords (u,v) pixels of source image > pixels of destination image Types of 2D Transformations Affine
More informationNonMetric ImageBased Rendering for Video Stabilization
NonMetric ImageBased Rendering for Video Stabilization Chris Buehler Michael Bosse Leonard McMillan MIT Laboratory for Computer Science Cambridge, MA 02139 Abstract We consider the problem of video stabilization:
More informationUnit 3 (Review of) Language of Stress/Strain Analysis
Unit 3 (Review of) Language of Stress/Strain Analysis Readings: B, M, P A.2, A.3, A.6 Rivello 2.1, 2.2 T & G Ch. 1 (especially 1.7) Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 11 THREE DIMENSIONAL GEOMETRY 111 Overview 1111 Direction cosines of a line are the cosines of the angles made by the line with positive directions of the coordinate axes 111 If l, m, n are the
More informationCS 534: Computer Vision 3D Modelbased recognition
CS 534: Computer Vision 3D Modelbased recognition Ahmed Elgammal Dept of Computer Science CS 534 3D Modelbased Vision  1 High Level Vision Object Recognition: What it means? Two main recognition tasks:!
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More information226332 Basic CAD/CAM. CHAPTER 5: Geometric Transformation
226332 Basic CAD/CAM CHAPTER 5: Geometric Transformation 1 Geometric transformation is a change in geometric characteristics such as position, orientation, and size of a geometric entity (point, line,
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationMonash University Clayton s School of Information Technology CSE3313 Computer Graphics Sample Exam Questions 2007
Monash University Clayton s School of Information Technology CSE3313 Computer Graphics Questions 2007 INSTRUCTIONS: Answer all questions. Spend approximately 1 minute per mark. Question 1 30 Marks Total
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationAffine Transformations
A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationA Fourstep Camera Calibration Procedure with Implicit Image Correction
A Fourstep Camera Calibration Procedure with Implicit Image Correction Janne Heikkilä and Olli Silvén Infotech Oulu and Department of Electrical Engineering University of Oulu FIN957 Oulu, Finland email:
More information3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)
3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis
More informationComputer Vision  part II
Computer Vision  part II Review of main parts of Section B of the course School of Computer Science & Statistics Trinity College Dublin Dublin 2 Ireland www.scss.tcd.ie Lecture Name Course Name 1 1 2
More informationCinema Projection Distortion
This paper was presented at: The 141st SMPTE Technical Conference and Exhibition Displays for the Theater and Home Session November 20, 1999 Cinema Projection Distortion Ronald A. Petrozzo Stuart W. Singer
More informationTWODIMENSIONAL TRANSFORMATION
CHAPTER 2 TWODIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationComputer Graphics Labs
Computer Graphics Labs Abel J. P. Gomes LAB. 2 Department of Computer Science and Engineering University of Beira Interior Portugal 2011 Copyright 20092011 All rights reserved. LAB. 2 1. Learning goals
More informationDaniel F. DeMenthon and Larry S. Davis. Center for Automation Research. University of Maryland
ModelBased Object Pose in 25 Lines of Code Daniel F. DeMenthon and Larry S. Davis Computer Vision Laboratory Center for Automation Research University of Maryland College Park, MD 20742 Abstract In this
More information521493S Computer Graphics. Exercise 2 & course schedule change
521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 1618 in TS128 Question 2.1 Given two nonparallel,
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationLecture 7. Matthew T. Mason. Mechanics of Manipulation. Lecture 7. Representing Rotation. Kinematic representation: goals, overview
Matthew T. Mason Mechanics of Manipulation Today s outline Readings, etc. We are starting chapter 3 of the text Lots of stuff online on representing rotations Murray, Li, and Sastry for matrix exponential
More informationSolution Guide IIIC. 3D Vision. Building Vision for Business. MVTec Software GmbH
Solution Guide IIIC 3D Vision MVTec Software GmbH Building Vision for Business Machine vision in 3D world coordinates, Version 10.0.4 All rights reserved. No part of this publication may be reproduced,
More informationVOLUMNECT  Measuring Volumes with Kinect T M
VOLUMNECT  Measuring Volumes with Kinect T M Beatriz Quintino Ferreira a, Miguel Griné a, Duarte Gameiro a, João Paulo Costeira a,b and Beatriz Sousa Santos c,d a DEEC, Instituto Superior Técnico, Lisboa,
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationCamera geometry and image alignment
Computer Vision and Machine Learning Winter School ENS Lyon 2010 Camera geometry and image alignment Josef Sivic http://www.di.ens.fr/~josef INRIA, WILLOW, ENS/INRIA/CNRS UMR 8548 Laboratoire d Informatique,
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationLecture 3: Coordinate Systems and Transformations
Lecture 3: Coordinate Systems and Transformations Topics: 1. Coordinate systems and frames 2. Change of frames 3. Affine transformations 4. Rotation, translation, scaling, and shear 5. Rotation about an
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationStudents will understand 1. use numerical bases and the laws of exponents
Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?
More informationTHE problem of visual servoing guiding a robot using
582 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 13, NO. 4, AUGUST 1997 A Modular System for Robust Positioning Using Feedback from Stereo Vision Gregory D. Hager, Member, IEEE Abstract This paper
More informationELEMENTS OF VECTOR ALGEBRA
ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 3 Linear Least Squares Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c 2002. Reproduction
More informationUsing Many Cameras as One
Using Many Cameras as One Robert Pless Department of Computer Science and Engineering Washington University in St. Louis, Box 1045, One Brookings Ave, St. Louis, MO, 63130 pless@cs.wustl.edu Abstract We
More information8. Linear leastsquares
8. Linear leastsquares EE13 (Fall 21112) definition examples and applications solution of a leastsquares problem, normal equations 81 Definition overdetermined linear equations if b range(a), cannot
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationImage Formation. 7year old s question. Reference. Lecture Overview. It receives light from all directions. Pinhole
Image Formation Reerence http://en.wikipedia.org/wiki/lens_(optics) Reading: Chapter 1, Forsyth & Ponce Optional: Section 2.1, 2.3, Horn. The slides use illustrations rom these books Some o the ollowing
More informationGeometric Transformation CS 211A
Geometric Transformation CS 211A What is transformation? Moving points (x,y) moves to (x+t, y+t) Can be in any dimension 2D Image warps 3D 3D Graphics and Vision Can also be considered as a movement to
More information4BA6  Topic 4 Dr. Steven Collins. Chap. 5 3D Viewing and Projections
4BA6  Topic 4 Dr. Steven Collins Chap. 5 3D Viewing and Projections References Computer graphics: principles & practice, Fole, vandam, Feiner, Hughes, SLEN 5.644 M23*;6 (has a good appendix on linear
More informationConverting Between Coordinate Systems
Converting Between Coordinate Systems David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: April 23, 2014 Last Modified: March 14, 2016
More informationSTUDIES IN ROBOTIC VISION, OPTICAL ILLUSIONS AND NONLINEAR DIFFUSION FILTERING
STUDIES IN ROBOTIC VISION, OPTICAL ILLUSIONS AND NONLINEAR DIFFUSION FILTERING HENRIK MALM Centre for Mathematical Sciences Mathematics Mathematics Centre for Mathematical Sciences Lund University Box
More informationA Flexible New Technique for Camera Calibration
A Flexible New Technique for Camera Calibration Zhengyou Zhang December 2, 1998 (updated on December 14, 1998) (updated on March 25, 1999) (updated on Aug. 1, 22; a typo in Appendix B) (updated on Aug.
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More information