Parameter estimation

Parameter estimation

• 2D homographyGiven a set of (xi,xi’), compute H (xi’=Hxi)

• 3D to 2D camera projectionGiven a set of (Xi,xi), compute P (xi=PXi)

• Fundamental matrixGiven 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

• At least as many independent equations as degrees of freedom required

• Example: Hxx'

11

λ

333231

232221

131211

y

x

hhh

hhh

hhh

y

x

2 independent equations / point8 degrees of freedom

4x2≥8

Approximate solutions

• Minimal solution4 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

• 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(DLT)

ii Hxx 0Hxx ii

i

i

i

i

xh

xh

xh

Hx3

2

1

T

T

T

iiii

iiii

iiii

ii

yx

xw

wy

xhxh

xhxh

xhxh

Hxx12

31

23

TT

TT

TT

0

h

h

h

0xx

x0x

xx0

3

2

1

TTT

TTT

TTT

iiii

iiii

iiii

xy

xw

yw

Tiiii wyx ,,x

0hA i

Direct Linear Transformation(DLT)

• Equations are linear in h

0

h

h

h

0xx

x0x

xx0

3

2

1

TTT

TTT

TTT

iiii

iiii

iiii

xy

xw

yw

0AAA 321 iiiiii wyx

0hA i

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

0

h

h

h

x0x

xx0

3

2

1

TTT

TTT

iiii

iiii

xw

yw

(only drop third row if wi’≠0)• Holds for any homogeneous

representation, e.g. (xi’,yi’,1)

Direct Linear Transformation(DLT)

• Solving for H

0Ah 0h

A

A

A

A

4

3

2

1

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

Trivial solution is h=09T is not interesting

pick for example the one with 1h

Direct Linear Transformation(DLT)

• Over-determined solution

No exact solution because of inexact measurementi.e. “noise”

0Ah 0h

A

A

A

n

2

1

Find approximate solution- Additional constraint needed to avoid 0, e.g.

- not possible, so minimize

1h Ah0Ah

Singular Value Decomposition

XXVT XVΣ T XVUΣ T

Homogeneous least-squares

TVUΣA

1X AXmin subject to nVX solution

DLT algorithmObjective

Given n≥4 2D to 2D point correspondences {xi↔xi’}, determine the 2D homography matrix H such that xi’=Hxi

Algorithm

(i) For each correspondence xi ↔xi’ compute Ai. Usually only two first rows needed.

(ii) Assemble n 2x9 matrices Ai into a single 2nx9 matrix A

(iii) Obtain SVD of A. Solution for h is last column of V

(iv) Determine H from h

Solutions from lines, etc.

ii lHl T 0Ah

2D homographies from 2D lines

Minimum of 4 lines

Minimum of 5 points or 5 planes

3D Homographies (15 dof)

2D affinities (6 dof)

Minimum of 3 points or lines

Conic provides 5 constraints

Mixed configurations? combination of points and lines ….

Cost functions

• Algebraic distance• Geometric distance• Reprojection error

Algebraic distanceAhDLT minimizes

Ahe residual vector

ie partial vector for each (xi↔xi’)algebraic error vector

2

22alg h

x0x

xx0Hx,x

TTT

TTT

iiii

iiiiiii

xw

ywed

algebraic distance

22

21

221alg x,x aad where 21

T321 xx,,a aaa

2222alg AhHx,x eed

ii

iii

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

Geometric distancemeasured coordinatesestimated coordinatestrue coordinates

xxx

2

HxH,xargminH ii

i

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

e.g. calibration pattern

221-

HHx,xxH,xargminH iiii

i

dd Symmetric transfer error

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

ii

iiiii

ii ddii

xHx subject to

x,xx,xargminx,x,H 22

x,xH,

Reprojection error

Note: we are minimizing over H AND corrected correspondence pair

Reprojection error

221- Hx,xxHx, dd

22 x,xxx, dd

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; 22 2/xx,

2πσ2

1xPr de

22i 2xH,x /

2iπσ2

1H|xPr ide

i

constantxH,xH|xPrlog 2i2i

σ2

1id

Error in one image: assume errors at each point independent

Maximum Likelihood Estimate: maximizes log likelihood or minimizes

2i xH,x id

Statistical cost function and Maximum Likelihood

Estimation• Optimal cost function related to noise

model• Assume zero-mean isotropic Gaussian

noise (assume outliers removed) 22 2/xx,2πσ2

1xPr de

22i

2i 2xH,xx,x /

2iπσ2

1H|xPr

ii dd

ei

Error in both images

Maximum Likelihood Estimate minimizes:

Over both H and corrected correspondence identical to minimizing reprojection error

2i

2i x,xx,x ii dd