Multiple View Geometry: Solution Sheet 7
Prof. Dr. Florian Bernard, Florian Hofherr, Tarun Yenamandra
Computer Vision Group, TU Munich
Link Zoom Room , Password: 307238
Exercise: June 9th, 2021
Part I: Theory
1. (a) l is in the coimage of L, and therefore l is a normal vector to the plane that is determined
by the camera position and L.
⇒
lTx1 = 0
lTx2 = 0.
⇒ l ∼ x1 × x2 = x̂1×2.
l1 and l2 are normal vectors to the planes through camera position and L1, L2 respectively.
⇒
lT1 x = 0
lT2 x = 0
⇒ x ∼ l1 × l2 = l̂1l2.
(b) i. l1 ∼ x̂u :
x is in the preimage of L1. ⇒ l>1 x = 0.
∃ point u 6= p in L1. ⇒ l>1 u = 0
⇒ l1 ∼ x̂u.
ii. l2 ∼ x̂v : analog to i.
iii. x1 ∼ l̂r :
x1 is in the preimage of L. ⇒ x>1 l = 0
∃ a line L′ through p1 with coimage r 6= l. ⇒ x>1 r = 0.
⇒ x1 ∼ l̂r.
iv. x2 ∼ l̂s : analog to iii.
2. rank
(
x̂1Π1
x̂2Π2
)
5 3
⇒ ∃X ∈ R4\{0} with
(
x̂1Π1
x̂2Π2
)
X = 0.
⇒ x̂1Π1X = 0 ∧ x̂2Π2X = 0,
⇒ x1 ×Π1X = 0 ∧ x2 ×Π2X = 0.
⇒ x1 and Π1X are linearly dependent; and x2 and Π2X are linearly dependent.
⇒ ∃λ1, λ2 ∈ R with Π1X = λ1×1 ∧ Π2X = λ2×2
⇒ x1 and x2 are projections of X.
1
https://tum-conf.zoom.us/s/62772800235?pwd=SUpZN2QrV0JpeXJyR2R1TWx5cHEwdz09
3. ∃λ ∈ R : [R′, T ′] = λ [R, T ]H = λ [R, T ]
[
I 0
v> v4
]
= λ [R+ Tv>, T v4]
E′ = T̂ ′R′
= (λ̂v4T ) · (λ(R+ Tv>))
= λ2v4T̂ (R+ Tv
>)
= λ2v4T̂R+ λ
2v4 T̂ T︸︷︷︸
=0
v>
= λ2v4T̂R
= λ2v4E with λ
2v4 ∈ R
⇒ E′ ∼ E
2