Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
A recap of the computer vision course taught at the university of central florida in fall 2005. The topics covered include estimation of camera parameters using corresponding world and image points, and rotation around an arbitrary axis. The document also includes mathematical equations and diagrams to illustrate the concepts.
Typology: Study notes
1 / 15
This page cannot be seen from the preview
Don't miss anything!
Univ. of Central Florida www.cs.ucf.edu/courses/cap5415/fall Office: CSB 250
Recap
Estimation of Camera Parameters
i i i z
i i i x i x x r X r Y r Z t
r X r Y r Z t x o f
3 , 1 3 , 2 3 , 3
1 , 1 1 , 2 1 , 3
i i i z
i i i y i y y r X r Y r Z t
r X r Y r Z t y o f
3 , 1 3 , 2 3 , 3
2 , 1 2 , 2 2 , 3
(A)
(B)
Recap
Estimation of Camera Parameters
z Given corresponding world and image points
z Divide (A) to (B), rearrange result
x (^) i Xiv 1 + xiYiv 2 + xiZiv 3 + xiv 4 − yiXiv 5 − yiYiv 6 − yiZiv 7 − yiv 8 = 0 (C) v 1 = r 21 v 2 = r 22 v 3 = r 23 v 4 = ty v 5 = α r 11 v 6 = α r 12 v 7 = α r 13 v 8 = α t x
z Rearrange into matrix and solve using SVD
z Estimate scale factor Î r (^) 2i and ty are there!!
z Compute α similar to scale factor
z Compute r (^) 3i from r (^) 1i and r (^) 2i.
z Estimate fx , fy and tz.
Recap
Rotation around arbitrary axis
V n
V −( Vn ) n
( Vn ) n
V ′
V ⊥′
n × V = n × ( V −( V. n ) n )
n × ( n × V )=( V. n ) n − V
V ′ = V ⊥′+( V ⋅ n ) n
V n n V
× + × × +
sin ( )
cos ( )
θ
θ
Row 1
Row q
(^1) 1 1
0 0 0 00
1
10 5 9
100
I ( x , y ) I ( x , y ) n ( x , y )
Î
2
2
2
σ
n
−
( )
( )
⎩ (^ )
⎨
⎧
s rs s p l
Ixy p l Ixy
min max min
, ˆ ,
x x f x f x
f x f x x
dx
df = ′ = ∆
∆ → ( )
( ) ( ) lim 0
speed accelerati on
dt
dv a
dt
ds v = =
3
2 4
2 x 4 x
dx
dy
y x x
= +
= +
x
x
x e
dx
dy
y x e
−
−
= + −
= +
cos ( 1 )
sin
( )
( ) ( ) lim 0 f x x
f x f x x
dx
df
x = ′ ∆
− −∆ = (^) ∆→
( ) 1
( ) ( 1 ) f x
f x f x
dx
df = ′
f ( x ) f ( x 1 ) f ( x ) dx
df = − − = ′
Discrete Derivative
Finite Difference
Forward difference
Central difference
Derivative masks f x
⎥
f y
I x
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
I y
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
10 10 20 20 20
10 10 20 20 20
10 10 20 20 20
10 10 20 20 20
10 10 20 20 20
I
x (^) x
f
h
( 1 , 1 ) ( 1 , 1 ) (, 1 )( 0 , 1 ) ( 1 , 1 )( 1 , 1 )
( 1 , ) ( 1 , 0 ) (, )( 0 , 0 ) ( 1 , ) ( 1 , 0 )
( 1 , 1 ) ( 1 , 1 ) ( , 1 ) ( 0 , 1 ) ( 1 , 1 ) ( 1 , 1 )
f x y h f xy h f x y h
f x yh f xyh f x yh
f h f x y h f xy h f x y h
− + − + + + + +
− − + + +
∗ = − − − − + − − + + − − +
∑ ∑ =− =−
∗ = − −
1
1
1
1
( , ) (, ) i j
f h f x iy ihi j
n
n
n
i
i n
=
n
wI
n
wI wI wI I
n
i
i i n n
=
