- 1. Shearlet Frames and Optimally Sparse Approximations Jakob Lemvig — Technical University of Denmark (DTU) joint work with Gitta Kutyniok & Wang-Q Lim (TU Berlin)
- 2. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 2/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 3. Outline 1 Applied Harmonic Analysis and Imaging Sciences Anisotropic Phenomena Image Model Goal for Today 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 3/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 4. Anisotropic Phenomena in Multivariate Data Many important multivariate problem classes are governed by anisotropic features, which require eﬃcient encoding strategies. The anisotropic structure can be given. . . • . . . explicitly • Image Processing: Edges • Seismology: Earth layers • • . . . implicitely • PDEs: Shock waves, Boundary layers 4/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 5. Modern Imaging Some important tasks: • Denoising • Inpainting • Feature Detection/Extraction • ... Imaging Sciences using Applied Harmonic Analysis: Exploit a carefully designed representation system (ψλ )λ ⊂ L2 (Rd ) : Image = Image, ψλ ψλ . λ • Sparse coeﬃcients • Approximation properties 5/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 6. Geometric Structures of Multidimensional Images • What are anisotropic features in a 2D Image? Curves Objects Cartoon- with like ‘smooth’ image boundaries 6/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 7. Geometric Structures of Multidimensional Images • What are anisotropic features in a 2D Image? Curves Objects Cartoon- with like ‘smooth’ image boundaries • What are anisotropic features in 3D data? Curves & surfaces Video: objects carve out 3D medical spatial- scans: temporal tubes 6/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 8. Geometric Structures of Multidimensional Images • What are anisotropic features in a 2D Image? Curves Objects Cartoon- with like ‘smooth’ image boundaries • What are anisotropic features in 3D data? Curves & surfaces Video: objects carve out 3D medical spatial- scans: temporal tubes • Most information is contained in lower dimensional structures! 6/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 9. Reasonable Model for 2D Images Deﬁnition (Donoho; 2001) The set of cartoon-like 2D images E2 (R2 ) is deﬁned by 2 E2 (R2 ) = {f ∈ L2 (R2 ) : f = f0 +f1 χB } fi ∈ C 2 (R2 ), supp fi ⊂ [0, 1]2 , 2 where B ⊂ [0, 1]2 and the boundary curve ∂B is a closed C 2 -curve with curvature bounded by ν. f0=0 7/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 10. Reasonable Model for 2D Images Deﬁnition (Donoho; 2001) The set of cartoon-like 2D images E2 (R2 ) is deﬁned by 2 E2 (R2 ) = {f ∈ L2 (R2 ) : f = f0 +f1 χB } fi ∈ C 2 (R2 ), supp fi ⊂ [0, 1]2 , 2 where B ⊂ [0, 1]2 and the boundary curve ∂B is a closed C 2 -curve with curvature bounded by ν. Theorem (Donoho; 2001) Let (ψλ )λ ⊂ L2 (R2 ). The optimal asymptotic approximation error of f ∈ E2 (R2 ) is 2 2 f − fN 2 N −2 , N → ∞, where fN = cλ ψλ . λ∈IN 7/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 11. Approaches for 2D Non-exhaustive list for 2D data: • Directional wavelets (Antoine, Murenzi, Vandergheynst; 1999) • Curvelets (Candès & Donoho; 2002) • Contourlets (Do & Vetterli; 2002) • Bandlets (LePennec & Mallat; 2003) • Shearlets (Guo, Kutyniok, Labate, Lim & Weiss; 2006) Known results (band-limited generators): Curvelets, Contourlets, and Shearlets provide (almost) optimally sparse expansions for 2D cartoon-like images f ∈ E2 (R2 ) with 2 asymptotic error 2 f − fN 2 = O(N −2 (log N)3 ), N → ∞. 8/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 12. Approaches for 2D Known results (band-limited generators): Curvelets, Contourlets, and Shearlets provide (almost) optimally sparse expansions for 2D cartoon-like images f ∈ E2 (R2 ) with 2 asymptotic error 2 f − fN 2 = O(N −2 (log N)3 ), N → ∞. Success of 2D shearlets: Kutyniok and Lim (2011) proved that this asymptotic rate can be achieved with compactly supported shearlets. Intuitive explanation (shearlets vs. wavelets): N −2 8/42 DTU Mathematics, Technical University of Denmark N −1 Shearlet Frames 26.3.2012
