Computer Vision, Laboratory session 1
|
|
- Wendy McKinney
- 6 years ago
- Views:
Transcription
1 Centre for Mathematical Sciences, january 200 Computer Vision, Laboratory session Overview In this laboratory session you are going to use matlab to look at images, study the representations of points, lines and conics used in projective geometry. You are going to learn more about the camera equation and about how images and features change under coordinate transformations. Some of the instructions here are matlab commands that you are supposed to try. To relieve you of some of the tedious typing, we have collected some of those commands in a script lab_cheats.m in the folder lab that you are going to download and unpack. Computer preparations Download the matlab files needed for the laboration using your favorite web-browser. The homepage for the course is at and the files needed for the laboratory session at Copy this file to your home directory, decompress and unpack it. In unix use: unzip datorseende.zip On PCs use winzip or some other alternative. During this course several directories of matlab routines are going to be downloaded and used. It is practical to have these placed in a common directory datorseende with subdirectories tools data lab lab2 lab3 lab4 These directories are created automatically when you unpack datorseende.zip. A matlab script startup.m is located in the directory datorseende. If you start your matlab session in that directory the search paths are automatically set to the directories above. This simplifies things. If you start matlab in some other way you have to set matlab search paths yourself. Image formats There is an enormous amount of different image formats. These are usually differentiated by their suffixes (.tif,.jpg,.gif,.pgm, etc.). Some of these formats use compression in order to save space.
2 There is also a number of programs that can show and manipulate images. It is convenient to know at least one such program, preferably one that can convert between many different image formats. In the course we are not going to focus so much on these issues. We are mainly going to use a raw format called PGM. Each intensity value of the image is stored either in ascii or using binary encoding. We have written a routine in matlab that can read such images. Matlab also has a image reading routine called imread, that can read images in the formats BMP, HDF, JPEG, PCX, TIFF and XWD. Loading and viewing images Load one of the images into matlab and display it using the matlab command imagesc. bild = readpgm( plc00.pgm ); figure(); colormap(gray(64)); imagesc(bild); Question: Try zooming in on different parts of the image by klicking on the figure. Do you see anything special? Try zooming in on the sharp edges. Is there anything strange with every 6 th column in the image? The errors are introduced in the frame grabber. This is quite typical. Each frame grabber or camera has its on quirks. We have implemented a special routine that removes these problems (for this specific frame grabber). Try bild = pgmlas( plc00.pgm,); figure(); colormap(gray(64)); imagesc(bild); Image points A point in the image can be represented by its Cartesian coordinates (x, y). Most often, however, we will use so called extended or homogeneous coordinates where denotes equality up to scale. Question: x x y, What Cartesian coordinates do the following points have: , , 00 23? 2 Load one of the images into matlab and display it using the matlab command imagesc. Then plot a point using a home made command rita. bild = pgmlas( plc00.pgm,); figure(); 2
3 colormap(gray(64)); imagesc(bild); hold on; u=[229.6; 25.6; ]; rita(u, * ); Question: Where are the coordinates (230,25) located? In the upper left corner of a pixel or in the middle of a pixel? What interpolation method does matlab use when displaying images? Image lines An image line can be written in affine form as {(x, y) x T l = ax + by + c = 0} i.e. all points that fulfill the constraint ax + by + c = 0. We will usually use the representation l=[a;b;c]; for this line. Notice that an image point u lies on the line l if and only if l T u = 0. Notice also that two lines are identical if and only if their coefficients differ by a non-zero scale factor. The notation is used to denote this equality up to scale Try l=[0.43; -0.9; 34.3]; rital(l); Image conics An image conic, such as an ellipse or a parabola, is defined as the solution set for a quadratic equation: {(x, y) C x 2 + 2C 2 xy + 2C 3 x + C 22 y 2 + 2C 23 y + C 33 = 0}. Similar to lines, the conic can be represented in homogeneous coordinates x = (x, y, ) T as the solutions of x T Cx = 0, where C is a symmetric matrix containing the coefficients of the conic: C C 2 C 3 C = C 2 C 22 C 23 C 3 C 23 C 33 Sometimes it is convenient to represent the conic using all of its tangent lines. The conic is then defined as the set of points which are touched by exactly one of the following (tangent) lines: {l = (a, b, c) l T Dl = D a 2 + 2D 2 ab + 2D 3 ac + D 22 b 2 + 2D 23 bc + D 33 c 2 = 0} This set of lines is called the dual conic because it is itself a conic in the variable l. The two matrices D and C are inverses of each other D = C. Try (remember that most matlab code is in the file lab_cheats.m.) 3
