Wavelets demystified Michael Unser Biomedical Imaging Group EPFL, Lausanne Switzerland Eindhoven, June 2006 THE WAVELET TRANSFORM Wavelet basis functions Dilation and translation of a single prototype ( x 2 ψ i,k = 2 i/2 i ) k ψ 2 i Motivation From legos to wavelets Signal processing perspective Mathematical properties 1-2
Motivation for using wavelets Remarkable wavelet properties Multi-scale decomposition Self-similarity One-to-one vs. redundant New computational paradigm Multi-resolution formulation Filterbank algorithms: O(N) complexity Regularization via sparsity constraints Decoupling: (bi-)orthogonality Vanishing moments Kills polynomials Sparse representation of piecewise-smooth functions Multi-scale differentiation Joint time-frequency localization Classes of problems Data compression: JPEG2000... Data processing: filtering, denoising, inverse problems Data analysis: singularities, texture, fractals... 1-3 Wavelets in medical imaging: Survey 1991-1999 References Unser and Aldroubi, Proc IEEE, 1996 Laine, Annual Rev Biomed Eng, 2000 Special issue, IEEE Trans Med Im, 2003 Image processing task Application / modality Principal Authors Image compression Filtering Feature extraction MRI Mammograms CT Angiograms, etc Image enhancement Digital radiograms MRI Mammograms Lung X-rays, CT Denoising MRI Ultrasound (speckle) SPECT Detection of micro-calcifications Mammograms Texture analysis and classification Ultrasound CT, MRI Mammograms Snakes and active contours Ultrasound Angelis 94; DeVore 95; Manduca 95; Wang 96; etc Laine 94, 95; Lu, 94; Qian 95; Guang 97; etc Weaver 91; Xu 94; Coifman 95; Abdel-Malek 97; Laine 98; Novak 98, 99 Qian 95; Yoshida 94; Strickland 96; Dhawan 96; Baoyu 96; Heine 97; Wang 98 Barman 93; Laine 94; Unser 95; Wei 95; Yung 95; Busch 97; Mojsilovic 97 Chuang-Kuo 96 Wavelet encoding Magnetic resonance imaging Weaver-Healy 92; Panych 94, 96; Geman 96; Shimizu 96; Jian 97 Image reconstruction Statistical data analysis Multi-scale Registration Computer tomography Limited angle data Optical tomography PET, SPECT Functional imaging PET fmri Motion correction fmri, angiography Multi-modality imaging CT, PET, MRI Olson 93, 94; Peyrin 94; Walnut 93; Delaney 95; Sahiner 96; Zhu 97; Kolaczyk 94; Raheja 99 Ruttimann 93, 94, 98; Unser 95; Feilner 99; Raz 99 Unser 93; Thévenaz 95, 98; Kybic 99 3D visualization CT, MRI Gross 95, 97; Muraki 95; Kamath 98; Horbelt 99 1-4
Wavelets: Haar transform revisited s 0 (x) Signal representation s 0 (x) = k Z c[k]ϕ(x k) Scaling function s 1 (x) ϕ(x) s 2 (x) Multi-scale signal representation s i (x) = k Z c i [k]ϕ i,k (x) s 3 (x) Multi-scale basis functions ( x 2 i ) k ϕ i,k (x) = ϕ 2 i 1-5 Wavelets: Haar transform revisited Wavelet: ψ(x) + - r i (x) = s i 1 (x) s i (x) + - + - 1-6
