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Derivation of Invariants by Tensor Methods
Tomáš Suk
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Motivation
Invariants to geometric transformations of 2D and 3D images
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Tensor Calculus
Gregorio Ricci, Tullio Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Mathematische Annalen (Springer) 54 (1–2): pp. 125–201, March 1900.
William Rowan Hamilton, On some extensions of Quaternions,Philosophical Magazine (4th series):vol. vii (1854), pp. 492-499,vol. viii (1854), pp. 125-137, 261-9,vol. ix (1855), pp. 46-51, 280-290.
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Tensor Calculus
David Cyganski, John A. Orr, Object Recognition and Orientation Determination by Tensor Methods, in Advances in Computer Vision and Image Processing, JAI Press, pp. 101—144, 1988.
Grigorii Borisovich Gurevich, Osnovy teorii algebraicheskikhinvariantov, (OGIZ, Moskva), 1937.Foundations of the Theory of Algebraic Invariants, (Nordhoff, Groningen), 1964.
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Tensor Calculus
• Enumeration
• Sum of products (dot product)
Tensor = multidimensional array + rules of multiplication
2 rules:
Example: product of 2 vectors
¡b1 b2 : : : bn
¢
0
BB@
a1a2...an
1
CCA =c
¡b1 b2 : : : bn
¢
0
BB@
a1a2...an
1
CCA =cSum of products:
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Tensor CalculusExample: product of 2 vectors aibk=cik
Enumeration:
.
0
BB@
a1a2...an
1
CCA¡b1 b2 : : : bm
¢=
=
0
BB@
c11 c12 : : : c1mc21 c22 : : : c2m...cn1 cn2 : : : cnm
1
CCA
.
0
BB@
a1a2...an
1
CCA¡b1 b2 : : : bm
¢=
=
0
BB@
c11 c12 : : : c1mc21 c22 : : : c2m...cn1 cn2 : : : cnm
1
CCA
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Tensor CalculusExample: product of 2 matrices aijbkl
I enumeration, j=k sum of products, l enumeration
.
0
BBBB@
a11 a12 : :: a1n1...ai1 ai2 : :: ain1...
an21 an22 : :: an2n1
1
CCCCA
0
BB@
b11 : : : b1` : : : b1n3b21 : : : b2` : : : b2n3...bn11 : : : bn1` : : : bn1n3
1
CCA
=
0
BBBB@
c11 : : : c1 : : : c1n1...ci1 : : : ci` : : : cin1...
cn21 : : : cn2 : : : cn2n3
1
CCCCA
.
0
BBBB@
a11 a12 : :: a1n1...ai1 ai2 : :: ain1...
an21 an22 : :: an2n1
1
CCCCA
0
BB@
b11 : : : b1` : : : b1n3b21 : : : b2` : : : b2n3...bn11 : : : bn1` : : : bn1n3
1
CCA
=
0
BBBB@
c11 : : : c1 : : : c1n1...ci1 : : : ci` : : : cin1...