4 C=[ ; ; ;... ]; ritac(c); Coordinate transformations Many computer vision calculations become better conditioned if one changes coordinate system from pixels to angles. This requires the knowledge of the so called intrinsic parameters. For these images the following change of image coordinates can be used. (u, v) (x, y) = ((u 348)/200, (v 286)/200). () Here (u, v) are pixel coordinates and (x, y) are image coordinates corrected for approximate internal calibration. We have subtracted (348, 286) from the pixel coordinates so that the middle of the image (the principal point) becomes the origin. We have also divided by the camera constant 200. This is the distance between the image plane and the focal point of the camera (measured in pixels). Using homogeneous coordinates the transformation can be represented by a 3 3 matrix K so that x y = K u v. (2) What should the matrix K be in this case? Enter this matrix into the matlab workspace. If we change the coordinates of the image points we have to change the coordinate representation of features such as points, lines and conics. Question: What are the coordinate transformation formuli? Compute the coordinate representations u, l and C, of the point u the line l and the conic C that we have studied in the laboratory session. Plot the image in the new coordinate system using: figure(); colormap(gray(64)); hold off; imagesc(([ 697]-348)/200,([ 573]-286)/200,bild) Now plot the point, line and conic in the new coordinate system. u_tilde = K*u; %... calculate l_tilde and C_tilde according to the formuli hold on; rita(u_tilde, * ); rital(l_tilde); ritac(c_tilde); Check that your transformations are correct. If they are now being plotted on the same place relative to the image all is well. Otherwise, check your calculations and do it again. 4
5 The camera equation Points in a world coordinate system is projected onto points in the image using the camera equation, x P X. Since we only get the image points up to scale it is necessary to normalize the third coordinate of x to one, before plotting them. This can be done with the command pflat. Try X = [randn(3,20);ones(,20)]; P = [eye(3) [0;0;0]]; x = pflat(p*x); figure(2); rita(x, * ); axis([ ]); This gives the projection of 20 points from one viewpoint. The next script generates a sequence of camera matrices P t, using a local parametrisation of rotations: Try r r 2 e r 3 0 r 3 r 2 r 3 0 r r 2 r 0. R=eye(3); r=0.05*randn(3,); for i=:00; R=expm([0 -r(3) r(2);r(3) 0 -r();-r(2) r() 0])*R; T=[R zeros(3,);zeros(,3) ]; Pt = P*T; xt = pflat(pt*x); rita(xt, * ); axis([ ]); drawnow; pause(0.); end; If you want to you can try the same script with hold on to see the paths of the points in the images. By using the commmand rita3, it is possible to plot 3D-points as well. Modify the above script such that the 3D-points X and the camera path are plotted in a separate figure in the for-loop. Hint: Use rita3(x, * ) to plot the 3D-points. The homogeneous coordinates of the camera centre is obtained by computing the nullspace of the camera matrix (null(pt)). Vanishing points Question: Consider two lines in the scene. The first line goes through points ( 0.5,, 0) and ( 0.5,, 00). The second line goes through points (0.5,, 0) and (0.5,, 00). Do the lines lie in a common plane? Do they intersect? In matlab plot the projection of these four points with the camera matrix P above. Xl = [ ] ; 5
6 Xl2 = [ ] ; Xl2 = [0.5-0 ] ; Xl22 = [ ] ; xl = pflat(p*xl); xl2 = pflat(p*xl2); xl2 = pflat(p*xl2); xl22 = pflat(p*xl22); hold off; rita(x, * ); hold on rita([xl xl2]); rita([xl2 xl22]); Question: Calculate the intersection of the two object lines. What are the homogeneous coordinates? What are the Cartesian coordinates? Calculate the projection of the intersection and plot it in the image. Xint = [0 0 0] ; xint = pflat(p*xint); rita(xint, r* ); Notice that the intersection of these lines in the scene is a strange point (a point at infinity), but that its projection in the image is an ordinary point. 6
Computer Vision, Laboratory session 1
Centre for Mathematical Sciences, january 2007 Computer Vision, Laboratory session 1 Overview In this laboratory session you are going to use matlab to look at images, study projective geometry representations
More informationComputer Vision, Assignment 1 Elements of Projective Geometry
Centre for Mathematical Sciences, February 05 Due study week Computer Vision, Assignment Elements of Projective Geometry Instructions In this assignment you will study the basics of projective geometry.
