Lobachevsky Geometry and Image Recognitionshape2015/slides/lychagin1.pdfLobachevsky Geometry and Image Recognition Metric invariants in image recognision Nadiia Konovenko & Valentin

Lobachevsky Geometry and Image RecognitionMetric invariants in image recognision

Nadiia Konovenko & Valentin Lychagin

NAFT, Odessa, Ukraine,IPU RAN, Moscow, Russia &

Department of Mathematics and Statistics,University of Tromsø, Norway

Workshop on“Infinite-dimensional Riemannian geometry”

Vienna, January 12 —16, 2015

KL (Institute) Lobachevsky Geometry and Image RecognitionWorkshop on “Infinite-dimensional Riemannian geometry” Vienna, January 12 — 16, 2015 1

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Why do we need Lobachevsky Geometry?

The Mumford - Sharon approach.

Poincaré model of Lobachevsky geometry

Figure: Escher’s circle limit iii


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Why do we need Lobachevsky Geometry?

The Mumford - Sharon approach.

Poincaré model of Lobachevsky geometry

Figure: Escher’s circle limit iii


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Lobachevsky Geometry

Structure group PSL2 (R) .Möbius transformations of the unit disk D : z 7→ e2πθı z−a

1−az ,where a ∈ D, θ ∈ R/Z .

Structure Lie algebra sl2 (R) :

x∂y − y∂x ,(1− x2 + y2

)∂x + 2xy∂y ,

(1− x2 + y2

)∂y − 2xy∂x

Invariant metric: g = dx 2+dy 2

(1−x 2−y 2)2 .

Invariant symplectic structure: Ω = dx∧dy(1−x 2−y 2)2 .

Invariant complex structure: I = ∂y ⊗ dx − ∂x ⊗ dy .


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x∂y − y∂x ,(1− x2 + y2

)∂x + 2xy∂y ,

(1− x2 + y2

)∂y − 2xy∂x


(1−x 2−y 2)2 .




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x∂y − y∂x ,(1− x2 + y2

)∂x + 2xy∂y ,

(1− x2 + y2

)∂y − 2xy∂x


(1−x 2−y 2)2 .




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x∂y − y∂x ,(1− x2 + y2

)∂x + 2xy∂y ,

(1− x2 + y2

)∂y − 2xy∂x


(1−x 2−y 2)2 .




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x∂y − y∂x ,(1− x2 + y2

)∂x + 2xy∂y ,

(1− x2 + y2

)∂y − 2xy∂x


(1−x 2−y 2)2 .




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Functions

Invariant coframe:

ω1 = u1dx + u2dy ,

ω2 = −u2dx + u1dy .

Invariant frame:

δ1 = |∇u|−2(u1ddx+ u2

ddy

),

δ2 = |∇u|−2(−u2

ddx+ u1

ddy

),

where T = |∇u|2 = u21 + u22 .Structure equations:

dω1 = 0, dω2 =∆u∇u2 ω1 ∧ω2,

or

[δ2, δ1] =∆u

|∇u|2δ2


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Functions

Invariant coframe:

ω1 = u1dx + u2dy ,

ω2 = −u2dx + u1dy .Invariant frame:

δ1 = |∇u|−2(u1ddx+ u2

ddy

),

δ2 = |∇u|−2(−u2

ddx+ u1

ddy

),

where T = |∇u|2 = u21 + u22 .

Structure equations:

dω1 = 0, dω2 =∆u∇u2 ω1 ∧ω2,

or

[δ2, δ1] =∆u

|∇u|2δ2


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Functions

Invariant coframe:

ω1 = u1dx + u2dy ,

ω2 = −u2dx + u1dy .Invariant frame:

δ1 = |∇u|−2(u1ddx+ u2

ddy

),

δ2 = |∇u|−2(−u2

ddx+ u1

ddy

),

where T = |∇u|2 = u21 + u22 .Structure equations:

dω1 = 0, dω2 =∆u∇u2 ω1 ∧ω2,

or

[δ2, δ1] =∆u

|∇u|2δ2


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Metric invariants of functions

0 - orderJ0 = u.

