DOCSLIB.ORG

Basic Stereo & Epipolar Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basic Stereo & Epipolar Geometry Basic Stereo & Epipolar Geometry EECS 598-08 Fall 2014! Instructor: Jason Corso (jjcorso)! Foundations of Computer Vision! web.eecs.umich.edu/~jjcorso/t/598F14! ! ! Readings: FP 7; SZ 11.1; TV 7! Date: 10/22/14! ! Materials on these slides have come from many sources in addition to myself; individual slides reference specific sources.! 2 Plan •" Basic Stereo! –" Why is more than one view useful?! –" Triangulation! •" Epipolar Geometry! On to Shape •" What cues help us to perceive 3d shape and depth? Shading [Figure from Prados & Faugeras 2006] Focus/Defocus [Figure from H. Jin and P. Favaro, 2002] Texture [From A.M. Loh. The recovery of 3-D structure using visual texture patterns. PhD thesis] Perspective effects Image credit: S. Seitz Motion Figures from L. Zhang http://www.brainconnection.com/teasers/?main=illusion/motion-shape Estimating scene shape •" Shape from X: Shading, Texture, Focus, Motion… •" Stereo: –" shape from “motion” between two views –" infer 3d shape of scene from two (multiple) images from different viewpoints 10 Can Structure Be Recovered from a Single View? P! p! Ow! Scene! C! Calibration rig! Camera K! From calibration rig! → location/pose of the rig, K! From points and lines at infinity ! + orthogonal lines and planes! → structure of the scene, K! Knowledge about scene (point correspondences, geometry of lines & planes, etc…! Slide source: S. Savarese.! 11 Can Structure Be Recovered from a Single View? Pinhole perspective projection! P! p! Ow! Scene! C! Calibration rig! Camera K! Why is it so difficult?! Intrinsic ambiguity of the mapping from 3D to image (2D)! Slide source: S. Savarese.! 12 Can Structure Be Recovered from a Single View? Intrinsic ambiguity of the mapping from 3D to image (2D)! Courtesy slide S. Lazebnik! Slide source: S. Savarese.! 13 Two Eyes Help! Slide source: S. Savarese.! Fixation, convergence Slide source: K. Grauman.! Human stereopsis: disparity Disparity occurs when eyes fixate on one object; others appear at different visual angles Human Stereopsis; Disparity d=0 Disparity: d = r-l = D-F. Adapted from M. Pollefeys Random dot stereograms •" Julesz 1960: Do we identify local brightness patterns before fusion (monocular process) or after (binocular)? •" To test: pair of synthetic images obtained by randomly spraying black dots on white objects Slide source: K. Grauman.! Random dot stereograms Forsyth & Ponce Random dot stereograms Slide source: K. Grauman.! Random dot stereograms From Palmer, “Vision Science”, MIT Press Slide source: K. Grauman.! Random dot stereograms •" When viewed monocularly, they appear random; when viewed stereoscopically, see 3d structure. •" Conclusion: human binocular fusion not directly associated with the physical retinas; must involve the central nervous system •" Imaginary “cyclopean retina” that combines the left and right image stimuli as a single unit Slide source: K. Grauman.! Autostereograms Exploit disparity as depth cue using single image (Single image random dot stereogram, Single image stereogram) Images from magiceye.com Slide source: K. Grauman.! Autostereograms Images from magiceye.com Slide