More informationExercise session using MATLAB: Quasiconvex Optimixation
Optimization in Computer Vision, May 2008 Exercise session using MATLAB: Quasiconvex Optimixation Overview In this laboratory session you are going to use matlab to study structure and motion estimation
More informationCSE 252B: Computer Vision II
CSE 252B: Computer Vision II Lecturer: Serge Belongie Scribe: Sameer Agarwal LECTURE 1 Image Formation 1.1. The geometry of image formation We begin by considering the process of image formation when a
More informationProjective geometry for Computer Vision
Department of Computer Science and Engineering IIT Delhi NIT, Rourkela March 27, 2010 Overview Pin-hole camera Why projective geometry? Reconstruction Computer vision geometry: main problems Correspondence
More informationCamera model and multiple view geometry
Chapter Camera model and multiple view geometry Before discussing how D information can be obtained from images it is important to know how images are formed First the camera model is introduced and then
More informationPerspective Projection in Homogeneous Coordinates
Perspective Projection in Homogeneous Coordinates Carlo Tomasi If standard Cartesian coordinates are used, a rigid transformation takes the form X = R(X t) and the equations of perspective projection are
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 More on Single View Geometry Lecture 11 2 In Chapter 5 we introduced projection matrix (which
More informationIntroduction to Homogeneous coordinates
Last class we considered smooth translations and rotations of the camera coordinate system and the resulting motions of points in the image projection plane. These two transformations were expressed mathematically
More informationComputer Vision I Name : CSE 252A, Fall 2012 Student ID : David Kriegman Assignment #1. (Due date: 10/23/2012) x P. = z
Computer Vision I Name : CSE 252A, Fall 202 Student ID : David Kriegman E-Mail : Assignment (Due date: 0/23/202). Perspective Projection [2pts] Consider a perspective projection where a point = z y x P
More informationCamera Model and Calibration
Camera Model and Calibration Lecture-10 Camera Calibration Determine extrinsic and intrinsic parameters of camera Extrinsic 3D location and orientation of camera Intrinsic Focal length The size of the
More informationCS-9645 Introduction to Computer Vision Techniques Winter 2019
Table of Contents Projective Geometry... 1 Definitions...1 Axioms of Projective Geometry... Ideal Points...3 Geometric Interpretation... 3 Fundamental Transformations of Projective Geometry... 4 The D
More informationComputer Vision I - Appearance-based Matching and Projective Geometry
Computer Vision I - Appearance-based Matching and Projective Geometry Carsten Rother 05/11/2015 Computer Vision I: Image Formation Process Roadmap for next four lectures Computer Vision I: Image Formation
More informationCamera Models and Image Formation. Srikumar Ramalingam School of Computing University of Utah
Camera Models and Image Formation Srikumar Ramalingam School of Computing University of Utah srikumar@cs.utah.edu Reference Most slides are adapted from the following notes: Some lecture notes on geometric
More informationCSE152a Computer Vision Assignment 1 WI14 Instructor: Prof. David Kriegman. Revision 0
CSE152a Computer Vision Assignment 1 WI14 Instructor: Prof. David Kriegman. Revision Instructions: This assignment should be solved, and written up in groups of 2. Work alone only if you can not find a
More informationECE 470: Homework 5. Due Tuesday, October 27 in Seth Hutchinson. Luke A. Wendt
ECE 47: Homework 5 Due Tuesday, October 7 in class @:3pm Seth Hutchinson Luke A Wendt ECE 47 : Homework 5 Consider a camera with focal length λ = Suppose the optical axis of the camera is aligned with
More informationAPPM 2360 Project 2 Due Nov. 3 at 5:00 PM in D2L
APPM 2360 Project 2 Due Nov. 3 at 5:00 PM in D2L 1 Introduction Digital images are stored as matrices of pixels. For color images, the matrix contains an ordered triple giving the RGB color values at each
More informationcalibrated coordinates Linear transformation pixel coordinates
1 calibrated coordinates Linear transformation pixel coordinates 2 Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial
More informationComputer Vision. Coordinates. Prof. Flávio Cardeal DECOM / CEFET- MG.
