Single View Geometry. Camera model & Orientation + Position estimation. What am I?
|
|
- Zoe Charles
- 5 years ago
- Views:
Transcription
1 Single View Geometr Camera model & Orientation + Position estimation What am I?
2 Ideal case: c Projection equation: x = f X / Z = f Y / Z c p f C x c Zx = f X Z = f Y Z = Z
3 Step 1: Camera projection matrix Zx = f X Z = f Y Z = Z c f f X c Y c Z c 1 = x 1 Z c c p x c f C P 0 X = x
4 Step 2: Intrinsic camera parameters: map camera coordinate to pixel coordinate c K α x s p x 0 α p (3x3 submatrix) x Optical world x = Pixel world c p x c f C α x, α is pixel scaling factor p x, p is the principle point (where optical axis hits image plane) s is the slant factor, when the image plane is not normal to the optical axis
5 Combine the Intrinsic camera parameters α x s p x 0 α p f f X c Y c Z c 1 = x 1 K (Calibration matrix) α x f s p x 0 0 α f p P 0 X = x x c c c 1 = x P = [K, 0] = K [I, 0]
6 Step 3: External parameters: rotation and translation map the world to camera coordinates 3x3 3x1 (R,t) Camera coordinate c World coordinate c p f C x c
7 Combining Internal and External parameters 1) (R,t) Camera coordinate c World coordinate c p f C 2) x c 1) Translate the world coordinate into the camera coordinate 2) Translate the camera coordinate into the pixel coordinate
8 Combining Internal and External parameters (R,t) Camera coordinate c World coordinate After simplication: c p f C x c x = K [R, t] X pixel world
9 Special case, planar world, homograph X= (xw,w,w=0,1) x = K [R, t] X Expand: x = K r1 r2 r3 t xw w w=0 1 World coordinate c (R,t) c p C f x c Camera coordinate
10 Special case, planar world, homograph X= (xw,w,w=0,1) (R,t) After simplication: x = K r1 r2 t xw w World coordinate c c p C f x c Camera coordinate 1 We have the homographic mapping: X = K H3x3 X
11 Special case: rotating camera x = K [R, t] X with t = 0 Expand: x = K [R] X World coordinate c (R,t) p f x c c C Camera coordinate X = xw w w This is also a homograph with H = R
12 Panoramas x = K [R] X
13 Visual Odometr: Overview 1) The transformation (R,t) tells us how to rotate + move the world coordinate to the camera coordinate. 2) In the following slides, we will see 1) how to get the world reference s frame orientation+position from (R,t), 2) how to get the pan/tilt/aw angles from R 3) If we want to know the camera orientation+position, we need to look at its inverse transform, which we should see is, 1) The computation of the camera pan/tilt/aw angle is the same as in the previous case World coordinate c (R,t) p f x c c C
14 Recover Camera orientation Case 1: single vanishing point (direction)
15 Pan/tilt/aw angles Pan : rotation around -axis Yaw :rotation around -axis Tilt : rotation around x-axis x x
16 Rotation: Pan/Tilt/Yaw Given the 3x3 rotation matrix R, we can recover, pan/tilt/aw angle, and vise. vica. cam world, (0,0,1) T x cam,
17 The basic idea is that if we can see the north star, we know how to oriented ourselves (minus aw angle) (wh?) And the ``north star is just the point of infinit in some direction (R,t) Camera coordinate c World coordinate c p f C x c
18 Seeing the vanishing point can tell us: Camera projection equation: K R t x=k [R,t] X, or x w w w t w = x c c c [R,t]= t r 1 r 2 r 3 Columns of R matrix = image of vanishing points of world-space axes! r 1,r 2,r 3 == image of x,, axis vanishing points r 1 r 2 r 3 t = r 3 -direction (need to normalie to norm =1) vanishing point
19 Recover Pan/Tilt angle from -vanishing point Geometric explanation: Pan Tilt The alignment between -axis of the world and image is defined onl b the pan/tilt angle: cam world, (0,0,1) T r3 = x cam, cam Note: tilt angle with rotation around x-axis is positive when pointing down in this figure
20 Sanit check Vanishing point in
21 In this example, we are walking down a track, the ground is tilted up(we are going up the hill). Assume we are b meter tall. We see two parallel lines to our left and right, with x = 1,-1. A line on the ground is described b equation = a - b, where a is the slope the ground. Image plane x x = 1 = a -b x = -1 Ground plane So we know: = a -b Tilt angle
