DiscreteScaleCWT

DiscreteScaleCWT - 1 Discrete Scale Continuous Wavelet...

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1 Discrete Scale Continuous Wavelet Transform Computations with the Continuous Wavelet Transform (CWT) has been hindered by numerical complexity especially in the evaluation of the inverse transform. In this note we show that a frequency domain approach provides a convenient tool for numerical and theoretical developments. A particularly significant result arising from our development is a modification in the transform that is required when a discrete set of scales is used. The modification is required if one needs to preserve the energy distribution property of the scalogram A Frequency Domain Formulation for the CWT Notation To simplify understanding, we first establish the notation followed in the paper. We have some wavelet ψ ( t ) L 2 ( < ) satisfying the admissibility condition C ψ = Z -∞ | Ψ( ω ) | 2 | ω | dω < and the family of its translations and dilations ψ ab ( t ) = 1 a ψ t - b a We have two function spaces that are used in addition to the L 2 ( < ) space: H = c ( a, b ) : C - 1 ψ Z a Z b | c ( a, b ) | 2 dadb a 2 < A = s ( a ) : C - 1 ψ Z a | s ( a ) | 2 da a 2 < We know that H = A ⊗ L 2 ( < ) and that the space of the continuous wavelet transforms ( CWT ) is a proper closed subspace, M ⊂ H . The continuous wavelet transform is the map Γ : L 2 ( < ) → H defined by c x ψ = Γ[ x ]; x L 2 ( < ) (1) c x ψ ( a, b ) = < x, ψ ab > L 2 ( < ) (2) = Z x ( t ) 1 a ψ t - b a dt (3) The adjoint transformation, Γ * : H → L 2 ( < ) , has the definition x c = Γ * [ c ]; c ∈ H = C - 1 ψ Z a Z b c ( a, b ) 1 a ψ t - b a dadb a 2 Moreover, the map ΓΓ * is an orthogonal projector in H with range M and Γ * Γ is the identity in L 2 . This second property is the one that allows the definition of | c x ψ ( a, b ) | 2 as a time frequency distribution since one has the identity Z | x ( t ) | 2 dt = C - 1 ψ Z a Z b | c x ( a, b ) | 2 dadb a 2
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Prepared by Dr. Jorge L. Aravena. Last revised on December 21, 1999 2 A Frequency Domain Formulation For a give scale the function c x ψ ( a, · ) L 2 . One can compute its Fourier transform and define the new function C x ψ ( a, ω ) = Z Z x ( t ) ψ ab ( t ) e - jωb dtdb = Z t x ( t ) Z b ψ ab ( t ) e - jωb dbdt = Z t x ( t ) a Ψ( ) e - jωt dt = X ( ω ) a Ψ( ) Since the wavelet is admissible, one has C - 1 ψ Z a a | Ψ( ) | 2 da a 2 = 1 Hence, one has a bounded linear map from the ’frequency space’, L 2 ( < , 2 π ) C x ( a, ω ) = Λ[ X ( ω )] = X ( ω ) a Ψ( ) The adjoint map can be easily determined and one can show Λ * [ C ( a, ω )]( ω ) = C - 1 ψ Z a C ( a, ω ) a Ψ( ) da a 2 With the definition of adjoint one can readily show Λ * Λ[ C x ( )] = X ( ω ) C - 1 ψ Z a Z ω | C x ( a, ω ) | 2 da a 2 2 π = Z ω | X ( ω ) | 2 2 π Hence, | C x ( a, ω ) | 2 can be interpreted as a scale-frequency energy distribution. Discrete Scale CWT We now study the practical situation when only a discrete, and finite, set of scales is used. Assume that the function C x ψ ( a, ω ), is computed on a discrete grid of scales, a i , i = 1 , 2 , . . . , M . The result is now an M-vector valued function of the frequency. In order to preserve the similarity with the continuous scale case, one can use a weighted inner product for M-vectors h x, y i = C - 1 ψ X i x i y i a i +1 - a i a 2 i by default, we assume a M +1 = 2 a
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