Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters

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Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters

Adrien Bartoli, Peter Sturm

To cite this version:

Adrien Bartoli, Peter Sturm. Three New Algorithms for Projective Bundle Adjustment with Minimum

Parameters. [Research Report] RR-4236, INRIA. 2001. �inria-00072351�

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ISRN INRIA/RR--4236--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 3

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters

Adrien Bartoli — Peter Sturm

N° 4236

Août 2001

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Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters

HAL Id: inria-00072351

https://hal.inria.fr/inria-00072351

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Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters

Adrien Bartoli, Peter Sturm

To cite this version:

Adrien Bartoli, Peter Sturm. Three New Algorithms for Projective Bundle Adjustment with Minimum

Parameters. [Research Report] RR-4236, INRIA. 2001. �inria-00072351�

ISRN INRIA/RR--4236--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 3

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters

Adrien Bartoli — Peter Sturm

N° 4236

Août 2001

×

P i

i ∈ 1 . . . n

n

Q j

j ∈ 1 . . . m

m

q i,j

q i,j

j

i

w i,j

w i,j =

1

Q j

P i

0

π P

π Q

P P

P Q

P 1 , . . . , P n −→ P P π P

Q 1 , . . . , Q m −→ P Q π Q .

P P − 1

P Q − 1

π P

π Q

argmin

π P , π Q C ( P P − 1

( π P ), P Q − 1 ( π Q )),

C

C (P 1 , . . . , P n , Q 1 , . . . , Q x m) = X n i =1

X m j =1

w i,j d 2 ( q i,j , P i Q j ),

d(., .)

C

√

2

(P, Q )

C (P, Q )

H

P

Q

PH − 1

HQ

PH − 1 HQ ∼ PQ,

C (PH − 1 , HQ) = C (P, Q).

H

#

= 12n + 4m − n − m − 15 = 11n + 3m − 15.

dim π P + dim π Q = #

P

Q

P

P i ∼ (I a i )

a i

×

×

P

P 0

P ⁱ

Q ^j

q ^i,j

q ^i,j

w ^i,j

w ^i,j =

Q ^j

P ⁱ

π _P

π _Q

P ^Q

P ¹ , . . . , P ⁿ −→ ^P ^P π _P

Q ¹ , . . . , Q ^m −→ ^P ^Q π _Q .

P ^Q ⁻ ¹

π _P

π _Q

π _P , π _Q C ( P P − 1

( π _P ), P ^Q ⁻ ¹ ( π _Q )),

C (P ¹ , . . . , P ⁿ , Q ¹ , . . . , Q ^x m) = X n i =1

w ^i,j d ² ( q ^i,j , P ⁱ Q ^j ),

PH ⁻ ¹

PH ⁻ ¹ HQ ∼ PQ,

C (PH ⁻ ¹ , HQ) = C (P, Q).

dim π _P + dim π _Q = #

P ⁱ ∼ (I a ⁱ )

a ⁱ

P ⁰

P ⁰ ∼ (H e ⁰ ),

e ⁰

e ⁰

H ^?

H ^? ∼ [ e ⁰ ] × F,

H ^?T e ⁰ = 0.

H ^?

F ∼ [e ⁰ ] × H

H ∼ H ^?

H ^?

e ⁰

P ⁰ ∼ (H ^? e ⁰ )

|| H ^? || ² = || e ⁰ || ² = 1.

H ^?

e ⁰

Q ∼ (Q 1 Q 2 Q 3 Q 4 ) ^T

P ⁿ

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v ∼ e v _v← 1

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