22 As a line gets far awa, -> infinit. If (1,,) is a point on this line, its image is f(1/,/). As -> infinit, 1/ -> 0. What about /? / = (a-b)/ = a - b/ -> a. So a point on the line appears at: (0,a). Image plane x VanishingPoint (0,a) = a - b Ground plane
23 world, (0,0,1) T r = = Now, we check x (0,a) = a - b
24 Recover Pan/Tilt angle from -vanishing point Geometric explanation: Pan Tilt The alignment between -axis of the world and image is defined onl b the pan/tilt angle: cam world, (0,0,1) T r 3 = x cam, cam
25 Recover Camera orientation Case 2: two vanishing points, x- direction
26 In some case, we can t measure the -vanishing point directl, but if we have x- vanishing points, we can imagine what the -vanishing point
27 Basic idea: 1) x- vanishing point gives us the first two columns of R, r1 and r2 2) -vanishing point, the third columns of R, can be computed b r3 = r1 x r2 Recall: vanishing points of x, direction, give us the, r1 and r2 r 1 r 2 r 3 t first two columns of R
28 Steps for Recovering R from Two vanishing Point 1. Measure X / Y direction vanishing point
29 Recovering R from Two vanishing Point (2) 2. Translate to the camera coordinates b intrinsic matrix K
30 Recovering R from Two vanishing Point (3) 3. Get r 1 and r 2 from vanishing point x vx and x vy vanishing points of x, direction, give us the, r1 and r2 r 1 r 2 r 3 t first two columns of R
31 4. Get r 3 Recovering R from Two vanishing Point (4) 5. Get R r3 = r1 x r2 r1 r2 r3
32 Example Recover Camera Pan /Tilt From r3
33 Recover Camera Pan /Tilt From r 3 (2) Examples of Camera Pan/Tilt angle - - x α x
34 R= Recovering all three angles: aw + pan/tilt r 1 r 2 r 3 Depends on pan/tilt Depends on aw/pan/tilt 1) Write out equation in aw/pan/tilt angles for r1 r2 2) Solve for aw angle given pan/tilt angles
35 Summar so far 1) If we have the vanishing point in -direction, we can recover pan/tilt angle 1) This is when the planar target is on the ground 2) If don t see the -direction vanishing point directl, ou would need two vanishing points of x- directions 3) Given x- directions vanishing point, we can cover the full rotation matrix, therefore its -direction (pan/tilt) 4) Given pan/tilt + x-direction vanishing point, we can recover aw angle
36 Recover Camera orientation + position Case 3: homograph matrix
37 Recall for planar surface (R,t) H3x3 x = K r1 r2 t xw w World coordinate c c p C f x c Camera coordinate 1 We have the homographic mapping: X = K H3x3 X
38 Recall for planar surface (R,t) H3x3 x = K r1 r2 t xw w World coordinate c c p C f x c Camera coordinate 1 This implies if we have H, 1) We can recover the full rotation matrix, R, 2) We can recover the position vector t
39 Example Recover [R,t] from H (1) 1. Get H from 4 points 2. Computer the norm of first column
40 3. Computer r 1, r 2, t Example Recover [R,t] from H (2) 4. Computer r 3
41 Procedure of recover R, and t from H 1) Compute the rotation of x-axis, set x = K 2) Compute the rotation of the -axis, set r1 r2 t xw w 1 3) Compute the translation vector t (the position of the robot) 4) Compute the rotation of the -axis, set 5) Use the previous case(2), to compute the pan/tilt/aw angle
42 Summar so far 1) Each vanishing point in the image gives ou one rotation vector 2) Given the -vanishing point, we can detect the pan/tilt angle. 1) This is the case when the planar target is on the ground. 3) Given x- vanishing point, we can compute the -vanishing point(therefore pan/tilt angle). 1) This is the case when we are facing the planar surface 2) Given x-rotation vector, pan-tilt angle, we can determine the aw angle 3) But we can t determine the position of the robot 4) If we are given the homograph, (using 4 corresponding points), we can recover both the rotational angles, and the position of the robot.
43 How to estimate the rotation and translation of the robot from the world point of view? In the case of moving robot(rather than moving target), we need to know the orientation/position of the robot in the world ==> we need to how to pan/tilt the world oriented to the robot. Note: pan/tilt of the camera is ver different from the pan/tilt of the world!