Computer Vision Coordinates Prof. Flávio Cardeal DECOM / CEFET- MG cardeal@decom.cefetmg.br Abstract This lecture discusses world coordinates and homogeneous coordinates, as well as provides an overview
More informationC / 35. C18 Computer Vision. David Murray. dwm/courses/4cv.
C18 2015 1 / 35 C18 Computer Vision David Murray david.murray@eng.ox.ac.uk www.robots.ox.ac.uk/ dwm/courses/4cv Michaelmas 2015 C18 2015 2 / 35 Computer Vision: This time... 1. Introduction; imaging geometry;
More informationPerspective Projection [2 pts]
Instructions: CSE252a Computer Vision Assignment 1 Instructor: Ben Ochoa Due: Thursday, October 23, 11:59 PM Submit your assignment electronically by email to iskwak+252a@cs.ucsd.edu with the subject line
More informationEXAM SOLUTIONS. Image Processing and Computer Vision Course 2D1421 Monday, 13 th of March 2006,
School of Computer Science and Communication, KTH Danica Kragic EXAM SOLUTIONS Image Processing and Computer Vision Course 2D1421 Monday, 13 th of March 2006, 14.00 19.00 Grade table 0-25 U 26-35 3 36-45
More informationStereo Vision. MAN-522 Computer Vision
Stereo Vision MAN-522 Computer Vision What is the goal of stereo vision? The recovery of the 3D structure of a scene using two or more images of the 3D scene, each acquired from a different viewpoint in
More informationModule 4F12: Computer Vision and Robotics Solutions to Examples Paper 2
Engineering Tripos Part IIB FOURTH YEAR Module 4F2: Computer Vision and Robotics Solutions to Examples Paper 2. Perspective projection and vanishing points (a) Consider a line in 3D space, defined in camera-centered
More informationPartial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives
In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. By the definition of a derivative, we have Then we are really
More information3D Computer Vision II. Reminder Projective Geometry, Transformations
3D Computer Vision II Reminder Projective Geometry, Transformations Nassir Navab" based on a course given at UNC by Marc Pollefeys & the book Multiple View Geometry by Hartley & Zisserman" October 21,
More informationCIS 580, Machine Perception, Spring 2016 Homework 2 Due: :59AM
CIS 580, Machine Perception, Spring 2016 Homework 2 Due: 2015.02.24. 11:59AM Instructions. Submit your answers in PDF form to Canvas. This is an individual assignment. 1 Recover camera orientation By observing
More information1 Projective Geometry
CIS8, Machine Perception Review Problem - SPRING 26 Instructions. All coordinate systems are right handed. Projective Geometry Figure : Facade rectification. I took an image of a rectangular object, and
More information3D Geometry and Camera Calibration
3D Geometry and Camera Calibration 3D Coordinate Systems Right-handed vs. left-handed x x y z z y 2D Coordinate Systems 3D Geometry Basics y axis up vs. y axis down Origin at center vs. corner Will often
More informationImage Manipulation in MATLAB Due Monday, July 17 at 5:00 PM
Image Manipulation in MATLAB Due Monday, July 17 at 5:00 PM 1 Instructions Labs may be done in groups of 2 or 3 (i.e., not alone). You may use any programming language you wish but MATLAB is highly suggested.