source: K. Grauman.! Stereo photography and stereo viewers Take two pictures of the same subject from two slightly different viewpoints and display so that each eye sees only one of the images. Invented by Sir Charles Wheatstone, 1838 Image courtesy of fisher-price.com Slide source: K. Grauman.! http://www.johnsonshawmuseum.org Slide source: K. Grauman.! http://www.johnsonshawmuseum.org Slide source: K. Grauman.! Public Library, Stereoscopic Looking Room, Chicago, by Phillips, 1923 Slide source: K. Grauman.! 28 Two Eyes Help! X = l ×lʹ X?! l' l x1! x2! K =known! K =known! R, T! O2! O1! This is called triangulation! Slide source: S. Savarese.! 29 Triangulation X! Y X Y! x2! y1! x1! y2! Although geometrically correct,! O2! O1! invalid correspondences lead ! to incorrect triangulation! 30 Triangulation • Find X that minimizes! 2 2 d (x1, M1 X ) + d (x2 , M 2 X ) X! x2! x1! O Slide source: S. Savarese.! O1! 2! 31 Triangulation Geometry in Simple Stereo X Z! u2! u1! c2! x c1! x2! 1! f! T! O2! O1! Depth from disparity image I(x,y) Disparity map D(x,y) image I´(x´,y´) (x´,y´)=(x+D(x,y), y) Slide source: K. Grauman.! 33 Core Problems in Stereo •" Correspondence: Given a point in one image, how can I find the corresponding point x’ in another one ?# ! •" Camera geometry: Given corresponding points in two images, find camera matrices, position and pose.! •" Scene geometry: Find coordinates of 3D point from its projection into 2 or multiple images.! Slide source: S. Savarese.! 34 Epipolar Geometry X! x1! x2! e1! e2! O1! O2! •" Epipolar Plane! •" Epipoles e1, e2! •" Baseline! = intersections of baseline with image planes ! = projections of the other camera center! •" Epipolar Lines! = vanishing points of camera motion direction! Slide source: S. Savarese.! Epipolar Geometry Terms •" Baseline: line joining the camera centers •" Epipole: point of intersection of baseline with the image plane •" Epipolar plane: plane containing baseline and world point •" Epipolar line: intersection of epipolar plane with the image plane •" All epipolar lines intersect at the epipole •" An epipolar plane intersects the left and right image planes in epipolar lines Slide source: K. Grauman.! 36 Example: Converging image planes Slide source: S. Savarese, K. Grauman; Figure from Hartley and Zisserman.! 37 Example: Parallel Image Planes X! e 2! x 1! x2! e2! O1! O2! •" Baseline intersects the image plane at infinity! •" Epipoles are at infinity! •" Epipolar lines are parallel to x axis! Slide source: S. Savarese.! 38 Example: Parallel Image Planes Slide source: S. Savarese, K Grauman; Figure from Hartley and Zisserman.! 39 Example: Forward translation e2! O2! e1! O2! •" The epipoles have same position in both images! •" Epipole called FOE (focus of expansion)! Slide source: S. Savarese, K Grauman; Figure from Hartley and Zisserman.! 40 The Epipolar Constraint -" Two views of the same object ! -" Suppose I know the camera positions and camera matrices! - Given a point on left image, how can I find the corresponding point on right image?! Slide source: S. Savarese.! Image Examples of the Epipolar Constraint Slide source: K. Grauman.! 