The Geometry Behind the Numerical
Reconstruction of Two Photos
Hellmuth Stachel
[email protected] — http://www.geometrie.tuwien.ac.at/stachel
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007
Table of contents
1. Remarks on linear images
2. Geometry of two images
3. Numerical reconstruction of two images
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 1
1. Remarks on linear images
linear image nonlinear (curved) image
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 2
Central projection
The central projection (according to A. Durer)
can be generalized by a central axonometry.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 3
Central axonometric principle
in space E3:
O
E1
E2
E3
U1
U2
U3
cartesian basis O; E1, E2, E3
and points at infinity U1, U2, U3
U c1
U c2
U c3
Ec1
Ec2
Ec3
Oc
in the image plane E2:
central axonometric reference systemOc; Ec
1, Ec2, E
c3;U
c1 , U c
2 , U c3
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 4
Definition of linear images
There is a unique collinear transformation
κ : E3 → E
2 mit O 7→ Oc, Ei 7→ Eci , Ui 7→ U c
i , i = 1, 2, 3.
Any two-dimensional image of E3 under a collinear transformation is called linear.
=⇒
{collinear points have collinear or coincident imagescross-ratios of any four collinear points are preserved.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 5
Definition of linear images
There is a unique collinear transformation
κ : E3 → E
2 mit O 7→ Oc, Ei 7→ Eci , Ui 7→ U c
i , i = 1, 2, 3.
Any two-dimensional image of E3 under a collinear transformation is called linear.
=⇒
{collinear points have collinear or coincident imagescross-ratios of any four collinear points are preserved.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 5
Central projection in coordinates
Notation:
Z . . . center
H . . . principal point
d . . . focal length
x1, x2, x3 . . .camera frame
x′1, x
′2 . . . imagecoordinate frame
image plane
vanishing planeΠΠ
v
x1
x2
x3
X
Z H
d
Xc
x′1
x′2
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 6
Central projection in coordinates
(x′
1
x′2
)
=d
x3
(x1
x2
)
, or homogeneous
ξ′
0
ξ′
1
ξ′
2
=
0 0 0 10 d 0 00 0 d 0
ξ0...
ξ3
.
Transformation from the camera frame (x1, x2, x3) into arbitrary world coordinates(x1, x2, x3) and translation from the particular image frame (x′
1, x′2) into arbitrary
(x′1, x
′2) gives in homogeneous form
ξ′0ξ′1ξ′2
=
1 0 0h′
1 d f1 0h′
2 0 d f2
0 0 0 10 1 0 00 0 1 0
1 0 0 0o1...
o3
R
︸ ︷︷ ︸
matrix A
ξ0...ξ3
.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 7
Central projection in coordinates
(x′
1
x′2
)
=d
x3
(x1
x2
)
, or homogeneous
ξ′
0
ξ′
1
ξ′
2
=
0 0 0 10 d 0 00 0 d 0
ξ0...
ξ3
.
Transformation from the camera frame (x1, x2, x3) into arbitrary world coordinates(x1, x2, x3) and translation from the particular image frame (x′
1, x′2) into arbitrary
(x′1, x
′2) gives in homogeneous form
ξ′0ξ′1ξ′2
=
1 0 0h′
1 d f1 0h′
2 0 d f2
0 0 0 10 1 0 00 0 1 0
1 0 0 0o1...
o3
R
︸ ︷︷ ︸
matrix A
ξ0...ξ3
.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 7
Central projection in coordinates
Left hand matrix: (h′1, h
′2) are image coordinates of the principal point H,
(f1, f2) are possible scaling factors, and d is the focal length.
These parameters are called the intrinsic calibration parameters.
Right hand matrix: R is an orthogonal matrix.
The position of the camera frame with respect to the world coordinates definesthe extrinsic calibration parameters.
Photos with known interior orientation are called calibrated images, others (likecentral axonometries) are uncalibrated.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 8
Central projection in coordinates
Left hand matrix: (h′1, h
′2) are image coordinates of the principal point H,
(f1, f2) are possible scaling factors, and d is the focal length.
These parameters are called the intrinsic calibration parameters.
Right hand matrix: R is an orthogonal matrix.
The position of the camera frame with respect to the world coordinates definesthe extrinsic calibration parameters.
