Geometric Camera Parameters
|
|
- Sydney Cannon
- 7 years ago
- Views:
Transcription
1 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 are measured in the camera s reference frame. (2) the image coordinates have their origin at the principal point. - In general, the world and pixel coordinate systems are related by a set of physical parameters such as: *the focal length of the lens *the size of the pixels *the position of the principal point *the position and orientation of the camera
2 Types of parameters (Trucco 2.4) -2- -Two types of parameters need to be recovered in order for us to reconstruct the 3D structure of a scene from the pixel coordinates of its image points: Extrinsic camera parameters: the parameters that define the location and orientation of the camera reference frame with respect to a known world reference frame. Intrinsic camera parameters: the parameters necessary to link the pixel coordinates of an image point with the corresponding coordinates in the camera reference frame. Object Coordinates (3D) World Coordinates (3D) Camera Coordinates (3D) extrinsic camera parameters Image Plane Coordinates (2D) intrinsic camera parameters Pixel Coordinates (2D, int)
3 -3- Extrinsic camera parameters -These are the parameters that identify uniquely the transformation between the unknown camera reference frame and the known world reference frame. -Typically, determining these parameters means: () finding the translation vector between the relative positions of the origins of the two reference frames. (2) finding the rotation matrix that brings the corresponding axes of the two frames into alignment (i.e., onto each other)
4 -4- - Using the extrinsic camera parameters, we can find the relation between the coordinates of a point P in world (P w )and camera (P c )coordinates: P c = R(P w T )where R = r r 2 r 3 r 2 r 22 r 32 r 3 r 23 r 33 -If P c = X c Y c Z c and P w = Y w,then X c Y c Z c = r r 2 r 3 r 2 r 22 r 32 r 3 r 23 r 33 T x Y w T y T z or X c = R T (P w T ) Y c = R T 2 (P w T ) Z c = R T 3 (P w T ) where R T i corresponds to the i-th row ofthe rotation matrix
5 -5- Intrinsic camera parameters - These are the parameters that characterize the optical, geometric, and digital characteristics of the camera: () the perspective projection (focal length f ). (2) the transformation between image plane coordinates and pixel coordinates. (3) the geometric distortion introduced by the optics. From Camera Coordinates to Image Plane Coordinates -Apply perspective projection: x = f X c = f RT (P w T ) Z c R T 3 (P w T ), y = f Y c Z c = f RT 2 (P w T ) R T 3 (P w T ) From Image Plane Coordinates to Pixel coordinates x im pixel frame P y im Camera Frame Yc Xc Z c center of perspective projection optical axis y o x,oy image plane frame x principal point World Frame Yw x = (x im o x )s x or x im = x/s x + o x y = (y im o y )s y or y im = y/s y + o y where (o x, o y )are the coordinates of the principal point (in pixels, e.g., o x = N/2, o y = M/2 if the principal point is the center of the image) and s x, s y correspond to the effective size of the pixels in the horizontal and vertical directions (in millimeters).
6 -6- -Using matrix notation: x im y im = /s x /s y o x o y x y Relating pixel coordinates to wor ld coordinates or (x im o x )s x = f RT (P w T ) R T 3 (P w T ), (y im o y )s y = f RT 2 (P w T ) R T 3 (P w T ) x im = fs x R T (P w T ) R T 3 (P w T ) + o x, y im = fs y R T 2 (P w T ) R T 3 (P w T ) + o y Image distortions due to optics Assuming radial distortion: x = x d ( + k r 2 + k 2 r 4 ) y = y d ( + k r 2 + k 2 r 4 ) where (x d, y d )are the coordinates of the distorted points (r 2 = x 2 d + y 2 d) k and k 2 are intrinsic parameters too but will not be considered here...