42 Epipolar Geometry X! How do we represent the epipolar x x geometry1! algebraically?2! e2! O1! O2! •" Epipolar Plane! •" Epipoles e1, e2! •" Baseline! = intersections of baseline with image planes ! = projections of the other camera center! •" Epipolar Lines! = vanishing points of camera motion direction! Slide source: S. Savarese.! 43 Epipolar Constraint! •" Potential matches for p have to lie on the corresponding ! epipolar line l’.! •" Potential matches for p’ have to lie on the corresponding ! epipolar line l.! Slide source: S. Savarese.! 44 The Epipolar Constraint P! p’! p! O! R, T! O’! M = K[I 0] M'= K[R T] ⎡u⎤ ⎡uʹ⎤ P M P ⎢v⎥ P → Mʹ P = ⎢vʹ⎥ → = ⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢ 1 ⎥ Slide source: S. Savarese.! ⎣ ⎦ ⎣ ⎦ 45 The Epipolar Constraint P! p’! p! O! R, T! O’! K is known! M = K[I 0] (canonical cameras)! M'= K[R T] ⎡1 0 0 ⎤ K ⎢0 1 0 ⎥ = ⎢ ⎥ M = [I 0] ⎣⎢0 0 1 ⎦⎥ M'= [R T] Slide source: S. Savarese.! 46 The Epipolar Constraint P! p’! p! O! R, T! O’! p’ in first camera reference system is! = R pʹ T ×(R pʹ) is perpendicular to epipolar plane! → pT ⋅ [T ×(R pʹ)]= 0 Slide source: S. Savarese.! 47 Cross product as matrix multiplication ⎡ 0 − az ay ⎤⎡bx ⎤ ⎢ ⎥ a×b = a 0 − a ⎢b ⎥ = [a ]b ⎢ z x ⎥⎢ y ⎥ × ⎢ a a 0 ⎥ b ⎣− y x ⎦⎣⎢ z ⎦⎥ Slide source: S. Savarese.! 48 The Epipolar Constraint P! p’! p! O! R, T! O’! T T p ⋅ [T×(R pʹ)]= 0 → p ⋅ [T× ]⋅R pʹ = 0 E = essential matrix! (Longuet-Higgins, 1981)! Slide source: S. Savarese.! 49 The Epipolar Constraint T P! p1 ⋅ E p2 = 0 p 1! p2! l1! l2! e e1! 2! O1! O2! •" E p2 is the epipolar line associated with p2 (l1 = E p2)! T T •" E p1 is the epipolar line associated with p1 (l2 = E p1)! T •" E e2 = 0 and E e1 = 0! •" E is 3x3 matrix; 5 DOF! •" E is singular (rank two)! ! Slide source: S. Savarese.! 50 The Epipolar Constraint P! p’! p! R, T! O! O’! K is unknown! M = K[I 0] M'= K[R T] Slide source: S. Savarese.! 51 The Epipolar Constraint P! p → K −1 p pʹ → K'−1 pʹ p’! p! O! O’! T (K−1 p)T T R K −1 p 0 p ⋅ [T× ]⋅R pʹ = 0 → ⋅ [ × ]⋅ ʹ ʹ = T −T −1 → pT F pʹ = 0 p K ⋅ [T× ]⋅R Kʹ pʹ = 0 Slide source: S. Savarese.! 52 The Epipolar Constraint P! p’! p! O! O’! pT F pʹ = 0 −T −1 F = K ⋅ [T× ]⋅ R Kʹ F = Fundamental Matrix! (Faugeras and Luong, 1992)! Slide source: S. Savarese.! 53 The Epipolar Constraint T P! p1 ⋅ F p2 = 0 p 1! p2! e e1! 2! O1! O2! •" F p2 is the epipolar line associated with p2 (l1 = F p2)! T T •" F p1 is the epipolar line associated with p1 (l2 = F p1)! T •" F e2 = 0 and F e1 = 0! •" F is 3x3 matrix; 7 DOF ! •" F is singular (rank two)! Slide source: S. Savarese.! 54 Why F is useful? T x! l’ = F x! -" Suppose F is known! -" No additional information about the scene and camera is given! - Given a point on left image, how can I find the corresponding point on right image?! Slide source: S. Savarese.! 55 Why is F Useful? •" F captures information about the epipolar geometry of 2 views + camera parameters ! •" MORE IMPORTANTLY: F gives constraints on how the scene changes under view point transformation (without reconstructing the scene!)! •" Powerful tool in:! •" 3D reconstruction! •" Multi-view object/scene matching ! Slide source: S. Savarese.! 56 Estimating F The Eight-Point Algorithm! (Longuet-Higgins, 1981)! (Hartley, 1995)! P! p’! p! O! O’! ⎡u⎤ ⎡uʹ⎤ P p ⎢v⎥ P pʹ ⎢vʹ⎥ T → = ⎢ ⎥ → = ⎢ ⎥ p F pʹ = 0 ⎣⎢1⎦⎥ ⎣⎢ 1 ⎦⎥ Slide source: S.