- 13. Reasonable Model for 3D Images Deﬁnition The set of 3D images E2 (R3 ) is deﬁned by 2 E2 (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C 2 -surface for which the principal curvatures are bounded by ν. f0=0 9/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 14. Reasonable Model for 3D Images Deﬁnition The set of 3D images E2 (R3 ) is deﬁned by 2 E2 (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C 2 -surface for which the principal curvatures are bounded by ν. Theorem (Kutyniok, L, Lim; 2011) Let (ψλ )λ ⊂ L2 (R3 ). The optimal asymptotic approximation error of f ∈ E2 (R3 ) is 2 2 f − fN 2 N −1 , N → ∞, where fN = cλ ψλ . λ∈IN 9/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 15. Most Approaches are for 2D Data Question: Why is the 3D situation so crucial? Answer: • Our world is 3-dimensional. • 3D data is essential for Biology, Seismology,... • Transition 2D ❀ 3D is special. • In 3D anisotropic features occur for the ﬁrst time in diﬀerent dimensions. 10/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 16. Success of 1D Wavelets • The wavelet system generated by ψ ∈ L2 (R) is ψj,k = 2j/2 ψ(2j · −m) : j ∈ Z, m ∈ Z . • Wavelets provide sparse expansions for ‘1D cartoon images’ (C 2 apart from point discontinuities) with asymptotic error 2 f − fN 2 = O(N −4 ), N → ∞. • In comparison, Fourier series only gives O(N −1 ). ❀ Wavelets are very good at detecting singularities in 1D! 11/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 17. Goal for Today Design a representation system which... • ...forms a orthonormal basis frame for L2 (R3 ), • ...captures anisotropic structures, e.g., singularities concentrated on lower dimensional manifolds, • ...provides optimally sparse approximation of cartoon images, • ...allows for compactly supported analyzing elements, • ...is associated with fast decomposition algorithms, • ...treats the continuum and digital ’world’ uniformly. Vision: Introduce a system for 3D data that is as powerful as wavelets are for 1D data! 12/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 18. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 1D Analysis: Fourier Series 1D Analysis: Wavelets Problem with Wavelets for 2D and 3D Data 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 13/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 19. Fourier Series • {e2πinx }n∈Z is an ONB for L2 ([0, 1]): f = n cn e2πinx • Modulation: En f (x) := e2πinx f (x), f ∈ L2 (R). • Translation: Tm f (x) := f (x − m), f ∈ L2 (R). • {En Tm χ[0,1] }n,m∈Z ONB for L2 (R): f = n,m cn,m En Tm χ[0,1] 14/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 20. Fourier Series • {e2πinx }n∈Z is an ONB for L2 ([0, 1]): f = n cn e2πinx • Modulation: En f (x) := e2πinx f (x), f ∈ L2 (R). • Translation: Tm f (x) := f (x − m), f ∈ L2 (R). • {En Tm χ[0,1] }n,m∈Z ONB for L2 (R): f = n,m cn,m En Tm χ[0,1] Simple 1D cartoon-like image (0 < x0 < x1 < 1): 1 x ∈ [x0 , x1 ] , f (x) = 0 otherwise Fourier Coeﬃcients: 1 x1 1 |cn,0 | = f (x) e−2πinx dx = e−2πinx dx ∼ 0 x0 n 14/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 21. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 22. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z • Best N-term approximation: fN = cn,m En Tm χ[0,1] , # |IN | = N (n,m)∈IN 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 23. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z • Best N-term approximation: fN = cn,m En Tm χ[0,1] , # |IN | = N (n,m)∈IN • By Parseval’s Equality: ∞ ∞ 2 1 1 1 f − fN 2 = |cn,m |2 ∼ ∼ dx ∼ (n,m)∈IN / n=N n2 N x2 N 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 24. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z • Best N-term approximation: fN = cn,m En Tm χ[0,1] , # |IN | = N (n,m)∈IN • By Parseval’s Equality: ∞ ∞ 2 1 1 1 f − fN 2 = |cn,m |2 ∼ ∼ dx ∼ (n,m)∈IN / n=N n2 N x2 N • Conclusion: 2 f − fN 2 ∼ N −1 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 25. 