cn21 : : : cn2 : : : cn2n3
1
CCCCA
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Einstein Notation
cik =nP
j =1aijbj k 8i;k =1;::: ;ncik =
nP
j =1aijbj k 8i;k =1;::: ;n
Instead of
we write
cik =aijbj kcik =aijbj k
other optionsvectors:matrices:
aibi; aibjaibi; aibj
aijbij ; aijbj i; aijbkj ; aijbik; aijbki; aijbk`aijbij ; aijbj i; aijbkj ; aijbik; aijbki; aijbk`
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Other Notations
Other symbols
Ð ² £ (n) ¤Ð ² £ (n) ¤
Penrose graphical notation
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Tensors in Affine Space
Affine transform in 2D
xi =pi®x® x®=q®i x
ixi =pi®x® x®=q®i x
i
Contravariant vector
u®=pi®uiu®=pi®ui
Covariant vector
A =µp11p
12
p21p22
¶A =
µp11p
12
p21p22
¶A ¡ 1=
µq11 q
12
q21 q22
¶A ¡ 1=
µq11 q
12
q21 q22
¶
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Tensors in Affine Space
Mixed tensor
ai1;i2;:::;ir =pi1®1pi2®2¢¢¢p
ir®ra
®1;®2;:::;®rai1;i2;:::;ir =pi1®1pi2®2¢¢¢p
ir®ra
®1;®2;:::;®r
Contravariant tensor
a®1;®2;:::;®r =pi1®1pi2®2¢¢¢p
ir®rai1;i2;:::;ira®1;®2;:::;®r =pi1®1p
i2®2¢¢¢p
ir®rai1;i2;:::;ir
Covariant tensor
a¯1; 2;:::; r2®1;®2;:::;®r1 =q¯1j1q
¯2j2 ¢¢¢q
¯r2j r2pi1®1p
i2®2¢¢¢p
ir1®r1a
j1;j2;:::;j r2i1;i2;:::;ir1
a¯1; 2;:::; r2®1;®2;:::;®r1 =q¯1j1q
¯2j2 ¢¢¢q
¯r2j r2pi1®1p
i2®2¢¢¢p
ir1®r1a
j1;j2;:::;j r2i1;i2;:::;ir1
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Tensors in Affine Space
aijbjk =cikaijbjk =cik
Multiplication
aij +bij =cijaij +bij =cijAdition
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Geometric Meaning
xi =pi®x®xi =pi®x®Point - contravariant vector
uixi =0 ui =pi®u®uixi =0 ui =pi®u®
Angle of straight line – covariant vector
aijxixj +2bixi +h=0aijxixj +2bixi +h=0
Conic section – covariant tensor
aij =pi®pj¯a
®aij =pi®pj¯a
®
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Properties
aij k =akj iaij k =akj i
Symmetric tensor
-To two indices-To several indices-To all indices
aij k =¡ akj iaij k =¡ akj i
Antisymmetric tensor (skew-symmetric)
-To two indices-To several indices-To all indices
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Other Tensor Operations
a(ij k) = 16(aij k+aikj +aj ki +aj ik+akij +akj i)
= 16
P
p2Paf ij kgp
a(ij k) = 16(aij k+aikj +aj ki +aj ik+akij +akj i)
= 16
P
p2Paf ij kgp
Symmetrization
Alternation (antisymmetrization)
a[ij k]= 16(aij k ¡ aikj +aj ki ¡ aj ik+akij ¡ akj i)a[ij k]= 16(aij k ¡ aikj +aj ki ¡ aj ik+akij ¡ akj i)
h[ijj jk]= 12(hij k ¡ hkj i)h[ijj jk]= 12(hij k ¡ hkj i)
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Other Tensor Operations
ki`m=hijj `m hijijm; hijj im; h
ijkij ;ki`m=hijj `m hijijm; h
