Agenda. Perspective projection. Rotations. Camera models
|
|
- Moris Martin
- 6 years ago
- Views:
Transcription
1 Image formation
2 Agenda Perspective projection Rotations Camera models
3 Light as a wave + particle
4 Light as a wave (ignore for now) Refraction Diffraction
5 Image formation Digital Image Film Human eye
6 Pixel brightness (More on light as psychics at end of semseter)
7 Pinhole optics
8 Camera Obscura
9 World s largest photograph El Toro Marine Corps, Irvine CA 2006
10 Accidental pinholes what s the dark stuff? (the view from Antonio s hotel room)
11
12 Accidental pinhole and pinspeck cameras: revealing the scene outside the picture CVPR 2012 Antonio Torralba, William T. Freeman Computer Science and Artificial Intelligence Laboratory (CSAIL) MIT
13 Perspective projection Closer objects appear larger Closer objects are lower in the image Parallel lines meet
14 Great reference
15 Pinhole Camera optical axis [Aside: right-handed coordinate system] How do we compute P? [on board]
16 Pinhole Camera
17 Image inversion
18 Image inversion Perplexed folks for a while. But software (or the brain) can simply invert this.
19 Physical model that avoids inversion easel COP = pinhole, camera center Distance of COP to easel = focal length
20 Visual angle (common unit in human vision) easel Note: math is easier for a spherical easel (e.g., retina) = L f L = length of projection on sphere theta = units of radians Human head is 9 inches high. At a distance of 9 feet, it subtends 1/12 radians = 4.8 degrees, regardless of focal length
21 Field of view (FOV) 24mm 50mm 135mm FOV = total sensor size (diagonal) focal length (in radians)
22 Increasing the focal length and stepping back What happens to apparant object size and FOV when we double distance to object and double the focal length? x new = 2fX 2Z = fx Z = x old sensor size FOV new = = 1 2f 2 FOV old
23 Decreasing the focal length and moving forward
24 Perspective projection Closer objects appear larger Closer objects are lower in the image Parallel lines meet All these can be simply derived with x = f X Z!
25 (parallel lines meet) Vanishing point: proof 2 3 X 4Y 5 = 2 4 A x B y D x 4D 5 y COP (x,y,f) (X,Y,Z) Z C z D z Compute projected point (x,y) as lambda approaches infinity [on board]: x = fx Z = f(a x + D x ) A z + D z! fd x D z as!1 y = fy Z = f(a x + D x ) A z + D z! fd y D z as!1 3D lines with identical direction vectors coverge to same 2D image location
26 VP 3 Special case: manhatten world Consider a city-block world where all lines follow one of 3 directions VP 1 VP 2
27 Special case: horizon line Claim: all 3D lines on ground plane meet at a horizon line
28 Horizon line: proof 2 3 X 4Y 5 = Z 2 A x 3 4B y 5 + C z 2 D x 3 4D y 5 D z (x, y)! ( fd x D z, fd y D z ) as!1 Equation of ground plane is Y = -h (x,y,f) (X,Y,Z) COP For all points A on ground plane (Ax,-h,Az) with a direction D along ground plane (Dx,0,Dz), where will vanishing points converge to? ( fd x D z, 0) Why is horizon line not always at center of image?
