The Hitchhiker s Guide to the Dual Tree Complex Wavelet

The Hitchhiker’s Guide to the Dual-Tree Complex Wavelet Transform —DON’T PANIC— October 26, 2007 Outline The Hilbert Transform Definition The Fourier Transform Definition Invertion Fourier Approximation The Wavelet Transform Definition Invertion Wavelet Approximation The Dual-Tree Complex Wavelet Transform Definition The Hilbert Transform Definition (Hilbert Transform) Given a real-valued function of a real variable, f : R → R, define Hf : R → R by Hf (y) = lim 1 ε→0 π f (y − x) dx x {|x|>ε} f (x) Hf (ξ) The Fourier Transform Definition (Fourier Transform) Given a complex-valued function of real variable f : R → C, define Ff : R → C by ∞ Ff (ξ) = −∞ f (x)e−2πixξ dx = f (x), e2πixξ . f (x) |Ff (ξ)| Invertion of the Fourier Transform Theorem For f “good enough”, ∞ ∞ f (x) = −∞ Ff (ξ)e2πixξ dξ = −∞ f (x), e2πixξ e2πixξ dξ. = n∈Z f (x), e2πixn e2πixn Hilbert Transform via Fourier Transform f (x) H Hf (y) Ff (ξ) −i sign(ξ)Ff (ξ) Hilbert Transform via Fourier Transform f (x) H Hf (y) F F Ff (ξ) −i sign(ξ)Ff (ξ) FHf (ξ) Hilbert Transform via Fourier Transform f (x) H Hf (y) F F Ff (ξ) −i sign(ξ)Ff (ξ) Hilbert Transform via Fourier Transform f (x) H Hf (y) F F Ff (ξ) −i sign(ξ)Ff (ξ) Hilbert Transform via Fourier Transform f (x) H Hf (y) F F −1 Ff (ξ) −i sign(ξ)Ff (ξ) Discretization of the Fourier Transform Theorem For f “good enough”, ∞ ∞ f (x) = −∞ Ff (ξ)e2πixξ dξ = −∞ f (x), e2πixξ e2πixξ dξ Ff (n)e2πixn n∈Z = n∈Z f (x), e2πixn e2πixn = Fourier Approximation N Use partial sums n=−N Ff (n)e2πixn to approximate f . f (x) Fourier Approximation N Use partial sums n=−N Ff (n)e2πixn to approximate f . f (x) and 3-term approximation Fourier Approximation N Use partial sums n=−N Ff (n)e2πixn to approximate f . f (x) and 5-term approximation Fourier Approximation N Use partial sums n=−N Ff (n)e2πixn to approximate f . f (x) and 11-term approximation Fourier Approximation N Use partial sums n=−N Ff (n)e2πixn to approximate f . f (x) and 33-term approximation Fourier Approximation N Use partial sums n=−N Ff (n)e2πixn to approximate f . Gibbs phenomenom f (x) and 33-term approximation Fourier Approximation N Use partial sums n=−N Ff (n)e2πixn to approximate f . f (x) and 65-term approximation The Wavelet Transform Definition (Wavelets) Functions ψ : R → R satisfying the admissibility condition: ∞ |Fψ(ξ)|2 −∞ dξ |ξ| = 1. Denote the shifts and dilations of the wavelet by ψa,b (x) = 1 |a|1/2 ψ x−b a , a, b ∈ R. The Wavelet Transform Definition (Wavelets) Functions ψ : R → R satisfying the admissibility condition: ∞ |Fψ(ξ)|2 −∞ dξ |ξ| = 1. Denote the shifts and dilations of the wavelet by ψa,b (x) = 1 |a|1/2 ψ x−b a , a, b ∈ R. Definition (Wavelet Transform) Given a real-valued function of real variable f : R → R, define Wf : R2 → R by ∞ Wf (a, b) = −∞ f (x)ψa,b (x) dx = f, ψa,b . Invertion of the Wavelet Transform Theorem For f “good enough”, ∞ ∞ −∞ f (x) = −∞ Wf (a, b)ψa,b (x) db da a2 = m∈Z n∈Z Wf (2−m , n2m ) ψ2−m ,n2m (x) Discretization of the Wavelet Transform Theorem For f “good enough”, ∞ ∞ −∞ f (x) = −∞ Wf (a, b)ψa,b (x) db da a2 = m∈Z n∈Z Wf (2−m , n2m ) ψ2−m ,n2m (x) Wavelet Approximation Use partial sums approximate f M −M N −N Wf (2−m , n2m ) ψ2−m ,n2m (x) to f (x) Wavelet Approximation Use partial sums approximate f M −M N −N Wf (2−m , n2m ) ψ2−m ,n2m (x) to f (x) and 1-term approximation Wavelet Approximation Use partial sums approximate f M −M N −N Wf (2−m , n2m ) ψ2−m ,n2m (x) to f (x) and 2-term approximation Wavelet Approximation Use partial sums approximate f M −M N −N Wf (2−m , n2m ) ψ2−m ,n2m (x) to f (x) and 4-term approximation