ijj im; h
ijkij ;
Contraction
Total contraction
aijk` ¡ ! aijij ; aijj iaijk` ¡ ! aijij ; aijj i
→ Affine invariant
a¯1; 2;:::; r2®1;®2;:::;®r1 =q¯1j1q
¯2j2 ¢¢¢q
¯r2j r2pi1®1p
i2®2¢¢¢p
ir1®r1a
j1;j2;:::;j r2i1;i2;:::;ir1
a¯1; 2;:::; r2®1;®2;:::;®r1 =q¯1j1q
¯2j2 ¢¢¢q
¯r2j r2pi1®1p
i2®2¢¢¢p
ir1®r1a
j1;j2;:::;j r2i1;i2;:::;ir1
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Unit polyvector
²12:::n =1²12:::n =1Completely antisymmetric tensor
- covariant
µ0 1
¡ 1 0
¶µ0 1
¡ 1 0
¶
n=2:
²12:::n =1²12:::n =1- contravariant
Also Levi-Civita symbol
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Unit polyvector
n=3:
note: vector cross product ¡!a £¡!b =²ij kajbk¡!a £¡!b =²ij kajbk
nn components
n! non-zero
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Affine Invariants• Products with total contraction
aij kbij kaij kbij k
• Using unit polyvectors
aij kbmn²ij ²k`²mnaij kbmn²ij ²k`²mn
aij kbmn²i`²jm²knaij kbmn²i`²jm²kn
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Example - moments2D Moment tensor
M i1i2¢¢¢ir =1R
¡ 1
1R
¡ 1xi1xi2¢¢¢xirf (x1;x2) dx1 dx2M i1i2¢¢¢ir =
1R
¡ 1
1R
¡ 1xi1xi2¢¢¢xirf (x1;x2) dx1 dx2
M i1i2¢¢¢ir =mpqM i1i2¢¢¢ir =mpq if p indices equals 1 and q indices equals 2
mpq=1R
¡ 1
1R
¡ 1xpyqf (x;y)dxdympq=
1R
¡ 1
1R
¡ 1xpyqf (x;y)dxdy
Geometric moment
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Example - moments
Relative oriented contravariant tensor with weight -1
M i1i2¢¢¢ir =jJ j¡ 1qi1®1qi2®2¢¢¢q
ir®rM
®1®2¢¢¢®rM i1i2¢¢¢ir =jJ j¡ 1qi1®1qi2®2¢¢¢q
ir®rM
®1®2¢¢¢®r
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Example - moments
M ijM klmM nop²ik²j n²lo²mp=
=2(m20(m21m03¡ m212)¡
¡ m11(m30m03¡ m21m12)++m02(m30m12¡ m2
21))
M ijM klmM nop²ik²j n²lo²mp=
=2(m20(m21m03¡ m212)¡
¡ m11(m30m03¡ m21m12)++m02(m30m12¡ m2
21))
! (2¹ 20¹ 21¹ 03¡ 2¹ 20¹ 212¡¡ ¹ 11¹ 30¹ 03+¹ 11¹ 21¹ 12++¹ 02¹ 30¹ 12¡ ¹ 02¹ 221)=¹
700
! (2¹ 20¹ 21¹ 03¡ 2¹ 20¹ 212¡¡ ¹ 11¹ 30¹ 03+¹ 11¹ 21¹ 12++¹ 02¹ 30¹ 12¡ ¹ 02¹ 221)=¹
700
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Example – 3D moments
M ijM klMmn²ikm²j ln =
=6(m200m020m002+2m110m101m011¡¡ m200m2
011¡ m020m2101¡ m002m2
110)
M ijM klMmn²ikm²j ln =
=6(m200m020m002+2m110m101m011¡¡ m200m2
011¡ m020m2101¡ m002m2
110)
M i1i2¢¢¢ik =
=1R
¡ 1
1R
¡ 1
1R
¡ 1xi1xi2¢¢¢xikf (x1;x2;x3) dx1 dx2 dx3
M i1i2¢¢¢ik =
=1R
¡ 1
1R
¡ 1
1R
¡ 1xi1xi2¢¢¢xikf (x1;x2;x3) dx1 dx2 dx3
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Invariants to other transformations
• Non-linear transforms
A =
Ã@g1(x1;x2)
@x1@g1(x1;x2)
@x2@g2(x1;x2)
@x1@g2(x1;x2)
@x2
!
A =
Ã@g1(x1;x2)
@x1@g1(x1;x2)
@x2@g2(x1;x2)
@x1@g2(x1;x2)
@x2
!
