Hilbert Transform Pairs of Wavelets. September 19, 2007

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1 Hilbert Transform Pairs of Wavelets September 19, 2007

2 Outline The Challenge Hilbert Transform Pairs of Wavelets The Reason The Dual-Tree Complex Wavelet Transform The Dissapointment Background Characterization by means of Frequency Responses A Workaround Approximate Hilbert Transform Pairs The Design Problem

3 Hilbert Transform Pairs 1 f(x y) Hf(x) = lim dy. ε 0 + π { y >ε} y equivalently, if f(ξ) is defined, F{Hf}(ξ) = i sign(ξ) f(ξ).

4 Hilber Pairs of Wavelets Given two mother wavelets ψ 1, ψ 2 : R R, find necessary and sufficient conditions so that ψ 2 = Hψ 1. If this is possible, find an example. If not, characterize how close we can get (and give an example).

5 The Dual-Tree Complex Wavelet Transform We are interested in approximants with the following properties: Fast algorithms Perfect Reconstruction Modeling geometric features Shift Invariance

6 The Dual-Tree Complex Wavelet Transform We are interested in approximants with the following properties: } Fast algorithms Orthogonal real-valued wavelets Perfect Reconstruction (MRA) Modeling geometric features Shift Invariance

7 The Dual-Tree Complex Wavelet Transform We are interested in approximants with the following properties: } Fast algorithms Orthogonal real-valued wavelets Perfect Reconstruction (MRA) } Modeling geometric features Complex valued wavelets ψ = u + ihu. Shift Invariance

8 The Dual-Tree Complex Wavelet Transform We are interested in approximants with the following properties: Fast algorithms Perfect Reconstruction Modeling geometric features Shift Invariance Dual-Tree Complex Wavelet Transform: Ψ = ψ + ihψ.

9 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1

10 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 L 2 (R) = W n ψ W 0 m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1

11 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 L 2 (R) = W n ψ W 0 h n = φ, φ 1,n g n = ψ, φ 1,n m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1

12 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 L 2 (R) = W n ψ W 0 h n = φ, φ 1,n g n = ψ, φ 1,n hn 2 = 1 hnhn+2k = δk gn = ( 1) n hn 1 m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1

13 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 L 2 (R) = W n ψ W 0 h n = φ, φ 1,n g n = ψ, φ 1,n hn 2 = 1 hnhn+2k = δk gn = ( 1) n hn 1 H(z) = h n z n G(z) = g n z n m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1

14 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 L 2 (R) = W n ψ W 0 h n = φ, φ 1,n g n = ψ, φ 1,n hn 2 = 1 hnhn+2k = δk gn = ( 1) n hn 1 H(z) = h n z n G(z) = g n z n H(z)H(1/z) + H( z)h( 1/z) = 2 G(z) = 1 z H( 1 z ) m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1

15 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 L 2 (R) = W n ψ W 0 h n = φ, φ 1,n g n = ψ, φ 1,n hn 2 = 1 hnhn+2k = δk gn = ( 1) n hn 1 H(z) = h n z n G(z) = g n z n m h (ξ) = h n e inξ H(z)H(1/z) + H( z)h( 1/z) = 2 G(z) = 1 z H( 1 ) z m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1

16 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 L 2 (R) = W n ψ W 0 h n = φ, φ 1,n g n = ψ, φ 1,n hn 2 = 1 hnhn+2k = δk gn = ( 1) n hn 1 H(z) = h n z n G(z) = g n z n m h (ξ) = h n e inξ H(z)H(1/z) + H( z)h( 1/z) = 2 G(z) = 1 z H( 1 ) z m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1 mg(ξ) = e iξ mh(ξ π)

17 Multiresolution Analysis V 2 V 1 V 0 V 1 φ V 0 φ(ξ) = mh(ξ/2) φ(ξ/2) ψ(ξ) = mg(ξ/2) φ(ξ/2) L 2 (R) = W n ψ W 0 h n = φ, φ 1,n g n = ψ, φ 1,n hn 2 = 1 hnhn+2k = δk gn = ( 1) n hn 1 H(z) = h n z n G(z) = g n z n m h (ξ) = h n e inξ H(z)H(1/z) + H( z)h( 1/z) = 2 G(z) = 1 z H( 1 ) z m g (ξ) = g n e inξ mh(ξ) 2 + mh(ξ + π) 2 = 1 mg(ξ) = e iξ mh(ξ π)