More informationPre-Calculus Guided Notes: Chapter 10 Conics. A circle is
Name: Pre-Calculus Guided Notes: Chapter 10 Conics Section Circles A circle is _ Example 1 Write an equation for the circle with center (3, ) and radius 5. To do this, we ll need the x1 y y1 distance formula:
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationCamera Model and Calibration. Lecture-12
Camera Model and Calibration Lecture-12 Camera Calibration Determine extrinsic and intrinsic parameters of camera Extrinsic 3D location and orientation of camera Intrinsic Focal length The size of the
More informationDD2423 Image Analysis and Computer Vision IMAGE FORMATION. Computational Vision and Active Perception School of Computer Science and Communication
DD2423 Image Analysis and Computer Vision IMAGE FORMATION Mårten Björkman Computational Vision and Active Perception School of Computer Science and Communication November 8, 2013 1 Image formation Goal:
More informationSingle View Geometry. Camera model & Orientation + Position estimation. What am I?
Single View Geometry Camera model & Orientation + Position estimation What am I? Vanishing point Mapping from 3D to 2D Point & Line Goal: Point Homogeneous coordinates represent coordinates in 2 dimensions
More informationAuto-calibration. Computer Vision II CSE 252B
Auto-calibration Computer Vision II CSE 252B 2D Affine Rectification Solve for planar projective transformation that maps line (back) to line at infinity Solve as a Householder matrix Euclidean Projective
More informationCIS 580, Machine Perception, Spring 2015 Homework 1 Due: :59AM
CIS 580, Machine Perception, Spring 2015 Homework 1 Due: 2015.02.09. 11:59AM Instructions. Submit your answers in PDF form to Canvas. This is an individual assignment. 1 Camera Model, Focal Length and
More informationMath 3 Coordinate Geometry Part 2 Graphing Solutions
Math 3 Coordinate Geometry Part 2 Graphing Solutions 1 SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY The solution of two linear equations is the point where the two lines intersect. For example, in the graph
More informationCamera Models and Image Formation. Srikumar Ramalingam School of Computing University of Utah
Camera Models and Image Formation Srikumar Ramalingam School of Computing University of Utah srikumar@cs.utah.edu VisualFunHouse.com 3D Street Art Image courtesy: Julian Beaver (VisualFunHouse.com) 3D
More informationGeometric Modeling of Curves
Curves Locus of a point moving with one degree of freedom Locus of a one-dimensional parameter family of point Mathematically defined using: Explicit equations Implicit equations Parametric equations (Hermite,
More informationComputer Vision Projective Geometry and Calibration. Pinhole cameras
Computer Vision Projective Geometry and Calibration Professor Hager http://www.cs.jhu.edu/~hager Jason Corso http://www.cs.jhu.edu/~jcorso. Pinhole cameras Abstract camera model - box with a small hole
More informationProduct information. Hi-Tech Electronics Pte Ltd
Product information Introduction TEMA Motion is the world leading software for advanced motion analysis. Starting with digital image sequences the operator uses TEMA Motion to track objects in images,
More informationCHAPTER 3. Single-view Geometry. 1. Consequences of Projection
CHAPTER 3 Single-view Geometry When we open an eye or take a photograph, we see only a flattened, two-dimensional projection of the physical underlying scene. The consequences are numerous and startling.
More informationComputer Vision cmput 428/615
Computer Vision cmput 428/615 Basic 2D and 3D geometry and Camera models Martin Jagersand The equation of projection Intuitively: How do we develop a consistent mathematical framework for projection calculations?