44 Converting rotation/translation from (world->camera) to (camera->world) (R,t) (world-> camera) Camera coordinate c World coordinate c p f C x c = With the fact: (camera->world)
45 Converting rotation/translation from (world->camera) to (camera->world) What does it mean: 1) The position of the camera in the world reference frame is: 2) The pan/tilt/aw of the world to align with camera is computed from the rows of the R instead of columns of R World coordinate c (R,t) (world-> camera) Camera coordinate c p C f x c (camera-> world)
46 Recovering World pan/tilt from Rotation matrix R^T 1. Compute the (world->camera) rotation R and translation t 2. Take the last row of the R, and recover pan/tilt angle of camera 3) Compute the aw angle if needed, from first two rows of R 4) Compute the position of the robot, b
47 Example Recover Pan/Tilt in World Frame From World-> Camera: [R, t] èfrom Camera -> World : [R T, -R T t] Tilt x Pan x
48 Example Recover Translation in World Frame (2) target x
49 Final notes 1) There are two rotation/translation we care about: 1) (R,t): world->camera transformation. It tells us parameters of the camera. The columns of R: rotation of camera so it aligns with the world. The vector t = coordinate of the world center in camera frame. 2) : camera-> world. It tells us parameters of the world. Columns of R^t: rotation of the world so it aligns with the camera. The vector R^t(-t) = coordinate of the camera in the world frame. 3) For the visual odometr task used in project 2, ou need to use the transformation. 4) The pan angle of the world, is the robot orientation in the world frame.
Single View Geometry. Camera model & Orientation + Position estimation. Jianbo Shi. What am I? University of Pennsylvania GRASP
Single View Geometry Camera model & Orientation + Position estimation Jianbo Shi What am I? 1 Camera projection model The overall goal is to compute 3D geometry of the scene from just 2D images. We will
More informationSingle View Geometry. Camera model & Orientation + Position estimation. What am I?
Single View Geometry Camera model & Orientation + Position estimation What am I? Vanishing points & line http://www.wetcanvas.com/ http://pennpaint.blogspot.com/ http://www.joshuanava.biz/perspective/in-other-words-the-observer-simply-points-in-thesame-direction-as-the-lines-in-order-to-find-their-vanishing-point.html
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 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 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 information3D Transformations. CS 4620 Lecture 10. Cornell CS4620 Fall 2014 Lecture Steve Marschner (with previous instructors James/Bala)
3D Transformations CS 4620 Lecture 10 1 Translation 2 Scaling 3 Rotation about z axis 4 Rotation about x axis 5 Rotation about y axis 6 Properties of Matrices Translations: linear part is the identity
More information3D Transformations. CS 4620 Lecture Kavita Bala w/ prior instructor Steve Marschner. Cornell CS4620 Fall 2015 Lecture 11
3D Transformations CS 4620 Lecture 11 1 Announcements A2 due tomorrow Demos on Monday Please sign up for a slot Post on piazza 2 Translation 3 Scaling 4 Rotation about z axis 5 Rotation about x axis 6
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 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 informationMidterm Exam Solutions
Midterm Exam Solutions Computer Vision (J. Košecká) October 27, 2009 HONOR SYSTEM: This examination is strictly individual. You are not allowed to talk, discuss, exchange solutions, etc., with other fellow
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 informationDetermining the 2d transformation that brings one image into alignment (registers it) with another. And
Last two lectures: Representing an image as a weighted combination of other images. Toda: A different kind of coordinate sstem change. Solving the biggest problem in using eigenfaces? Toda Recognition
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 informationCS770/870 Spring 2017 Transformations
CS770/870 Spring 2017 Transformations Coordinate sstems 2D Transformations Homogeneous coordinates Matrices, vectors, points 01/29/2017 1 Coordinate Sstems Coordinate sstems used in graphics Screen coordinates:
More information3-D D Euclidean Space - Vectors
3-D D Euclidean Space - Vectors Rigid Body Motion and Image Formation A free vector is defined by a pair of points : Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Coordinates of the vector : 3D Rotation
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 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 informationImage Formation I Chapter 2 (R. Szelisky)
Image Formation I Chapter 2 (R. Selisky) Guido Gerig CS 632 Spring 22 cknowledgements: Slides used from Prof. Trevor Darrell, (http://www.eecs.berkeley.edu/~trevor/cs28.html) Some slides modified from
More informationAutonomous Navigation for Flying Robots
Computer Vision Group Prof. Daniel Cremers Autonomous Navigation for Flying Robots Lecture 3.1: 3D Geometry Jürgen Sturm Technische Universität München Points in 3D 3D point Augmented vector Homogeneous
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 informationGeometric Transformations
Geometric Transformations CS 4620 Lecture 9 2017 Steve Marschner 1 A little quick math background Notation for sets, functions, mappings Linear and affine transformations Matrices Matrix-vector multiplication
More informationCS223b Midterm Exam, Computer Vision. Monday February 25th, Winter 2008, Prof. Jana Kosecka
CS223b Midterm Exam, Computer Vision Monday February 25th, Winter 2008, Prof. Jana Kosecka Your name email This exam is 8 pages long including cover page. Make sure your exam is not missing any pages.