Load more

Recommended publications
  • Camera Array Image Rectification and Calibration for Stereoscopic and Autostereoscopic Displays Vincent Nozick
    Camera array image rectification and calibration for stereoscopic and autostereoscopic displays Vincent Nozick To cite this version: Vincent Nozick. Camera array image rectification and calibration for stereoscopic and autostereoscopic displays. Annals of Telecommunications - annales des télécommunications, Springer, 2013, 68 (11), pp.581-596. hal-00908586 HAL Id: hal-00908586 https://hal-upec-upem.archives-ouvertes.fr/hal-00908586 Submitted on 25 Nov 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) Camera array image rectification and calibration for stereoscopic and autostereoscopic displays Vincent Nozick Received: date / Accepted: date Abstract This paper presents an image rectification ically, this paper presents an extension of an image rec- method for an arbitrary number of views with aligned tification method [26] to perform a camera calibration camera center. This paper also describes how to extend for an arbitrary number of views with aligned camera this method to easily perform a robust camera cali- center. This technique can be used for stereoscopic ren- bration. These two techniques can be used for stereo- dering to enhance the perception comfort or for depth scopic rendering to enhance the perception comfort or from stereo.
    [Show full text]
  • Essential Matrix” • Two Uncalibrated Cameras “Fundamental Matrix” • Gives Epipolar Lines
    Two Views Part 2: Essential CS 4495 Computer Vision – A. Bobick and Fundamental Matrices CS 4495 Computer Vision N-Views (2) – Essential and Fundamental Matrices Aaron Bobick School of Interactive Computing Two Views Part 2: Essential CS 4495 Computer Vision – A. Bobick and Fundamental Matrices Administrivia • Today: Second half of N-Views (n = 2) • PS 3: Will hopefully be out by Thursday • Will be due October 6th. • Will be based upon last week and today’s material • We may revisit the logistics – suggestions? Two Views Part 2: Essential CS 4495 Computer Vision – A. Bobick and Fundamental Matrices Two views…and two lectures • Projective transforms from image to image • Some more projective geometry • Points and lines and planes • Two arbitrary views of the same scene • Calibrated – “Essential Matrix” • Two uncalibrated cameras “Fundamental Matrix” • Gives epipolar lines Two Views Part 2: Essential CS 4495 Computer Vision – A. Bobick and Fundamental Matrices Last time • Projective Transforms: Matrices that provide transformations including translations, rotations, similarity, affine and finally general (or perspective) projection. • When 2D matrices are 3x3; for 3D they are 4x4. Two Views Part 2: Essential CS 4495 Computer Vision – A. Bobick and Fundamental Matrices Last time: Homographies • Provide mapping between images (image planes) taken from same center of projection; also mapping between any images of a planar surface. PP2 PP1 Two Views Part 2: Essential CS 4495 Computer Vision – A. Bobick and Fundamental Matrices Last time: Projective geometry • A line is a plane of rays through origin – all rays (x,y,z) satisfying: ax + by + cz = 0 x in vector notation : 0 = [a b c]y z l p • A line is also represented as a homogeneous 3-vector l Two Views Part 2: Essential CS 4495 Computer Vision – A.