1D Wavelet Approximations • The wavelet system generated by ψ ∈ L2 (R) is ψj,m = Dj Tm ψ = 2j/2 ψ(2j · −m) : j ∈ Z, m ∈ Z . • ONB e.g., Haar Wavelet: 1 1 x ∈ [0, 2 ), ψ(x) = −1 x ∈ [ 1 , 1), 2 0 otherwise. • Haar Scaling function Tm φ, m ∈ Z (replaces ψj,m , j < 0, m ∈ Z): φ(x) = χ[0,1] (x) 16/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 26. 1D Wavelet Approximations • At ﬁxed scale j there are exactly two non-zero wavelet coeﬃcients! |cj,k | = | f , ψj,k | ∼ 2−j/2 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 27. 1D Wavelet Approximations • At ﬁxed scale j there are exactly two non-zero wavelet coeﬃcients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 28. 1D Wavelet Approximations • At ﬁxed scale j there are exactly two non-zero wavelet coeﬃcients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 • For piecewise C 2 functions (1D cartoon-like images): 2 f − fN 2 = O(N −4 ), N → ∞. 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 29. 1D Wavelet Approximations • At ﬁxed scale j there are exactly two non-zero wavelet coeﬃcients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 • For piecewise C 2 functions (1D cartoon-like images): 2 f − fN 2 = O(N −4 ), N → ∞. • In comparison, Fourier series only gave O(N −1 ). 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 30. 1D Wavelet Approximations • At ﬁxed scale j there are exactly two non-zero wavelet coeﬃcients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 • For piecewise C 2 functions (1D cartoon-like images): 2 f − fN 2 = O(N −4 ), N → ∞. • In comparison, Fourier series only gave O(N −1 ). ❀ Wavelets are very good at detecting singularities in 1D! 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 31. Why Not nD Wavelets? • Isotropic scaling (same scaling in all directions): 2j 0 0 Aj = 0 2j 0. 0 0 2j • The wavelet system generated by ψ ∈ L2 (R3 ) is ψj,m = 23j/2 ψ(Aj · −m) : j ∈ Z, m ∈ Z3 . • Wavelets do not provide optimal sparse expansions for 3D cartoon images. For f ∈ E2 (R3 ) wavelets only provide an aymptotic error 2 of: f − fN 2 ∼ N −1/2 , 2 N → ∞. ❀ Wavelets are not good at eﬀectively capturing 2D singularities in 3D: Far from the optimal rate N −1 ! 18/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 32. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory Discrete Shearlet Systems Pyramid-adapted Systems 4 Sparse Approximation using 3D Shearlets 5 Conclusion 19/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 33. Introducing 2D Shearlets • Anisotropic scaling Aj : 2j 0 Aj = , 0 2j/2 • Shearing Sk (direction parameter ↔ rotations): 1 k Sk = 0 1 • The shearlet system generated by ψ ∈ L2 (R2 ) is ψj,k,m = DSk Aj Tm ψ = 23j/4 ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z, m ∈ Z2 20/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 34. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 0, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 35. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 36. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 2, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 37. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 38. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (1, −1) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 39. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 40. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = −1, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 41. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = −2, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 42. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = −3, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 43. Introducing 3D Shearlets • Anisotropic scaling Aj : 2j 0 0 Aj = 0 2j/2 0 , 0 0 2j/2 • Shearing Sk , k = (k1 , k2 ) (direction parameter ↔ rotations): 1 k1 k2 Sk = 0 1 0 0 0 1 • The shearlet system generated by ψ ∈ L2 (R3 ) is ψj,k,m = 2j ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z2 , m ∈ Z3 22/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 44. Introducing 3D Shearlets • Anisotropic scaling Aj (α ∈ (1, 2]): 2jα/2 0 0 Aj = 0 2j/2 0 , 0 0 2j/2 • Shearing Sk , k = (k1 , k2 ) (direction parameter ↔ rotations): 1 k1 k2 Sk = 0 1 0 0 0 1 • The shearlet system generated by ψ ∈ L2 (R3 ) is 1 ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z2 , m ∈ Z3 α 22/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 45. Compactly Supported Generators Idea: We want generators of the form ψ(x1 , x2 , x3 ) = η(x1 )φ(x2 )φ(x3 ) for the shearlets system associated with the pyramids P1 , where η is a 1D wavelet and φ is a scaling/bump function. The wavelet η will then point in the ‘short’ direction of the plate-like elements. Class of generators: We consider generators with weak directional vanishing moments and essential support (in Frequency domain) as: 23/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 46. Example: 3D Action of Anisotropic scaling and Shearing Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter m = (0, 0, 0) be ﬁxed. Scaling For k = (0, 0): supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the shearlet becomes a small, plate-like element as j → ∞. 24/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 47. Example: 3D Action of Anisotropic scaling and Shearing Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter m = (0, 0, 0) be ﬁxed. Scaling For k = (0, 0): supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the shearlet becomes a small, plate-like element as j → ∞. Shearing For j > 0: ψ(Sk A2j x) = ψ(2j x1 + k1 2j/2 x2 + k2 2j/2 x3 , 2j/2 x2 , 2j/2 x3 ) 24/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 48. Example: 3D Action of Anisotropic scaling and Shearing Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter m = (0, 0, 0) be ﬁxed. Scaling For k = (0, 0): supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the shearlet becomes a small, plate-like element as j → ∞. Shearing For j > 0: ψ(Sk A2j x) = ψ(2j x1 + k1 2j/2 x2 + k2 2j/2 x3 , 2j/2 x2 , 2j/2 x3 ) Problem To get horizontal plate-elements require very large shear parameters |ki | = ∞ ❀ non-uniform treatment of directions. 24/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 49. 3D Action of Anisotropic Scaling and Shearing Spatial Domain: Frequency Domain: 25/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 50. Pyramid-adapted Shearlet Systems Idea: We partition the frequency space R3 into 3 pairs of pyramids: {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ1 | ≥ 1, |ξ2 /ξ1 | ≤ 1, |ξ3 /ξ1 | ≤ 1} : ι = 1, Pι = {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ2 | ≥ 1, |ξ1 /ξ2 | ≤ 1, |ξ3 /ξ2 | ≤ 1} : ι = 2, {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ3 | ≥ 1, |ξ1 /ξ3 | ≤ 1, |ξ2 /ξ3 | ≤ 1} : ι = 3, and a centered cube C ={(ξ1 , ξ2 , ξ3 ) ∈ R3 : (ξ1 , ξ2 , ξ3 ) ∞ < 1}. P3 C P1 015 P2 016 017 ❀ For each pair of pyramids we construct a shearlet frame. 26/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 51. Pyramid-adapted Shearlet Systems ˜ ˘ The (pyramid-adapted) shearlet system SH(φ, ψ, ψ, ψ) generated ˜ ψ ∈ L2 (R3 ) is the union of by φ, ψ, ψ, ˘ {φm = φ(· − m) : m ∈ Z3 }, {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 }, ˜ ˜ ˜ ˜ {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 }, ˘ ˘ ˘ ˘ {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 }, ˜ where j ∈ N0 and k ∈ Z2 . We call φ a scaling function and ψ, ψ, ˘ shearlets. and ψ 27/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 52. Pyramid-adapted Shearlet Systems ˜ ˘ The (pyramid-adapted) shearlet system SH(φ, ψ, ψ, ψ; α) generated ˜ ψ ∈ L2 (R3 ) is the union of by φ, ψ, ψ, ˘ {φm = φ(· − m) : m ∈ Z3 }, α 1 {ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 }, α 1 {ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 }, ˜ ˜ ˜ ˜ α 1 ˘ ˘ ˘ ˘ {ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 }, ˜ where j ∈ N0 and k ∈ Z2 . We call φ a scaling function and ψ, ψ, ˘ shearlets. and ψ 27/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 53. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets Optimally Theorem Sketch of proof 5 Conclusion 28/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 54. Revisiting 3D Cartoon-like Images Deﬁnition 2 Let 1 < α ≤ 2. The set of 3D images Eα (R3 ) is deﬁned by Eα (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C α -surface with ‘curvature’ bounded by ν. ❇➯ ✶ ✺✼ ✳✵ ✵✺ ✳✵ ✺✷✳✵ ✶ ✺✼✳✵ ✵ ✵ ✵✺ ✳✵ ✺✷ ✳✵ ✵✺✳✵ ✺✷✳✵ ✺✼ ✳✵ ✵ ✶ 29/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 55. Revisiting 3D Cartoon-like Images Deﬁnition Let 1 < α ≤ 2. The set of 3D images Eα (R3 ) is deﬁned by 2 Eα (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C α -surface with ‘curvature’ bounded by ν. Theorem (Kutyniok, L, Lim; 2011) Let (ψλ )λ ⊂ L2 (R3 ). The optimal asymptotic approximation error of f ∈ Eα (R3 ) is 2 2 f − fN 2 N −α/2 , N → ∞, where fN = cλ ψλ . λ∈IN 29/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 56. Main result: Optimal Sparsity of 3D shearlet Theorem (Kutyniok, L, Lim; 2011) ˜ ˘ Fix α ∈ (1, 2]. Let φ, ψ, ψ, ψ ∈ L2 (R3 ) be compactly supported. Assume: ˆ (i) |ψ(ξ)| min(1, |ξ1 |δ ) · 3 min(1, |ξi |−γ ) i=1 ∂ ˆ |ξ2 | −γ |ξ3 | −γ (ii) ∂ξi ψ(ξ) ≤ |h(ξ1 )| · 1 + |ξ1 | 1+ |ξ1 | , i = 2, 3, ˜ ˘ where δ > 8, γ ≥ 4, h ∈ L1 (R), and similar for ψ and ψ. Further, ˜ ψ; α) forms a frame for L2 (R3 ). For suppose that SH(φ, ψ, ψ, ˘ f ∈ Eα (R3 ), 2 2 O(N −α/2+µ ), if α < 2, f − fN 2 = −1 2 as N → ∞, O(N (log N) ), if α = 2, where 3(2−α)(α−1)(α+2) µ = µ(α) = 2(9α2 +17α−10) , 30/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 57. Optimality Gap of Main result If α < 2, we have for f ∈ Eα (R3 ), 2 2 f − fN 2 = O(N −α/2+µ ), as N → ∞, where • 0 < µ(α) < 0.037 for α (1, 2), • µ(α) → 0 for α → 1+ or α → 2− . 1 0.9 0.8 0.7 0.6 0.5 1 1.2 1.4 1.6 1.8 2 α ❀ Shearlets provide nearly optimally sparse approximations of Eα (R3 ). 2 31/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 58. Sketch of proof: Frames and Non-linear approximations Deﬁnition A sequence (ψλ )λ ⊂ L2 (Rd ) is a frame for L2 (Rd ) if 2 ∃A, B > 0 : A f ≤ | f , ψλ |2 ≤ B f 2 for all f ∈ L2 (Rd ). λ ˜ Fact from frame theory: There exists a canonical dual frame {ψλ }λ such that f = ˜ f , ψλ ψλ for all f ∈ L2 (Rd ). λ 32/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 59. Sketch of proof: Frames and Non-linear approximations Deﬁnition A sequence (ψλ )λ ⊂ L2 (Rd ) is a frame for L2 (Rd ) if 2 ∃A, B > 0 : A f ≤ | f , ψλ |2 ≤ B f 2 for all f ∈ L2 (Rd ). λ ˜ Fact from frame theory: There exists a canonical dual frame {ψλ }λ such that f = ˜ f , ψλ ψλ for all f ∈ L2 (Rd ). λ As N-term approximation (not necessarily ‘best’) we take: fN = ˜ f , ψλ ψλ , λ∈IN where (| f , ψ |)λ λ∈IN are the N largest coeﬃcients. 32/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 60. Sketch of proof: Outline ˜ • Estimates of f , ψj,k,m for f ∈ Eα (R3 ). Similar for ψj,k,m and 2 ˘ ψj,k,m . • WLOG assume j ≥ 0 is suﬃciently large. • Let |θ(f )|n denote the nth largest shearlet coeﬃcient | f , ψj,k,m |: 2 1 f − fN 2 ≤ |θ(f )|2 . n A n>N • Hence, it suﬃces to prove: |θ(f )|2 ≤ C N −α/2 n as N → ∞. n>N 33/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 61. Sketch of proof: Heuristic argument of N −α/2 Recall: The shearlet is of the form ψ(x) = η(x1 )ϕ(x2 )ϕ(x3 ), where η is a wavelet and ϕ a bump (or a scaling) function. We consider three cases of coeﬃcients f , ψj,k,m : (a) Shearlets whose support do not overlap with the boundary ∂B. (b) Shearlets whose support overlap with ∂B and are nearly tangent. (c) Shearlets whose support overlap with ∂B, but tangentially. (a) (b) (c) 34/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 62. Sketch of proof: Heuristic argument of N −α/2 Case (b): • At scale j > 0: At most O(2j ) coeﬃcients since the plate-like element is of size 2−j/2 × 2−j/2 (and “thickness” 2−jα/2 ). Moreover, α 1 α 1 −j( 4 + 2 ) | f , ψj,k,m | ≤ f ∞ ψj,k,m 1 ≤ f ∞ ψ 12 ≤C 2−j( 4 + 2 ) 1 −j( 2 + α ) ❀ we have O(2j ) coeﬃcients bounded by C · 2 4 . 