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 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 informationAn idea which can be used once is a trick. If it can be used more than once it becomes a method
An idea which can be used once is a trick. If it can be used more than once it becomes a method - George Polya and Gabor Szego University of Texas at Arlington Rigid Body Transformations & Generalized
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 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 informationCamera Geometry II. COS 429 Princeton University
Camera Geometry II COS 429 Princeton University Outline Projective geometry Vanishing points Application: camera calibration Application: single-view metrology Epipolar geometry Application: stereo correspondence
More informationMETRIC PLANE RECTIFICATION USING SYMMETRIC VANISHING POINTS
METRIC PLANE RECTIFICATION USING SYMMETRIC VANISHING POINTS M. Lefler, H. Hel-Or Dept. of CS, University of Haifa, Israel Y. Hel-Or School of CS, IDC, Herzliya, Israel ABSTRACT Video analysis often requires
More informationTD2 : Stereoscopy and Tracking: solutions
TD2 : Stereoscopy and Tracking: solutions Preliminary: λ = P 0 with and λ > 0. If camera undergoes the rigid transform: (R,T), then with, so that is the intrinsic parameter matrix. C(Cx,Cy,Cz) is the point
More informationStructure from motion
Structure from motion Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R 1,t 1 R 2,t 2 R 3,t 3 Camera 1 Camera
More informationDD2429 Computational Photography :00-19:00
. Examination: DD2429 Computational Photography 202-0-8 4:00-9:00 Each problem gives max 5 points. In order to pass you need about 0-5 points. You are allowed to use the lecture notes and standard list
More informationProblem 2. Problem 3. Perform, if possible, each matrix-vector multiplication. Answer. 3. Not defined. Solve this matrix equation.
Problem 2 Perform, if possible, each matrix-vector multiplication. 1. 2. 3. 1. 2. 3. Not defined. Problem 3 Solve this matrix equation. Matrix-vector multiplication gives rise to a linear system. Gaussian
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 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 informationLecture 4: Viewing. Topics:
Lecture 4: Viewing Topics: 1. Classical viewing 2. Positioning the camera 3. Perspective and orthogonal projections 4. Perspective and orthogonal projections in OpenGL 5. Perspective and orthogonal projection
More informationTake Home Exam # 2 Machine Vision
1 Take Home Exam # 2 Machine Vision Date: 04/26/2018 Due : 05/03/2018 Work with one awesome/breathtaking/amazing partner. The name of the partner should be clearly stated at the beginning of your report.
More informationHand-Eye Calibration from Image Derivatives
Hand-Eye Calibration from Image Derivatives Abstract In this paper it is shown how to perform hand-eye calibration using only the normal flow field and knowledge about the motion of the hand. The proposed
More information521466S Machine Vision Exercise #1 Camera models
52466S Machine Vision Exercise # Camera models. Pinhole camera. The perspective projection equations or a pinhole camera are x n = x c, = y c, where x n = [x n, ] are the normalized image coordinates,
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 Project-1
University of Utah, School Of Computing Computer Vision Project- Singla, Sumedha sumedha.singla@utah.edu (00877456 February, 205 Theoretical Problems. Pinhole Camera (a A straight line in the world space
More informationCS 351: Perspective Viewing
CS 351: Perspective Viewing Instructor: Joel Castellanos e-mail: joel@unm.edu Web: http://cs.unm.edu/~joel/ 2/16/2017 Perspective Projection 2 1 Frustum In computer graphics, the viewing frustum is the
More informationh(x) and r(x). What does this tell you about whether the order of the translations matters? Explain your reasoning.