    [Show full text]
  • Journal of Global Research in Computer Science RENEWED
    Volume 2, No. 7, July 2011 Journal of Global Research in Computer Science RESEARCH PAPER Available Online at www.jgrcs.info RENEWED APPROACH FOR IMAGE RECTIFICATION Sahil Bansal* and Seema Baghla *Computer Engineering Section, Yadavindra College of Engineering, Talwandi Sabo [email protected] Computer Engineering Section, Yadavindra College of Engineering, Talwandi Sabo [email protected] Abstract: This paper proposed a new method for image rectification ,it is the process by which the pairs of stereo images of same solid scene taken from different viewpoints in order to produce a pair of “matched epipolar projections” and become parallel to the x-axis of image. A stereo rectified images are helpful for matching algorithms.It restricts that each line parallel to x-axis.The stereo rectification is not unique and actually lead to undesirable distortions. To overcome the drawback of the relative distortion between left image and right image an epipolar line rectification technique is used for point detection and reduce the distortion by minimized the camera rotation angle at each step.By comparative experiments show that the algorithm has an accuracy where other methods fail, namely when the epipolar lines are far from horizontal. Keywords: Rectification, stereovision, epipolar, distortion, camera rotation. INTRODUCTION alignment. Image rectification is usually performed Image Rectification is a transformation process used to regardless of camera precision due to two reasons:- project two-or-more images onto a common image plane. It . Impractically or Impossibility of perfectly aligning corrects image distortion by transforming the image into a cameras. standard coordinate system. Image Rectification is used . Perfectly aligned cameras may become misaligned over in computer stereo vision to simplify the problem of finding time.
    [Show full text]
  • Stereopsis Exam #1 5 Pts "-2.5 Any Computational Error, Error with Units, Etc"
    Stereopsis Exam #1 5 pts "-2.5 any computational error, error with units, etc" #2 2pts each all or nothing (tend to give credit unless answer is very far off) i. (answers for 'perspective projection' were uniformly weak, so I was unusually liberal with credit) iii must mention parallel lines iv. Partial credit unavoidable here. -.5 per missed vi. Must mention neighbors, or a neighborhood, etc. x. must say power per unit area, (Can use meters^2 instead of area, etc.) I write OK if their answer is less than great, but I'm not taking off credit #3 4 pts formula for convolution of filter H with image I. Don't care about if they put limits on the summation. 8 pts the actual output. 1 pt per pixel. (there are 8 pixels fully covered.) #4 9 pts -0 If they choose mask 1, regardless of reasoning' -4 if they choose mask 2 or mask 3, and give some reasonably plausible reasoning' -9 if they choose mask 4, or choose (2 or 3) with very bad reasoning 60 50 40 30 Series1 20 10 0 1 3 5 7 9 111315171921 Mark Twain at Pool Table", no date, UCR Museum of Photography Woman getting eye exam during immigration procedure at Ellis Island, c. 1905 - 1920 , UCR Museum of Phography Why Stereo Vision? • 2D images project 3D points into 2D: P Q P’’=Q’’ O • 3D Points on the same viewing line have the same 2D image: – 2D imaging results in depth information loss Stereo • Assumes (two) cameras. • Known positions.
    [Show full text]
  • Computer Vision ECE 661 Homework 10
    Computer Vision ECE 661 Homework 10 Rohan Sarkar ([email protected]) November 29, 2016 1. Introduction In this assignment, we perform a 3D scene reconstruction with projective distortion using two stereo images captured by an uncalibrated camera. Such a 3D reconstruction is called projective reconstruction. because the reconstruction is projectively distorted visually and it is related by a 4 4 homography to the real scene. If the scene is visually rich and we £ have prior knowledge about the objects that appear in the scene, we can remove projective distortion, affine and similarity distortion successively. The following steps are performed for the 3D reconstruction : 1. An initial estimate of the fundamental matrix F is found, using the manually selected correspondences (minimum of 8) between the 2 images by linear least squares optimization and refine the fundamental matrix F using the Levenberg Marquadt Method. 2. Using this initial estimate of F, we rectify the images to send the epipoles to infinity T such that e £1 0 0¤ and thereby find the Homographies for the two cameras for Æ this purpose, represented by H,H0. 3. Using the rectified images, the interest points on the rectified images are found using SURF and canny edge detector. 4. After finding the interest points in the rectified images, we calculate the euclidean distance geometric error to calculate the cost function and apply the non-linear least squares optimization to minimize it and refine the fundamental matrix, camera matrices and the 3D world points. 5. Finally we shall use triangulation to project the point correspondences to world 3D and compute the 3D Scene structure from the set of correspondences for the given camera pair (P,P0).