Photos with known interior orientation are called calibrated images, others (likecentral axonometries) are uncalibrated.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 8
Positive and negative central pespective
DGDGDGimage plane
vanishing plane
negative plane
ΠΠ Πv
x1
x2
x3
X
Z
H
H
dd
Xc
Xc
x′1
x′2
x′1
x′2
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 9
Photo versus linear image
Photo (= central perspective) or photo of a photo (= linear image) ?
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 10
unknown interior calibration parameters
ZZZZZZZZZZZZZZZZZ
collinear
bundle tran
sformation
ZZZZZZZZZZZZZZZZZ
the bundles Z and Zof the rays of sight arecollinear
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 11
2. Geometry of two images
Given: Two linear images or two photographs.
Wanted: Dimensions of the depicted 3D-object.
Historical ‘Stadtbahn’ station Karlsplatz in Vienna (Otto Wagner, 1897)
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 12
2. Geometry of two images
The geometry of two images is a classical subject of Descriptive Geometry.Its results have become standard (Finsterwalder, Kruppa, Krames,Wunderlich, Hohenberg, Tschupik, Brauner, Havlicek, H.S., . . . ).
Why now ? Advantages of digital images:
• less distorsion, because no paper prints are needed,
• exact boundary is available, and
• precise coordinate measurements are possible using standard software.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 13
2. Geometry of two images
The geometry of two images is a classical subject of Descriptive Geometry.Its results have become standard (Finsterwalder, Kruppa, Krames,Wunderlich, Hohenberg, Tschupik, Brauner, Havlicek, H.S., . . . ).
Why now ? Advantages of digital images:
• less distorsion, because no paper prints are needed,
• exact boundary is available, and
• precise coordinate measurements are possible using standard software.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 13
Geometry of two images (epipolar geometry)
viewing situation
collinear transformations
two images
π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1
π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2
Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2 Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1
Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21
Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12
zzzzzzzzzzzzzzzzz
X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1
X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2
XXXXXXXXXXXXXXXXX
δXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδX
l1l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2
π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1
π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2
γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1
γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2
X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′
X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′
l′l′l′l′l′l′l′l′l′l′l′l′l′l′l′l′l′
l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′Z′
2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2
Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 14
Geometry of two images (epipolar geometry)
Notations:
line z = Z1Z2 . . . baseline,
Z ′2, Z
′′1 . . . epipoles
(German: Kernpunkte),
δX . . . epipolar plane (it is twiceprojecting),
l′, l′′ . . . pair of epipolar lines(German: Kernstrahlen),
(X ′, X ′′) . . . corresponding views.
π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1
π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2
Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2 Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1
Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21
Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12
zzzzzzzzzzzzzzzzz
X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1
X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2
XXXXXXXXXXXXXXXXX
δXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδX
l1l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2
π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1
π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2
γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1γ1
γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2γ2
X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′X ′
X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′X ′′
l′l′l′l′l′l′l′l′l′l′l′l′l′l′l′l′l′
l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′l′′Z′
2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2Z′2
Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1Z′′1
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 15
Epipolar constraint
Theorem (synthetic version): For any two linear images of a scene, there is aprojectivity between two line pencils
Z ′2(δ
′X) ∧− Z ′′
1 (δ′′X)
such that the points X ′,X ′′ are corresponding ⇐⇒ they are located on(corresponding=) epipolar lines.
Theorem (analytic version): Using homogeneous coordinates for both images,there is a bilinear form β of rank 2 such that two points X ′ = x
′R = (ξ′0 : ξ′1 : ξ′2)
and X ′′ = x′′R = (ξ′′0 : ξ′′1 : ξ′′2 ) are corresponding
⇐⇒ β(x′,x′′) =2∑
i,j=0
bij ξ′i ξ′′j = (ξ′0 ξ′1 ξ′2)·(bij
)
0
@
ξ′′0
ξ′′1
ξ′′2
1
A = x′T · B · x′′ = 0 .
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 16
Epipolar constraint
Theorem (synthetic version): For any two linear images of a scene, there is aprojectivity between two line pencils
Z ′2(δ
′X) ∧− Z ′′
1 (δ′′X)
such that the points X ′,X ′′ are corresponding ⇐⇒ they are located on(corresponding=) epipolar lines.