1 -st order

J1 =(1− x2 − y2

)2 |∇u|2 or |∇gu|2 ,δ1 (J0) = 1, δ2 (J0) = 0.

2 -nd order

J2 =∆u

|∇u|2, or ∆gu

J11 = δ1 (J1) , J12 = δ2 (J2) .

k− th orderinvariant derivatives of J1 and J2.


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0 - orderJ0 = u.

1 -st order

J1 =(1− x2 − y2

)2 |∇u|2 or |∇gu|2 ,δ1 (J0) = 1, δ2 (J0) = 0.

2 -nd order

J2 =∆u

|∇u|2, or ∆gu

J11 = δ1 (J1) , J12 = δ2 (J2) .



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0 - orderJ0 = u.

1 -st order

J1 =(1− x2 − y2

)2 |∇u|2 or |∇gu|2 ,δ1 (J0) = 1, δ2 (J0) = 0.

2 -nd order

J2 =∆u

|∇u|2, or ∆gu

J11 = δ1 (J1) , J12 = δ2 (J2) .



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0 - orderJ0 = u.

1 -st order

J1 =(1− x2 − y2

)2 |∇u|2 or |∇gu|2 ,δ1 (J0) = 1, δ2 (J0) = 0.

2 -nd order

J2 =∆u

|∇u|2, or ∆gu

J11 = δ1 (J1) , J12 = δ2 (J2) .



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TheoremThe field of rational metric differential invariants for functions given on theunit disk is generated by invariants J0, J1, J2 and invariant derivationsδ1, δ2.This field separates regular PSL2-orbits.

TheoremThe field of rational metric differential invariants for functions given on theunit disk is generated by invariants J0, J1, J11, J12, J2 and Tresse derivations

DDJ0

,DDJ1

.

This field separates regular PSL2-orbits.

Tresse derivationsDDJ0

= δ1 −J11J12

δ2,DDJ1

=1J12

δ2.


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Summary

Given D ⊂ CP1 -a proper simply connected domain and a function f ,with df 6= 0 and J12 (f ) 6= 0.

gD -the metric, defined by the standard metric on D through theRiemann theorem.Basic invariants:

J0 (f ) = f , J1 (f ) = |∇gDf |2 , J2 (f ) = ∆gDf , J11 (f ) , J12 (f ) .

Coframe (or frame):df , Idf

Invariantization map:

Jf : D→ R2

Jf =(f , |∇gDf |

2),

and functions

∆gDf = F2(f , |∇gDf |

2), J11 (f ) = F11

(f , |∇gDf |

2), J12 (f ) = F12

(f , |∇gDf |

2).


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Summary

Given D ⊂ CP1 -a proper simply connected domain and a function f ,with df 6= 0 and J12 (f ) 6= 0.gD -the metric, defined by the standard metric on D through theRiemann theorem.

Basic invariants:




Jf : D→ R2

Jf =(f , |∇gDf |

2),

and functions

∆gDf = F2(f , |∇gDf |

2), J11 (f ) = F11

(f , |∇gDf |

2), J12 (f ) = F12

(f , |∇gDf |

2).


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Summary

Given D ⊂ CP1 -a proper simply connected domain and a function f ,with df 6= 0 and J12 (f ) 6= 0.gD -the metric, defined by the standard metric on D through theRiemann theorem.Basic invariants:




Jf : D→ R2

Jf =(f , |∇gDf |

2),

and functions

∆gDf = F2(f , |∇gDf |

2), J11 (f ) = F11

(f , |∇gDf |

2), J12 (f ) = F12

(f , |∇gDf |

2).


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Summary





Jf : D→ R2

Jf =(f , |∇gDf |

2),

and functions

∆gDf = F2(f , |∇gDf |

2), J11 (f ) = F11

(f , |∇gDf |

2), J12 (f ) = F12

(f , |∇gDf |

2).


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Summary





Jf : D→ R2

Jf =(f , |∇gDf |

2),

and functions

∆gDf = F2(f , |∇gDf |

2), J11 (f ) = F11

(f , |∇gDf |

2), J12 (f ) = F12

(f , |∇gDf |

2).


