Approaching NSP-EMD with B-Splines Laslo Hunhold Oberseminar in SS - PowerPoint PPT Presentation

Approaching NSP-EMD with B-Splines Laslo Hunhold ‘Oberseminar’ in SS 2018 Mathematisches Institut Universität zu Köln hosted by Prof. Dr. Angela Kunoth 7th June 2018 1 / 14

introduction intrinsic mode function (IMF) (1/2) 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) if and only if for all t ∈ R ◮ a ( t ) > 0, φ ′ ( t ) > 0, 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) if and only if for all t ∈ R ◮ a ( t ) > 0, φ ′ ( t ) > 0, ◮ | a ′ ( t ) | ≤ ε | φ ′ ( t ) | , 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) if and only if for all t ∈ R ◮ a ( t ) > 0, φ ′ ( t ) > 0, ◮ | a ′ ( t ) | ≤ ε | φ ′ ( t ) | , ◮ | φ ′′ ( t ) | ≤ ε | φ ′ ( t ) | . 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) if and only if for all t ∈ R ◮ a ( t ) > 0, φ ′ ( t ) > 0, ◮ | a ′ ( t ) | ≤ ε | φ ′ ( t ) | , ◮ | φ ′′ ( t ) | ≤ ε | φ ′ ( t ) | . heuristic 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) if and only if for all t ∈ R ◮ a ( t ) > 0, φ ′ ( t ) > 0, ◮ | a ′ ( t ) | ≤ ε | φ ′ ( t ) | , ◮ | φ ′′ ( t ) | ≤ ε | φ ′ ( t ) | . heuristic ◮ Physical reasonability. 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) if and only if for all t ∈ R ◮ a ( t ) > 0, φ ′ ( t ) > 0, ◮ | a ′ ( t ) | ≤ ε | φ ′ ( t ) | , ◮ | φ ′′ ( t ) | ≤ ε | φ ′ ( t ) | . heuristic ◮ Physical reasonability. ◮ Slowly varying amplitude. 2 / 14

introduction intrinsic mode function (IMF) (1/2) definition Let a , φ ∈ C 2 b ( R ) with bounded derivatives. a ( t ) cos ( φ ( t )) : ∈ I is an IMF with accuracy ε ∈ (0 , 1) if and only if for all t ∈ R ◮ a ( t ) > 0, φ ′ ( t ) > 0, ◮ | a ′ ( t ) | ≤ ε | φ ′ ( t ) | , ◮ | φ ′′ ( t ) | ≤ ε | φ ′ ( t ) | . heuristic ◮ Physical reasonability. ◮ Slowly varying amplitude. ◮ Slowly varying frequency. 2 / 14

introduction intrinsic mode function (IMF) (2/2) example 3 / 14

introduction intrinsic mode function (IMF) (2/2) example a ( t ) cos ( φ ( t )) a ( t ) φ ( t ) 3 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input signal s : [ p , q ] → R 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input signal s : [ p , q ] → R decomposition 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input signal s : [ p , q ] → R decomposition w � s ( t ) = s k ( t ) + r w +1 ( t ) , s k ∈ I k =0 s k ( t ) = a k ( t ) cos ( φ k ( t )) 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input signal s : [ p , q ] → R decomposition w � s ( t ) = s k ( t ) + r w +1 ( t ) , s k ∈ I k =0 s k ( t ) = a k ( t ) cos ( φ k ( t )) iterative 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input signal s : [ p , q ] → R decomposition w � s ( t ) = s k ( t ) + r w +1 ( t ) , s k ∈ I k =0 s k ( t ) = a k ( t ) cos ( φ k ( t )) iterative r 0 ( t ) = s ( t ) r k +1 ( t ) = r k ( t ) − s k ( t ) 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input signal s : [ p , q ] → R decomposition w � s ( t ) = s k ( t ) + r w +1 ( t ) , s k ∈ I k =0 s k ( t ) = a k ( t ) cos ( φ k ( t )) iterative r 0 ( t ) = s ( t ) r k +1 ( t ) = r k ( t ) − s k ( t ) goal 4 / 14

