Representation: Fractional Splines, Wavelets and related Basis Function Expansions. Felix Herrmann and Jonathan Kane, ERL-MIT

Transcription

1 Representation: Fractional Splines, Wavelets and related Basis Function Expansions Felix Herrmann and Jonathan Kane, ERL-MIT

2 Objective: Build a representation with a regularity that is consistent with data. Means: Generalize ordinary splines to fractional splines: put algebraic singularities at the knots useful references [5, 1, 4, 3]

3 Scaling analysis on well data Well-data are multifractal! scale different everywhere, i.e. f(z + ) f(z) C α(z). typically with α [ 0.2, 2.5]. seismic reflectivity entails a differentiation. singularities are accumulated, they are everywhere...

4 Starting point f(x) n N c n +χ α n + (x x n ) + c n χ α n (x x n ), with χ α +(x) 0 x 0 x α Γ(α+1) x > 0, χα (x) 0 x x x α Γ(α+1)

5 Splines Simplify to at integers: f(x) = k Z a[k]χ α (x k) For α integer we get splines Alg. singularities are unbounded [5]!

6 B-splines α degree B-splines are α convolutions of the boxcar function [5]. * = α determines thee Hölder regularity of the spline. Boxcar in Fourier domain: ˆβ 0 (ω) = 1 0 e jωx dx = 1 e jω jω

7 Causal fractional B-splines: Fractional B-splines ˆβ α (ω) = ( 1 e jω jω ) α+1 Anti-Causal: ˆβ α (ω) = ( e jω 1 jω ) α+1

8 Causal B-splines α = 0 : 0.2 :

9 Symmetric B-Splines Non-causal, symmetric version: ˆβ α (ω) = sin(ω/2) ω/2 1/2 ˆβ (ω) 0 = e jωx dx = 1/2 α+1 as α then β α (x) Gaussian. sin(ω/2) ω/2

10 Symmetric B-splines α = 0 : 0.2 :

11 Relation alg. singularities [5] β α +(x) d = α+1 + χ α +(x) = ( 1) k α + 1 χ α +(x k) k 0 k where α +f(x) = k 0 ( 1) k α + 1 f(x k) k

12 and ˆ α +(ω) = ( 1 e jω) α = ( 1) k k 0 α + 1 e jωk k

13 and Fractional B-splines β (x) α = d α+1 χ α (x) = β+( x) α β α (x) = β α β α 1 2 = F 1 sin( ω 2 ) β α 1 + β α 2 + = β α 1+α D γ +β α + = γ +β α γ + ω 2 α+1

14 Cubic B-splines

15 Cardinal Splines [5] ˆβ α car.(ω) = ˆβ α (ω) k ˆβ α (ω + 2πk). denominator sampled version of B-spline sampled values are zero everywhere except at origin α = sinc-function

16 Orthogonal splines ˆφ α (ω) = ˆβ α (ω) ( k ˆβ α (ω + 2πk) 2 ) 1/2 with A(e jω ) = k ˆβ α (ω+2πk) 2 = k β 2α+1 (k)e jωk auto-correlation of B-splines a[k] = β α ( )β( k.

17 Orthogonal splines

18 Two-scale relation: Fractional Spline Wavelets ˆφ α (2ω) = 1 2 ĥ α (ω) ˆφ α (ω) so ĥ α (ω) = 2 ˆφ α (2ω) ˆφ α (ω).

19 Fractional Spline Wavelets Now we have ĥ α +(ω) = 2 ( 1 + e iω 2 ) α+1 â 2α+1 (ω) â 2α+1 (2ω) ĥ α (ω) = 2 ([ 1 + e iω 2 ] )α+1 â 2α+1 (ω) â 2α+1 (2ω) ĥ α (ω) = e iω 2 α+1 â 2α+1 (ω) â 2α+1 (2ω).

20 Fractional Spline Wavelets From father wavelet [5, 1, 4]: ˆφ α (2ω) = 1 2 ĥ α (ω) ˆφ α (ω) construct [3] ˆψ α (2ω) = 1 2 ĝ α (ω) ˆψ α (ω) with ĝ α (ω) = e iω [ ĥ α (ω + π)].