7 -7- Combine extrinsic with intrinsic camera parameters -The matrix containing the intrinsic camera parameters: f /s x M in = f /s y o x o y -The matrix containing the extrinsic camera parameters: M ex = r r 2 r 3 r 2 r 22 r 32 r 3 r 23 r 33 R T T R T 2 T R T 3 T -Using homogeneous coordinates: x h y h w = M in M ex Y w = M Y w = m m 2 m 3 m 2 m 22 m 32 m 3 m 23 m 33 m 4 m 24 m 34 Y w -Homogenization is needed to obtain the pixel coordinates: x im = x h w = m + m 2 Y w + m 3 + m 4 m 3 + m 32 Y w + m 33 + m 34 y im = y h w = m 2 + m 22 Y w + m 23 + m 24 m 3 + m 32 Y w + m 33 + m 34 - M is called the projection matrix (it is a 3 x 4 matrix). Note: the relation of 3D points and their 2D projections can be seen as a linear transformation from the projective space (, Y w,,) T to the projective plane (x h, y h, w) T.
8 -8- The perspective camera model (using matrix notation) -Assuming o x = o y = and s x = s y = fr M p = fr 2 r 3 fr 2 fr 22 r 32 fr 3 fr 23 r 33 fr T T fr T 2 T R T 3 T -Let s verify the correctness of the above matrix: fr T p = M p P w = fr T 2 R T 3 fr T T fr T 2 T R T 3 T P w fr T (P w T ) = fr T 2 (P w T ) R T 3 (P w T ) -After homogenization (we get the same equations as in page 23): x = f RT (P w T ) R T 3 (P w T ) y = f RT 2 (P w T ) R T 3 (P w T ) The weak perspective camera model (using matrix notation) fr M wp = fr 2 fr 2 fr 22 fr 3 fr 23 fr T T fr T 2 T R T 3 (P T ) where P is the centroid of the object (i.e., object s average distance from the camera) -Wecan verify the correctness of the above matrix: fr T p = M wp P w = fr T 2 fr T T fr T 2 T R T 3 (P T ) P w fr T (P w T ) = fr T 2 (P w T ) R T 3 (P T ) -After homogenization: x = f RT (P w T ) R T 3 (P T ) y = f RT 2 (P w T ) R T 3 (P T )
9 -9- The affine camera model -The entries of the projection matrix are totally unconstrained: M a = a a 2 a 2 a 22 a 3 a 23 a 4 a 24 a 34 -The affine model does not appear to correspond to any physical camera. -Leads to simple equations and appealing geometric properties. -Does not preserve angles but does preserve parallelism.
The 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 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 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 16-18 in TS128 Question 2.1 Given two nonparallel,
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 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 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 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 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 informationRotation Matrices and Homogeneous Transformations
Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n
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 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 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 informationA System for Capturing High Resolution Images
A System for Capturing High Resolution Images G.Voyatzis, G.Angelopoulos, A.Bors and I.Pitas Department of Informatics University of Thessaloniki BOX 451, 54006 Thessaloniki GREECE e-mail: pitas@zeus.csd.auth.gr
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 informationLecture 2: Geometric Image Transformations
Lecture 2: Geometric Image Transformations Harvey Rhody Chester F. Carlson Center for Imaging Science Rochester Institute of Technology rhody@cis.rit.edu September 8, 2005 Abstract Geometric transformations
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics 3D views and projection Adapted from notes by Yong Cao 1 Overview of 3D rendering Modeling: *Define object in local coordinates *Place object in world coordinates (modeling transformation)
More information226-332 Basic CAD/CAM. CHAPTER 5: Geometric Transformation
226-332 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 informationDICOM Correction Item
Correction Number DICOM Correction Item CP-626 Log Summary: Type of Modification Clarification Rationale for Correction Name of Standard PS 3.3 2004 + Sup 83 The description of pixel spacing related attributes
More information8. Linear least-squares
8. Linear least-squares EE13 (Fall 211-12) definition examples and applications solution of a least-squares problem, normal equations 8-1 Definition overdetermined linear equations if b range(a), cannot
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 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 real-valued matrix A is said to be an orthogonal
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, S-LEN 5.644 M23*;-6 (has a good appendix on linear
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 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 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 three-dimensional space, we also examine the
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 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 informationwith functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
More informationSolution Guide III-C. 3D Vision. Building Vision for Business. MVTec Software GmbH
Solution Guide III-C 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 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 informationCS 534: Computer Vision 3D Model-based recognition
CS 534: Computer Vision 3D Model-based recognition Ahmed Elgammal Dept of Computer Science CS 534 3D Model-based Vision - 1 High Level Vision Object Recognition: What it means? Two main recognition tasks:!