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Classification

We say that function f is regular if

J1 (f ) 6= 0 and J12 (f ) 6= 0.

For such a function find functions F2,F11,F12 and consider the abovePDEs system. This is

Frobenius type systemIntegrablePSL2 - automorphicSolution space ⇔ PSL2 - orbit of the function f .

Functions F2,F11,F12 are not arbitrary they satisfy two relations(=integrability conditions of the above system)

Theorem

The regular functions f and f are PSL2-equivalent if and only if they havethe same representive functions F2,F11,F12 and Im Jf = Im Jf .


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Classification


J1 (f ) 6= 0 and J12 (f ) 6= 0.




Theorem



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Classification


J1 (f ) 6= 0 and J12 (f ) 6= 0.


Frobenius type system

IntegrablePSL2 - automorphicSolution space ⇔ PSL2 - orbit of the function f .


Theorem



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Classification


J1 (f ) 6= 0 and J12 (f ) 6= 0.


Frobenius type systemIntegrable

PSL2 - automorphicSolution space ⇔ PSL2 - orbit of the function f .


Theorem



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Classification


J1 (f ) 6= 0 and J12 (f ) 6= 0.


Frobenius type systemIntegrablePSL2 - automorphic

Solution space ⇔ PSL2 - orbit of the function f .


Theorem



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Classification


J1 (f ) 6= 0 and J12 (f ) 6= 0.




Theorem



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Classification


J1 (f ) 6= 0 and J12 (f ) 6= 0.




Theorem



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Classification


J1 (f ) 6= 0 and J12 (f ) 6= 0.




Theorem



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J11 =2T(u1u11 + 2u1u2u12 + u2u22 + 2t (xu1 + yu2)) ,

J12 =2T

(u1u2 (u22 − u11) + u12

(u22 − u21

)+ 2t (yu1 − xu2)

),

wheret = 1− x2 − y2,T = u21 + u22 .


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Differential forms

Differential 1-form

θ = a (x , y) dx + b (x , y) dy

in a proper simply connected domain D ⊂ CP1.

Section

sθ : M → T∗M,sθ : (x , y) 7→ (x , y , u = a (x , y) , v = b (x , y)) ,

of the cotangent bundle τ∗ : T∗M → M, where (x , y , u, v) are thecanonical coordinates in T∗M.Let

ω = udx + vdy

be the universal Liouville 1−form on T∗M.Then

θ = s∗θ (ω) .


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Differential forms

Differential 1-form

θ = a (x , y) dx + b (x , y) dy

in a proper simply connected domain D ⊂ CP1.Section


of the cotangent bundle τ∗ : T∗M → M, where (x , y , u, v) are thecanonical coordinates in T∗M.

Letω = udx + vdy


θ = s∗θ (ω) .


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Differential forms

Differential 1-form

θ = a (x , y) dx + b (x , y) dy

in a proper simply connected domain D ⊂ CP1.Section


of the cotangent bundle τ∗ : T∗M → M, where (x , y , u, v) are thecanonical coordinates in T∗M.Let

ω = udx + vdy


θ = s∗θ (ω) .


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Actions

sl2 (R)-action on D :

X = −y∂x + x∂y ,

Y =(1− x2 + y2

)∂x + 2xy∂y ,

Z =(1− x2 + y2

)∂y − 2xy∂x .

sl2-action on T∗D :

X = X − u∂v + v∂u ,

Y = Y − 2(xu + yv)∂u − 2 (xv − yu) ∂v ,

Z = Z + 2 (xv − yu) ∂u − 2 (xu + yv) ∂v .


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Actions

sl2 (R)-action on D :

X = −y∂x + x∂y ,

Y =(1− x2 + y2

)∂x + 2xy∂y ,

Z =(1− x2 + y2

)∂y − 2xy∂x .

sl2-action on T∗D :

X = X − u∂v + v∂u ,

Y = Y − 2(xu + yv)∂u − 2 (xv − yu) ∂v ,

Z = Z + 2 (xv − yu) ∂u − 2 (xu + yv) ∂v .