.6 Combinations of Transformations An anamorphosis is an image that can onl be seen correctl when viewed from a certain perspective. For example, the face in the photo can onl be seen correctl in the side
More informationMultiple Views Geometry
Multiple Views Geometry Subhashis Banerjee Dept. Computer Science and Engineering IIT Delhi email: suban@cse.iitd.ac.in January 2, 28 Epipolar geometry Fundamental geometric relationship between two perspective
More informationImage Formation I Chapter 1 (Forsyth&Ponce) Cameras
Image Formation I Chapter 1 (Forsyth&Ponce) Cameras Guido Gerig CS 632 Spring 215 cknowledgements: Slides used from Prof. Trevor Darrell, (http://www.eecs.berkeley.edu/~trevor/cs28.html) Some slides modified
More informationCamera Parameters Estimation from Hand-labelled Sun Sositions in Image Sequences
Camera Parameters Estimation from Hand-labelled Sun Sositions in Image Sequences Jean-François Lalonde, Srinivasa G. Narasimhan and Alexei A. Efros {jlalonde,srinivas,efros}@cs.cmu.edu CMU-RI-TR-8-32 July
More informationCalibrating an Overhead Video Camera
Calibrating an Overhead Video Camera Raul Rojas Freie Universität Berlin, Takustraße 9, 495 Berlin, Germany http://www.fu-fighters.de Abstract. In this section we discuss how to calibrate an overhead video
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 informationJacobians. 6.1 Linearized Kinematics. Y: = k2( e6)
Jacobians 6.1 Linearized Kinematics In previous chapters we have seen how kinematics relates the joint angles to the position and orientation of the robot's endeffector. This means that, for a serial robot,
More informationComputer Graphics: Geometric Transformations
Computer Graphics: Geometric Transformations Geometric 2D transformations By: A. H. Abdul Hafez Abdul.hafez@hku.edu.tr, 1 Outlines 1. Basic 2D transformations 2. Matrix Representation of 2D transformations
More informationImage Warping, Linear Algebra CIS581
Image Warping, Linear Algebra CIS581 From Plane to Plane Degree of freedom Translation: # correspondences? How many correspondences needed for translation? How many Degrees of Freedom? What is the transformation
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 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 informationRobotics - Projective Geometry and Camera model. Marcello Restelli
Robotics - Projective Geometr and Camera model Marcello Restelli marcello.restelli@polimi.it Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano Ma 2013 Inspired from Matteo
More informationME5286 Robotics Spring 2014 Quiz 1 Solution. Total Points: 30
Page 1 of 7 ME5286 Robotics Spring 2014 Quiz 1 Solution Total Points: 30 (Note images from original quiz are not included to save paper/ space. Please see the original quiz for additional information and
More informationSo we have been talking about 3D viewing, the transformations pertaining to 3D viewing. Today we will continue on it. (Refer Slide Time: 1:15)
Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture - 8 3D Viewing So we have been talking about 3D viewing, the
More information55:148 Digital Image Processing Chapter 11 3D Vision, Geometry
55:148 Digital Image Processing Chapter 11 3D Vision, Geometry Topics: Basics of projective geometry Points and hyperplanes in projective space Homography Estimating homography from point correspondence
More informationImage Transformations & Camera Calibration. Mašinska vizija, 2018.