    [Show full text]
  • Three-Step Image Rectification 1
    MONASSE et al.: THREE-STEP IMAGE RECTIFICATION 1 Three-step image rectification Pascal MONASSE1 1 IMAGINE, LIGM/Université [email protected] Paris-Est/École des Ponts ParisTech Jean-Michel MOREL2 Marne-la-Vallée, France [email protected] 2 CMLA, ENS-Cachan Zhongwei TANG2 Cachan, France [email protected] Abstract Image stereo-rectification is the process by which two images of the same solid scene undergo homographic transforms, so that their corresponding epipolar lines coincide and become parallel to the x-axis of image. A pair of stereo-rectified images is helpful for dense stereo matching algorithms. It restricts the search domain for each match to a line parallel to the x-axis. Due to the redundant degrees of freedom, the solution to stereo- rectification is not unique and actually can lead to undesirable distortions or be stuck in a local minimum of the distortion function. In this paper a robust geometric stereo- rectification method by a three-step camera rotation is proposed and mathematically ex- plained. Unlike other methods which reduce the distortion by explicitly minimizing an empirical measure, the intuitive geometric camera rotation angle is minimized at each step. For un-calibrated cameras, this method uses an efficient minimization algorithm by optimizing only one natural parameter, the focal length. This is in contrast with all former methods which optimize between 3 and 6 parameters. Comparative experiments show that the algorithm has an accuracy comparable to the state-of-art, but finds the right minimum in cases where other methods fail, namely when the epipolar lines are far from horizontal.
    [Show full text]
  • Three-Step Image Rectification Pascal Monasse, Jean-Michel Morel, Zhongwei Tang
    Three-step image rectification Pascal Monasse, Jean-Michel Morel, Zhongwei Tang To cite this version: Pascal Monasse, Jean-Michel Morel, Zhongwei Tang. Three-step image rectification. BMVC 2010 - British Machine Vision Conference, Aug 2010, Aberystwyth, United Kingdom. pp.89.1–89.10, 10.5244/C.24.89. hal-00654415 HAL Id: hal-00654415 https://hal-enpc.archives-ouvertes.fr/hal-00654415 Submitted on 21 Dec 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MONASSE et al.: THREE-STEP IMAGE RECTIFICATION 1 Three-step image rectification Pascal MONASSE1 1 IMAGINE, LIGM/Université [email protected] Paris-Est/École des Ponts ParisTech Jean-Michel MOREL2 Marne-la-Vallée, France [email protected] 2 CMLA, ENS-Cachan Zhongwei TANG2 Cachan, France [email protected] Abstract Image stereo-rectification is the process by which two images of the same solid scene undergo homographic transforms, so that their corresponding epipolar lines coincide and become parallel to the x-axis of image. A pair of stereo-rectified images is helpful for dense stereo matching algorithms. It restricts the search domain for each match to a line parallel to the x-axis.
