Gabor wavelet analysis and the fractional Hilbert transform Kunal Narayan Chaudhury and Michael Unser (presented by Dimitri Van De Ville) Biomedical Imaging Group, Ecole Polytechnique Fédérale de Lausanne (EPFL) WAVELETS XIII, 2009
Outline 1 Introduction The dual-tree transform Multiresolution Gabor-like analysis 2 Amplitude-Shift Representation The fractional Hilbert transform Signal reconstruction Dual-tree Gabor analysis 3 Multi-D Extension Bivariate dual-tree wavelets Signal representation 4 Concluding Remarks
The dual-tree transform Hilbert transform (HT) pairs of wavelet bases Dual-tree transform introduced by Kingsbury to improve the shiftability of the decimated discrete wavelet transforms [Kingsbury, 2001]. HT pairs of wavelet bases {ψ α(x)} and {ψ α(x)} [Selesnick, 2001]. Formally, given a primary wavelet basis {ψ α(x)}, one constructs a secondary wavelet basis {ψ α(x)} with the correspondence where H denotes the HT operator, ψ α(x) = H ψ α(x) (for every α) dh f(ω) = jsign(ω) ˆf(ω). H maps cos(ω 0x) into sin(ω 0x), and acts as a orthogonal transform on finite-energy signals.
The dual-tree transform The invariance trick The primary (dyadic) basis generated through dilations and translations: ψ i,k (x) = Ξ i,k ψ(x) (i, k Z) where Ξ i,k is the discrete dilation-translation operator. Invariances of the HT [Chaudhury and Unser, 2009]: (i) H is unitary; maps a basis into a basis, (ii) H commutes with dilations and translations, particularly with Ξ i,k. Setting ψ (x) = H ψ(x) suffices; the functions ψ i,k(x) = Ξ i,k ψ (x) constitute the secondary wavelet basis. Identical argument holds for the dual bases { ψ i,k (x)} and { ψ i,k(x)}.
The dual-tree transform Signal analysis: local amplitude-phase factors Input signal f(x) is analyzed in both the bases: (P (i,k) Z a i[k]ψ f(x) = 2 i,k (x). P (i,k) Z b i[k]ψ i,k(x). 2 Notion of a complex analysis wavelet Ψ(x) = 1 ψ(x) + j 2 ψ (x). Complex wavelet coefficients c i[k] = f, Ψ i,k = 1 (ai[k] + jbi[k]), 2 and the associated amplitude-phase factors c i[k] e jφ i[k].
Multiresolution Gabor-like analysis Multiresolution Gabor-like transform realized within the framework of the dual-tree transform [C. and Unser, 2009]. The family of fractional (semi-orthogonal) spline wavelets 1 ψ(x; α, τ) is closed w.r.t the HT: H {ψ(x; α, τ)} = ψ(x; α, τ + 1/2). Moreover, the complex spline wavelet Ψ(x) = ψ(x; α, τ) + jψ(x; α, τ + 1/2) asymptotically converges to a Gabor function [Unser et al.]: Ψ(x; α) ϕ(x) exp`jω 0x + ξ 0 (α + ). (Movie1) Evolution of ψ(x; α, τ) with the increase in α. 1 indexed by the approximation order α + 1 and shift τ.
The fractional Hilbert transform Representation of f(x) in terms of the amplitude-shift factors? The group of fractional Hilbert transforms 2 (fht) H τ = cos(πτ) I sin(πτ) H (τ R) comprising the identity (I ) and the HT (H ) operator. Interpolates the phase-shift property of the HT: H τ {cos(ω 0x)} = cos(ω 0x + πτ). The fhts inherit the invariances of the HT: (i) Preserves (norm) energy. (ii) Invariant to translations and dilations. 2 linear shift τ = φ/π corresponding to the phase factor φ
Signal reconstruction Signal reconstruction from the amplitude-phase factors: f(x) = 1 X a i[k]ψ i,k (x) + b i[k]ψ i,k(x) 2 = X (i,k) Z 2 The fractionally-shifted wavelets ψi,k (x) c i[k] H φi [k]/π (i,k) Z 2 = X c i[k] Ξ i,k ψ(x; τi[k]). (i,k) Z 2 ψ(x; τ i[k]) = H τi [k]ψ(x) have identical norms. = c i[k] indicates the strength of local wavelet correlation. Local signal displacement encoded in the shift τ i[k]. = specifies the most appropriate wavelet within the family {H τ ψ i,k } τ R.
Dual-tree Gabor analysis fht of a modulated wavelet The case when ψ(x) is a modulated wavelet of the form ψ(x) = ϕ(x) cos `ω 0x + ξ 0. The phase-shift action of the fht is preserved in the presence of the window (under appropriate conditions): Proposition (Extension of the Bedrosian theorem) Let ϕ(x) be bandlimited to ( ω 0, ω 0). Then the following holds H τ ϕ(x) cos(ω0x) = ϕ(x) cos(ω 0x + πτ). = The fht acts only on the phase of the oscillation while the window remains fixed.