Wavelets: Haar transform revisited r 1 (x) = d 1 [k]ψ 1,k k + r 2 (x) = d 2 [k]ψ 2,k k + r 3 (x) = d 3 [k]ψ 3,k k + = Wavelet: s(x) = k ψ(x) c[k]ϕ(x k) s 3 (x) = k c 3 [k]ϕ 3,k 1-7 Signal processing perspective Splitting and putting together again... Perfect reconstruction filterbank ave 2 2 ave diff 2 2 diff Identity operator: Tree-structured wavelet transform Subband decomposition 1 Wavelet filterbank 1 2 2 3 3 3 3 2 1 ω π 2 π 1-8
Multi-rate operations Filtering z-transform: x[k] z X(z) = k Z x[k]z k H(z) (h x)[k] = l Z h[l]x[k l] z H(z) X(z) Down-sampling 2 (x) 2 [k] = x[2k] z 1 ( ) X(z 1/2 ) + X( z 1/2 ) 2 Up-sampling 2 (x) 2 [k] = { 0, k odd x[l], 2l = k even z X(z 2 ) Down-sampling and up-sampling 2 2 (x) 2 2 [k] z 1 (X(z) + X( z)) 2 1-9 Perfect reconstruction filterbanks x[k] 2 H (z "1 ) 2 2 2H(z) x out [k] 2 G (z "1 ) 2 2 2G(z) ( X out (z) = H(z) H(z 1 )X(z) + H( z ) ( 1 )X( z) +G(z) G(z 1 )X(z) + G( z ) 1 )X( z) Perfect reconstruction conditions (PR-1) H(z 1 )H(z) + G(z 1 )G(z) = 1 (distortion-free) (PR-2) H( z 1 )H(z) + G( z 1 )G(z) = 0 (aliasing-free) Wavelet transform design Construct 4 filters such that (PR-1) and (PR-2) are satisfied. 1-10
CONSTRUCTION OF WAVELET BASES Scaling functions Multiresolution analysis From scaling functions to wavelets The lego revisited Fractional B-splines Wavelet bases of L2 1-11 Scaling function Definition: ϕ(x) is an admissible scaling function of L 2 iff: Riesz basis condition c l 2, A c l2 c[k]ϕ(x k) B c l2 L 2 k Z Two-scale relation ϕ(x/2) = 2 k Z h[k]ϕ(x k) 1 1 Partition of unity ϕ(x k) = 1 k Z 1-12
Multiresolution analysis of L2 s V (0) ( ) Multiresolution basis functions: ϕ i,k (x) = 2 i/2 x 2 ϕ i k 2 i Subspace at resolution i: V (i) = span {ϕ i,k } k Z s 1 V (1) s 2 V (2) s 3 V (3) Two-scale relation V (i) V (j), for i j Partition of unity i Z V (i) = L 2 (R) 1-13 From scaling functions to wavelets Wavelet bases of L 2 [Mallat-Meyer, 1989] Theorem: For any given admissible scaling function of L 2, ϕ(x), there exists a wavelet ψ(x/2) = 2 k Z g[k]ϕ(x k) such that the family of functions { ( 2 i/2 x 2 ψ i k 2 )}i Z,k Z i forms a Riesz basis of L 2. 1-1 Constructive approach: perfect reconstruction filterbank s i (k) 2 H (z "1 ) 2 s i+1 (k) 2 2H(z) s i (k) 2 G (z "1 ) 2 d i +1 (k) 2 2G(z) 1-14
The lego revisited Continuous derivative D{ } = d dx F Discrete version (finite difference) + { } jω F 1 e jω 0 1 Construction of the B-spline of degreestep 0 function: x+ = D {δ(x)} Step function: The lego 0 β+ (x) = x0+ (x 1)0+ = 1+ x0+ x0+ = D 1 {δ(x)} n n+1 + x+ 0 n+1 (n+1) n β+ (x) = x0+ (x 1)0+ = 1+ D 1 {δ(x)}d β+ (x) = D {δ(x)} = F n D{ } = jω+ n+1 n dx " n+1 x + n (n+1) 1 e jω (1 e jω )n+1 β+ (x) = n+1 {δ(x)} = + n + D β (ω) = = n + F jω (jω)n+1 Fourier domain formula " + { } 1 e jω jω n+1 jω n+1 operator (finite 1 e (1 e Discrete ) m 1 " difference) n f S, m Dm f (x) β + (ω) = = + f (x) = β+ jω n+1 jω (jω) 1 e 0 n m β + (ω) = n Dm β+ (x) = m (x) m 1 + β+ " m f S, + f (x) =jω β+ Dm f (x) Continuous operator (derivative) n m n Dm β+ (x) = m (x) + β+ 1-15 4 5 4 Beyond legos: fractional splines Fractional B-splines 0 β+ (x) = + x0+ 1 e jω jω F M α β+ (x) = α α+1 + x+ Γ(α + 1) M F One-sided power function: xα + = 1 e jω jω # "α+1 xα, x 0 0, x<0 Properties (Unser & Blu, SIAM Rev, 2000) " α β+ (x k) k Z is a valid Riesz basis for α < 21 α1 α2 α1 +α2 +1 Convolution property: β+ β+ = β+ 1-16
Binomial refinement filter Dilation by a factor of 2 α β+ (x/2) =2 α hα + [k]β+ (x k Z Hint for the proof: k) α jω H+ (e ) = α β + (2ω) α (ω) β + = 1 + e jω 2 "α+1 Generalized binomial filter hα + [k] = where 1 2α+1 #u$ v = α+1 k " z α H+ (z) = 1 + z 1 2 "α+1 Γ(u + 1) Γ(v + 1)Γ(u v + 1) Example: piecewise linear splines: α = 1 1 1/2 1/2 1-17 Fractional B-spline wavelets α ψ+ (x/2) = ( 1)k " α + 1 # k Z 2α $ n N n %& α β 2α+1 (n + k 1) β+ (x k) 2g[k] ' (Unser and Blu, SIAM Review, 2000) " Remarkable property Each of these wavelets generates a Riesz basis of L2 1-18
Wavelet basis of L 2 Family of wavelet templates (basis functions) ( x 2 i ) k ψ i,k = 2 i/2 ψ α + 2 i Semi-orthogonal wavelet basis ψ i,k, ψ j,l = δ i j,k l f(x) L 2, f(x) = f, ψ i,k ψ i,k i Z k Z Orthogonal wavelet basis (Generalized Battle-Lemarié) ψ i,k, ψ j,l = δ i j,k l f(x) L 2, f(x) = f, ψ i,k ψ i,k i Z k Z 1-19 Extension to higher dimension: separability Split rows Split columns and iterate... Tensor-product basis functions: ψ k1,,k p (x 1,, x p ) = ψ k1 (x 1 ) ψ k2 (x 2 ) ψ kp (x p ) 1-20
2D wavelet decomposition: example Wavelet transform Inverse wavelet transform Discarding small coefficients 1-21 Other popular wavelet families Basic paradigm: PR filterbank design Implicit definition: solution of two-scale relation ϕ(x/2) = 2 k Z h[k]ϕ(x k) Daubechies wavelet: L=2 Daubechies wavelets Shortest orthogonal filter such that: H(z) = (1 + z 1 ) L Q(z) (order constraint) Biorthogonal splines (Cohen-Daubechies-Feauveau) 9/7 scaling function JPEG2000 wavelets biorthogonal 5/3: linear spline (2 vanishing moments) biorthogonal 9/7: symmetric, near-orthogonal (4 vanishing moments) 1-22
WAVELET THEORY Order of approximation Factorization theorem Reproduction of polynomials Vanishing moments Multi-scale differentiation Smoothness 1-23 Order of approximation General shift-invariant space at scale a { V a (ϕ) = s a (x) = } ( x ) c[k]ϕ a k : c l 2 k Z a = 1 1 2 3 4 5 Projection operator f L 2, P a f = arg min s a V a f s a L2 a = 2 2 4 Order of approximation Definition A scaling/generating function ϕ has order of approximation γ iff. f W γ 2, f P af L2 C a γ f (γ) L2 1-24