Wavelet Approximation Use partial sums approximate f M −M N −N Wf (2−m , n2m ) ψ2−m ,n2m (x) to f (x) and 8-term approximation Wavelet Approximation Use partial sums approximate f M −M N −N Wf (2−m , n2m ) ψ2−m ,n2m (x) to f (x) and 16-term approximation Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels 1 wavelet coefficient Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels 1 + 4 = 5 wavelet coefficients Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels 1 + 4 + 16 = 21 wavelet coefficients Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels 1 + 4 + 16 + 64 = 85 wavelet coefficients Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels 1 + 4 + 16 + 64 + 256 = 341 wavelet coefficients Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels 1 + 4 + 16 + 64 + 256 + 1024 = 1365 wavelet coefficients Wavelet Approximation 1, 024 × 1, 024 = 1, 048, 576 pixels 1 + 4 + 16 + 64 + 256 + 1024 + 4096 = 5461 wavelet coefficients Problems with Real Wavelets Poor Directional Selectivity The standard tensor-product construction of multi-variate wavelets produces a checkerboard pattern that is simultaneously oriented along several directions. This lack of directional selectivity complicates processing of geometric image features like ridges and edges. N. Kingsbury: “Complex Wavelets for Shift Invariant Analysis and Filtering of Signals” Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Curvelets (Cand´s & Donoho, 1999) e Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Curvelets (Cand´s & Donoho, 1999) e Wedgelets (Donoho, 1999) Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Curvelets (Cand´s & Donoho, 1999) e Wedgelets (Donoho, 1999) Beamlets (Donoho & Huo, 2001) Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Curvelets (Cand´s & Donoho, 1999) e Wedgelets (Donoho, 1999) Beamlets (Donoho & Huo, 2001) Surflets (Baraniuk, 2004) Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Curvelets (Cand´s & Donoho, 1999) e Wedgelets (Donoho, 1999) Beamlets (Donoho & Huo, 2001) Surflets (Baraniuk, 2004) Shearlets (G. Kutyniok, 2005) Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Curvelets (Cand´s & Donoho, 1999) e Wedgelets (Donoho, 1999) Beamlets (Donoho & Huo, 2001) Surflets (Baraniuk, 2004) Shearlets (G. Kutyniok, 2005) Needlets (P. Petrushev, 2005) Attempts to solve this problem in the last 10 years Brushlets (Meyer & Coifman, 1997) Ridgelets (Cand´s, 1998) e Curvelets (Cand´s & Donoho, 1999) e Wedgelets (Donoho, 1999) Beamlets (Donoho & Huo, 2001) Surflets (Baraniuk, 2004) Shearlets (G. Kutyniok, 2005) Needlets (P. Petrushev, 2005) (Your name here)-lets The Dual-Tree Complex Wavelet Transform N. Kingsbury: “Complex Wavelets for Shift Invariant Analysis and Filtering of Signals” The key: “Hilbert Transform pairs” Use complex-valued functions Ψ : R → C satisfying Ψ(x) = ψ(x) + iHψ(x). where both ψ and Hψ are real-valued. That’s a neat idea! If both ψ and Hψ are wavelets, we perform two different wavelet transforms, one with ψ, one with Hψ. For each choice a, b ∈ R, combine the corresponding real-valued wavelet coefficients f, ψa,b and f, Hψa,b to form a single complex-valued coefficient: f, Ψa,b = f, ψa,b + i f, Hψa,b . That’s a neat idea! If both ψ and Hψ are wavelets, we perform two different wavelet transforms, one with ψ, one with Hψ. For each choice a, b ∈ R, combine the corresponding real-valued wavelet coefficients f, ψa,b and f, Hψa,b to form a single complex-valued coefficient: f, Ψa,b = f, ψa,b + i f, Hψa,b . WARNING! The Dual-Tree Complex Wavelet Transform is not a transform per se. It is a smart way to combine the information we obtain from two transforms.