Generalized Wavelets from a Representation Theory Viewpoint Vignon S. Oussa Saint Louis University October 2011 Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 1 / 25
Content Signals on sets, measures and group structures. 1 Representation theory. 2 Admissibility of representations and generalized wavelets. 3 Example of generalized wavelets on Z n and D 2 n . 4 Admissibility of representation on the circle and the real line. 5 Euclidean motion group. 6 Continuous wavelets on groups of matrices. 7 Conclusion. 8 Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 2 / 25
Signals on a set Important questions in signal analysis How do we represent data on some arbitrary set? How do we use the structure of the set to our advantage? Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 3 / 25
Sets will be groups endowed with some natural measure. Signals here will be square integrable functions. (Hilbert space) To represent signals: bases, discrete wavelets, continuous wavelets. Examples Sets Group structures measures R n abelian Lebesgue measure T abelian Lebesgue measure abelian counting measure Z n D 2 n non abelian counting measure GL ( n , R ) non commutative Haar measure Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 4 / 25
What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L 2 ( R ) such that the system n o � � f j , k = 2 j / 2 f 2 j x � k : j , k 2 Z forms an orthonormal basis in L 2 ( R ) . Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25
What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L 2 ( R ) such that the system n o � � f j , k = 2 j / 2 f 2 j x � k : j , k 2 Z forms an orthonormal basis in L 2 ( R ) . Given g 2 L 2 ( R ) , have unique representation. g ( x ) = ∑ h g , f j , k i f j , k j , k 2 Z Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25
What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L 2 ( R ) such that the system n o � � f j , k = 2 j / 2 f 2 j x � k : j , k 2 Z forms an orthonormal basis in L 2 ( R ) . Given g 2 L 2 ( R ) , have unique representation. g ( x ) = ∑ h g , f j , k i f j , k j , k 2 Z Example . The Haar wavelet 8 if x 2 [ 0 , 1 / 2 ) < 1 f ( x ) = � 1 if x 2 [ 1 / 2 , 1 ) : 0 elsewhere Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25
What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L 2 ( R ) such that the system n o � � f j , k = 2 j / 2 f 2 j x � k : j , k 2 Z forms an orthonormal basis in L 2 ( R ) . Given g 2 L 2 ( R ) , have unique representation. g ( x ) = ∑ h g , f j , k i f j , k j , k 2 Z Example . The Haar wavelet 8 if x 2 [ 0 , 1 / 2 ) < 1 f ( x ) = � 1 if x 2 [ 1 / 2 , 1 ) : 0 elsewhere How do we generalize wavelets to other groups? Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25
Representation Theory A unitary representation is a strongly continuous homomorphism π from a group into a group of unitary operators on some Hilbert space H π : G ! U ( H ) A unitary representation π is reducible: existence non trivial closed subspace of H 1 � H s.t π ( G ) H 1 � H 1 . π is irreducible if all invariant subspaces of H are trivial. b G = unitary dual set of unitary irreducible rep. of G up to equivalence . Fact b G is needed for Fourier analysis on G . Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 6 / 25
Examples of unitary representations Trivial representation π : C ! U ( C ) = T . π ( z ) = 1 Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25
Examples of unitary representations Trivial representation π : C ! U ( C ) = T . π ( z ) = 1 Left regular representation on R . ( L ( x ) F ) ( y ) = F ( y � x ) for any x 2 R . Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25
Examples of unitary representations Trivial representation π : C ! U ( C ) = T . π ( z ) = 1 Left regular representation on R . ( L ( x ) F ) ( y ) = F ( y � x ) for any x 2 R . Irreducible representation of R . χ ( x ) z = e ix z for all x 2 R . Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25