B-spline factorization " Factorization theorem A valid scaling function ϕ(x) has order of approximation γ iff ϕ(x) = ( β+ α ) ϕ 0 (x) where β+ α with α = γ 1: regular, B-spline part ϕ 0 S : irregular, distributional part (Unser-Blu, IEEE-SP, 2003) = " " Refinement filter: general case ( ) 1 + z 1 γ H(z) = 2 }{{} spline part Q(z) }{{} distributional part with Q(e jω ) < + 1-25 Reproduction of polynomials B-splines reproduce polynomials of degree N = α β+(x α k) = 1 k Z. k n β+(x α k) = x n + a 1 x n 1 + + a n k Z α = 0 10 5 10 5 5 10 15 5 10 10 α = 1/2 5 10 5 5 10 15 α = 3 10 5 +10 k= 10 k β α +(x k) α = 1 5 10 10 5 10 5 5 10 15 5 10 5 5 10 15 5 10 10 1-26
Reproduction of polynomials (Contd) Proposition: If ϕ(x) = (β α + ϕ 0 )(x) with ˆϕ 0 (0) = 1, then ϕ(x) reproduces the polynomials of degree N = α. 1.5 1.25 1 0.75 0.5 0.25 8 ϕ(x k) 1 k=0 10 8 6 4 8 k ϕ(x k) x + b k=0-0.25-0.5 2 4 6 8 10 2 2 4 6 8 10 Argument: ϕ 0 x n = x n + b 1 x n 1 + + b n k n ϕ(x k) = ϕ 0 k n β+(x α k) = x n + c 1 x n 1 + + c n k Z k Z Strang-Fix (1971): Conversely, if ϕ(x) reproduces the polynomials of degree N and is compactly supported, then ϕ(x) = (β N + ϕ 0 )(x). 1-27 Vanishing moments Proposition If ϕ(x) reproduces the polynomials of degree N, then the analysis wavelet ψ(x) has L = N + 1 vanishing moments: x n ψ(x)dx = 0, n = 0,, N x R ψ kills all polynomials of degree n N p(x) π N, p(x) ψ(x/a b)dx = 0 x R B-spline wavelets: L = 1 L = 2 L = 3 Argument { Polynomial reproduction p(x) span{ϕ(x k)} k Z ψ(x) is perpendicular to V (ϕ) by construction L = 4 ψ is perpendicular to p(x) 1-28
Differentiation and regularity Fractional differential operators Derivative of order s: s F (jω) s Finite difference of order s: s + F (1 e jω ) s B-spline differentiation formula s β α +(x) = s +β α s + (x) Sketch of the proof: s β α +(x) F (jω) s ( 1 e jω jω ) α+1 = (1 e jω ) s ( 1 e jω jω ) α s+1 Continuity and differentiability Hölder smoothness: β α + Ċα α β α + bounded Sobolev smoothness: β α + W r 2, r < α + 1 2 r β α + L 2 (R) 1-29 Multi-scale differentiation Perfect reconstruction conditions ( ) ( H(z) G(z) H(z 1 ) H( z 1 ) H( z) G( z) G(z 1 ) G( z 1 ) ) = I Proposition: For a stable filterbank, the order constraint is equivalent to G(z) = (1 z) γ P (z) with P (e jω ) < +. Theorem: Let ϕ and ϕ be two valid biorthogonal scaling functions. Then, ϕ is of order γ (i.e., ϕ = β γ 1 + ϕ 0 ) iff ˆ ψ(ω) = O( ω γ ). Wavelet transform as a multi-scale differentiator f(x), ψ(x u) = γ {φ f} (u) Smoothing kernel: ˆφ(ω) = ˆ ψ (ω)/(jω) γ 1-30
Wavelet regularity: peeling Daubechies Theorem: If ϕ γ (x) = β r 1 + ϕ γ r (x) with ϕ γ r L p, then r ϕ γ L p ; i.e., ϕ γ has r derivatives in L p -sense. B-spline peeling mechanism: ϕ γ = β γ 1 + ϕ 0 = β r 1 Note: β 1 + (x) = δ(x) + ϕ γ r {}}{ (β γ r 1 + ϕ 0 ) r ϕ γ = ( r β r 1 + ) ϕ γ r = r +ϕ γ r Theorem: If ϕ is a valid scaling function such that r ϕ L 2, then ϕ(x) = β r 1 + ϕ 0 (x) with ϕ 0 L 2. 1-31 Splines: The key to wavelet theory Sobolev smoothness s ϕ L 2 γ s B-spline factorization: Approximation order: ϕ = β γ 1 + ϕ 0 f Pa f L2 = O(a γ ) Multi-scale differentiator ˆ ψ(ω) ( jω) γ, ω 0 Polynomial reproduction degree: N = γ 1 general case: N < γ N + 1 compact support: γ = N + 1 (Strang-Fix) c Vanishing moments: x n ψ(x)dx = 0, n = 0,, N (Unser and Blu, IEEE-SP, 2003) 1-32