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Differential Invariants

Invariant coframe

ω1 = udx + vdy ,

ω2 = −vdx + udy .

Invariant frame

δ1 =1

u2 + v2

(uddx+ v

ddy

),

δ2 =1

u2 + v2

(−v d

dx+ u

ddy

).


dω1 =−u2 + v1u2 + v2

ω1 ∧ω2,

dω2 =u1 + v2u2 + v2

ω1 ∧ω2.


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Invariant coframe

ω1 = udx + vdy ,


Invariant frame

δ1 =1

u2 + v2

(uddx+ v

ddy

),

δ2 =1

u2 + v2

(−v d

dx+ u

ddy

).


dω1 =−u2 + v1u2 + v2

ω1 ∧ω2,

dω2 =u1 + v2u2 + v2

ω1 ∧ω2.


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Invariant coframe

ω1 = udx + vdy ,


Invariant frame

δ1 =1

u2 + v2

(uddx+ v

ddy

),

δ2 =1

u2 + v2

(−v d

dx+ u

ddy

).


dω1 =−u2 + v1u2 + v2

ω1 ∧ω2,

dω2 =u1 + v2u2 + v2

ω1 ∧ω2.


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Differntial invariants

0 -order

J0 =(1− x2 − y2

)2 (u2 + v2

)= g (ω1,ω1) ,

and

g =ω21 +ω2

2

J0.

1 -st order

J1,1 =−u2 + v1u2 + v2

, J1,2 =u1 + v2u2 + v2

,

δ1 (J0) , δ2 (J0) .

k -th order

invariant derivatives of J0 and J1,1 , J1,2.


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0 -order

J0 =(1− x2 − y2

)2 (u2 + v2

)= g (ω1,ω1) ,

and

g =ω21 +ω2

2

J0.

1 -st order

J1,1 =−u2 + v1u2 + v2

, J1,2 =u1 + v2u2 + v2

,

δ1 (J0) , δ2 (J0) .

k -th order



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0 -order

J0 =(1− x2 − y2

)2 (u2 + v2

)= g (ω1,ω1) ,

and

g =ω21 +ω2

2

J0.

1 -st order

J1,1 =−u2 + v1u2 + v2

, J1,2 =u1 + v2u2 + v2

,

δ1 (J0) , δ2 (J0) .

k -th order



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Summary

Given D ⊂ CP1 -a proper simply connected domain and a differential1-form θ, with θ 6= 0.

gD -the metric, defined by the standard metric on D through theRiemann theorem.basic invariants:

J0 (θ) = gD (θ, θ)

and J1,1 J1,2, where

dθ = J1,1 (θ) θ ∧ I θ, dI θ = J1,2 (θ) θ ∧ I θ.Invariantization map:

Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,

and function J0 (θ) .Classification data: on the image of Jθ two differenatial 1 -forms θand I θ and function J0 (θ) .


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Summary

Given D ⊂ CP1 -a proper simply connected domain and a differential1-form θ, with θ 6= 0.gD -the metric, defined by the standard metric on D through theRiemann theorem.

basic invariants:J0 (θ) = gD (θ, θ)



Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,



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Summary

Given D ⊂ CP1 -a proper simply connected domain and a differential1-form θ, with θ 6= 0.gD -the metric, defined by the standard metric on D through theRiemann theorem.basic invariants:

J0 (θ) = gD (θ, θ)


dθ = J1,1 (θ) θ ∧ I θ, dI θ = J1,2 (θ) θ ∧ I θ.


Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,



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Summary


J0 (θ) = gD (θ, θ)



Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,

and function J0 (θ) .

Classification data: on the image of Jθ two differenatial 1 -forms θand I θ and function J0 (θ) .


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Summary


J0 (θ) = gD (θ, θ)



Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,



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Foliations

Foliation⇔ λωwhere ω is a non vanishing differential 1 -form and λ is a nonvanishing smooth function, defined on a domain D.

