Multiscale Geometric Analysis: Thoughts and Applications (a summary)

Transcription

1 Multiscale Geometric Analysis: Thoughts and Applications (a summary) Anestis Antoniadis, University Joseph Fourier Assimage 2005,Chamrousse, February 2005

2 Classical Multiscale Analysis Wavelets: Enormous impact Theory Applications Many success stories Deep understanding of the fact that wavelets are not good for all purposes Consequent constructions of new systems lying beyond wavelets Overview Other multiscale constructions Problems classical multiscale ideas do not address effectively

3 Fourier Analysis Diagonal representation of shift-invariant linear transformations; e.g. solution operator to the heat equation t u = a 2 u Fourier, Théorie Analytique de la Chaleur, Truncated Fourier series provide very good approximations of smooth functions if f C k. f S n (f) L2 C n k,

4 Limitations of Fourier Analysis Does not provide any sparse decomposition of differential equations with variable coefficients (sinusoids are no longer eigenfunctions) Provides poor representations of discontinuous objects (Gibbs phenomenon)

5 Wavelet Analysis Almost eigenfunctions of differential operators (Lf)(x) = a(x) x f(x) Sparse representations of piecewise smooth functions

6 Wavelets and Piecewise Smooth Objects 1-dimensional example: g(t) = 1 {t>t0 }e (t t 0) 2. Fourier series: g(t) = c k e ikt Fourier coefficients have slow decay: c (n) c 1/n. Wavelet series: g(t) = θ λ ψ λ (t) Wavelet coefficients have fast decay: θ (n) c (1/n) r for any r > 0. as if the object were non-singular

7 Sparse representations of point-singularities Sparse representation of certain matrices Simultaneously Applications Approximation theory Data compression Statistical estimation Scientific computing Donoho's-Candès Viewpoint More importantly: new mathematical architecture where information is organized by scale and location

8 Images Limitations of existing image representations Curvelets: geometry and tilings in Phase-Space Representation of functions, signals Potential applications

9 Three Anomalies Inefficiency of Existing Image Representations Limitations of Existing Pyramid Schemes Limitations of Existing Scaling Concepts

10 I: Inefficient Image Representations Edge Model: Object f(x 1, x 2 ) with discontinuity along generic C 2 smooth curve; smooth elsewhere. Fourier is awful Best m-term trigonometric approximation f m f f m 2 2 m 1/2, m Wavelets are bad Best m-term approximation by wavelets: f f m 2 2 m 1, m

11 Optimal Behavior There is a dictionary of atoms with best m-term approximant f m f f m 2 2 m 2, m No basis can do better than this. No depth-search limited dictionary can do better. No pre-existing basis does anything near this well.

12 (a) Wavelets (b) Triangulations

13 Traditional Scaling Limitations of Existing Scaling Concepts f a (x 1, x 2 ) = f(ax 1, ax 2 ), a > 0. Curves exhibit different kinds of scaling Anisotropic Locally Adaptive If f(x 1, x 2 ) = 1 {y x2 } then f a (x 1, x 2 ) = f(a x 1, a 2 x 2 ) In Harmonic Analysis this is called Parabolic Scaling.

14 x4 Fine Scale Rectangle x2 Identical Copies of Planar Curve Figure 1: Curves are Invariant under Anisotropic Scaling

15 Curvelets Candès and Guo, New multiscale pyramid: Multiscale Multi-orientations Parabolic (anisotropy) scaling width length 2 Earlier construction, Candès and Donoho (2000)

16 Philosophy (Slightly Inaccurate) Start with a waveform ϕ(x) = ϕ(x 1, x 2 ). oscillatory in x 1 lowpass in x 2 Parabolic rescaling D j ϕ(d j x) = 2 3j/4 ϕ(2 j x 1, 2 j/2 x 2 ), D j = 2j 0, j j/2 Rotation (scale dependent) 2 3j/4 ϕ(d j R θjl x), θ jl = 2π l2 j/2 Translation (oriented Cartesian grid with spacing 2 j 2 j/2 ); 2 3j/4 ϕ(d j R θjl x k), k Z 2

17 1 2 -j Rotate Translate 2 -j/2 Parabolic Scaling 2 -j/2 2 -j

18 In the frequency domain In the spatial domain Curvelet: Space-side Viewpoint ˆϕ j,0,k (ξ) = 2 3j/4 2π W j,0(ξ)e i k,ξ, k Λ j ϕ j,0,k (x) = 2 3j/4 ϕ j (x k), W j,0 = 2π ˆϕ j and more generally ϕ j,l,k (x) = 2 3j/4 ϕ j (R θj,l (x R 1 θ j,l k)), All curvelets at a given scale are obtained by translating and rotating a single mother curvelet.