Dual-tree Gabor analysis Characterization of the Gabor-like transform Mutiresolution windowed-fourier-like representation, f(x) X (i,k) Z 2 fixed window z } { n variable amp phase oscillation z } { ϕ i,k (x) Ξ i,k ci[k] cos `ω0x + ξ 0 + πτ i[k] o. ϕ i,k (x): fixed Gaussian window at scale i and translation k. c i[k]: measures the local signal energy. τ i[k]: shift applied to the modulating sinusoid the oscillation is shifted to fit the underlying signal singularities/transitions.
Dual-tree Gabor analysis Action of the fht on the Gabor-like wavelet (Movie2) Visualization of the action of the fhts on the Gabor-like wavelet. (a) τ = 1 (b) τ = 3 4 (c) τ = 1 4 (d) τ = 0 (e) τ = 1 3 (f) τ = 1 2 Figure: Quadrature pairs of Gabor-like spline wavelets obtained by the action of fht. Blue: H τ ψ(x; 8, 0), Red: H τ+ 1 2 ψ(x; 8, 0), Black: The fixed Gaussian-like localization window.
Bivariate dual-tree wavelets Wavelet Construction Kingsbury constructed direction-selective wavelets by appropriately combining the positive and negative frequency bands of analytic wavelets [Kingsbury, 2001]. Four separable multiresolutions, total of 3 4 = 12 separable wavelets. Direction-selective complex wavelets Ψ 1(x),..., Ψ 6(x) realized through linear combinations of the 12 wavelets. Similarly, one has the six complex duals Ψ 1(x),..., Ψ 6(x).
Bivariate dual-tree wavelets Directional Gabor-like wavelets Tensor products involving B-spline scaling function and B-spline wavelets [C. and Unser, 2009]. Dual-tree wavelets resemble Gaussian-windowed plane waves. Figure: Left: Real component of the complex wavelets, Right: Magnitude envelope of the complex wavelets.
Bivariate dual-tree wavelets Directional HT (dht) Correspondence between the real and imaginary components? Directional extension of the HT: Ĥ θ f(ω) = jsign(u T θ ω) ˆf(ω), where u θ is the unit vector along direction θ. The dht correspondences for the complex wavelets [C. and Unser, 2009]: Ψ l (x) = ψ l (x) + jh θl ψ l (x) (l = 1,..., 6) where θ 1 = θ 2 = 0; θ 3 = θ 4 = π/2; θ 5 = π/4; and θ 6 = 3π/4.
Bivariate dual-tree wavelets Fractional directional HT (Notion of direction-selective phase shifts) Fractional extensions of the directional HT: H θ,τ = cos(πτ) I sin(πτ) H θ (τ R). They are unitary, and commute with translations and (uniform) dilations. Action on windowed plane waves of the form ϕ(x) cos(ωu T θ x): Proposition Suppose that ϕ(x) be bandlimited to the disk {ω : ω < Ω}. Then H θ,τ ϕ(x) cos(ωu T θ x) = ϕ(x) cos(ωu T θ x + πτ).
Signal representation Complex wavelet coefficients c l i[k] = 1 4 f, Ψl,i,k. Signal representation for the bivariate dual-tree transform [4]: f(x) = X c l i[k] Ξ i,k ψl`x; τ l i [k]. (l,i,k) The explicit form for the Gabor-like transform: f(x) = X (l,i,k) fixed window z } { ϕ l,i,k (x) variable amp phase directional wave n z } { o Ξ i,k cl i [k] cos Ω l u T θ l x + πτi l [k]. = Superposition of direction-selective plane waves affected with appropriate phase-shifts (locally).
Remarks 1 A mathematical framework linking the reconstructed signal to the processed complex wavelet coefficients. 2 Applicable to generic modulated wavelets, e.g., Shannon wavelet. 3 Straightforward explanation of the improved shift-invariance of the dual-tree transform (cf. C. and Unser, 2009). 4 Potential interest in applications involving the dual-tree transform (e.g., signal denoising, texture analysis/synthesis).
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References N.G. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals, Journal of Applied and Computational Harmonic Analysis, 2001. K. N. Chaudhury and M. Unser, Construction of Hilbert transform pairs of wavelet bases and Gabor-like transforms, IEEE Trans. on Signal Processing, 2009. I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, The dual-tree complex wavelet transform, IEEE Signal Processing Magazine, 2005. K. N. Chaudhury and M. Unser, On the shiftability of dual-tree complex wavelet transforms, IEEE Trans. on Signal Processing, 2009.