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Lobachevsky Geometry and Image Recognition Metric invariants in image recognision Nadiia Konovenko & Valentin Lychagin NAFT, Odessa, Ukraine, IPU RAN, Moscow, Russia & Department of Mathematics and Statistics,University of Troms,

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Lobachevsky Geometry and Image Recognition

Metric 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 Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Why do we need Lobachevsky Geometry?

The Mumford - Sharon approach.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Lobachevsky Geometry

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

1−az ,

where a ∈ D, θ ∈ R/Z .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Lobachevsky Geometry

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Lobachevsky Geometry

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Lobachevsky Geometry

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Lobachevsky Geometry

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Functions

Invariant coframe: ω1 = u1dx + u2dy, ω2 = −u2dx + u1dy.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Functions

Invariant coframe: ω1 = u1dx + u2dy, ω2 = −u2dx + u1dy. Invariant frame: δ1 = |∇u|−2

  • u1

d dx + u2 d dy

  • ,

δ2 = |∇u|−2

  • −u2

d dx + u1 d dy

  • ,

where T = |∇u|2 = u2

1 + u2 2.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Functions

Invariant coframe: ω1 = u1dx + u2dy, ω2 = −u2dx + u1dy. Invariant frame: δ1 = |∇u|−2

  • u1

d dx + u2 d dy

  • ,

δ2 = |∇u|−2

  • −u2

d dx + u1 d dy

  • ,

where T = |∇u|2 = u2

1 + u2 2.

Structure equations:

  • dω1 = 0,
  • dω2 = ∆u

∇u2 ω1 ∧ ω2,

  • r

[δ2, δ1] = ∆u |∇u|2 δ2

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Metric invariants of functions

0 - order J0 = u.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Metric invariants of functions

0 - order J0 = u. 1 -st order J1 =

  • 1 − x2 − y22 |∇u|2 or

|∇gu|2 , δ1 (J0) = 1, δ2 (J0) = 0.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Metric invariants of functions

0 - order J0 = u. 1 -st order J1 =

  • 1 − x2 − y22 |∇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) .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Metric invariants of functions

0 - order J0 = u. 1 -st order J1 =

  • 1 − x2 − y22 |∇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 order invariant derivatives of J1 and J2.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Theorem

The field of rational metric differential invariants for functions given on the unit disk is generated by invariants J0, J1, J2 and invariant derivations δ1, δ2. This field separates regular PSL2-orbits.

Theorem

The field of rational metric differential invariants for functions given on the unit disk is generated by invariants J0, J1, J11, J12, J2 and Tresse derivations D DJ0 , D DJ1 . This field separates regular PSL2-orbits. Tresse derivations D DJ0 = δ1 − J11 J12 δ2, D DJ1 = 1 J12 δ2.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

Given D ⊂ CP1 -a proper simply connected domain and a function f , with df = 0 and J12 (f ) = 0. gD -the metric, defined by the standard metric on D through the Riemann theorem. Basic invariants: J0 (f ) = f , J1 (f ) = |∇gDf |2 , J2 (f ) = ∆gDf , J11 (f ) , J12 (f ) .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

Given D ⊂ CP1 -a proper simply connected domain and a function f , with df = 0 and J12 (f ) = 0. gD -the metric, defined by the standard metric on D through the Riemann theorem. Basic invariants: J0 (f ) = f , J1 (f ) = |∇gDf |2 , J2 (f ) = ∆gDf , J11 (f ) , J12 (f ) . Coframe (or frame): df , Idf

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

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

  • KL (Institute)

Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0. For such a function find functions F2, F11, F12 and consider the above PDEs system. This is

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0. For such a function find functions F2, F11, F12 and consider the above PDEs system. This is

Frobenius type system

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0. For such a function find functions F2, F11, F12 and consider the above PDEs system. This is

Frobenius type system Integrable

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0. For such a function find functions F2, F11, F12 and consider the above PDEs system. This is

Frobenius type system Integrable PSL2 - automorphic

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0. For such a function find functions F2, F11, F12 and consider the above PDEs system. This is

Frobenius type system Integrable PSL2 - automorphic Solution space ⇔ PSL2 - orbit of the function f .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0. For such a function find functions F2, F11, F12 and consider the above PDEs system. This is

Frobenius type system Integrable PSL2 - automorphic Solution space ⇔ PSL2 - orbit of the function f .

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Classification

We say that function f is regular if J1 (f ) = 0 and J12 (f ) = 0. For such a function find functions F2, F11, F12 and consider the above PDEs system. This is

Frobenius type system Integrable PSL2 - automorphic Solution 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 have the same representive functions F2, F11, F12 and Im Jf = Im J

f .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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J11 = 2 T (u1u11 + 2u1u2u12 + u2u22 + 2t (xu1 + yu2)) , J12 = 2 T

  • u1u2 (u22 − u11) + u12
  • u2

2 − u2 1

+ 2t (yu1 − xu2)

  • ,

where t = 1 − x2 − y2, T = u2

1 + u2 2.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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) → (x, y, u = a (x, y) , v = b (x, y)) ,

  • f the cotangent bundle τ∗ : T∗M → M, where (x, y, u, v) are the

canonical coordinates in T∗M.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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) → (x, y, u = a (x, y) , v = b (x, y)) ,

  • f the cotangent bundle τ∗ : T∗M → M, where (x, y, u, v) are the

canonical coordinates in T∗M. Let ω = udx + vdy be the universal Liouville 1−form on T∗M. Then θ = s∗

θ (ω) .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Differential Invariants

Invariant coframe ω1 = udx + vdy, ω2 = −vdx + udy.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Differential Invariants

Invariant coframe ω1 = udx + vdy, ω2 = −vdx + udy. Invariant frame δ1 = 1 u2 + v2

  • u d

dx + v d dy

  • ,

δ2 = 1 u2 + v2

  • −v d

dx + u d dy

  • .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Differential Invariants

Invariant coframe ω1 = udx + vdy, ω2 = −vdx + udy. Invariant frame δ1 = 1 u2 + v2

  • u d

dx + v d dy

  • ,

δ2 = 1 u2 + v2

  • −v d

dx + u d dy

  • .