Theorem (analytic version): Using homogeneous coordinates for both images,there is a bilinear form β of rank 2 such that two points X ′ = x
′R = (ξ′0 : ξ′1 : ξ′2)
and X ′′ = x′′R = (ξ′′0 : ξ′′1 : ξ′′2 ) are corresponding
⇐⇒ β(x′,x′′) =2∑
i,j=0
bij ξ′i ξ′′j = (ξ′0 ξ′1 ξ′2)·(bij
)
0
@
ξ′′0
ξ′′1
ξ′′2
1
A = x′T · B · x′′ = 0 .
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 16
Epipolar constraint in the calibrated case
Theorem: In the calibrated casethe essential matrix B = (bij) is theproduct of a skew symmetric matrixand an orthogonal one, i.e.,
B = S ·R .
π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1
π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2
Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2 Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21
Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12
X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2
XXXXXXXXXXXXXXXXXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδX
l1l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2x
′x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
Proof: We use both camera frames and the homogeneous coordinates
x′ =
−−−→Z1X
′, x′′ =
−−−→Z2X
′′.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 17
Epipolar constraint in the calibrated case
For transforming the coordinates from the second camera frame into the first one,there is an orthogonal matrix R such that
x′′1 = z
′ + R·x′′ with RT = R−1 and z′ = (z′1, z′2, z′3)
T =−−−→Z1Z2.
The points X1, X2, Z1,Z2 are coplanar ⇐⇒ the tripleproduct of the vectors x
′, z′ and
x′′1 = Z1X2 vanishes, i.e.,
det(x′, z′,x′′1) = x
′ · (z′×x′′1) = 0.
π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1
π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2
Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2 Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21
Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12
X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2
XXXXXXXXXXXXXXXXXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδX
l1l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2x
′x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 18
Epipolar constraint in the calibrated case
For transforming the coordinates from the second camera frame into the first one,there is an orthogonal matrix R such that
x′′1 = z
′ + R·x′′ with RT = R−1 and z′ = (z′1, z′2, z′3)
T =−−−→Z1Z2.
The points X1, X2, Z1, Z2
are coplanar ⇐⇒ the tripleproduct of the vectors x
′, z′ and
x′′1 = Z1X2 vanishes, i.e.,
det(x′, z′,x′′1) = x
′ · (z′×x′′1) = 0.
π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1
π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2
Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2 Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
z′
Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21
Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12
X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2
XXXXXXXXXXXXXXXXXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδX
l1l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2x
′x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
x′′
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 18
Epipolar constraint in the calibrated case
We replace the vector product (z′×x′′1) by
z′×(z′ + R·x′′) = z
′×R·x′′ = S ·R·x′′ mit S =
0
@
0 −z′3 z′
2
z′3 0 −z′
1
−z′2 z′
1 0
1
A.
Matrix S is skew symmetric and R is orthogonal.
Hence, the coplanarity of x′, x
′′ and z′ is equivalent to
0 = x′ · (z′×x
′′1) = x
′T · S ·R︸︷︷︸B
·x′′, also B = S ·R .
The decomposition of the fundamental matrix B into these two factors definesthe relative position of the second camera frame against the first one !
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 19
Epipolar constraint in the calibrated case
We replace the vector product (z′×x′′1) by
z′×(z′ + R·x′′) = z
′×R·x′′ = S ·R·x′′ mit S =
0
@
0 −z′3 z′
2
z′3 0 −z′
1
−z′2 z′
1 0
1
A.
Matrix S is skew symmetric and R is orthogonal.
Hence, the coplanarity of x′, x
′′ and z′ is equivalent to
0 = x′ · (z′×x
′′1) = x
′T · S ·R︸︷︷︸B
·x′′, also B = S ·R .
The decomposition of the fundamental matrix B into these two factors definesthe relative position of the second camera frame against the first one !
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 19
Singular value decomposition (SVD)
LinAlg
LinAlg
a0a1
a2 xA
α(a0)
α(a1)
α(a2)
α(x)
A′
U ·D·V T
A−→
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 20
Singular value decomposition (SVD)
LinAlg
LinAlg
a0a1
a2 xA
α(a0)
α(a1)
α(a2)
α(x)
A′
U ·D·V T
A−→
rotation ↓ V T rotation ↑ U
LinAlgLinAlg
D−→
scaling
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 20
Singular value decomposition (SVD)
Theorem: [Singular value decomposition]
Any matrix A ∈ M(m, n; R) can be decomposed into a product
A = U ·D ·V T with orthogonal U, V and D = diag(σ1, . . . , σp)
with D ∈ M(m,n; R), σi ≥ 0, and p = min{m, n}.