    [Show full text]
  • Structure from Motion Based on Monocular Image Sequences with a Virtual Stereo Camera
    FREIE UNIVERSITÄT BERLIN MASTER’S THESIS Structure from Motion Based on Monocular Image Sequences with a Virtual Stereo Camera Author: Supervisor & 1st Examiner: Johannes SAUER Prof. Dr. Raúl Matr.Nr. 4473561 ROJAS GONZÁLEZ Advisor: 2nd Examiner: Tobias LANGNER Prof. Dr. Daniel GÖHRING A thesis submitted in fulfillment of the requirements for the degree of Master of Science at the Department of Mathematics and Computer Science 10th April, 2019 iii Selbstständigkeitserklärung Name Sauer Vornamen Johannes Theo geb. am 08.02.1990 Matr.Nr. 4473561 Ich erkläre gegenüber der Freien Universität Berlin, dass ich die vorliegende Mas- terarbeit selbstständig und ohne Benutzung anderer als der angegebenen Quellen und Hilfsmittel angefertigt habe. Die vorliegende Arbeit ist frei von Plagiaten. Alle Ausführungen, die wörtlich oder inhaltlich aus anderen Schriften entnommen sind, habe ich als solche kenntlich gemacht. Diese Arbeit wurde in gleicher oder ähnlicher Form noch bei keiner anderen Uni- versität als Prüfungsleistung eingereicht. Datum Unterschrift v FREIE UNIVERSITÄT BERLIN Abstract Department of Mathematics and Computer Science Master of Science Structure from Motion Based on Monocular Image Sequences with a Virtual Stereo Camera by Johannes SAUER Stereo matching (SM) is a well researched method for generating depth information from camera images. Efficient implementations of SM algorithms exist as part of widely used computer vision libraries, such as OpenCV. Typically, SM is being performed on pairs of images from a stereo camera in which intrinsic and extrinsic parameters are fixed and determined in advance by calibration. Furthermore, the images are usually taken at approximately the same time by triggering the shutters simultaneously. In this thesis a different approach is being pursued: stereo pairs are selected from a video se- quence of a monocular camera, which is mounted on a moving vehicle.
    [Show full text]
  • Epipolar Geometry and Stereo Vision
    Epipolar Geometry and Stereo Vision Computer Vision Jia-Bin Huang, Virginia Tech Many slides from S. Seitz and D. Hoiem Last class: Image Stitching • Two images with rotation/zoom but no translation .X x x' f f' This class: Two-View Geometry •Epipolar geometry • Relates cameras from two positions •Stereo depth estimation • Recover depth from two images Depth from Stereo • Goal: recover depth by finding image coordinate x’ that corresponds to x X X x z x x’ f f x' C Baseline C’ B Depth from Stereo • Goal: Recover depth by finding image coordinate x’ that corresponds to x • Sub-Problems 1. Calibration: What’s the relation of the two cameras? 2. Correspondence: Where is the matching point x’? X x x' Correspondence Problem x ? • Two images taken from cameras with different intrinsic and extrinsic parameters • How do we match a point in the first image to a point in the second? • How can we constrain our search? Key idea: Epipolar constraint Key idea: Epipolar constraint X X X x x’ x’ x’ Potential matches for x have to lie on the corresponding line l’. Potential matches for x’ have to lie on the corresponding line l. Epipolar geometry: notation X x x’ • Baseline – line connecting the two camera centers • Epipoles = intersections of baseline with image planes = projections of the other camera center • Epipolar Plane – plane containing baseline (1D family) Epipolar geometry: notation X x x’ • Baseline – line connecting the two camera centers • Epipoles = intersections of baseline with image planes = projections of the other camera center • Epipolar Plane – plane containing baseline (1D family) • Epipolar Lines - intersections of epipolar plane with image planes (always come in corresponding pairs) Example: Converging cameras Example: Motion parallel to image plane Example: Forward motion What would the epipolar lines look like if the camera moves directly forward? Example: Forward motion e’ e Epipole has same coordinates in both images.
    [Show full text]
  • Wide Baseline Stereo Image Rectification and Matching
    To the Graduate Council: I am submitting herewith a dissertation written by Wei Hao entitled "Wide Baseline Stereo Image Rectification and Matching." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Electrical Engineering. Mongi A. Abidi, Major Professor We have read this dissertation and recommend its acceptance: Seddik M. Djouadi, Ohannes Karakashian, Andreas Koschan Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) Wide Baseline Stereo Image Rectification and Matching A Dissertation Presented for Doctor of Philosophy Degree The University of Tennessee, Knoxville Wei Hao December 2011 Copyright © 2011 by Wei Hao All rights reserved ii Acknowledgement First and foremost, I am deeply indebted to my parents, Ms. Jinxia Wang and Mr. Guoniu Hao, whose love and encouragement has always been inspiring me during the pursuit for higher education. I would like to express my sincerest gratitude to my academic advisor, Dr. Mongi Abidi, who fired up my interest in computer vision and taught me the fundamentals of good research, writing, and presentation. His profound knowledge and deep insights in the imaging and vision area has always been enlightening during the whole course of pursuing Ph. D. Special thanks go to Dr. Andreas Koschan, whose comments and suggestions provided valuable insights into stereo vision and have had tremendous impacts on this research. Also, I would like to thank Dr.