introduction empirical mode decomposition (EMD) (N.E. Huang et al., 1998) input signal s : [ p , q ] → R decomposition w � s ( t ) = s k ( t ) + r w +1 ( t ) , s k ∈ I k =0 s k ( t ) = a k ( t ) cos ( φ k ( t )) iterative r 0 ( t ) = s ( t ) r k +1 ( t ) = r k ( t ) − s k ( t ) goal Find a k and φ k . 4 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (1/2) 5 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (1/2) idea 5 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (1/2) idea Let P k = P k ( a k , φ k ), Q k = Q k ( a k , φ k ), R k = R k ( a k , φ k ). 5 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (1/2) idea Let P k = P k ( a k , φ k ), Q k = Q k ( a k , φ k ), R k = R k ( a k , φ k ). Find differential operator D P k , Q k , R k analytically such that D P k , Q k , R k s k = 0 . 5 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (1/2) idea Let P k = P k ( a k , φ k ), Q k = Q k ( a k , φ k ), R k = R k ( a k , φ k ). Find differential operator D P k , Q k , R k analytically such that D P k , Q k , R k s k = 0 . Solve ( P k , Q k , R k ) = arg min �Q 1 ( r k − ˜ s ) � (˜ P , ˜ Q , ˜ R ) s.t. D ˜ R ˜ s = 0 P , ˜ Q , ˜ �Q 2 ˜ P � ≤ τ �Q 3 ˜ Q � ≤ τ �Q 4 ˜ R � ≤ τ with τ > 0, Q 1 , Q 2 , Q 3 , Q 4 ∈ { D 0 , D 1 , . . . } regularization operators, ˜ a , ˜ φ ), ˜ a , ˜ φ ), ˜ a , ˜ P := P k (˜ Q := Q k (˜ R := R k (˜ φ ) and a ( t ) cos (˜ ˜ s := ˜ φ ( t )). 5 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (2/2) 6 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (2/2) unconstrained formulation 6 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (2/2) unconstrained formulation ( P k , Q k , R k ) = arg min �D ˜ R ˜ s � + P , ˜ Q , ˜ (˜ P , ˜ Q , ˜ R ) λ �Q 1 ( r k − ˜ s ) � + µ ( �Q 2 ˜ P � + �Q 3 ˜ Q � + �Q 4 ˜ R � ) 6 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (2/2) unconstrained formulation ( P k , Q k , R k ) = arg min �D ˜ R ˜ s � + P , ˜ Q , ˜ (˜ P , ˜ Q , ˜ R ) λ �Q 1 ( r k − ˜ s ) � + µ ( �Q 2 ˜ P � + �Q 3 ˜ Q � + �Q 4 ˜ R � ) problem 6 / 14

introduction operator-based signal separation (OSS) (S. Peng et al., 2008) (2/2) unconstrained formulation ( P k , Q k , R k ) = arg min �D ˜ R ˜ s � + P , ˜ Q , ˜ (˜ P , ˜ Q , ˜ R ) λ �Q 1 ( r k − ˜ s ) � + µ ( �Q 2 ˜ P � + �Q 3 ˜ Q � + �Q 4 ˜ R � ) problem Want to control ˜ s , and thus s k , in some way. 6 / 14

introduction null space pursuit (NSP) (S. Peng, W.-L. Hwang et al., 2010) 7 / 14

introduction null space pursuit (NSP) (S. Peng, W.-L. Hwang et al., 2010) solution 7 / 14

introduction null space pursuit (NSP) (S. Peng, W.-L. Hwang et al., 2010) solution Introduce leakage factor γ . 7 / 14

introduction null space pursuit (NSP) (S. Peng, W.-L. Hwang et al., 2010) solution Introduce leakage factor γ . ( P k , Q k , R k ) = arg min �D ˜ R ˜ s � + P , ˜ Q , ˜ (˜ P , ˜ Q , ˜ R ) λ ( �Q 1 ( r k − ˜ s ) � + γ � ˜ s � )+ µ ( �Q 2 ˜ P � + �Q 3 ˜ Q � + �Q 4 ˜ R � ) 7 / 14

introduction null space pursuit (NSP) (S. Peng, W.-L. Hwang et al., 2010) solution Introduce leakage factor γ . ( P k , Q k , R k ) = arg min �D ˜ R ˜ s � + P , ˜ Q , ˜ (˜ P , ˜ Q , ˜ R ) λ ( �Q 1 ( r k − ˜ s ) � + γ � ˜ s � )+ µ ( �Q 2 ˜ P � + �Q 3 ˜ Q � + �Q 4 ˜ R � ) Controls retainment of information in residual signal. 7 / 14

introduction null space pursuit (NSP) (S. Peng, W.-L. Hwang et al., 2010) solution Introduce leakage factor γ . ( P k , Q k , R k ) = arg min �D ˜ R ˜ s � + P , ˜ Q , ˜ (˜ P , ˜ Q , ˜ R ) λ ( �Q 1 ( r k − ˜ s ) � + γ � ˜ s � )+ µ ( �Q 2 ˜ P � + �Q 3 ˜ Q � + �Q 4 ˜ R � ) Controls retainment of information in residual signal. remaining problems (OSS, NSP) 7 / 14