Killing infinite dimensional pseudogroup : if ω = u dx + vdy , then

w =uv

is a function defined the foliation.Action sl2 :

−y∂x + x∂y − ∂w ,(1− x2 + y2

)∂x − 2xy∂y + 2y∂w ,(

1+ x2 − y2)

∂y − 2xy∂x − 2x∂w .


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Foliations

Foliation⇔ λωwhere ω is a non vanishing differential 1 -form and λ is a nonvanishing smooth function, defined on a domain D.Killing infinite dimensional pseudogroup : if ω = u dx + vdy , then

w =uv

is a function defined the foliation.

Action sl2 :

−y∂x + x∂y − ∂w ,(1− x2 + y2

)∂x − 2xy∂y + 2y∂w ,(

1+ x2 − y2)

∂y − 2xy∂x − 2x∂w .


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Foliations

Foliation⇔ λωwhere ω is a non vanishing differential 1 -form and λ is a nonvanishing smooth function, defined on a domain D.Killing infinite dimensional pseudogroup : if ω = u dx + vdy , then

w =uv

is a function defined the foliation.Action sl2 :

−y∂x + x∂y − ∂w ,(1− x2 + y2

)∂x − 2xy∂y + 2y∂w ,(

1+ x2 − y2)

∂y − 2xy∂x − 2x∂w .


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Frames

Coframe

ω1 =sinw dx + cosw dy

1− x2 − y2 ,

ω2 =− cosw dx + sinw dy

1− x2 − y2 .

Frame

δ1 = (1− x2 − y2)(cosw ddx+ sinw

ddy),

δ2 =(1− x2 − y2

)(− sinw d

dx+ cosw

ddy).


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Frames

Coframe

ω1 =sinw dx + cosw dy

1− x2 − y2 ,

ω2 =− cosw dx + sinw dy

1− x2 − y2 .

Frame

δ1 = (1− x2 − y2)(cosw ddx+ sinw

ddy),

δ2 =(1− x2 − y2

)(− sinw d

dx+ cosw

ddy).


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Invariants

Structure equations

dω1 = J1,1 ω1 ∧ω2,

dω2 = J1,2 ω1 ∧ω2,

where J1,1 and J1,2 are the following 1 -st order invariants:

J1,1 = (−w1 sinw − w2 cosw)(1− x2 − y2

)+ 2x cosw − 2y sinw ,

J1,2 = (−w2 sinw + w1 cosw)(1− x2 − y2

)+ 2y cosw + 2x sinw .

k-th orderinvariant derivatives of J1,1 and J1,2.


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Invariants

Structure equations

dω1 = J1,1 ω1 ∧ω2,

dω2 = J1,2 ω1 ∧ω2,

where J1,1 and J1,2 are the following 1 -st order invariants:

J1,1 = (−w1 sinw − w2 cosw)(1− x2 − y2

)+ 2x cosw − 2y sinw ,

J1,2 = (−w2 sinw + w1 cosw)(1− x2 − y2

)+ 2y cosw + 2x sinw .

k-th orderinvariant derivatives of J1,1 and J1,2.


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Summary

Given D ⊂ CP1 -a proper simply connected domain and a foliationdefined by differential 1-form θ, with θ 6= 0.

gD -the metric, defined by the standard metric on D through theRiemann theorem. Normalize θ :

θ 7→ θ

|θ|gD.


Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,

Classification data: on the image of Jθ two differenatial 1 -forms θand I θ.


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Summary

Given D ⊂ CP1 -a proper simply connected domain and a foliationdefined by differential 1-form θ, with θ 6= 0.gD -the metric, defined by the standard metric on D through theRiemann theorem. Normalize θ :

θ 7→ θ

|θ|gD.


Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,



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Summary


θ 7→ θ

|θ|gD.


Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,



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Summary


θ 7→ θ

|θ|gD.


Jθ : D→ R2,

Jθ = (J1,1 (θ) , J1,2 (θ)) ,



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Thank you for your attention


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Documents

Lobachevsky Geometry and Image Recognitionshape2015/slides/lychagin1.pdfLobachevsky Geometry and Image Recognition Metric invariants in image recognision Nadiia Konovenko & Valentin