29 Image y position: proof Equation of ground plane is Y = -h A point on ground plane will have y-coordinate=? y = -fh/z Z2 Z3 Z1
30 Image height: proof Bottom of tree: (X,-h,Z) Top of tree: (X,L-h,Z) y top y bot = f(l h) Z fh Z = fl Z
31 Consequence of derivations for image height and parallel lines distances and angles aren t preserved in camera projection
32 Orthographic projection COP (x,y,f) (X,Y,Z) x = fx/z y = fy/z (x,y,f) (X,Y,Z) x = X y = Y Life would be much simpler; we could trust angles and distances 32
33 Scaled orthographic projection Consider two points (A,B) at different depths that are far away from camera: 2 3 A x 4A 5 y Z 2 B x 4 B y Z + Z 3 5 if Z >> deltaz, what happens to their image projections (e.g., ax and bx)? a x = fa x Z = A x b x = fb x Z + Z COP fb x Z = B x for Z Z We can approximate sets of such points with a scaled orthographic model 33
34 Perspective vs Orthogrpahic Wide angle Standard Telephoto
35 Scaled orthographic
36 Scaled orthographic
37 Perspective tends to matter for large objects (change in depth of object large relative to distance from camera)
38 A look back: dominant effects of perspective Parallel lines meet at vanishing points Objects further away are smaller Foreshortening
39 Fronto-parallel view Foreshortened view Perspective view Rotation of far-away plane Affine linear warp Rotation of close-by plane Homography nonlinear warp
40 2D Geometric Transformations y translation similarity projective Euclidean affine x Transformation Matrix # DoF Preserves Icon translation rigid (Euclidean) similarity affine projective h I t h R t h sr t h A h H i i i i 2 3 i orientation 3 lengths S S 4 angles S S 6 parallelism 8 straight lines `` Let s define families of transformations by the properties that they preserve
41 but first, we ll need tools from geometry Where we are headed. Euclidean (trans + rot) preserves lengths + angles Affine: preserves parallel lines Projective: preserves lines Projective Affine Euclidean
42 Agenda Perspective projection Rotations Camera models
43 Orthogonal transformations Defn: Orthogonal transformations are linear transformations that preserve distances and angles a T b = F (a) T F (b) where F (a) =Aa, a 2 R n,a2 R 2 2 n n a T b = a T A T Ab () A T A = I [can conclude by setting a,b = coordinate vectors] Defn: A is a rotation matrix if A T A = I, det(a) = 1 Defn: A is a reflection matrix if A T A = I, det(a) = -1
44 2D Rotations R = apple cos sin sin cos 1 DOF
45 3D Rotations R 2 3 X 4Y 5 = Z 2 3 r 11 r 12 r 13 4r 21 r 22 r 23 5 r 31 r 32 r X 4Y 5 Z Think of as change of basis where ri = r(i,:) are orthonormal basis vectors r2 rotated coordinate frame r1 r3 How many DOFs? 3 = (2 to point r1 + 1 to rotate along r1)
46 Euler s rotation theorm Any rotation of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis
47 3D Rotations Lots of parameterizations that try to capture 3 DOFs Helpful ones for vision: orthonormal matrix, axis-angle, exponential maps Represent a 3D rotation with a unit vector pointed along the axis of rotation, and an angle of rotation about that vector -vs- 2D 3D
48 Review: dot and cross products Dot product: a b = a b cos Cross product: a b = 2 3 a 2 b 3 a 3 b 2 4b 1 a 3 a 1 b 3 5 a 1 b 2 a 2 b 1 Cross product matrix: a b = âb = a 3 a 2 b 1 a 3 0 a 1 5 4b 2 5 a 2 a 1 0 b 3
49 Approach! 2 R 3,! =1 x
50 Rodrigues' rotation formula 2 R 3,! =1 x k x? x 1. Write as x as sum of parallel and perpindicular component to omega 2. Rotate perpindicular component by 2D rotation of theta in plane orthogonal to omega R = I +ŵ sin +ŵŵ(1 cos ) [Rx can simplify to cross and dot product computations]
51 Exponential map representation! 2 R 3,! =1 x k x? x R =exp(ˆv), where v =! = I +ˆv + 1 2! ˆv [standard Taylor series expansion of x=0 as 1 + x + (1/2!)x 2 + ] [reduces to Rodrigous formula with Taylor series expansion of sine + cosine] Implies that we can approximate change in position of x due to a small rotation v as: v x,
52 Agenda Perspective projection Rotations Camera models
53 Recall perspective projection y x (x,y,1) (X,Y,Z) COP z x = f Z X y = f Z Y
54 Perspective projection revisited 2 3 x 4y5 = 1 2 f f X 4Y 5 Z Given (X,Y,Z) and f, compute (x,y) and lambda: x = fx = Z x = x = fx Z
