Lobachevsky Geometry and Image Recognition Metric invariants in - PowerPoint PPT Presentation

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

Why do we need Lobachevsky Geometry? The Mumford - Sharon approach. Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Why do we need Lobachevsky Geometry? The Mumford - Sharon approach. Poincaré model of Lobachevsky geometry Figure: Escher’s circle limit iii Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Lobachevsky Geometry Structure group PSL 2 ( R ) . z �→ e 2 πθ ı z − a Möbius transformations of the unit disk D : 1 − az , where a ∈ D , θ ∈ R / Z . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Lobachevsky Geometry Structure group PSL 2 ( R ) . z �→ e 2 πθ ı z − a Möbius transformations of the unit disk D : 1 − az , where a ∈ D , θ ∈ R / Z . Structure Lie algebra sl 2 ( R ) : � 1 − x 2 + y 2 � � 1 − x 2 + y 2 � x ∂ y − y ∂ x , ∂ x + 2 xy ∂ y , ∂ y − 2 xy ∂ x Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Lobachevsky Geometry Structure group PSL 2 ( R ) . z �→ e 2 πθ ı z − a Möbius transformations of the unit disk D : 1 − az , where a ∈ D , θ ∈ R / Z . Structure Lie algebra sl 2 ( R ) : � 1 − x 2 + y 2 � � 1 − x 2 + y 2 � x ∂ y − y ∂ x , ∂ x + 2 xy ∂ y , ∂ y − 2 xy ∂ x dx 2 + dy 2 Invariant metric: g = ( 1 − x 2 − y 2 ) 2 . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Lobachevsky Geometry Structure group PSL 2 ( R ) . z �→ e 2 πθ ı z − a Möbius transformations of the unit disk D : 1 − az , where a ∈ D , θ ∈ R / Z . Structure Lie algebra sl 2 ( R ) : � 1 − x 2 + y 2 � � 1 − x 2 + y 2 � x ∂ y − y ∂ x , ∂ x + 2 xy ∂ y , ∂ y − 2 xy ∂ x dx 2 + dy 2 Invariant metric: g = ( 1 − x 2 − y 2 ) 2 . dx ∧ dy Invariant symplectic structure: Ω = ( 1 − x 2 − y 2 ) 2 . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Lobachevsky Geometry Structure group PSL 2 ( R ) . z �→ e 2 πθ ı z − a Möbius transformations of the unit disk D : 1 − az , where a ∈ D , θ ∈ R / Z . Structure Lie algebra sl 2 ( R ) : � 1 − x 2 + y 2 � � 1 − x 2 + y 2 � x ∂ y − y ∂ x , ∂ x + 2 xy ∂ y , ∂ y − 2 xy ∂ x dx 2 + dy 2 Invariant metric: g = ( 1 − x 2 − y 2 ) 2 . dx ∧ dy Invariant symplectic structure: Ω = ( 1 − x 2 − y 2 ) 2 . Invariant complex structure: I = ∂ y ⊗ dx − ∂ x ⊗ dy . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Functions Invariant coframe: ω 1 = u 1 dx + u 2 dy , ω 2 = − u 2 dx + u 1 dy . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Functions Invariant coframe: ω 1 = u 1 dx + u 2 dy , ω 2 = − u 2 dx + u 1 dy . Invariant frame: � � d d |∇ u | − 2 δ 1 = u 1 dx + u 2 , dy � � d d |∇ u | − 2 δ 2 = − u 2 dx + u 1 , dy where T = |∇ u | 2 = u 2 1 + u 2 2 . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Functions Invariant coframe: ω 1 = u 1 dx + u 2 dy , ω 2 = − u 2 dx + u 1 dy . Invariant frame: � � d d |∇ u | − 2 δ 1 = u 1 dx + u 2 , dy � � d d |∇ u | − 2 δ 2 = − u 2 dx + u 1 , dy where T = |∇ u | 2 = u 2 1 + u 2 2 . Structure equations: d ω 2 = ∆ u � � d ω 1 = 0 , ∇ u 2 ω 1 ∧ ω 2 , or ∆ u [ δ 2 , δ 1 ] = |∇ u | 2 δ 2 Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Metric invariants of functions 0 - order J 0 = u . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Metric invariants of functions 0 - order J 0 = u . 1 -st order � 1 − x 2 − y 2 � 2 |∇ u | 2 or |∇ g u | 2 , J 1 = δ 1 ( J 0 ) = 1 , δ 2 ( J 0 ) = 0 . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Metric invariants of functions 0 - order J 0 = u . 1 -st order � 1 − x 2 − y 2 � 2 |∇ u | 2 or |∇ g u | 2 , J 1 = δ 1 ( J 0 ) = 1 , δ 2 ( J 0 ) = 0 . 2 -nd order ∆ u J 2 = |∇ u | 2 , or ∆ g u J 11 = δ 1 ( J 1 ) , J 12 = δ 2 ( J 2 ) . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Metric invariants of functions 0 - order J 0 = u . 1 -st order � 1 − x 2 − y 2 � 2 |∇ u | 2 or |∇ g u | 2 , J 1 = δ 1 ( J 0 ) = 1 , δ 2 ( J 0 ) = 0 . 2 -nd order ∆ u J 2 = |∇ u | 2 , or ∆ g u J 11 = δ 1 ( J 1 ) , J 12 = δ 2 ( J 2 ) . k − th order invariant derivatives of J 1 and J 2 . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Theorem The field of rational metric differential invariants for functions given on the unit disk is generated by invariants J 0 , J 1 , J 2 and invariant derivations δ 1 , δ 2 . This field separates regular PSL 2 -orbits. Theorem The field of rational metric differential invariants for functions given on the unit disk is generated by invariants J 0 , J 1 , J 11 , J 12 , J 2 and Tresse derivations D D , . DJ 0 DJ 1 This field separates regular PSL 2 -orbits. Tresse derivations D = δ 1 − J 11 D = 1 δ 2 , δ 2 . DJ 0 J 12 DJ 1 J 12 Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Summary Given D ⊂ C P 1 -a proper simply connected domain and a function f , with df � = 0 and J 12 ( f ) � = 0 . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Summary Given D ⊂ C P 1 -a proper simply connected domain and a function f , with df � = 0 and J 12 ( f ) � = 0 . g D -the metric, defined by the standard metric on D through the Riemann theorem. Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Summary Given D ⊂ C P 1 -a proper simply connected domain and a function f , with df � = 0 and J 12 ( f ) � = 0 . g D -the metric, defined by the standard metric on D through the Riemann theorem. Basic invariants: J 0 ( f ) = f , J 1 ( f ) = |∇ g D f | 2 , J 2 ( f ) = ∆ g D f , J 11 ( f ) , J 12 ( f ) . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Summary Given D ⊂ C P 1 -a proper simply connected domain and a function f , with df � = 0 and J 12 ( f ) � = 0 . g D -the metric, defined by the standard metric on D through the Riemann theorem. Basic invariants: J 0 ( f ) = f , J 1 ( f ) = |∇ g D f | 2 , J 2 ( f ) = ∆ g D f , J 11 ( f ) , J 12 ( f ) . Coframe (or frame): df , Idf Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Summary Given D ⊂ C P 1 -a proper simply connected domain and a function f , with df � = 0 and J 12 ( f ) � = 0 . g D -the metric, defined by the standard metric on D through the Riemann theorem. Basic invariants: J 0 ( f ) = f , J 1 ( f ) = |∇ g D f | 2 , J 2 ( f ) = ∆ g D f , J 11 ( f ) , J 12 ( f ) . Coframe (or frame): df , Idf Invariantization map: D → R 2 J f : � f , |∇ g D f | 2 � = J f , and functions � f , |∇ g D f | 2 � � f , |∇ g D f | 2 � � ∆ g D f = F 2 , J 11 ( f ) = F 11 , J 12 ( f ) = F 12 Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Classification We say that function f is regular if J 1 ( f ) � = 0 and J 12 ( f ) � = 0 . Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Classification We say that function f is regular if J 1 ( f ) � = 0 and J 12 ( f ) � = 0 . For such a function find functions F 2 , F 11 , F 12 and consider the above PDEs system. This is Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Classification We say that function f is regular if J 1 ( f ) � = 0 and J 12 ( f ) � = 0 . For such a function find functions F 2 , F 11 , F 12 and consider the above PDEs system. This is Frobenius type system Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19

Classification We say that function f is regular if J 1 ( f ) � = 0 and J 12 ( f ) � = 0 . For such a function find functions F 2 , F 11 , F 12 and consider the above PDEs system. This is Frobenius type system Integrable Workshop on “Infinite-dimensional Riemannian KL (Institute) Lobachevsky Geometry and Image Recognition / 19