    [Show full text]
  • Computer Vision Lecture 19
    Announcement Homework 4 has been posted online and in Blackboard. Due: 11:59pm EST, Thursday, April 1st Announcement Quiz #5 is available in Blackboard. Due date: 11:59pm EST, Wednesday, March. 24th Open book and open notes Reminder: Final Project Presentation Schedule Presentation days: April 20 and 22 A single presentation is required for a project Each presentation has 12 + 3 minutes Q&A Send me an email ([email protected]) that includes: • Your name and ranking of the dates starting from the most preferred • Earlier email has higher priority in choosing the date Today Early vision on multiple images • Epipolar geometry Passive Stereo General strategy: • Camera calibration • Correspondence pages.cs.wisc.edu • Detect a set of tokens (points, lines, and conic curves) • Build correspondence for the tokens in multiple (at least two) images • 3D reconstruction from matched 2D points • Image rectification • 3D triangulation from rectified images courtesy of Dr. Qiang Ji A Simple Case: 3D Triangulation – Rectified Geometry Rectified geometry: • two image planes are coplanar and parallel to the line 푂1푂2 courtesy of Dr. Qiang Ji A Simple Case: 3D Triangulation – Rectified Geometry Rectified geometry: Hence, triangle 푋푈1푈2 is similar to triangle 푋푂1푂2 푍 − 푓 푏 − 푢 + 푢 = 1 2 푍 푏 푓푏 푍 = 푢1 − 푢2 courtesy of Dr. Qiang Ji 3D Triangulation – Rectified Geometry 푓푏 푍 = 푢1 − 푢2 b: base distance 푢1 − 푢2: disparity 푓푏 푢 − 푢 = 1 2 푍 Disparity is inversely proportional to the depth. Establish Correspondence Problem: there are multiple images taken for the same object. We need to determine which point in one image is corresponding to the point in the others.
    [Show full text]
  • Lecture 10 Notes
    AA274A: Principles of Robot Autonomy I Course Notes The previous chapter developed a mathematical relationship between the position of a point P in a scene (expressed in world frame coordinates PW ), and the corresponding point p in pixel coordinates that gets projected onto the image plane of the camera. This relationship was derived based on the pinhole camera model, and required knowledge about the camera's intrinsic and extrinsic parameters. Nonetheless, even in the case where all of these camera parameters are known it is still impossible to reconstruct the depth of P with a single image (without additional information). However, in the context of robotics, recovering 3D information about the structure of the robot's environment through computer vision is often a very important task (e.g. for obstacle avoidance). Two approaches for using cameras to gather 3D information are therefore presented in this chapter, namely stereo vision and structure from motion. 10 Stereo Vision and Structure from Motion Recovering scene structure from images is extremely important for mobile robots to safely operate in their environment and successfully perform tasks. While a number of other sensors can also be used to recover 3D scene information, such as ultrasonic sensors or laser rangefinders, cameras capture a broad range of information that goes beyond depth sensing. Additionally, cameras are a well developed technology and can be an attractive option for robotics based on cost or size. Unfortunately, unlike sensors that are specifically designed to measure depth like laser rangefinders, the camera's projection of 3D data onto a 2D image makes it impossible to gather some information from a single image1.
    [Show full text]