The positive entries in the main diagonal of D are called singular values of A.
The singular values of A can be seen as principal distortion factors of the affinetransformation represented by A, i.e., the semiaxes of the affine image of the unitsphere.
E.g., the singular values of an orthogonal projection are (0, 1, 1) as the unit sphereis mapped onto a unit disk.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 21
Singular value decomposition (SVD)
Theorem: [Singular value decomposition]
Any matrix A ∈ M(m, n; R) can be decomposed into a product
A = U ·D ·V T with orthogonal U, V and D = diag(σ1, . . . , σp)
with D ∈ M(m,n; R), σi ≥ 0, and p = min{m, n}.
The positive entries in the main diagonal of D are called singular values of A.
The singular values of A can be seen as principal distortion factors of the affinetransformation represented by A, i.e., the semiaxes of the affine image of the unitsphere.
E.g., the singular values of an orthogonal projection are (0, 1, 1) as the unit sphereis mapped onto a unit disk.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 21
Singular values of the essential matrix
Theorem:The essential matrix B has two equalsingular values σ := σ1 = σ2.
Proof: We have B = S ·R withorthogonal R. The vector
S ·x = z′×x
is orthogonal zu the orthogonal viewx
n, where
‖z′×x‖ = | sinϕ| ‖x‖ ‖z′‖ =
= ‖xn‖ ‖z′‖ = σ ‖xn‖.
z′
x
xn
z′×x
ϕ
Π ⊥ z′
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 22
What means ‘reconstruction’
Given: Two either calibratedor uncalibrated images.
π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1π′1 π′′
2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2π′′2
X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1X ′1 X ′′
1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1X ′′1
X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2X ′2
X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2X ′′2
Wanted: ‘viewing situation’,i.e., determine
• the relative position of thetwo camera frames, and
• the location of any spacepoint X from its images(X ′, X ′′).
π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1π1
π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2π2
Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2Z2 Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1Z1
Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21Z21
Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12Z12
zzzzzzzzzzzzzzzzz
X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X1X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2
XXXXXXXXXXXXXXXXXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδXδX
l1l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2l2
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 23
The fundamental theorems
Theorem 1:From two uncalibrated images with given projectivity between epipolar lines thedepicted object can be reconstructed up to a collinear transformation.
Theorem 2 (S. Finsterwalder, 1899):From two calibrated images with given projectivity between epipolar lines thedepicted object can be reconstructed up to a similarity.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 24
Determination of epipoles — geometric meaning
Problem of Projectivity:
Given: 7 pairs of corresponding points (X ′1,X
′′1 ), . . . , (X ′
7, X′′7 ).
Wanted: A pair of points (S′, S′′) (= epipoles) such that there is a projectivity
S′([S′X ′1], . . . , [S
′X ′7]) ∧− S′′([S′X ′′
1 ], . . . , [S′′X ′′7 ]).
X ′1 X ′
2
X ′3X ′
4
X ′5
X ′6
X ′7
X ′′1
X ′′2
X ′′3
X ′′4
X ′′5
X ′′6
X ′′7π′
π′′
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 25
Determination of epipoles — geometric meaning
Problem of Projectivity:
Given: 7 pairs of corresponding points (X ′1,X
′′1 ), . . . , (X ′
7, X′′7 ).
Wanted: A pair of points (S′, S′′) (= epipoles) such that there is a projectivity
S′([S′X ′1], . . . , [S
′X ′7]) ∧− S′′([S′X ′′
1 ], . . . , [S′′X ′′7 ]).
X ′1 X ′
2
X ′3X ′
4
X ′5
X ′6
X ′7
X ′′1
X ′′2
X ′′3
X ′′4
X ′′5
X ′′6
X ′′7
S′
S′′π′
π′′
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 25
Determination of epipoles — analytic solution
Theorem: If 7 pairs of corresponding points (X ′1, X
′′1 ), . . . , (X ′
7, X′′7 ) are given,
the determination of the epipoles is a cubic problem.