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 informationFace detection is a process of localizing and extracting the face region from the
Chapter 4 FACE NORMALIZATION 4.1 INTRODUCTION Face detection is a process of localizing and extracting the face region from the background. The detected face varies in rotation, brightness, size, etc.
More informationPennsylvania System of School Assessment
Pennsylvania System of School Assessment The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, each structured with a common labeling system that can be read
More informationLIST OF CONTENTS CHAPTER CONTENT PAGE DECLARATION DEDICATION ACKNOWLEDGEMENTS ABSTRACT ABSTRAK
vii LIST OF CONTENTS CHAPTER CONTENT PAGE DECLARATION DEDICATION ACKNOWLEDGEMENTS ABSTRACT ABSTRAK LIST OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF NOTATIONS LIST OF ABBREVIATIONS LIST OF APPENDICES
More informationPhotogrammetry Metadata Set for Digital Motion Imagery
MISB ST 0801.5 STANDARD Photogrammetry Metadata Set for Digital Motion Imagery 7 February 014 1 Scope This Standard presents the Key-Length-Value (KLV) metadata necessary for the dissemination of data
More informationRealtime 3D Computer Graphics Virtual Reality
Realtime 3D Computer Graphics Virtual Realit Viewing and projection Classical and General Viewing Transformation Pipeline CPU Pol. DL Pixel Per Vertex Texture Raster Frag FB object ee clip normalized device
More informationThe Image Deblurring Problem
page 1 Chapter 1 The Image Deblurring Problem You cannot depend on your eyes when your imagination is out of focus. Mark Twain When we use a camera, we want the recorded image to be a faithful representation
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
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 information2-View Geometry. Mark Fiala Ryerson University Mark.fiala@ryerson.ca
CRV 2010 Tutorial Day 2-View Geometry Mark Fiala Ryerson University Mark.fiala@ryerson.ca 3-Vectors for image points and lines Mark Fiala 2010 2D Homogeneous Points Add 3 rd number to a 2D point on image
More informationPHYSIOLOGICALLY-BASED DETECTION OF COMPUTER GENERATED FACES IN VIDEO
PHYSIOLOGICALLY-BASED DETECTION OF COMPUTER GENERATED FACES IN VIDEO V. Conotter, E. Bodnari, G. Boato H. Farid Department of Information Engineering and Computer Science University of Trento, Trento (ITALY)
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 informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
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 informationTWO-DIMENSIONAL TRANSFORMATION
CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationPASSENGER SAFETY is one of the most important. Calibration of an embedded camera for driver-assistant systems
1 Calibration of an embedded camera for driver-assistant systems Mario Bellino, Frédéric Holzmann, Sascha Kolski, Yuri Lopez de Meneses, Jacques Jacot Abstract One of the main technical goals in the actual
More informationGeometric Optics Converging Lenses and Mirrors Physics Lab IV
Objective Geometric Optics Converging Lenses and Mirrors Physics Lab IV In this set of lab exercises, the basic properties geometric optics concerning converging lenses and mirrors will be explored. The
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
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 informationCLOUD DIGITISER 2014!