More informationAssignment 2 : Projection and Homography
TECHNISCHE UNIVERSITÄT DRESDEN EINFÜHRUNGSPRAKTIKUM COMPUTER VISION Assignment 2 : Projection and Homography Hassan Abu Alhaija November 7,204 INTRODUCTION In this exercise session we will get a hands-on
More informationCS6670: Computer Vision
CS6670: Computer Vision Noah Snavely Lecture 7: Image Alignment and Panoramas What s inside your fridge? http://www.cs.washington.edu/education/courses/cse590ss/01wi/ Projection matrix intrinsics projection
More informationReview Exercise. 1. Determine vector and parametric equations of the plane that contains the
Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A11, 2, 12, B12, 1, 12, and C13, 1, 42. 2. In question 1, there are a variety of different answers possible,
More informationPin Hole Cameras & Warp Functions
Pin Hole Cameras & Warp Functions Instructor - Simon Lucey 16-423 - Designing Computer Vision Apps Today Pinhole Camera. Homogenous Coordinates. Planar Warp Functions. Motivation Taken from: http://img.gawkerassets.com/img/18w7i1umpzoa9jpg/original.jpg
More informationCSCI 5980: Assignment #3 Homography
Submission Assignment due: Feb 23 Individual assignment. Write-up submission format: a single PDF up to 3 pages (more than 3 page assignment will be automatically returned.). Code and data. Submission
More informationEpipolar Geometry Prof. D. Stricker. With slides from A. Zisserman, S. Lazebnik, Seitz
Epipolar Geometry Prof. D. Stricker With slides from A. Zisserman, S. Lazebnik, Seitz 1 Outline 1. Short introduction: points and lines 2. Two views geometry: Epipolar geometry Relation point/line in two
More informationComputer Vision I - Appearance-based Matching and Projective Geometry
Computer Vision I - Appearance-based Matching and Projective Geometry Carsten Rother 01/11/2016 Computer Vision I: Image Formation Process Roadmap for next four lectures Computer Vision I: Image Formation
More informationOutline. ETN-FPI Training School on Plenoptic Sensing
Outline Introduction Part I: Basics of Mathematical Optimization Linear Least Squares Nonlinear Optimization Part II: Basics of Computer Vision Camera Model Multi-Camera Model Multi-Camera Calibration
More informationGraphics and Interaction Transformation geometry and homogeneous coordinates
433-324 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationCOMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationRigid Body Motion and Image Formation. Jana Kosecka, CS 482
Rigid Body Motion and Image Formation Jana Kosecka, CS 482 A free vector is defined by a pair of points : Coordinates of the vector : 1 3D Rotation of Points Euler angles Rotation Matrices in 3D 3 by 3
More informationEM225 Projective Geometry Part 2
EM225 Projective Geometry Part 2 eview In projective geometry, we regard figures as being the same if they can be made to appear the same as in the diagram below. In projective geometry: a projective point
More informationMatlab notes Matlab is a matrix-based, high-performance language for technical computing It integrates computation, visualisation and programming usin
Matlab notes Matlab is a matrix-based, high-performance language for technical computing It integrates computation, visualisation and programming using familiar mathematical notation The name Matlab stands
More informationMatlab Primer. Lecture 02a Optical Sciences 330 Physical Optics II William J. Dallas January 12, 2005
Matlab Primer Lecture 02a Optical Sciences 330 Physical Optics II William J. Dallas January 12, 2005 Introduction The title MATLAB stands for Matrix Laboratory. This software package (from The Math Works,
More informationPerspective Mappings. Contents
Perspective Mappings David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy
More informationOverview. By end of the week:
Overview By end of the week: - Know the basics of git - Make sure we can all compile and run a C++/ OpenGL program - Understand the OpenGL rendering pipeline - Understand how matrices are used for geometric
More informationMATRIX REVIEW PROBLEMS: Our matrix test will be on Friday May 23rd. Here are some problems to help you review.
MATRIX REVIEW PROBLEMS: Our matrix test will be on Friday May 23rd. Here are some problems to help you review. 1. The intersection of two non-parallel planes is a line. Find the equation of the line. Give
More informationStructure from Motion. Prof. Marco Marcon
Structure from Motion Prof. Marco Marcon Summing-up 2 Stereo is the most powerful clue for determining the structure of a scene Another important clue is the relative motion between the scene and (mono)
More informationMetric Rectification for Perspective Images of Planes
789139-3 University of California Santa Barbara Department of Electrical and Computer Engineering CS290I Multiple View Geometry in Computer Vision and Computer Graphics Spring 2006 Metric Rectification
More informationMidterm Exam Fundamentals of Computer Graphics (COMP 557) Thurs. Feb. 19, 2015 Professor Michael Langer
Midterm Exam Fundamentals of Computer Graphics (COMP 557) Thurs. Feb. 19, 2015 Professor Michael Langer The exam consists of 10 questions. There are 2 points per question for a total of 20 points. You
More informationIntroduction.