Proof: 7 pairs of corresponding points give 7 linear homogeneous equations
β(x′i,x
′′i ) = x
Ti · B · x′′
i = 0, i = 1, . . . , 7,
for the 9 entries in the (3×3)-matrix B = (bij) — called essential matrix.
det(bij) = 0 gives an additional cubic equation which fixes all bij up to a commonfactor.
For noisy image points it is recommended to use more than 7 points and methodsof least square approximation for obtaining the ‘best fitting matrix’ B:
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 26
Determination of epipoles — analytic solution
Theorem: If 7 pairs of corresponding points (X ′1, X
′′1 ), . . . , (X ′
7, X′′7 ) are given,
the determination of the epipoles is a cubic problem.
Proof: 7 pairs of corresponding points give 7 linear homogeneous equations
β(x′i,x
′′i ) = x
Ti · B · x′′
i = 0, i = 1, . . . , 7,
for the 9 entries in the (3×3)-matrix B = (bij) — called essential matrix.
det(bij) = 0 gives an additional cubic equation which fixes all bij up to a commonfactor.
For noisy image points it is recommended to use more than 7 points and methodsof least square approximation for obtaining the ‘best fitting matrix’ B:
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 26
Determination of epipoles — analytic solution
1) Let A denote the coefficient matrix in the linear system for the entries of B.Then the ‘least square fit’ for this overdetermined system is an eigenvector forthe smallest eigenvalue of the symmetric matrix AT · A.
2) As an essential matrix needs to have rank 2, we use the ’projection into theessential space’. This means, the singular value decomposition of B gives arepresentation
B = U · diag(σ1, σ2, σ3) · VT with orthogonal U, V and σ1 ≥ σ2 ≥ σ3 .
Then in the uncalibrated case B = U ·diag(σ1, σ2, 0) ·V is optimal (with respectto the Frobenius norm) and in the calibrated case
B = U · diag(σ, σ, 0) · V T with σ1 = (σ1 + σ2)/2.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 27
Determination of epipoles — analytic solution
1) Let A denote the coefficient matrix in the linear system for the entries of B.Then the ‘least square fit’ for this overdetermined system is an eigenvector forthe smallest eigenvalue of the symmetric matrix AT · A.
2) As an essential matrix needs to have rank 2, we use the ’projection into theessential space’. This means, the singular value decomposition of B gives arepresentation
B = U · diag(σ1, σ2, σ3) · VT with orthogonal U, V and σ1 ≥ σ2 ≥ σ3 .
Then in the uncalibrated case B = U · diag(σ1, σ2, 0) · V is optimal (withrespect to the Frobenius norm) and in the calibrated case
B = U · diag(σ, σ, 0) · V T with σ1 = (σ1 + σ2)/2.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 27
3. Numerical reconstruction of two images
Step 1: Specify at least 7 reference points
11111111111111111
22222222222222222
33333333333333333 44444444444444444
55555555555555555
6666666666666666677777777777777777
88888888888888888
99999999999999999
1010101010101010101010101010101010
1111111111111111111111111111111111
1212121212121212121212121212121212
13131313131313131313131313131313131414141414141414141414141414141414
1515151515151515151515151515151515
1616161616161616161616161616161616
1717171717171717171717171717171717
1818181818181818181818181818181818
1919191919191919191919191919191919
202020202020202020202020202020202011111111111111111
22222222222222222
33333333333333333 44444444444444444
55555555555555555
66666666666666666
7777777777777777788888888888888888
99999999999999999
1010101010101010101010101010101010
1111111111111111111111111111111111
1212121212121212121212121212121212
13131313131313131313131313131313131414141414141414141414141414141414
1515151515151515151515151515151515
1616161616161616161616161616161616
1717171717171717171717171717171717
1818181818181818181818181818181818
1919191919191919191919191919191919
2020202020202020202020202020202020
. . . manually — or automatically by methods of pattern recognition
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 28
Step 2: Compute the essential matrix
Step 2: Compute the essential matrix B — including the pairs of epipolar lines
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 29
Step 3: Factorize B = S.R
Theorem: There are exactly two ways of decomposing B = U ·D ·V T withD = diag(σ, σ, 0) into a product S ·R with skew-symmetric S and orthogonal R :
S = ±U ·R+·D ·UT and R = ±U ·RT+·V
T with R+ =
0
@
0 −1 0
1 0 0
0 0 1
1
A.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 30
Step 4: Intersecting corresponding rays
In one of the frames compute the approximate point of intersection betweencorresponding rays.