Examples of unitary representations Trivial representation π : C ! U ( C ) = T . π ( z ) = 1 Left regular representation on R . ( L ( x ) F ) ( y ) = F ( y � x ) for any x 2 R . Irreducible representation of R . χ ( x ) z = e ix z for all x 2 R . R = R , b b T = Z and b Z = T . �� a � � x G = : x 2 R , a > 0 , 0 1 b G = R [ f 1 , � 1 g Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25
Examples of unitary representations Trivial representation π : C ! U ( C ) = T . π ( z ) = 1 Left regular representation on R . ( L ( x ) F ) ( y ) = F ( y � x ) for any x 2 R . Irreducible representation of R . χ ( x ) z = e ix z for all x 2 R . R = R , b b T = Z and b Z = T . �� a � � x G = : x 2 R , a > 0 , 0 1 b G = R [ f 1 , � 1 g 8 2 3 9 1 x z < = 4 5 : x , y , z 2 R G = 0 1 y ; , : 0 0 1 R 2 [ R � b G = Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25
Admissible representations and generalized wavelets Admissibility A representation π of G acting in H is admissible if W ψ : H ! L 2 ( G ) W ψ φ ( x ) = h φ , π ( x ) ψ i is an isometry � � � W ψ φ � = k φ k H L 2 ( G ) ψ is a generalized wavelet or admissible vector. Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 8 / 25
Admissible representations and generalized wavelets Admissibility A representation π of G acting in H is admissible if W ψ : H ! L 2 ( G ) W ψ φ ( x ) = h φ , π ( x ) ψ i is an isometry � � � W ψ φ � = k φ k H L 2 ( G ) ψ is a generalized wavelet or admissible vector. W ψ is called a wavelet transform. Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 8 / 25
Admissible representations and generalized wavelets Admissibility A representation π of G acting in H is admissible if W ψ : H ! L 2 ( G ) W ψ φ ( x ) = h φ , π ( x ) ψ i is an isometry � � � W ψ φ � = k φ k H L 2 ( G ) ψ is a generalized wavelet or admissible vector. W ψ is called a wavelet transform. Reconstruction of functions. f ( t ) = R G h f , π ( x ) ψ i π ( x ) ψ ( t ) dx . Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 8 / 25
Admissibility of Left regular representation (sketch) General idea Given L the left regular representation on a group G . Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25
Admissibility of Left regular representation (sketch) General idea Given L the left regular representation on a group G . W ψ φ ( x ) = φ � ψ ( x � 1 ) (convolution). Put ψ � ( x ) = ψ ( x � 1 ) . Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25
Admissibility of Left regular representation (sketch) General idea Given L the left regular representation on a group G . W ψ φ ( x ) = φ � ψ ( x � 1 ) (convolution). Put ψ � ( x ) = ψ ( x � 1 ) . Fourier transform . F : L 2 ( G ) ! L 2 � � b G Z � x � 1 � F f ( π ) = G f ( x ) π dx Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25
Admissibility of Left regular representation (sketch) General idea Given L the left regular representation on a group G . W ψ φ ( x ) = φ � ψ ( x � 1 ) (convolution). Put ψ � ( x ) = ψ ( x � 1 ) . Fourier transform . F : L 2 ( G ) ! L 2 � � b G Z � x � 1 � F f ( π ) = G f ( x ) π dx � � � = k φ k for all φ 2 L 2 ( G ) . � W ψ φ We want � � Z Z z }| { 2 � � k φ � ψ � k 2 = F ψ � ( π ) � � G kF φ ( π ) k 2 � F φ ( π ) � d π = HS d π . � b b G HS The way we choose F ψ � ( π ) characterizes the construction of wavelets. Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25
Discrete circle Z 3 Problem How do we obtain an orthonormal basis in l 2 ( Z 3 ) ' C 3 by simply shifting the components of a single vector u = ( u 1 , u 2 , u 3 ) ? Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 10 / 25
Wavelets on the discrete circle Z 3 n o 2 Z 3 = f 0 , 1 , 2 g with addition mod 3 and b 2 π ik Z 3 = e 3 k = 0 Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 11 / 25
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