recovering metric and affine properties from images
DESCRIPTION
Recovering metric and affine properties from images. Affine preserves: Parallelism Parallel length ratios Similarity preserves: Angles Length ratios. Will show: Projective distortion can be removed once image of line at infinity is specified - PowerPoint PPT PresentationTRANSCRIPT
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Recovering metric and affine properties from images
• Affine preserves:• Parallelism• Parallel length ratios
• Similarity preserves:• Angles • Length ratios
Will show:• Projective distortion can be removed once image of line at infinity is specified• Affine distortion removed once image of circular points is specified• Then the remaining distortion is only similarity
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The line at infinity
l
1
0
0
1t
0ll
A
AH
TT
A
The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity
Note: not fixed pointwise
For an affine transformation line at infinity maps onto line at infinity
A point on line at infinity is mapped to ANOTHER point on the line at infinity, not necessarily the same point
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Affine properties from imagesprojection rectification
APA
lll
HH
321
010
001
0,l 3321 llll T
Euclidean planeTwo step process:1.Find l the image of line at infinity in plane 22. Transform l to its canonical position(0,0,1) T by plugging into HPA and applying it to the entire image to get a “rectified” image3. Make affine measurements on the rectified image
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Affine rectificationv1 v2
l1
l2 l4
l3
l∞
21 vvl
211 llv
432 llv
c
a b
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The circular points
“circular points”
0=++++ 233231
22
21 fxxexxdxxx 0=+ 2
221 xx
l∞
T
T
0,-,1J
0,,1I
i
i
03 x
Two points on l_inf: Every circle intersects l_inf at circular points
Line at infinity
Circle:
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The circular points
0
1
I i
0
1
J i
I
0
1
0
1
100
cossin
sincos
II
iseitss
tssi
y
x
S
H
The circular points I, J are fixed points under the projective transformation H iff H is a similarity
Circular points are fixed under any similarity transformation
Canonical coordinates of circular points
Identifying circular points allows recovery of similarity properties i.e. anglesratios of lengths
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Conic dual to the circular points
TT JIIJ* C
000
010
001*C
TSS HCHC **
The dual conic is fixed conic under the projective transformation H iff H is a similarity
*C
Note: has 4DOF (3x3 homogeneous; symmetric, determinant is zero)
l∞ is the nullvector
*C
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Angles
22
21
22
21
2211cosmmll
mlml
T321 ,,l lll T321 ,,m mmm
Euclidean Geometry: not invariant under
projective transformation
Projective: mmll
mlcos
**
*
CC
CTT
T
0ml * CT (orthogonal)
Once is identified in the projective plane, then Euclidean angles may be measured by equation (1)
*C
The above equation is invariant under projective transformation can be applied after projective transformation of the plane.
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Length ratios
sin
sin
),(
),(
cad
cbd
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Metric properties from images
vvv
v
'
*
*
**
TT
TT
T
TT
T
K
KKK
HHCHH
HHHCHHH
HHHCHHHC
APAP
APSSAP
SAPSAP
TUUC
000
010
001
'* UH
• Upshot: projective (v) and affine (K) components directly determined from the image of C*
∞
• Once C*∞ is identified on the projective plane then projective distortion may be
rectified up to a similarity• Can show that the rectifying transformation is obtained by applying SVD to
• Apply U to the pixels in the projective plane to rectify the image up to a Similarity
'*C
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SVD Decompose to get U
'*C
Euclidean
Perspectivetransformation
Rectifying Transformation: U
Similarity transformation
C*∞
Rectified imageProjectivelyDistortedImage
'*C
Recovering up to a similarity from Projective
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Recovering up to a similarity from Affine
'*C
Euclidean
affinetransformation
Rectifying Transformation: U
Similarity transformation
C*∞
Rectified imageAffinely DistortedImage
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Metric from affine
000
0
3
2
1
321
m
m
m
lllTKK
0,,,, 2221211
212
21122122111 T
kkkkkmlmlmlml
Affine rectified
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Metric from projective
0c,,,,, 332332133122122111 5.05.05.0 mlmlmlmlmlmlmlmlml
0vvv
v
3
2
1
321
m
m
m
lllTT
TT
K
KKK
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Projective 3D geometry
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Singular Value Decomposition
Tnnnmmmnm VΣUA
IUU T
021 n
IVV T
nm
XXVT XVΣ T XVUΣ T
TTTnnn VUVUVUA 222111
000
00
00
00
Σ
2
1
n
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Singular Value Decomposition
• Homogeneous least-squares
• Span and null-space
• Closest rank r approximation
• Pseudo inverse