Structure equations:

  • dω1

= −u2 + v1 u2 + v2 ω1 ∧ ω2,

  • dω2

= u1 + v2 u2 + v2 ω1 ∧ ω2.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Differntial invariants

0 -order J0 =

  • 1 − x2 − y22

u2 + v2 = g (ω1, ω1) , and g = ω2

1 + ω2 2

J0 .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Differntial invariants

0 -order J0 =

  • 1 − x2 − y22

u2 + v2 = g (ω1, ω1) , and g = ω2

1 + ω2 2

J0 . 1 -st order J1,1 = −u2 + v1 u2 + v2 , J1,2 = u1 + v2 u2 + v2 , δ1 (J0) , δ2 (J0) .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Differntial invariants

0 -order J0 =

  • 1 − x2 − y22

u2 + v2 = g (ω1, ω1) , and g = ω2

1 + ω2 2

J0 . 1 -st order J1,1 = −u2 + v1 u2 + v2 , J1,2 = u1 + v2 u2 + v2 , δ1 (J0) , δ2 (J0) . k -th order invariant derivatives of J0 and J1,1 , J1,2.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

Given D ⊂ CP1 -a proper simply connected domain and a differential 1-form θ, with θ = 0. gD -the metric, defined by the standard metric on D through the Riemann theorem. basic invariants: J0 (θ) = gD (θ, θ) and J1,1 J1,2, where dθ = J1,1 (θ) θ ∧ Iθ, dIθ = J1,2 (θ) θ ∧ Iθ.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

Given D ⊂ CP1 -a proper simply connected domain and a differential 1-form θ, with θ = 0. gD -the metric, defined by the standard metric on D through the Riemann theorem. basic invariants: J0 (θ) = gD (θ, θ) and J1,1 J1,2, where dθ = J1,1 (θ) θ ∧ Iθ, dIθ = J1,2 (θ) θ ∧ Iθ. Invariantization map: Jθ : D → R2, Jθ = (J1,1 (θ) , J1,2 (θ)) , and function J0 (θ) .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

Given D ⊂ CP1 -a proper simply connected domain and a differential 1-form θ, with θ = 0. gD -the metric, defined by the standard metric on D through the Riemann theorem. basic invariants: J0 (θ) = gD (θ, θ) and J1,1 J1,2, where dθ = J1,1 (θ) θ ∧ Iθ, dIθ = J1,2 (θ) θ ∧ Iθ. 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 θ and Iθ and function J0 (θ) .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Foliations

Foliation ⇔ {λω} where ω is a non vanishing differential 1 -form and λ is a non vanishing smooth function, defined on a domain D.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Foliations

Foliation ⇔ {λω} where ω is a non vanishing differential 1 -form and λ is a non vanishing smooth function, defined on a domain D. Killing infinite dimensional pseudogroup : if ω = u dx + vdy, then w = u v is a function defined the foliation.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Foliations

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Frames

Coframe ω1 = sin w dx + cos w dy 1 − x2 − y2 , ω2 = − cos w dx + sin w dy 1 − x2 − y2 .

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Frames

Coframe ω1 = sin w dx + cos w dy 1 − x2 − y2 , ω2 = − cos w dx + sin w dy 1 − x2 − y2 . Frame δ1 = (1 − x2 − y2)(cos w d dx + sin w d dy ), δ2 =

  • 1 − x2 − y2 (− sin w d

dx + cos w d dy ).

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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 sin w − w2 cos w)

  • 1 − x2 − y2 + 2x cos w − 2y sin w,

J1,2 = (−w2 sin w + w1 cos w)

  • 1 − x2 − y2 + 2y cos w + 2x sin w.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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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 sin w − w2 cos w)

  • 1 − x2 − y2 + 2x cos w − 2y sin w,

J1,2 = (−w2 sin w + w1 cos w)

  • 1 − x2 − y2 + 2y cos w + 2x sin w.

k-th order invariant derivatives of J1,1 and J1,2.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

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

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

Given D ⊂ CP1 -a proper simply connected domain and a foliation defined by differential 1-form θ, with θ = 0. gD -the metric, defined by the standard metric on D through the Riemann theorem. Normalize θ : θ → θ |θ|gD . Invariantization map: Jθ : D → R2, Jθ = (J1,1 (θ) , J1,2 (θ)) ,

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Summary

Given D ⊂ CP1 -a proper simply connected domain and a foliation defined by differential 1-form θ, with θ = 0. gD -the metric, defined by the standard metric on D through the Riemann theorem. Normalize θ : θ → θ |θ|gD . Invariantization map: Jθ : D → R2, Jθ = (J1,1 (θ) , J1,2 (θ)) , Classification data: on the image of Jθ two differenatial 1 -forms θ and Iθ.

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19

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Thank you for your attention

KL (Institute) Lobachevsky Geometry and Image Recognition Workshop on “Infinite-dimensional Riemannian / 19