derivation of invariants by tensor methods tomáš suk

Derivation of Invariants by Tensor Methods

Tomáš Suk

Motivation

Invariants to geometric transformations of 2D and 3D images

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.

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.

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:

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

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

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`

Other Notations

Other symbols

Ð ² £ (n) ¤Ð ² £ (n) ¤

Penrose graphical notation

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

¶

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

Tensors in Affine Space

aijbjk =cikaijbjk =cik

Multiplication

aij +bij =cijaij +bij =cijAdition

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

®

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

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)

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

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

Unit polyvector

n=3:

note: vector cross product ¡!a £¡!b =²ij kajbk¡!a £¡!b =²ij kajbk

nn components

n! non-zero

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

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

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

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

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

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

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

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

Cartesian Tensors• Rotation invariants by total contraction

aij kbij kaij kbij k

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:

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

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:

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

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

Thank you for your attention