Image Transformations & Camera Calibration Mašinska vizija, 2018. Image transformations What ve we learnt so far? Example 1 resize and rotate Open warp_affine_template.cpp Perform simple resize
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 informationStereo Observation Models
Stereo Observation Models Gabe Sibley June 16, 2003 Abstract This technical report describes general stereo vision triangulation and linearized error modeling. 0.1 Standard Model Equations If the relative
More informationMidterm Examination CS 534: Computational Photography
Midterm Examination CS 534: Computational Photography November 3, 2016 NAME: Problem Score Max Score 1 6 2 8 3 9 4 12 5 4 6 13 7 7 8 6 9 9 10 6 11 14 12 6 Total 100 1 of 8 1. [6] (a) [3] What camera setting(s)
More informationGeometric transformations in 3D and coordinate frames. Computer Graphics CSE 167 Lecture 3
Geometric transformations in 3D and coordinate frames Computer Graphics CSE 167 Lecture 3 CSE 167: Computer Graphics 3D points as vectors Geometric transformations in 3D Coordinate frames CSE 167, Winter
More informationComputer Vision I - Algorithms and Applications: Multi-View 3D reconstruction
Computer Vision I - Algorithms and Applications: Multi-View 3D reconstruction Carsten Rother 09/12/2013 Computer Vision I: Multi-View 3D reconstruction Roadmap this lecture Computer Vision I: Multi-View
More informationCOMP 175 COMPUTER GRAPHICS. Ray Casting. COMP 175: Computer Graphics April 26, Erik Anderson 09 Ray Casting
Ray Casting COMP 175: Computer Graphics April 26, 2018 1/41 Admin } Assignment 4 posted } Picking new partners today for rest of the assignments } Demo in the works } Mac demo may require a new dylib I
More informationViewing. Reading: Angel Ch.5
Viewing Reading: Angel Ch.5 What is Viewing? Viewing transform projects the 3D model to a 2D image plane 3D Objects (world frame) Model-view (camera frame) View transform (projection frame) 2D image View
More informationCOMP 558 lecture 19 Nov. 17, 2010
COMP 558 lecture 9 Nov. 7, 2 Camera calibration To estimate the geometry of 3D scenes, it helps to know the camera parameters, both external and internal. The problem of finding all these parameters is
More informationGeometry of a single camera. Odilon Redon, Cyclops, 1914
Geometr o a single camera Odilon Redon, Cclops, 94 Our goal: Recover o 3D structure Recover o structure rom one image is inherentl ambiguous??? Single-view ambiguit Single-view ambiguit Rashad Alakbarov
More information3D Viewing. CMPT 361 Introduction to Computer Graphics Torsten Möller. Machiraju/Zhang/Möller
3D Viewing CMPT 361 Introduction to Computer Graphics Torsten Möller Reading Chapter 4 of Angel Chapter 6 of Foley, van Dam, 2 Objectives What kind of camera we use? (pinhole) What projections make sense
More informationMAN-522: COMPUTER VISION SET-2 Projections and Camera Calibration
MAN-522: COMPUTER VISION SET-2 Projections and Camera Calibration Image formation How are objects in the world captured in an image? Phsical parameters of image formation Geometric Tpe of projection Camera
More informationSingle-view 3D Reconstruction
Single-view 3D Reconstruction 10/12/17 Computational Photography Derek Hoiem, University of Illinois Some slides from Alyosha Efros, Steve Seitz Notes about Project 4 (Image-based Lighting) You can work
More informationImage Formation I Chapter 1 (Forsyth&Ponce) Cameras
Image Formation I Chapter 1 (Forsyth&Ponce) Cameras Guido Gerig CS 632 Spring 213 cknowledgements: Slides used from Prof. Trevor Darrell, (http://www.eecs.berkeley.edu/~trevor/cs28.html) Some slides modified
More informationCompositing a bird's eye view mosaic
Compositing a bird's eye view mosaic Robert Laganiere School of Information Technology and Engineering University of Ottawa Ottawa, Ont KN 6N Abstract This paper describes a method that allows the composition
More informationProjective Geometry and Camera Models
Projective Geometry and Camera Models Computer Vision CS 43 Brown James Hays Slides from Derek Hoiem, Alexei Efros, Steve Seitz, and David Forsyth Administrative Stuff My Office hours, CIT 375 Monday and
More informationJane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute
Jane Li Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute We know how to describe the transformation of a single rigid object w.r.t. a single
More informationCS4670: Computer Vision
CS467: Computer Vision Noah Snavely Lecture 13: Projection, Part 2 Perspective study of a vase by Paolo Uccello Szeliski 2.1.3-2.1.6 Reading Announcements Project 2a due Friday, 8:59pm Project 2b out Friday
More informationMERGING POINT CLOUDS FROM MULTIPLE KINECTS. Nishant Rai 13th July, 2016 CARIS Lab University of British Columbia