Product information Image Systems AB Main office: Ågatan 40, SE-582 22 Linköping Phone +46 13 200 100, fax +46 13 200 150 info@imagesystems.se, Introduction TEMA Automotive is the world leading system
More informationCamera Calibration. Schedule. Jesus J Caban. Note: You have until next Monday to let me know. ! Today:! Camera calibration
Camera Calibration Jesus J Caban Schedule! Today:! Camera calibration! Wednesday:! Lecture: Motion & Optical Flow! Monday:! Lecture: Medical Imaging! Final presentations:! Nov 29 th : W. Griffin! Dec 1
More informationProjective 2D Geometry
Projective D Geometry Multi View Geometry (Spring '08) Projective D Geometry Prof. Kyoung Mu Lee SoEECS, Seoul National University Homogeneous representation of lines and points Projective D Geometry Line
More information9.3 Hyperbolas and Rotation of Conics
9.3 Hyperbolas and Rotation of Conics Copyright Cengage Learning. All rights reserved. What You Should Learn Write equations of hyperbolas in standard form. Find asymptotes of and graph hyperbolas. Use
More informationChapter 5. Projections and Rendering
Chapter 5 Projections and Rendering Topics: Perspective Projections The rendering pipeline In order to view manipulate and view a graphics object we must find ways of storing it a computer-compatible way.
More informationCSE/Math 485 Matlab Tutorial and Demo
CSE/Math 485 Matlab Tutorial and Demo Some Tutorial Information on MATLAB Matrices are the main data element. They can be introduced in the following four ways. 1. As an explicit list of elements. 2. Generated
More informationHomogeneous Coordinates. Lecture18: Camera Models. Representation of Line and Point in 2D. Cross Product. Overall scaling is NOT important.
Homogeneous Coordinates Overall scaling is NOT important. CSED44:Introduction to Computer Vision (207F) Lecture8: Camera Models Bohyung Han CSE, POSTECH bhhan@postech.ac.kr (",, ) ()", ), )) ) 0 It is
More informationThree-Dimensional Viewing Hearn & Baker Chapter 7
Three-Dimensional Viewing Hearn & Baker Chapter 7 Overview 3D viewing involves some tasks that are not present in 2D viewing: Projection, Visibility checks, Lighting effects, etc. Overview First, set up
More informationWe have already studied equations of the line. There are several forms:
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationnotes13.1inclass May 01, 2015
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationTopics and things to know about them:
Practice Final CMSC 427 Distributed Tuesday, December 11, 2007 Review Session, Monday, December 17, 5:00pm, 4424 AV Williams Final: 10:30 AM Wednesday, December 19, 2007 General Guidelines: The final will
More informationform. We will see that the parametric form is the most common representation of the curve which is used in most of these cases.
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 36 Curve Representation Welcome everybody to the lectures on computer graphics.
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Projective 3D Geometry (Back to Chapter 2) Lecture 6 2 Singular Value Decomposition Given a
More informationKEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila
KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila January 26, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic
More informationProject report Augmented reality with ARToolKit
Project report Augmented reality with ARToolKit FMA175 Image Analysis, Project Mathematical Sciences, Lund Institute of Technology Supervisor: Petter Strandmark Fredrik Larsson (dt07fl2@student.lth.se)
More informationProjective spaces and Bézout s theorem
Projective spaces and Bézout s theorem êaû{0 Mijia Lai 5 \ laimijia@sjtu.edu.cn Outline 1. History 2. Projective spaces 3. Conics and cubics 4. Bézout s theorem and the resultant 5. Cayley-Bacharach theorem
More informationHomogeneous Coordinates and Transformations of the Plane
2 Homogeneous Coordinates and Transformations of the Plane 2. Introduction In Chapter planar objects were manipulated by applying one or more transformations. Section.7 identified the problem that the
More informationTECHNISCHE UNIVERSITÄT BERLIN WS2005 Fachbereich 3 - Mathematik Prof. Dr. U. Pinkall / Charles Gunn Abgabe:
TECHNISCHE UNIVERSITÄT BERLIN WS2005 Fachbereich 3 - Mathematik Prof. Dr. U. Pinkall / Charles Gunn Abgabe: 27.10.2005 1. Übung Geometrie I (Lines in P 2 (R), Separation, Desargues) 1. Aufgabe Line equations
More informationAll human beings desire to know. [...] sight, more than any other senses, gives us knowledge of things and clarifies many differences among them.
