Single View Metrology
3D photography course schedule Topic Feb 21 Introduction Feb 28 Lecture: Geometry, Camera Model, Calibration Mar 7 Lecture: Features & Correspondences Mar 14 Project Proposals Mar 21 Lecture: Epipolar Geometry Mar 28 Depth Estimation + 2 papers Apr 4 Single View Geometry + 2 papers Apr 11 Active Ranging and Structured Light + 2 papers Apr 18 Project Updates Apr. 25 --- Easter --- May 2 SLAM + 2 papers May 9 3D & Registration + 2 papers May 16 Structure from Motion + 2 papers May 23 Shape from Silhouettes + 2 papers May 30 Final Projects (if not demo day)
Single View Metrology Pictures from “Single View Metrology” by A. Criminisi et al.
Measuring in a plane Need to compute H as well as uncertainty
Direct Linear Transformation (DLT) (wrap-up, compare lect. 3) T 1 h x i x T x Hx Hx 0 T 2 x x , y , w Hx h x i i i i i i i i i i T 3 h x T T 3 2 y h x w h x i i i i i T T 1 3 x Hx w h x x h x i i i i i i T T 2 1 x h x y h x i i i i T T T 1 0 w x y x h i i i i T T T 2 w x 0 x x h 0 i i i i T T T 3 y x x x 0 h i i i i A h 0 Normalize coordinates ! i
Gold Standard algorithm Objective Given n≥4 2D to 2D point correspondences {x i ↔x i ’}, determine the Maximum Likelyhood Estimation of H (this also implies computing optimal x i ’=Hx i ) Algorithm (i) Initialization: compute an initial estimate using normalized DLT or RANSAC (ii) Geometric minimization of reprojection error: ● Minimize using Levenberg-Marquardt over 9 entries of h or Gold Standard error: ● compute initial estimate for optimal {x i } 2 2 ● minimize cost over {H,x 1 ,x 2 ,…,x n } ˆ ˆ d x , x d x , x i i i i ● if many points, use sparse method
Using covariance matrix in point transfer Error in one image J T J x h h h Error in two images (or image and scene) T T J J J J x h h h x x x (if h and x independent, i.e. new points)
Example: s =1 pixel =0.5cm (Criminisi’97)
Example: s =1 pixel =0.5cm (Criminisi’97)
Example: (Criminisi’97)
Monte Carlo estimation of covariance • To be used when previous assumptions do not hold (e.g. non-flat within variance) or to complicate to compute. • Simple and general, but expensive • Generate samples according to assumed noise distribution, carry out computations, observe distribution of result
Single view measurements: 3D scene
Background: Projective geometry of 1D H x ' x 3DOF (2x2-1) 2 2 The cross ratio x , x x , x x , x ; x , x 1 2 3 4 x , x x , x 1 2 3 4 1 3 2 4 Invariant under projective transformations
Vanishing points • Under perspective projection points at infinity can have a finite image • The projection of 3D parallel lines intersect at vanishing points in the image
Basic geometry
Basic geometry • Allows to relate height of point to height of camera
Homology mapping between parallel planes • Allows to transfer point from one plane to another
Single view measurements
Single view measurements
Forensic applications 190.6±4.1 cm 190.6±2.9 cm A. Criminisi, I. Reid, and A. Zisserman. Computing 3D euclidean distance from a single view. Technical Report OUEL 2158/98, Dept. Eng. Science, University of Oxford, 1998.
Example courtesy of Antonio Criminisi
La Flagellazione di Cristo (1460) Galleria Nazionale delle Marche by Piero della Francesca (1416-1492) http://www.robots.ox.ac.uk/~vgg/projects/SingleView/
More interesting stuff • Criminisi demo http://www.robots.ox.ac.uk/~vgg/presentations/ spie98/criminis/index.html • work by Derek Hoiem on learning single view 3D structure and apps http://www.cs.cmu.edu/~dhoiem/ • similar work by Ashutosh Saxena on learning single view depth http://ai.stanford.edu/~asaxena/learningdepth/
Administrative • Projects and Papers assigned !? • Hardware? Mobile Kinects? • Forum: Share Experiences • In two weeks: Project updates Each team presents (5-10 min.): - intermediate results - solved/unsolved/unforeseen things - adaption of goals - … anything else relevant …
Presentations • Automatic Photo Popup: Classify in Ground/Verticals/Sky and reconstruct • Video Compass: Estimate Vanishing Points and Camera Calibration
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