MERGING POINT CLOUDS FROM MULTIPLE KINECTS Nishant Rai 13th July, 2016 CARIS Lab University of British Columbia Introduction What do we want to do? : Use information (point clouds) from multiple (2+) Kinects
More informationSingle-view metrology
Single-view metrology Magritte, Personal Values, 952 Many slides from S. Seitz, D. Hoiem Camera calibration revisited What if world coordinates of reference 3D points are not known? We can use scene features
More information3D Viewing. Introduction to Computer Graphics Torsten Möller. Machiraju/Zhang/Möller
3D Viewing Introduction to Computer Graphics Torsten Möller Machiraju/Zhang/Möller Reading Chapter 4 of Angel Chapter 13 of Hughes, van Dam, Chapter 7 of Shirley+Marschner Machiraju/Zhang/Möller 2 Objectives
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 informationAgenda. Rotations. Camera models. Camera calibration. Homographies
Agenda Rotations Camera models Camera calibration Homographies D Rotations R Y = Z r r r r r r r r r Y Z Think of as change of basis where ri = r(i,:) are orthonormal basis vectors r rotated coordinate
More 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 informationProjective Geometry and Camera Models
/2/ Projective Geometry and Camera Models Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem Note about HW Out before next Tues Prob: covered today, Tues Prob2: covered next Thurs Prob3:
More informationCameras and Radiometry. Last lecture in a nutshell. Conversion Euclidean -> Homogenous -> Euclidean. Affine Camera Model. Simplified Camera Models
Cameras and Radiometry Last lecture in a nutshell CSE 252A Lecture 5 Conversion Euclidean -> Homogenous -> Euclidean In 2-D Euclidean -> Homogenous: (x, y) -> k (x,y,1) Homogenous -> Euclidean: (x, y,
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 informationCS 2770: Intro to Computer Vision. Multiple Views. Prof. Adriana Kovashka University of Pittsburgh March 14, 2017
CS 277: Intro to Computer Vision Multiple Views Prof. Adriana Kovashka Universit of Pittsburgh March 4, 27 Plan for toda Affine and projective image transformations Homographies and image mosaics Stereo
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 informationLinear Algebra Simplified
Linear Algebra Simplified Readings http://szeliski.org/book/drafts/szeliskibook_20100903_draft.pdf -2.1.5 for camera geometry, -2.1.3, 2.1.4 for rotation representation Inner (dot) Product v w α 3 3 2
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 informationFunctions. Name. Use an XY Coordinate Pegboard to graph each line. Make a table of ordered pairs for each line. y = x + 5 x y.
Lesson 1 Functions Name Use an XY Coordinate Pegboard to graph each line. Make a table of ordered pairs for each line. 1. = + = + = 2 3 = 2 3 Using an XY Coordinate Pegboard, graph the line on a coordinate
More informationA Fast Linear Registration Framework for Multi-Camera GIS Coordination
A Fast Linear Registration Framework for Multi-Camera GIS Coordination Karthik Sankaranarayanan James W. Davis Dept. of Computer Science and Engineering Ohio State University Columbus, OH 4320 USA {sankaran,jwdavis}@cse.ohio-state.edu
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 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 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 informationCS 325 Computer Graphics
CS 325 Computer Graphics 02 / 29 / 2012 Instructor: Michael Eckmann Today s Topics Questions? Comments? Specifying arbitrary views Transforming into Canonical view volume View Volumes Assuming a rectangular
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 informationInverse Kinematics of a Rhino Robot
Inverse Kinematics of a Rhino Robot Rhino Robot (http://verona.fi-p.unam.mx/gpocontrol/images/rhino1.jpg) A Rhino robot is very similar to a 2-link arm with the exception that The base can rotate, allowing
More informationCoordinate Transformations for VERITAS in OAWG - Stage 4
Coordinate Transformations for VERITAS in OAWG - Stage 4 (11 June 2006) Tülün Ergin 1 Contents 1 COORDINATE TRANSFORMATIONS 1 1.1 Rotation Matrices......................... 1 1.2 Rotations of the Coordinates...................
More informationLecture 9: Epipolar Geometry
Lecture 9: Epipolar Geometry Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F (Problem Set 2
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 information3D Vision Real Objects, Real Cameras. Chapter 11 (parts of), 12 (parts of) Computerized Image Analysis MN2 Anders Brun,
3D Vision Real Objects, Real Cameras Chapter 11 (parts of), 12 (parts of) Computerized Image Analysis MN2 Anders Brun, anders@cb.uu.se 3D Vision! Philisophy! Image formation " The pinhole camera " Projective
More information