All human beings desire to know. [...] sight, more than any other senses, gives us knowledge of things and clarifies many differences among them. - Aristotle University of Texas at Arlington Introduction
More informationFor each question, indicate whether the statement is true or false by circling T or F, respectively.
True/False For each question, indicate whether the statement is true or false by circling T or F, respectively. 1. (T/F) Rasterization occurs before vertex transformation in the graphics pipeline. 2. (T/F)
More informationPerspective projection and Transformations
Perspective projection and Transformations The pinhole camera The pinhole camera P = (X,,) p = (x,y) O λ = 0 Q λ = O λ = 1 Q λ = P =-1 Q λ X = 0 + λ X 0, 0 + λ 0, 0 + λ 0 = (λx, λ, λ) The pinhole camera
More informationAnswers to practice questions for Midterm 1
Answers to practice questions for Midterm Paul Hacking /5/9 (a The RREF (reduced row echelon form of the augmented matrix is So the system of linear equations has exactly one solution given by x =, y =,
More informationAugmented Reality II - Camera Calibration - Gudrun Klinker May 11, 2004
Augmented Reality II - Camera Calibration - Gudrun Klinker May, 24 Literature Richard Hartley and Andrew Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, 2. (Section 5,
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationWe have already studied equations of the line. There are several forms:
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationUnit 3 Multiple View Geometry
Unit 3 Multiple View Geometry Relations between images of a scene Recovering the cameras Recovering the scene structure http://www.robots.ox.ac.uk/~vgg/hzbook/hzbook1.html 3D structure from images Recover
More informationMachine vision. Summary # 11: Stereo vision and epipolar geometry. u l = λx. v l = λy
1 Machine vision Summary # 11: Stereo vision and epipolar geometry STEREO VISION The goal of stereo vision is to use two cameras to capture 3D scenes. There are two important problems in stereo vision:
More informationN-Views (1) Homographies and Projection
CS 4495 Computer Vision N-Views (1) Homographies and Projection Aaron Bobick School of Interactive Computing Administrivia PS 2: Get SDD and Normalized Correlation working for a given windows size say
More informationCS201 Computer Vision Camera Geometry
CS201 Computer Vision Camera Geometry John Magee 25 November, 2014 Slides Courtesy of: Diane H. Theriault (deht@bu.edu) Question of the Day: How can we represent the relationships between cameras and the
More informationCOMP30019 Graphics and Interaction Three-dimensional transformation geometry and perspective
COMP30019 Graphics and Interaction Three-dimensional transformation geometry and perspective Department of Computing and Information Systems The Lecture outline Introduction Rotation about artibrary axis
More informationPin Hole Cameras & Warp Functions
Pin Hole Cameras & Warp Functions Instructor - Simon Lucey 16-423 - Designing Computer Vision Apps Today Pinhole Camera. Homogenous Coordinates. Planar Warp Functions. Example of SLAM for AR Taken from:
More informationExponential Maps for Computer Vision
Exponential Maps for Computer Vision Nick Birnie School of Informatics University of Edinburgh 1 Introduction In computer vision, the exponential map is the natural generalisation of the ordinary exponential
More informationMultivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
More informationCamera Projection Models We will introduce different camera projection models that relate the location of an image point to the coordinates of the
Camera Projection Models We will introduce different camera projection models that relate the location of an image point to the coordinates of the corresponding 3D points. The projection models include:
More informationContents. 1 Introduction Background Organization Features... 7
Contents 1 Introduction... 1 1.1 Background.... 1 1.2 Organization... 2 1.3 Features... 7 Part I Fundamental Algorithms for Computer Vision 2 Ellipse Fitting... 11 2.1 Representation of Ellipses.... 11
More informationCSE 4392/5369. Dr. Gian Luca Mariottini, Ph.D.
University of Texas at Arlington CSE 4392/5369 Introduction to Vision Sensing Dr. Gian Luca Mariottini, Ph.D. Department of Computer Science and Engineering University of Texas at Arlington WEB : http://ranger.uta.edu/~gianluca
More information