21 Causal Orthogonal Spline Wavelets α = 0 : 0.2 :

22 Fractional Spline Wavelets Dyadic family of orthogonal smoothing functions with prescribed regularity: φ α (x/2) = 2 k h α [k]φ α (x k) and orthogonal wavelets ψ α (x/2) = 2 k g α [k]ψ α (x k)

23 Fix a j = J: Discrete Wavelet Transform f(x) = 2 J 1 u J,k φ J,k (x) + 2 j 1 v j,k ψ j,k k=0 j=j k=0 with u J,k = f, φ J,k v j,k = f, ψ j,k d = d = f(x)φ J,k (x)dx f(x)ψ j,k (x)dx

24 and φ J,k (x) = 1 2 J/2 φ( x 2 J k) ψ j,k (x) = 1 2 j/2 ψ( x 2 j k).

25 Emperical Discrete Wavelet Transform DWT computed from functions not from sampled data. Practical viewpoint: Consider the data to be sampled after applying an anti-aliasing low-pass filter which has sufficient db per octave decay in the Fourier domain. Theoretical viewpoint: Not so simple (see e.g. [2]). Corresponds to transforming a finite data sequence into approximate empirical wavelet coefficients with Deslauriers-Dubuc interpolation. Coverges when N. Leaves the choice (smoothness) of the interpolation function!

26 Discrete Wavelet Transform For discrete data (f = f(x i ) N 1 i=0 with N = 2n ) sample ψ, φ f = W T w w = Wf with W 1 = W T and where w = [c J,k, d j,k ] T are the empirical wavelet coefficients for discrete data.

27 Discrete Wavelet Transform Sampling of the wavelets: N Wj,k (i) 1 2 j/2 ψ( x 2 j ) with x = i/n k2j. approximation improves for larger N [2]. difference between continous and discrete orthogonality conditions.

28 Discrete Wavelet Transform Wavelet transform decomposes f into a multiresolution basis implemented recursively by filtering steps (not by matrix multiplication). orthogonal smoothings a collection of orthogonal details each step splits into detail and smoothing and downsamples. non-data adaptive

29 Discrete Wavelet Transform Show some Examples with Wavelab: http: //www-stat.stanford.edu/ donoho WaveLab802/Workouts/Toons/

30 Frames Non-adaptive wavelets are not shift-invariant. Beneficial to decompose f into a redundant frame: f Uf[n] = f, g n with {g n } n Γ a family of vectors which is a frame if A f 2 n Γ f, g n B f 2 when A = B a tight frame orthogonal when A = B = 1.

31 Reconstruct via pseudo inverse f = (U U) 1 U Uf = n Γ f, g n g n with the duals g n = (U U) 1 g n.

32 Wavelet Packets: Examples of Frames split f into both smoothings and details n n log 2 n. Cosine Packets: like Wavelet Packets in the Fourier domain. Shift-invariant DWT: not down-sample Organized in Packet tables which are binary trees.

33 Examples Show some Examples with Wavelab: http: //www-stat.stanford.edu/ donoho WaveLab802/Workouts/Toons/

34 Wavelets Wavelet s miracle is that it represents piece-wise continuous functios almost as well as smooth functions... without having to know where the singularities are...non-adaptive From that a whole suite of non-linear denoising and inversion techniques are derived... Lack shift-invariance and characterization!

35 References [1] T. Blu and M. Unser. The fractional spline wavelet transform: Definition and implementation. volume I, pages IEEE, In Proceedings, [2] D. L. Donoho and I. M. Johnstone. Minimax estimation via wavelet shrinkage. Annals of Statistics, 26(3): , URL citeseer.nj.nec.com/ donoho92minimax.html. [3] S. G. Mallat. A wavelet tour of signal processing. Academic Press, [4] M. Unser and T. Blu. Fractional splines. URL 34-1

36 fractsplines/. [5] M. Unser and T. Blu. Fractional splines and wavelets. SIAM Review, 42(1):43 67,