X
photo 2
photo 1x′′
x′
z2
z1
s
For the center of the common perpendicular line segment the sum of squareddistances is minimal.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 31
Summary of algorithm
1) Specify n > 7 pairs (X ′i,X
′′i ), i = 1, . . . , n.
2) Set up linear system of equations for the essential matrix B and seek bestfitting matrix (eigenvector of the smallest eigenvalue).
3) Compute the closest rank 2 matrix B with two equal singular values.
4) Factorize B = S · R ; this reveals the relative position of the two cameraframes.
5) In one of the frames compute the approximate point of intersection betweencorresponding rays.
6) Transform the recovered coordinates into world coordinates.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 32
Remaining problems
• Analysis of precision,
• automated calibration (autofocus and zooming change the focal distance d),
• critical configurations.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 33
The solution
11111111111111111
22222222222222222
33333333333333333 44444444444444444
55555555555555555
6666666666666666677777777777777777
88888888888888888
99999999999999999
1010101010101010101010101010101010
1111111111111111111111111111111111
1212121212121212121212121212121212
13131313131313131313131313131313131414141414141414141414141414141414
1515151515151515151515151515151515
1616161616161616161616161616161616
1717171717171717171717171717171717
1818181818181818181818181818181818
1919191919191919191919191919191919
2020202020202020202020202020202020
original image
1
2
3 4
56
78
9
10
11
121314
15
16
17
18
19
20
the reconstruction (M ∼ 1 : 100)
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 34
1
2
3 4
56
78
9
9
10
11
12
12
13
1314
15
16
17
18
18
19
20
Z1
Z1
Z2
Z2
Position of centers
relative to the depicted object
front view
top viewPhoto 1
Photo 2
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 35
Literatur
• H. Brauner: Lineare Abbildungen aus euklidischen Raumen. Beitr. AlgebraGeom. 21, 5–26 (1986).
• O. Faugeras: Three-Dimensional Computer Vision. A Geometric Viewpoint.MIT Press, Cambridge, Mass., 1906 .
• O. Faugeras, Q.-T. Luong: The Geometry of Multiple Images. MITPress, Cambridge, Mass., 2001.
• R. Harley, A. Zisserman: Multiple View Geometry in Computer Vision.Cambridge University Press 2000.
• H. Havlicek: On the Matrices of Central Linear Mappings. Math. Bohem.121, 151–156 (1996).
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 36
• E. Kruppa: Zur achsonometrischen Methode der darstellenden Geometrie.Sitzungsber., Abt. II, osterr. Akad. Wiss., Math.-Naturw. Kl. 119, 487–506(1910).
• Yi Ma, St. Soatto, J. Kosecka, S. Sh. Sastry: An Invitation to 3-DVision. Springer-Verlag, New York 2004.
• H. Stachel: Zur Kennzeichnung der Zentralprojektionen nach H. Havlicek.Sitzungsber., Abt. II, osterr. Akad. Wiss., Math.-Naturw. Kl. 204, 33–46(1995).
• H. Stachel: Descriptive Geometry Meets Computer Vision — The Geometryof Two Images. J. Geometry Graphics 10, 137–153 (2006).
• J. Szabo, H. Stachel, H. Vogel: Ein Satz uber die Zentralaxonometrie.Sitzungsber., Abt. II, osterr. Akad. Wiss., Math.-Naturw. Kl. 203, 3–11 (1994).
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 37
• J. Tschupik, F. Hohenberg: Die geometrische Grundlagen derPhotogrammetrie. In Jordan, Eggert, Kneissl (eds.): Handbuch derVermessungskunde III a/3. 10. Aufl., Metzlersche Verlagsbuchhandlung,Stuttart 1972, 2235–2295.
ICEGD 2007, The 2nd Internat. Conf. on Eng’g Graphics and Design, Galati/Romania, June 7–10, 2007 38
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