55 Special case: f = 1 Natural geometric intuition: 3D point is obtained by scaling ray pointed at image coordinate Scale factor = true depth of point (x,y,1) (X,Y,Z) COP Z 2 3 x 4y5 = X 4Y 5 Z [Aside: given an image with a focal length f, resize by 1/f to obtain unit-focal-length image]
56 Homogenous notation For now, think of above as shorthand notation for 2 4 x y z X Y Z x y z X Y Z s.t. 2 4 x y z 3 5 = 2 4 X Y Z 3 5
57 Camera projection 2 3 x 4y5 = f 0 0 r 11 r 12 r 13 t x 40 f 05 4r 21 r 22 r 23 t 5 y Camera instrinsic matrix K (can include skew & non-square pixel size) r 31 r 32 r 33 t z Camera extrinsics (rotation and translation) X 6Y 4Z D point in world coordinates r2 r1 camera r3 T world coordinate frame Aside: homogenous notation is shorthand for x = x
58 Fancier intrinsics x s = s x x y s = s y y x 0 = x s + o x y 0 = y s + o y x =x 0 + s y 0 } } non-square pixels shifted origin y skewed image axes x K = 2 3 s x s o x 4 0 s y o 5 y f f 05 = fs x fs o x 4 0 fs y o 5 y 0 0 1
59 Notation [Using Matlab s rows x columns] X x fs x fs o x r 11 r 12 r 13 t x 4y5 = 4 0 fs y o y 5 4r 21 r 22 r 23 t y 5 6Y 7 4Z r 31 r 32 r 33 t z X = K 3 3 R3 3 T 3 1 6Y 7 4Z X = M 3 4 6Y 7 4Z 5 1 Claims (without proof): 1. A 3x4 matrix M can be a camera matrix iff det(m) is not zero 2. M is determined only up to a scale factor
60 Notation (more) M X 6Y 4Z = A 3 3 b 3 1 = A X 6Y 4Z X 4Y 5 + b 3 1 Z M = 2 m T 1 4m T 2 m T 3 3 5, A = 2 a T 1 4a T 2 a T 3 3 5, b = 2 3 b 1 4b 5 2 b 3
61 Applying the projection matrix x = 1 ( X Y Z a 1 + b 1 ) y = 1 ( X Y Z a 2 + b 2 ) = X Y Z a 3 + b3 Set of 3D points that project to x = 0: Set of 3D points that project to y = 0: X Y Z a1 + b 1 =0 X Y Z a2 + b 2 =0 Set of 3D points that project to x = inf or y = inf: X Y Z a3 + b 3 =0
62 Rows of the projection matrix describe the 3 planes defined by the image coordinate system a 3 y a 1 COP a 2 x image plane
63 Other geometric properties (x,y) COP (X,Y,Z) Draw plane infront of pinhole. Write (x,y) for normalized coordinate and (u,v) for image coordinates? What s set of (X,Y,Z) points that project to same (x,y)? X x 4Y 5 = w + b where w = A 1 4y5,b= A 1 b Z 1 What s the position of COP / pinhole? 2 3 X A 4Y 5 + b =0 ) Z 2 3 X 4Y 5 = A 1 b Z
64 Affine cameras perspective m T 3 = weak perspective
65 Affine cameras Captures 3D affine transformation + orthographic projection + 2D affine transformation apple x y = = = apple X a 11 a 12 a 13 b 1 4a 21 a 22 a 23 b 2 5 6Y 7 4Z apple X apple a11 a 12 a 13 4Y 5 b1 + a 21 a 22 a 23 b Z 2 x = AX + b X Y Z Projection defined by 8 parameters Parallel lines project to parallel lines 2D points = linear projection of 3D points (+ 2D translation)
66 Affine Cameras m T 3 = x = X Y Z a 1 + b 1 y = X Y Z a 2 + b 1 Image coordinates (x,y) are an affine function of world coordinates (X,Y,Z) Example: Weak-perspective projection model Projection defined by 8 parameters Parallel lines project to parallel lines The transformation can be written as a direct linear transformation plus an offset
67 Geometric Transformations Euclidean (trans + rot) preserves lengths + angles Affine: preserves parallel lines Projective: preserves lines Projective Affine Euclidean
Agenda. Rotations. Camera calibration. Homography. Ransac
Agenda Rotations Camera calibration Homography Ransac Geometric Transformations y x Transformation Matrix # DoF Preserves Icon translation rigid (Euclidean) similarity affine projective h I t h R t h sr
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 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 informationCOSC579: Scene Geometry. Jeremy Bolton, PhD Assistant Teaching Professor
COSC579: Scene Geometry Jeremy Bolton, PhD Assistant Teaching Professor Overview Linear Algebra Review Homogeneous vs non-homogeneous representations Projections and Transformations Scene Geometry The
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 informationIntroduction to Computer Vision
Introduction to Computer Vision Michael J. Black Nov 2009 Perspective projection and affine motion Goals Today Perspective projection 3D motion Wed Projects Friday Regularization and robust statistics