CLOUD DIGITISER 2014 Interactive measurements of point cloud sequences July 2014 Cloud Digitiser Manual 1 CLOUD DIGITISER Interactive measurement of point clouds Bill Sellers July 2014 Introduction Photogrammetric
More informationProjection Center Calibration for a Co-located Projector Camera System
Projection Center Calibration for a Co-located Camera System Toshiyuki Amano Department of Computer and Communication Science Faculty of Systems Engineering, Wakayama University Sakaedani 930, Wakayama,
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationUniversidad de Cantabria Departamento de Tecnología Electrónica, Ingeniería de Sistemas y Automática. Tesis Doctoral
Universidad de Cantabria Departamento de Tecnología Electrónica, Ingeniería de Sistemas y Automática Tesis Doctoral CONTRIBUCIONES AL ALINEAMIENTO DE NUBES DE PUNTOS 3D PARA SU USO EN APLICACIONES DE CAPTURA
More informationTracking Moving Objects In Video Sequences Yiwei Wang, Robert E. Van Dyck, and John F. Doherty Department of Electrical Engineering The Pennsylvania State University University Park, PA16802 Abstract{Object
More information4. CAMERA ADJUSTMENTS
4. CAMERA ADJUSTMENTS Only by the possibility of displacing lens and rear standard all advantages of a view camera are fully utilized. These displacements serve for control of perspective, positioning
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 3-space, as well as define the angle between two non-parallel planes, and determine the distance
More informationGrazing incidence wavefront sensing and verification of X-ray optics performance
Grazing incidence wavefront sensing and verification of X-ray optics performance Timo T. Saha, Scott Rohrbach, and William W. Zhang, NASA Goddard Space Flight Center, Greenbelt, Md 20771 Evaluation of
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 information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationSolving Equations Involving Parallel and Perpendicular Lines Examples
Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
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 informationSYNTHESIZING FREE-VIEWPOINT IMAGES FROM MULTIPLE VIEW VIDEOS IN SOCCER STADIUM
SYNTHESIZING FREE-VIEWPOINT IMAGES FROM MULTIPLE VIEW VIDEOS IN SOCCER STADIUM Kunihiko Hayashi, Hideo Saito Department of Information and Computer Science, Keio University {hayashi,saito}@ozawa.ics.keio.ac.jp
More information3D MODEL DRIVEN DISTANT ASSEMBLY
3D MODEL DRIVEN DISTANT ASSEMBLY Final report Bachelor Degree Project in Automation Spring term 2012 Carlos Gil Camacho Juan Cana Quijada Supervisor: Abdullah Mohammed Examiner: Lihui Wang 1 Executive
More informationCabri Geometry Application User Guide
Cabri Geometry Application User Guide Preview of Geometry... 2 Learning the Basics... 3 Managing File Operations... 12 Setting Application Preferences... 14 Selecting and Moving Objects... 17 Deleting
More informationDaniel F. DeMenthon and Larry S. Davis. Center for Automation Research. University of Maryland
Model-Based 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 informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
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 informationUnderstanding astigmatism Spring 2003
MAS450/854 Understanding astigmatism Spring 2003 March 9th 2003 Introduction Spherical lens with no astigmatism Crossed cylindrical lenses with astigmatism Horizontal focus Vertical focus Plane of sharpest
More informationSpatial location in 360 of reference points over an object by using stereo vision
EDUCATION Revista Mexicana de Física E 59 (2013) 23 27 JANUARY JUNE 2013 Spatial location in 360 of reference points over an object by using stereo vision V. H. Flores a, A. Martínez a, J. A. Rayas a,
More informationAffine Object Representations for Calibration-Free Augmented Reality
Affine Object Representations for Calibration-Free Augmented Reality Kiriakos N. Kutulakos kyros@cs.rochester.edu James Vallino vallino@cs.rochester.edu Computer Science Department University of Rochester