1X AXmin subject to nVX solution
nrrdiag ,,,,,, 121 0 ,, 0 ,,,,~
21 rdiag TVΣ~
UA~ TVUΣA
0000
0000
000
000
Σ 2
1
4321 UU;UU LL NS 4321 VV;VV RR NS
TUVΣA 0 ,, 0 ,,,, 112
11 rdiag
TVUΣA
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Projective 3D Geometry
• Points, lines, planes and quadrics
• Transformations
• П∞, ω∞ and Ω ∞
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3D points
TT
1 ,,,1,,,X4
3
4
2
4
1 ZYXX
X
X
X
X
X
in R3
04 X
TZYX ,,
in P3
XX' H (4x4-1=15 dof)
projective transformation
3D point
T4321 ,,,X XXXX
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Planes
0ππππ 4321 ZYX
0ππππ 44332211 XXXX
0Xπ T
Dual: points ↔ planes, lines ↔ lines
3D plane
0X~
.n d T321 π,π,πn TZYX ,,X~
14 Xd4π
Euclidean representation
n/d
XX' Hππ' -TH
Transformation
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Planes from points
0π
X
X
X
3
2
1
T
T
T
2341DX T123124134234 ,,,π DDDD
0det
4342414
3332313
2322212
1312111
XXXX
XXXX
XXXX
XXXX
0πX 0πX 0,πX π 321 TTT andfromSolve
(solve as right nullspace of )π
T
T
T
3
2
1
X
X
X
0XXX Xdet 321
Or implicitly from coplanarity condition
124313422341 DXDXDX 01234124313422341 DXDXDXDX 13422341 DXDX
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Points from planes
0X
π
π
π
3
2
1
T
T
T
xX M 321 XXXM
0π MT
0Xπ 0Xπ 0,Xπ X 321 TTT andfromSolve
(solve as right nullspace of )X
T
T
T
3
2
1
π
π
π
Representing a plane by its span
• M is 4x3 matrix. Columns of M are null space of• X is a point on plane• x ( a point on projective plane P2 ) parameterizes
points on the plane π • M is not unique
Tπ
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Lines
T
T
B
AW μBλA
2×2** 0=WW=WWTT
0001
1000W
0010
0100W*
Example: X-axis:
(4dof)
Representing a line by its span: two vectors A, B for two space points
T
T
Q
PW* μQλPDual representation: P and Q are planes; line
is span of row space of W*
join of (0,0,0) and (1,0,0) points Intersection of y=0 and z=0
planes
• Line is either joint of two points or intersection of two planes
• Span of WT is the pencil of points on the line• Span of the 2D right null space of W is the pencil of the planes with the line as axis
• Span of W*T is the pencil of Planes with the line as axis
2x4
2x4
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Points, lines and planes
TX
WM 0π M
Tπ
W*
M 0X M
W
X
*W
π
Plane π defined by the join of the point X and line W is obtained from the null space of M:
• Point X defined by the intersection of line W with plane is the null space of Mπ
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Quadrics and dual quadrics
(Q : 4x4 symmetric matrix)0QXX T
1. 9 d.o.f.
2. in general 9 points define quadric
3. det Q=0 ↔ degenerate quadric
4. (plane ∩ quadric)=conic
5. transformation
Q
QMMC T MxX:π -1-TQHHQ'
0πQπ * T
-1* QQ 1. relation to quadric (non-degenerate)
2. transformation THHQQ' **
• Dual Quadric: defines equation on planes: tangent planes π to the point quadric Q satisfy:
Quadratic surface in P3 defined by:
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Quadric classification
Rank Sign.
Diagonal Equation Realization
4 4 (1,1,1,1) X2+ Y2+ Z2+1=0 No real points
2 (1,1,1,-1) X2+ Y2+ Z2=1 Sphere
0 (1,1,-1,-1) X2+ Y2= Z2+1 Hyperboloid (1S)
3 3 (1,1,1,0) X2+ Y2+ Z2=0 Single point
1 (1,1,-1,0) X2+ Y2= Z2 Cone
2 2 (1,1,0,0) X2+ Y2= 0 Single line
0 (1,-1,0,0) X2= Y2 Two planes
1 1 (1,0,0,0) X2=0 Single plane
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Quadric classificationProjectively equivalent to sphere:
hyperboloid of two sheets
paraboloidsphere ellipsoid
Degenerate ruled quadrics:
cone two planes
Ruled quadrics:
hyperboloids of one sheet
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Hierarchy of transformations
vTv
tAProjective15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
The absolute conic Ω∞
Volume
10
tAT
10
tRT
s
10
tRT
group transform distortion invariants properties
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The plane at infinity
π
1
0
0
0
1t
0ππ
A
AH
TT
A
The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity
1. canonical position2. contains directions 3. two planes are parallel line of intersection in π∞
4. line // line (or plane) point of intersection in π∞
T1,0,0,0π
T0,,,D 321 XXX
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The absolute conic
The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity
04
23
22
21
X
XXX
The absolute conic Ω∞ is a (point) conic on π. In a metric frame:
1. Ω∞ is only fixed as a set, not pointwise2. Circle intersect Ω∞ in two points3. All Spheres intersect π∞ in Ω∞
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The absolute conic
2211
21
dddd
ddcos
TT
T
2211
21
dddd
ddcos
TT
T
0dd 21 T
Euclidean:
Projective:
(orthogonality=conjugacy)
d1 and d2 are lines
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The absolute dual quadric
00
0I*TQ
The absolute conic Q*∞ is a fixed conic under the
projective transformation H iff H is a similarity
1. 8 dof2. plane at infinity π∞ is the nullvector of 3. Angles:
2*
21*
1
2*
1
ππππ
ππcos
QTT
T
*Q
Absolute dual Quadric = All planes tangent to Ω∞