Jacobi matrix
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Invariants to other transformations• Projective transform
x =p11x +p12y+p13p31x +p32y+p33
y=p21x +p22y+p23p31x +p32y+p33
x =p11x +p12y+p13p31x +p32y+p33
y=p21x +p22y+p23p31x +p32y+p33
0
@x1
x2
x3
1
A =
0
@p11p
12p
13
p21p22p
23
p31p32p
33
1
A
0
@x1
x2
x3
1
A
0
@x1
x2
x3
1
A =
0
@p11p
12p
13
p21p22p
23
p31p32p
33
1
A
0
@x1
x2
x3
1
A
Homogeneous coordinates
x =x1
x3y=
x2
x3x =
x1
x3y=
x2
x3
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Rotation Invariants• Cartesian tensors
Cartesian tensor
a®1;®2;:::;®r =p®1i1p®2i2¢¢¢p®rirai1;i2;:::;ira®1;®2;:::;®r =p®1i1p®2i2¢¢¢p®rirai1;i2;:::;ir
Affine tensor
a¯1; 2;:::; r2®1;®2;:::;®r1 =q¯1j1q
¯2j2 ¢¢¢q
¯r2j r2pi1®1p
i2®2¢¢¢p
ir1®r1a
j1;j2;:::;j r2i1;i2;:::;ir1
a¯1; 2;:::; r2®1;®2;:::;®r1 =q¯1j1q
¯2j2 ¢¢¢q
¯r2j r2pi1®1p
i2®2¢¢¢p
ir1®r1a
j1;j2;:::;j r2i1;i2;:::;ir1
pij =µcosµ ¡ sinµsinµ cosµ
¶pij =
µcosµ ¡ sinµsinµ cosµ
¶
Orthonormal matrix
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Cartesian Tensors• Rotation invariants by total contraction
aij kbij kaij kbij k
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Rotation Invariants
M ii !M ii !
2D:
M11+M22+M33 !(¹ 200+¹ 020+¹ 002)=¹
5=3000
M11+M22+M33 !(¹ 200+¹ 020+¹ 002)=¹
5=3000
M11+M22 ! (¹ 20+¹ 02)=¹ 200M11+M22 ! (¹ 20+¹ 02)=¹ 200
3D:
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Rotation InvariantsM ijM ij !M ijM ij !2D:
M11M11+M12M12+M13M13++M21M21+M22M22+M23M23++M31M31+M32M32+M33M33
M11M11+M12M12+M13M13++M21M21+M22M22+M23M23++M31M31+M32M32+M33M33
M11M11+M12M12+M21M21+M22M22
! (¹ 220+¹ 202+2¹ 211)=¹400
M11M11+M12M12+M21M21+M22M22
! (¹ 220+¹ 202+2¹ 211)=¹400
3D:
! (¹ 2200+¹ 2020+¹ 2002+2¹ 2110++2¹ 2101+2¹ 2011)=¹
10=3000
! (¹ 2200+¹ 2020+¹ 2002+2¹ 2110++2¹ 2101+2¹ 2011)=¹
10=3000
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Rotation InvariantsM ijM j kMki !M ijM j kMki !
2D:
(¹ 3200+3¹ 200¹ 2110+3¹ 200¹ 2101++3¹ 2110¹ 020+3¹ 2101¹ 002+¹ 3020++3¹ 020¹ 2011+3¹ 2011¹ 002+¹ 3002++6¹ 110¹ 101¹ 011)=¹ 5000
(¹ 3200+3¹ 200¹ 2110+3¹ 200¹ 2101++3¹ 2110¹ 020+3¹ 2101¹ 002+¹ 3020++3¹ 020¹ 2011+3¹ 2011¹ 002+¹ 3002++6¹ 110¹ 101¹ 011)=¹ 5000
(¹ 320+3¹ 20¹ 211+3¹ 211¹ 02+¹ 302)=¹600(¹ 320+3¹ 20¹ 211+3¹ 211¹ 02+¹ 302)=¹600
3D:
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Alternative Methods
1R
¡ 1¢¢¢
1R
¡ 1(x1y2¡ x2y1)2f (x1;y1)f (x2;y2)
dx1dy1dx2dy2
1R
¡ 1¢¢¢
1R
¡ 1(x1y2¡ x2y1)2f (x1;y1)f (x2;y2)
dx1dy1dx2dy2
• Method of geometric primitives
• Solution of Equations
Transformation decompositionSystem of linear equations for unknown coefficients
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Alternative Methods
cpq=1R
0
2¼R
0rp+q+1e¡ i(p¡ q)®f (r;µ)drdµcpq=
1R
0
2¼R
0rp+q+1e¡ i(p¡ q)®f (r;µ)drdµ
• Complex moments
• Normalization
2D:
3D:cms`=1R
0
¼R
0
2¼R
0rs+1Ym
` (#;' )f (r;#;' )drd#d'cms`=1R
0
¼R
0
2¼R
0rs+1Ym
` (#;' )f (r;#;' )drd#d'
Transformation decomposition
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