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 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 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 informationVisual Recognition: Image Formation
Visual Recognition: Image Formation Raquel Urtasun TTI Chicago Jan 5, 2012 Raquel Urtasun (TTI-C) Visual Recognition Jan 5, 2012 1 / 61 Today s lecture... Fundamentals of image formation You should know
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 informationPinhole Camera Model 10/05/17. Computational Photography Derek Hoiem, University of Illinois
Pinhole Camera Model /5/7 Computational Photography Derek Hoiem, University of Illinois Next classes: Single-view Geometry How tall is this woman? How high is the camera? What is the camera rotation? What
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 informationCMPSCI 670: Computer Vision! Image formation. University of Massachusetts, Amherst September 8, 2014 Instructor: Subhransu Maji
CMPSCI 670: Computer Vision! Image formation University of Massachusetts, Amherst September 8, 2014 Instructor: Subhransu Maji MATLAB setup and tutorial Does everyone have access to MATLAB yet? EdLab accounts
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 informationPerspective projection. A. Mantegna, Martyrdom of St. Christopher, c. 1450
Perspective projection A. Mantegna, Martyrdom of St. Christopher, c. 1450 Overview of next two lectures The pinhole projection model Qualitative properties Perspective projection matrix Cameras with lenses
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 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 informationComputer Vision CS 776 Fall 2018
Computer Vision CS 776 Fall 2018 Cameras & Photogrammetry 1 Prof. Alex Berg (Slide credits to many folks on individual slides) Cameras & Photogrammetry 1 Albrecht Dürer early 1500s Brunelleschi, early
More informationCS6670: Computer Vision
CS6670: Computer Vision Noah Snavely Lecture 5: Projection Reading: Szeliski 2.1 Projection Reading: Szeliski 2.1 Projection Müller Lyer Illusion http://www.michaelbach.de/ot/sze_muelue/index.html Modeling
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 informationImage formation. Thanks to Peter Corke and Chuck Dyer for the use of some slides
Image formation Thanks to Peter Corke and Chuck Dyer for the use of some slides Image Formation Vision infers world properties form images. How do images depend on these properties? Two key elements Geometry
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 informationSpecifying Complex Scenes
Transformations Specifying Complex Scenes (x,y,z) (r x,r y,r z ) 2 (,,) Specifying Complex Scenes Absolute position is not very natural Need a way to describe relative relationship: The lego is on top
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 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 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 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 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 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 informationScene Modeling for a Single View
Scene Modeling for a Single View René MAGRITTE Portrait d'edward James CS194: Image Manipulation & Computational Photography with a lot of slides stolen from Alexei Efros, UC Berkeley, Fall 2014 Steve
More informationImage Formation. Antonino Furnari. Image Processing Lab Dipartimento di Matematica e Informatica Università degli Studi di Catania
Image Formation Antonino Furnari Image Processing Lab Dipartimento di Matematica e Informatica Università degli Studi di Catania furnari@dmi.unict.it 18/03/2014 Outline Introduction; Geometric Primitives
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 informationGeometric camera models and calibration
Geometric camera models and calibration http://graphics.cs.cmu.edu/courses/15-463 15-463, 15-663, 15-862 Computational Photography Fall 2018, Lecture 13 Course announcements Homework 3 is out. - Due October
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 informationHumanoid Robotics. Projective Geometry, Homogeneous Coordinates. (brief introduction) Maren Bennewitz
