Upload
others
View
50
Download
0
Embed Size (px)
Citation preview
3Dcamerapose
⎡ ⎤⎢ ⎥⎣ ⎦
x =uv1
11
P1 ∈ !3×4 P2
PF2Dcorrespondences
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
X=XYZ
Optical Flow: 2D point correspondences
Optical Flow: 2D point correspondences
Optical Flow: 2D point correspondences
I(x) J(x)
t = 0 t = 1
d
I(x) J(x)
I(x) = J(x+ d)
Whend=0
=-
mindE(d) = ||J(x+ d)� I(x)||2
E(d=0)
Three steps for solving this problem
1: Solve for
�E
�d
����d⇤
= 0
3: Solve for d, warp image, iterate
2: Taylor expansion on J(x+ d)
Step1:Solvefor
E(d) = ||J(x+ d)� I(x)||2
�E
�d
����d⇤
= 0
E(d⇤ = 0)
Step1:Solvefor
E(d) = ||J(x+ d)� I(x)||2
�E
�d
����d⇤
= 0
E(d⇤ = 0)
�E
�d
����d⇤
= 2�J(x+ d)
�d
T
(J(x+ d)� I(x)) = 0
Step1:Solvefor
E(d) = ||J(x+ d)� I(x)||2
�E
�d
����d⇤
= 0
E(d⇤ = 0)
�E
�d
����d⇤
= 2�J(x+ d)
�d
T
(J(x+ d)� I(x)) = 0
�E
�d
����d⇤
= 2�J(x)
�x
T
(J(x+ d)� I(x)) = 0
E(d⇤ = 0)
�E
�d
����d⇤
= 2�J(x)
�x
T
(J(x+ d)� I(x)) = 0
�J(x)
�x
=
�J(x)
�y=
= +
Step2:TaylorexpansionJ(x+ d)
J(x+ d) = J(x) +�J(x)
�xd
dy⇤dx
⇤ +
Putting all together
J(x+ d) = J(x) +�J(x)
�xd
�E
�d
����d⇤
= 2�J(x)
�x
T
(J(x+ d)� I(x)) = 0
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
2Dunknownsflowvectorperpixel2equations
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
Alsoknownassecondmomentmatrix
�J(x)
�x
T �J(x)
�x
�J(x)
�x
2
�J(x)
�x
�J(x)
�y
�J(x)
�x
�J(x)
�y
�J(x)
�y
2
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
dx
dy
=
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
dx
dy =+
min
d
E(d) =X
x
||J(x+ d)� I(x)||2
X
x
�J(x)
�x
T �J(x)
�xd =
X
x
�J(x)
�x
T
(I(x)� J(x))
min
d
E(d) =X
x
||J(x+ d)� I(x)||2
2×2matrix
Summingoverpixels
=
2×1matrix
Summingoverpixels
=
dx
dy =+
X
x
�J(x)
�x
T �J(x)
�xd =
X
x
�J(x)
�x
T
(I(x)� J(x))
3: Solve for d, warp image, iterate
I(x)J(x) Error
d = (�7,�9)
3: Solve for d, warp image, iterate
I(x)J(x) Error
d = (�7,�9)
J
t=1(x) = J(x+ d)
Error
d=(-7.1,-8.8)
d=(-6.8,-8.9)
d=(-1.4,-3.0)
I(x) J(x)
t = 0 t = 1
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
�J(x)
�x
=
�J(x)
�y=
�J(x)
�x
T �J(x)
�xd =
�J(x)
�x
T
(I(x)� J(x))
dx
dy
=
3: Solve for d, warp image, iterate
I(x)J(x) Error
J
t=1(x) = J(x+ d)
d=(-4.9,-0.4)
Error
d=(-4.9,-0.4)
d=(-0.1,-5.8)
d=(0, -3.7)