FURTHER (ANTHROPOMORPHIC) ARGUMENTS... Localization properties Wavelets and the uncertainty principle 1-33 Localization properties " frequency " x = min x 0 (x # x 0 )$(x) L2 $ L2 " x # " $ = Const (# $# 0 )% ˆ (#) L2 " # = min # 0 ˆ % L2 x time or space "Heisenbergs uncertainty relation " x # " $ % 1 2 with equality iff. ψ(x) = a e b(x x 0) 2 +jω 0 x Question: are there Gabor-like wavelet bases? 1-34
Are there optimally localized wavelet bases? Theorem The B-spline wavelets converge (in L p -norm) to modulated Gaussians as the degree goes to infinity: lim % " + (x) " #$ { } = C & e '(x'x " )2 2 / 2( " lim % " + (x) " #$ { } = & " # = # +1 12 C ' e ((x( x & " ) 2 / 2 & 14 2443 * cos + x +, 0 " 1 42 44 3 Gaussian ) " 2 #" $ = B %# $ with B & 2.59 ( ) sinusoid " = 3 Cubic B-spline wavelets: within 2% of the uncertainty limit (Unser et al., IEEE-IT, 1992) 1-35 Are there wavelets in my brain? 1-36
CONCLUSION Important wavelet features Simple, fast implementation: Mallats filterbank algorithm Mathematical properties: Riesz basis, vanishing moments, polynomial reproduction, order of approximation,... Fundamental connection between splines and wavelets Simulates the organization of the primary visual system Many successful applications Data compression Filtering, denoising (non-linear) Detection and feature extraction Inverse problems: wavelet regularization Current topics in wavelet research Non-separable multidimensional wavelets: isotropic vs. directional Complex wavelets, Bandelets Wavelet frames, ridgelets, curvelets, etc... 1-37 Acknowledgments Many thanks to # Dr. Thierry Blu # Prof. Akram Aldroubi # Prof. Murray Eden # Dr. Philippe Thévenaz # Dr. Dimitri Van de Ville # Annette Unser, Artist + many other researchers, and graduate students 1-38
Bibliography The lego connection M. Unser, M. Unser, "Wavelet Games," Wavelet Digest, vol. 11, no. 4, April 1, 2003 (http://www.wavelet.org/phpbb2/viewtopic.php?t=5129). Wavelet theory S.G. Mallat, "A theory of multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-11, no. 7, pp. 674-693, 1989. M. Unser, T. Blu, "Wavelets Theory Demystified," IEEE Trans. Signal Processing, vol. 51, no. 2, pp. 470-483, 2003. M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, 2000. Wavelets in medicine and biology M. Unser, A. Aldroubi, "A Review of Wavelets in Biomedical Applications," Proc. IEEE, vol. 84, no. 4, pp. 626-638, 1996. Preprints and demos: http://bigwww.epfl.ch/ 1-39