Humanoid Robotics Projective Geometry, Homogeneous Coordinates (brief introduction) Maren Bennewitz Motivation Cameras generate a projected image of the 3D world In Euclidian geometry, the math for describing
More informationScene Modeling for a Single View
Scene Modeling for a Single View René MAGRITTE Portrait d'edward James with a lot of slides stolen from Steve Seitz and David Brogan, 15-463: Computational Photography Alexei Efros, CMU, Fall 2005 Classes
More informationScene Modeling for a Single View
on to 3D Scene Modeling for a Single View We want real 3D scene walk-throughs: rotation translation Can we do it from a single photograph? Reading: A. Criminisi, I. Reid and A. Zisserman, Single View Metrology
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 informationGEOMETRIC TRANSFORMATIONS AND VIEWING
GEOMETRIC TRANSFORMATIONS AND VIEWING 2D and 3D 1/44 2D TRANSFORMATIONS HOMOGENIZED Transformation Scaling Rotation Translation Matrix s x s y cosθ sinθ sinθ cosθ 1 dx 1 dy These 3 transformations are
More informationHow to achieve this goal? (1) Cameras
How to achieve this goal? (1) Cameras History, progression and comparisons of different Cameras and optics. Geometry, Linear Algebra Images Image from Chris Jaynes, U. Kentucky Discrete vs. Continuous
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 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 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 informationVision Review: Image Formation. Course web page:
Vision Review: Image Formation Course web page: www.cis.udel.edu/~cer/arv September 10, 2002 Announcements Lecture on Thursday will be about Matlab; next Tuesday will be Image Processing The dates some
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 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 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 informationINTRODUCTION TO COMPUTER GRAPHICS. It looks like a matrix Sort of. Viewing III. Projection in Practice. Bin Sheng 10/11/ / 52
cs337 It looks like a matrix Sort of Viewing III Projection in Practice / 52 cs337 Arbitrary 3D views Now that we have familiarity with terms we can say that these view volumes/frusta can be specified
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 informationCOS429: COMPUTER VISON CAMERAS AND PROJECTIONS (2 lectures)
COS429: COMPUTER VISON CMERS ND PROJECTIONS (2 lectures) Pinhole cameras Camera with lenses Sensing nalytical Euclidean geometry The intrinsic parameters of a camera The extrinsic parameters of a camera
More informationCamera Calibration. COS 429 Princeton University
Camera Calibration COS 429 Princeton University Point Correspondences What can you figure out from point correspondences? Noah Snavely Point Correspondences X 1 X 4 X 3 X 2 X 5 X 6 X 7 p 1,1 p 1,2 p 1,3
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 informationCapturing Light: Geometry of Image Formation
Capturing Light: Geometry of Image Formation Computer Vision James Hays Slides from Derek Hoiem, Alexei Efros, Steve Seitz, and David Forsyth Administrative Stuff My Office hours, CoC building 35 Monday
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 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 informationInstance-level recognition I. - Camera geometry and image alignment
Reconnaissance d objets et vision artificielle 2011 Instance-level recognition I. - Camera geometry and image alignment Josef Sivic http://www.di.ens.fr/~josef INRIA, WILLOW, ENS/INRIA/CNRS UMR 8548 Laboratoire
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 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 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 informationGeometric transformations assign a point to a point, so it is a point valued function of points. Geometric transformation may destroy the equation
Geometric transformations assign a point to a point, so it is a point valued function of points. Geometric transformation may destroy the equation and the type of an object. Even simple scaling turns a
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 informationLecture 11 MRF s (conbnued), cameras and lenses.
