Single View Metrology Class 3

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Single View MetrologyClass 3
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3D photography course schedule(tentative)
Lecture Exercise
Sept 26 Introduction -
Oct. 3 Geometry Camera model Camera calibration
Oct. 10 Single View Metrology Measuring in images
Oct. 17 Feature Tracking/matching (Friedrich Fraundorfer) Correspondence computation
Oct. 24 Epipolar Geometry F-matrix computation
Oct. 31 Shape-from-Silhouettes (Li Guan) Visual-hull computation
Nov. 7 Stereo matching Project proposals
Nov. 14 Structured light and active range sensing Papers
Nov. 21 Structure from motion Papers
Nov. 28 Multi-view geometry and self-calibration Papers
Dec. 5 Shape-from-X Papers
Dec. 12 3D modeling and registration Papers
Dec. 19 Appearance modeling and image-based rendering Final project presentations
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Single View Metrology
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Measuring in a plane

• Need to compute H as well as uncertainty

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Direct Linear Transformation(DLT)
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Direct Linear Transformation(DLT)

• Equations are linear in h

• Only 2 out of 3 are linearly independent
• (indeed, 2 eq/pt)

(only drop third row if wi?0)

• Holds for any homogeneous representation, e.g.
  (xi,yi,1)

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Direct Linear Transformation(DLT)

• Solving for H

size A is 8x9 or 12x9, but rank 8
Trivial solution is h09T is not interesting
1-D null-space yields solution of interest pick
for example the one with
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Direct Linear Transformation(DLT)

• Over-determined solution

No exact solution because of inexact
measurement i.e. noise

• Find approximate solution
• Additional constraint needed to avoid 0, e.g.
• not possible, so minimize

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DLT algorithm

• Objective
• Given n4 2D to 2D point correspondences
  xi?xi, determine the 2D homography matrix H
  such that xiHxi
• Algorithm
• For each correspondence xi ?xi compute Ai.
  Usually only two first rows needed.
• Assemble n 2x9 matrices Ai into a single 2nx9
  matrix A
• Obtain SVD of A. Solution for h is last column of
  V
• Determine H from h

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Importance of normalization
102
102
102
102
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104
102
1
1
orders of magnitude difference!
Monte Carlo simulation for identity computation
based on 5 points (not normalized ? normalized)
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Normalized DLT algorithm

• Objective
• Given n4 2D to 2D point correspondences
  xi?xi, determine the 2D homography matrix H
  such that xiHxi
• Algorithm
• Normalize points
• Apply DLT algorithm to
• Denormalize solution

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Geometric distance
d(.,.) Euclidean distance (in image)
e.g. calibration pattern
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Reprojection error
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Statistical cost function and Maximum Likelihood
Estimation

• Optimal cost function related to noise model
• Assume zero-mean isotropic Gaussian noise (assume
  outliers removed)

Error in one image
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Statistical cost function and Maximum Likelihood
Estimation

• Optimal cost function related to noise model
• Assume zero-mean isotropic Gaussian noise (assume
  outliers removed)

Error in both images
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Gold Standard algorithm

• Objective
• Given n4 2D to 2D point correspondences
  xi?xi, determine the Maximum Likelyhood
  Estimation of H
• (this also implies computing optimal xiHxi)
• Algorithm
• Initialization compute an initial estimate using
  normalized DLT or RANSAC
• Geometric minimization of reprojection error
• ? Minimize using Levenberg-Marquardt over 9
  entries of h
• or Gold Standard error
• ? compute initial estimate for optimal xi
• ? minimize cost
  over H,x1,x2,,xn
• ? if many points, use sparse method

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Uncertainty error in one image

1. Estimate the transformation from the data
2. Compute Jacobian , evaluated at
3. The covariance matrix of the estimated is
   given by

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Uncertainty error in both images
separate in homography and point parameters
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Using covariance matrix in point transfer
Error in one image
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Example
s1 pixel S0.5cm
(Criminisi97)
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Example
s1 pixel S0.5cm
(Criminisi97)
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Example
(Criminisi97)
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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

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Single view measurements3D scene
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Background Projective geometry of 1D
3DOF (2x2-1)
The cross ratio
Invariant under projective transformations
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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

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Basic geometry
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Basic geometry

• Allows to relate height of point to height of
  camera

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Homology mapping between parallel planes

• Allows to transfer point from one plane to another

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Single view measurements
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Single view measurements
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Forensic applications
190.64.1 cm

• 190.62.9 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.
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Example
courtesy of Antonio Criminisi
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La Flagellazione di Cristo (1460) Galleria
Nazionale delle Marche by Piero della Francesca
(1416-1492)
http//www.robots.ox.ac.uk/vgg/projects/SingleVie
w/
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More interesting stuff

• Criminisi demo http//www.robots.ox.ac.uk/vgg/pre
  sentations/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/

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Next class

• Feature tracking and matching