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Parameter estimation. Parameter estimation. 2D homography Given a set of (x i ,x i ’), compute H (x i ’=Hx i ) 3D to 2D camera projection Given a set of (X i ,x i ), compute P (x i =PX i ) Fundamental matrix Given a set of (x i ,x i ’), compute F (x i ’ T Fx i =0) Trifocal tensor

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parameter estimation1
Parameter estimation
  • 2D homography

Given a set of (xi,xi’), compute H (xi’=Hxi)

  • 3D to 2D camera projection

Given a set of (Xi,xi), compute P (xi=PXi)

  • Fundamental matrix

Given a set of (xi,xi’), compute F (xi’TFxi=0)

  • Trifocal tensor

Given a set of (xi,xi’,xi”), compute T

number of measurements required
Number of measurements required
  • At least as many independent equations as degrees of freedom required
  • Example:

2 independent equations / point

8 degrees of freedom

4x2≥8

approximate solutions
Approximate solutions
  • Minimal solution

4 points yield an exact solution for H

  • More points
    • No exact solution, because measurements are inexact (“noise”)
    • Search for “best” according to some cost function
    • Algebraic or geometric/statistical cost
gold standard algorithm
Gold Standard algorithm
  • Cost function that is optimal for some assumptions
  • Computational algorithm that minimizes it is called “Gold Standard” algorithm
  • Other algorithms can then be compared to it
direct linear transformation dlt1
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)
direct linear transformation dlt2
Direct Linear Transformation(DLT)
  • Solving for H

size A is 8x9 or 12x9, but rank 8

Trivial solution is h=09T is not interesting

pick for example the one with

direct linear transformation dlt3
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
singular value decomposition
Singular Value Decomposition

Homogeneous least-squares

dlt algorithm
DLT algorithm
  • Objective
  • Given n≥4 2D to 2D point correspondences {xi↔xi’}, determine the 2D homography matrix H such that xi’=Hxi
  • 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
solutions from lines etc

3D Homographies (15 dof)

Minimum of 5 points or 5 planes

2D affinities (6 dof)

Minimum of 3 points or lines

Solutions from lines, etc.

2D homographies from 2D lines

Minimum of 4 lines

Conic provides 5 constraints

Mixed configurations?

combination of points and lines ….

cost functions
Cost functions
  • Algebraic distance
  • Geometric distance
  • Reprojection error
algebraic distance

algebraic distance

where

Algebraic distance

DLT minimizes

residual vector

partial vector for each (xi↔xi’)

algebraic error vector

Not geometrically/statistically meaningfull, but given good normalization it works fine and is very fast (use for initialization)

geometric distance

measured coordinates

estimated coordinates

true coordinates

Error in one image: assumes points in the first image are measured perfectly

Symmetric transfer error

Reprojection error

Geometric distance

d(.,.) Euclidean distance (in image)

e.g. calibration pattern

Note: we are minimizing over H AND corrected correspondence pair

statistical cost function and maximum likelihood estimation

Maximum Likelihood Estimate: maximizes log likelihood or minimizes

Statistical cost function and Maximum Likelihood Estimation
  • Optimal cost function related to noise model; assume in the absence of noise , perfect match
  • Assume zero-mean isotropic Gaussian noise (assume outliers removed); x = measurement;

Error in one image: assume errors at each point independent

statistical cost function and maximum likelihood estimation1

Maximum Likelihood Estimate minimizes:

Over both H and corrected correspondence 

identical to minimizing reprojection error

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