6.869 Advances in Computer Vision Bill Freeman and Antonio Torralba Spring 2011 Lecture 11 MRF s (conbnued), cameras and lenses. remember correction on Gibbs sampling Motion application image patches image
More informationMath background. 2D Geometric Transformations. Implicit representations. Explicit representations. Read: CS 4620 Lecture 6
Math background 2D Geometric Transformations CS 4620 Lecture 6 Read: Chapter 2: Miscellaneous Math Chapter 5: Linear Algebra Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector
More informationVector Algebra Transformations. Lecture 4
Vector Algebra Transformations Lecture 4 Cornell CS4620 Fall 2008 Lecture 4 2008 Steve Marschner 1 Geometry A part of mathematics concerned with questions of size, shape, and relative positions of figures
More informationIntroduction to Computer Vision. Introduction CMPSCI 591A/691A CMPSCI 570/670. Image Formation
Introduction CMPSCI 591A/691A CMPSCI 570/670 Image Formation Lecture Outline Light and Optics Pinhole camera model Perspective projection Thin lens model Fundamental equation Distortion: spherical & chromatic
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 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 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 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 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 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 informationComputer Vision Projective Geometry and Calibration
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 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 informationUnderstanding Variability
Understanding Variability Why so different? Light and Optics Pinhole camera model Perspective projection Thin lens model Fundamental equation Distortion: spherical & chromatic aberration, radial distortion
More informationINTRODUCTION TO COMPUTER GRAPHICS. cs123. It looks like a matrix Sort of. Viewing III. Projection in Practice 1 / 52
It looks like a matrix Sort of Viewing III Projection in Practice 1 / 52 Arbitrary 3D views } view volumes/frusta spec d by placement and shape } Placement: } Position (a point) } look and up vectors }
More informationViewing. Announcements. A Note About Transformations. Orthographic and Perspective Projection Implementation Vanishing Points
Viewing Announcements. A Note About Transformations. Orthographic and Perspective Projection Implementation Vanishing Points Viewing Announcements. A Note About Transformations. Orthographic and Perspective
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 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 information2D/3D Geometric Transformations and Scene Graphs
2D/3D Geometric Transformations and Scene Graphs Week 4 Acknowledgement: The course slides are adapted from the slides prepared by Steve Marschner of Cornell University 1 A little quick math background
More informationLecture 5: Transforms II. Computer Graphics and Imaging UC Berkeley CS184/284A
Lecture 5: Transforms II Computer Graphics and Imaging UC Berkeley 3D Transforms 3D Transformations Use homogeneous coordinates again: 3D point = (x, y, z, 1) T 3D vector = (x, y, z, 0) T Use 4 4 matrices
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 informationProjective geometry, camera models and calibration
Projective geometry, camera models and calibration Subhashis Banerjee Dept. Computer Science and Engineering IIT Delhi email: suban@cse.iitd.ac.in January 6, 2008 The main problems in computer vision Image
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 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 information(Refer Slide Time: 00:01:26)
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 9 Three Dimensional Graphics Welcome back everybody to the lecture on computer
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 informationScene Modeling for a Single View
Scene Modeling for a Single View René MAGRITTE Portrait d'edward James with a lot of slides stolen from Steve Seitz and David Brogan, Breaking out of 2D now we are ready to break out of 2D And enter the
More informationMysteries of Parameterizing Camera Motion - Part 1
Mysteries of Parameterizing Camera Motion - Part 1 Instructor - Simon Lucey 16-623 - Advanced Computer Vision Apps Today Motivation SO(3) Convex? Exponential Maps SL(3) Group. Adapted from: Computer vision:
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 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 informationStereo CSE 576. Ali Farhadi. Several slides from Larry Zitnick and Steve Seitz
Stereo CSE 576 Ali Farhadi Several slides from Larry Zitnick and Steve Seitz Why do we perceive depth? What do humans use as depth cues? Motion Convergence When watching an object close to us, our eyes
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 informationViewing. Part II (The Synthetic Camera) CS123 INTRODUCTION TO COMPUTER GRAPHICS. Andries van Dam 10/10/2017 1/31
Viewing Part II (The Synthetic Camera) Brownie camera courtesy of http://www.geh.org/fm/brownie2/htmlsrc/me13000034_ful.html 1/31 The Camera and the Scene } What does a camera do? } Takes in a 3D scene
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 informationStructure from Motion
Structure from Motion Outline Bundle Adjustment Ambguities in Reconstruction Affine Factorization Extensions Structure from motion Recover both 3D scene geoemetry and camera positions SLAM: Simultaneous
More informationECE-161C Cameras. Nuno Vasconcelos ECE Department, UCSD
ECE-161C Cameras Nuno Vasconcelos ECE Department, UCSD Image formation all image understanding starts with understanding of image formation: projection of a scene from 3D world into image on 2D plane 2
More informationCS 664 Slides #9 Multi-Camera Geometry. Prof. Dan Huttenlocher Fall 2003
CS 664 Slides #9 Multi-Camera Geometry Prof. Dan Huttenlocher Fall 2003 Pinhole Camera Geometric model of camera projection Image plane I, which rays intersect Camera center C, through which all rays pass
More informationCSE528 Computer Graphics: Theory, Algorithms, and Applications
CSE528 Computer Graphics: Theory, Algorithms, and Applications Hong Qin Stony Brook University (SUNY at Stony Brook) Stony Brook, New York 11794-2424 Tel: (631)632-845; Fax: (631)632-8334 qin@cs.stonybrook.edu
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 information