35/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 63. Sketch of proof: Heuristic argument of N −α/2 Case (b): • At scale j > 0: At most O(2j ) coeﬃcients since the plate-like element is of size 2−j/2 × 2−j/2 (and “thickness” 2−jα/2 ). Moreover, α 1 α 1 −j( 4 + 2 ) | f , ψj,k,m | ≤ f ∞ ψj,k,m 1 ≤ f ∞ ψ 12 ≤C 2−j( 4 + 2 ) 1 −j( 2 + α ) ❀ we have O(2j ) coeﬃcients bounded by C · 2 4 . • Assuming, case (a) and (c) coeﬃcients are negligible, the nth 1 largest coeﬃcient |θ(f )|n is bounded by |θ(f )|n ≤ C n−( 2 + 4 ) : α |θ(f )|2 n n−1−α/2 N −α/2 . n>N n>N 35/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 64. Heuristic argument of N −1/2 for Wavelets Wavelets: • At scale j > 0: At most O(2jα ) coeﬃcients since the cube-like element is of size 2−jα/2 × 2−jα/2 (and “thickness” 2−jα/2 ). Moreover, 3α 3 −j 4 | f , ψj,k,m | ≤ f ∞ ψj,k,m 1 ≤ f ∞ ψ 12 ≤C 2jα(− 4 ) jα(− 3 ) ❀ we have O(2jα ) coeﬃcients bounded by C · 2 . 4 • Assuming, case (a) and (c) coeﬃcients are negligible, the nth 3 largest coeﬃcient |θ(f )|n is bounded by |θ(f )|n ≤ C n− 4 : |θ(f )|2 ≤ n n−3/2 ≤ C · N −1/2 . n>N n>N 36/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 65. Generalization: Optimal Sparsity of 3D shearlets Question: • The above result shows that the ‘plate-like’ shearlet system is optimal for describing 2D singularities. But not all singularities in 3D images are located on C α -smooth surfaces! 37/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 66. Generalization: Optimal Sparsity of 3D shearlets Question: • The above result shows that the ‘plate-like’ shearlet system is optimal for describing 2D singularities. But not all singularities in 3D images are located on C α -smooth surfaces! • ❀ We would also like to consider point, curve and surface singularities: B f0=0 f1=1 37/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 67. Generalization: Optimal Sparsity of 3D shearlets Theorem (Kutyniok, L, Lim; 2011) The sparsity result extends to a class of cartoon-like 3D images with only piecewise smooth C α boundary ∂B. 38/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 68. Generalization: Optimal Sparsity of 3D shearlets Theorem (Kutyniok, L, Lim; 2011) The sparsity result extends to a class of cartoon-like 3D images with only piecewise smooth C α boundary ∂B. Observations: 1 The optimal rate is still the same when introducing “corner points” and “curves” 2 Pyramid-adapted ‘plate-like’ shearlets still achieve the optimal rate. 38/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 69. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 39/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 70. What To Take Home? • Anisotropic features in multivariate data require special eﬃcient encoding strategies • The shearlet theory is perfectly suited to this problem • One main advantage of shearlets is that they provide a uniﬁed treatment of the continuum and digital setting. • The Shearlet Theory for 2D and 3D data provides: • Compactly supported shearlet frames • Explicit estimates for frame bounds • Optimal sparse approximation of cartoon-like images. 40/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 71. Thank You! Preprints: http://www.shearlets.org http://www2.mat.dtu.dk/people/J.Lemvig/ Software: http://www.shearlab.org 41/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
- 72. References D. L. Donoho, Sparse components of images and optimal atomic decomposition, Constr. Approx. 17 (2001), 353–382. E. J. Candés and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities, Comm. Pure and Appl. Math. 56 (2004), 216–266. K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets, SIAM J. Math Anal. 39 (2007), 298–318. M. N. Do and M. Vetterli, The contourlet transform: an eﬃcient directional multiresolution image representation, IEEE Trans. Image Process. 14 (2005), 2091–2106. G. Kutyniok and W.-Q Lim, Compactly supported shearlets are optimally sparse, J. Approx. Theory 163 (2011), 1564–1589. S. Dahlke, G. Steidl, and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. Appl. 16 (2010), 340–364. G. Kutyniok, J. Lemvig, and W.-Q Lim, Compactly supported shearlets, in Approximation Theory XIII (San Antonio, TX, 2011), Springer. 42/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012