Recovering metric and affine properties from images Affine preserves: Parallelism Parallel length...
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Transcript of Recovering metric and affine properties from images Affine preserves: Parallelism Parallel length...
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
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
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
Affine rectificationv1 v2
l1
l2 l4
l3
l∞
21 vvl
211 llv
432 llv
c
a b
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:
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
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
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.
Length ratios
sin
sin
),(
),(
cad
cbd
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
SVD Decompose to get U
'*C
Euclidean
Perspectivetransformation
Rectifying Transformation: U
Similarity transformation
C*∞
Rectified imageProjectivelyDistortedImage
'*C
Recovering up to a similarity from Projective
Recovering up to a similarity from Affine
'*C
Euclidean
affinetransformation
Rectifying Transformation: U
Similarity transformation
C*∞
Rectified imageAffinely DistortedImage
Metric from affine
000
0
3
2
1
321
m
m
m
lllTKK
0,,,, 2221211
212
21122122111 T
kkkkkmlmlmlml
Affine rectified
Metric from projective
0c,,,,, 332332133122122111 5.05.05.0 mlmlmlmlmlmlmlmlml
0vvv
v
3
2
1
321
m
m
m
lllTT
TT
K
KKK
Projective 3D geometry
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
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
Projective 3D Geometry
• Points, lines, planes and quadrics
• Transformations
• П∞, ω∞ and Ω ∞
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
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
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
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
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
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
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:
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
Quadric classificationProjectively equivalent to sphere:
hyperboloid of two sheets
paraboloidsphere ellipsoid
Degenerate ruled quadrics:
cone two planes
Ruled quadrics:
hyperboloids of one sheet
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
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
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∞
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
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∞