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 informationProcedure: Geometrical Optics. Theory Refer to your Lab Manual, pages 291 294. Equipment Needed
Theory Refer to your Lab Manual, pages 291 294. Geometrical Optics Equipment Needed Light Source Ray Table and Base Three-surface Mirror Convex Lens Ruler Optics Bench Cylindrical Lens Concave Lens Rhombus
More informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
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 informationThe Point-Slope Form
7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More information2. SPATIAL TRANSFORMATIONS
Digitales Video 1 2. SPATIAL TRANSFORMATIONS This chapter describes common spatial transformations derived for digital image warping applications in computer vision and computer graphics. A spatial transformation
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 2009-2011 All rights reserved. LAB. 2 1. Learning goals
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationSUPER RESOLUTION FROM MULTIPLE LOW RESOLUTION IMAGES
SUPER RESOLUTION FROM MULTIPLE LOW RESOLUTION IMAGES ABSTRACT Florin Manaila 1 Costin-Anton Boiangiu 2 Ion Bucur 3 Although the technology of optical instruments is constantly advancing, the capture of
More informationLeast-Squares Intersection of Lines
Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A
More informationHigh-resolution Imaging System for Omnidirectional Illuminant Estimation
High-resolution Imaging System for Omnidirectional Illuminant Estimation Shoji Tominaga*, Tsuyoshi Fukuda**, and Akira Kimachi** *Graduate School of Advanced Integration Science, Chiba University, Chiba
More information12.510 Introduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 04/30/2008 Today s
More information404 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 3, JUNE 1998. Moving Obstacle Detection From a Navigating Robot
404 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 3, JUNE 1998 Moving Obstacle Detection From a Navigating Robot Dinesh Nair, Member, IEEE, and Jagdishkumar K. Aggarwal, Fellow, IEEE Abstract
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationEXPLORING THE TRUE GEOMETRY OF THE INELASTIC INSTANTANEOUS CENTER METHOD FOR ECCENTRICALLY LOADED BOLT GROUPS
EXPLORING THE TRUE GEOMETRY OF THE INELASTIC INSTANTANEOUS CENTER METHOD FOR ECCENTRICALLY LOADED BOLT GROUPS L.S. Muir, P.E., Cives Steel Company, The United States W.A. Thornton, P.E., PhD, Cives Steel
More informationInteractive 3D Scanning Without Tracking
Interactive 3D Scanning Without Tracking Matthew J. Leotta, Austin Vandergon, Gabriel Taubin Brown University Division of Engineering Providence, RI 02912, USA {matthew leotta, aev, taubin}@brown.edu Abstract
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationGeorgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade
Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade
More informationDESIGNING A HIGH RESOLUTION 3D IMAGING DEVICE FOR FOOTPRINT AND TIRE TRACK IMPRESSIONS AT CRIME SCENES 1 TR-CIS-0416-12. Ruwan Egoda Gamage
DESIGNING A HIGH RESOLUTION 3D IMAGING DEVICE FOR FOOTPRINT AND TIRE TRACK IMPRESSIONS AT CRIME SCENES 1 TR-CIS-0416-12 Ruwan Egoda Gamage Abhishek Joshi Jiang Yu Zheng Mihran Tuceryan A Technical Report
More informationPHOTOGRAMMETRIC TECHNIQUES FOR MEASUREMENTS IN WOODWORKING INDUSTRY
PHOTOGRAMMETRIC TECHNIQUES FOR MEASUREMENTS IN WOODWORKING INDUSTRY V. Knyaz a, *, Yu. Visilter, S. Zheltov a State Research Institute for Aviation System (GosNIIAS), 7, Victorenko str., Moscow, Russia
More informationIntroduction to 2D and 3D Computer Graphics Mastering 2D & 3D Computer Graphics Pipelines
Introduction to 2D and 3D Computer Graphics Mastering 2D & 3D Computer Graphics Pipelines CS447 3-1 Mastering 2D & 3D Graphics Overview of 2D & 3D Pipelines What are pipelines? What are the fundamental
More information