19 Further Properties Tight frame f = j,l,k f, ϕ j,l,k ϕ j,l,k f 2 2 = j,l,k f, ϕ j,l,k 2 Geometric Pyramid structure Dyadic scale Dyadic location Direction New tiling of phase space

20 Needle-shaped Scaling laws length 2 j/2 width 2 j width length 2 #Directions = 2 j/2 Doubles angular resolution at every other scale Unprecedented combination

21 Digital Curvelets Source: Digital Curvelet Transform, Candès and Donoho (2004).

22 Digital Curvelets: Frequency Localization Time domain Frequency domain (modulus) Source: Digital Curvelet Transform, Candès and Donoho (2004).

23 Digital Curvelets: : Frequency Localization Time domain Frequency domain (real part) Source: Digital Curvelet Transform, Candès and Donoho (2004).

24 Curvelets and Edges

25 Optimality Suppose f is smooth except for discontinuity on C 2 curve Curvelet m-term approximations, naive thresholding f f curve m 2 2 C m 2 (log m) 3 Near-optimal rate of m-term approximation (wavelets m 1 ).

26 Idea of the Proof s f(x,y) Edge fragment -s Scale 2 Coefficient ~ 0 Decomposition of a Subband

27 Curvelets and Warpings C 2 change of coordinates preserves sparsity (Candès 2002). Let ϕ : R 2 R 2 be a one to one C 2 function such that J ϕ is bounded away from zero and infinity. Curvelet expansion f = µ θ µ (f)γ µ, θ µ (f) = f, γ µ Likewise, f ϕ = µ θ µ (f ϕ)γ µ The coefficient sequences of f and f ϕ are equally sparse. Theorem 1 Then, for each p > 2/3, we have θ(f ϕ) lp C p θ(f) lp.

28 42 x µ(t) ξ µ(t) 2 j/2 ξ µ x µ 2 j Curvelets are nearly invariant through a smooth change of coordinates

29 Curvelets and Curved Singularities f smooth except along a C 2 curve f n, n-term approximation obtained by naive thresholding f f n 2 L 2 C (log n) 3 n 2 Why? 1. True for a straight edge 2. Deformation preserves sparsity φ Optimal arbitrary singularity straight singularity

30 Importance of Parabolic Scaling Consider arbitrary scaling (anisotropy incresases as α decreases) width 2 j, length 2 jα, 0 α 1. ridgelets α = 0 (very anisotropic), curvelets α = 1/2 (parabolic anisotropy), wavelets α = 1 (roughly isotropic). For wave-like behavior, need width length 2 For particle-like behavior, need width length 2 For both (simultaneously), need width length 2 Only working scaling: α = 1/2

31 Examples I 2 -j t = 0 t = T

32 Examples II

33 Examples III 2 -j 2 -αj 2 -(1-α)j t = 0 t = T

34 Summary New geometric multiscale ideas Key insight: geometry of Phase-Space New mathematical architecture Addresses new range of problems effectively Promising potential

35 Numerical Experiments

36 (a) Noisy Phantom (b) Curvelets

37 (c) Curvelets and TV (d) Curvelets and TV: Residuals

38 (e) Wavelets (f) Curvelets

39

40 (l) Noisy (m) Curvelets & TV: Residuals

41 (n) Noisy Scanline (o) True Scanline and Curvelets and TV Reconstruction

42 (a) Noisy Scanline (b) Curvelets (c) Curvelets and TV (d) True Scanline and Curvelets and TV Reconstruction Figure 10: Scanline Plots

43 (p) Original (q) Noisy

44 (r) Curvelets (s) Curvelets and TV

45 (t) Noisy Detail (u) Curvelets (v) Curvelets and TV