18 The Phase Condition Assume there exists a 2π-periodic function θ : R R so that m h2 (ξ) = e iθ(ξ) m h1 (ξ). φ2 (ξ) = e i P k=1 θ(ξ/2k ) φ1 (ξ). m g2 (ξ) = e iθ(ξ π) m g1 (ξ). { ψ2 (ξ) = e i θ(ξ/2 π) P } k=1 θ(ξ/2k+1 ) ψ 1 (ξ). Therefore, { i sign(ξ) = e i θ(ξ/2 π) P } k=2 θ(ξ/2k ) θ(ξ/2 + π) implies ψ 2 = Hψ 1. θ(ξ/2 k+1 ) = π 2 sign(ξ). k=1 e.g. θ(ξ) = ξ/2 for π < ξ < π.

19 Approximate Hilbert Transform Pairs The Phase condition is equivalent to a half-sample delay : h (2) n = h (1) n+1/2. Design approximate Hilbert transform pairs, by making error function small near ξ = 0: Equivalently, near z = 1, E(ξ) = m h2 (2ξ) e iξ m h1 (2ξ). E(z) = G(z 2 ) 1 z H(z2 ).

20 The Design Problem Construct shortest FIR filters m h1, m h2, with specified number of zero moments K 1, K 2 ; and m h2 (ξ) e iξ/2 m h1 (ξ).

21 The Design Problem Construct shortest FIR filters m h1, m h2, with specified number of zero moments K 1, K 2 ; and m h2 (ξ) e iξ/2 m h1 (ξ). The CoCoA Method The Flat-Delay Allpass Filter Method.

22 Computational Commutative Algebra (CoCoA) δ k = h (1) n h (1) n+2k δ k = h (2) n+2k N 1 /2 conditions N 2 /2 conditions H 1 (z) = Q(z) ( z ) K1 K1 conditions H 2 (z) = Q(z) ( z ) K2 K2 conditions H 2 (z 2 ) 1 z H 1(z) = Q(z) ( 1 1 z ) L L conditions

23 Computational Commutative Algebra (CoCoA) Each condition gives a polynomial p k Q [ X 0,..., X N1, Y 0,..., Y N2 ]. ) Find Gröbner basis of I = (p 1,..., p s Q[X, Y ] with Lex order. ( s = N1 /2 + N 2 /2 + K 1 + K 2 + L ) From Gröbner basis, easier to choose a set of solutions.

24 Computational Commutative Algebra (CoCoA)

25 Flat-Delay Allpass Filter N 1 = N 2, K 1 = K 2 Design filters with z-transforms H 1 and H 2 satisfying H 1 (z) = F (z)d(z), H 2 (z) = F (z)z L D(z).

26 Flat-Delay Allpass Filter N 1 = N 2, K 1 = K 2 Design filters with z-transforms H 1 and H 2 satisfying H 1 (z) = F (z)d(z), H 2 (z) = F (z)z L D(z). D is chosen to achieve the (approximate) half-sample delay. F is chosen to achieve the vanishing moments condition.

27 Flat-Delay Allpass Filter N 1 = N 2, K 1 = K 2 Design of D: ( ) L ( d n = ( 1) n 1 2 L) ( 1 2 L + n) n ( ) ( n + 1). ( ) n = 0, 1,..., L

28 Flat-Delay Allpass Filter N 1 = N 2, K 1 = K 2 Design of D: ( ) L ( d n = ( 1) n 1 2 L) ( 1 2 L + n) n ( ) ( n + 1). ( ) n = 0, 1,..., L Design of F : F (z) = Q(z) ( z ) K

29 Flat-Delay Allpass Filter N 1 = N 2, K 1 = K 2 Design of D: ( ) L ( d n = ( 1) n 1 2 L) ( 1 2 L + n) n ( ) ( n + 1). ( ) n = 0, 1,..., L Design of F : F (z) = Q(z) ( z ) K Design of Q: Generate (r n ) symmetric, minimal length such that R(z) ( z z ) KD(z)D(1/z) is halfband. Q is a spectral factor of R: R(z) = Q(z)Q(1/z).

SDP APPROXIMATION OF THE HALF DELAY AND THE DESIGN OF HILBERT PAIRS. Bogdan Dumitrescu

SDP APPROXIMATION OF THE HALF DELAY AND THE DESIGN OF HILBERT PAIRS Bogdan Dumitrescu Tampere International Center for Signal Processing Tampere University of Technology P.O.Box 553, 3311 Tampere, FINLAND