Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Applications Conclusions Compactly Supported Shearlets: Construction, Optimally Sparse Approximation, and Applications Gitta Kutyniok (University of Osnabr¨uck, Germany) joint with Jakob Lemvig & Wang-Q Lim (University of Osnabr¨uck, Germany) Illinois/Missouri Applied Harmonic Analysis Seminar University of Illinois, October 16, 2010 Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Applications Conclusions Outline Applied Harmonic Analysis and Imaging Sciences Anisotropic Phenomena Goal for Today Compactly Supported Shearlet Systems Review: Discrete Shearlets in 2D Shearlet Systems in 3D Optimal Sparse Approximation Frame Properties Applications Denoising Image Separation Conclusions Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions 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: Layers of earth. ◮ ...implicitely. ◮ Transport Equations: Shock fronts. Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Modern Imaging Some important tasks: ◮ Denoising. ◮ Inpainting. ◮ Feature Detection/Extraction. ◮ ... Imaging Sciences using Applied Harmonic Analysis: Exploit a carefully designed representation system (ψλ)λ: Image = Image, ψλ ψλ. λ ◮ Sparse coefficients! ◮ Approximation properties! Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions What is an Image? Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions What is an Image? Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions What is an Image? Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Reasonable Model for Images Definition (Donoho; 2001): 2 The set of cartoon-like 2D images E2 (ν) is defined by 2 2 R2 E2 (ν)= {f ∈ L ( ) : f = f0 + f1 χB }, 2 2 2 2 where f0, f1 ∈ C (R ) with supp fi ⊂ [0, 1] and B ⊂ [0, 1] with ∂B a closed C 2-curve whose curvature is bounded by ν. Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Reasonable Model for Images Definition (Donoho; 2001): 2 The set of cartoon-like 2D images E2 (ν) is defined by 2 2 R2 E2 (ν)= {f ∈ L ( ) : f = f0 + f1 χB }, 2 2 2 2 where f0, f1 ∈ C (R ) with supp fi ⊂ [0, 1] and B ⊂ [0, 1] with ∂B a closed C 2-curve whose curvature is bounded by ν. Theorem (Donoho; 2001): 2 2 Let (ψλ)λ ⊆ L (R ). Allowing only polynomial depth search, the 2 optimal asymptotic approximation error of f ∈E2 (ν) is 2 −2 f − fN 2 ≤ C · N , N →∞, where fN = cλψλ. λ∈IN Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions What can Wavelets do? Problem: ◮ 2 2 −1 For E2 (ν), Wavelets only achieve f − fN 2 ≤ C · N , N →∞. ◮ Isotropic structure of wavelets: −j/2 −j 2 {ψj,k = 2 ψ(2 · −k) : j ∈ Z, k ∈ Z } ◮ Wavelets cannot sparsely represent curves. Intuitive explanation: Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Main Goal in Geometric Multiscale Analysis Design a representation system which... ◮ ...is generated by one ‘mother function’, ◮ ...provides optimally sparse approximation of cartoons, ◮ ...allows for compactly supported analyzing elements, ◮ ...is associated with fast decomposition algorithms, ◮ ...treats the continuum and digital ‘world’ uniformly. Vision: Introduce a system for 2D data as powerful as wavelets for 1D data! Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Previous Approaches Non-exhaustive list: ◮ Directional wavelets (Antoine, Murenzi, Vandergheynst; 1999) ◮ Ridgelets (Cand`es and Donoho; 1999) ◮ Curvelets (Cand`es and Donoho; 2002) ◮ Contourlets (Do and Vetterli; 2002) ◮ Bandlets (LePennec and Mallat; 2003) ◮ Wavelets with composite dilations (Guo, Labate, Lim, Weiss, and Wilson; 2004) ◮ Shearlets (K, Labate, Lim, and Weiss; 2005), (Guo, K, and Labate; 2006) ◮ . Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Most Approaches are for 2D Data Question: Why is the 3D situation such crucial? Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Most Approaches are for 2D Data Question: Why is the 3D situation such crucial? Answer: ◮ Our world is 3-dimensional. ◮ 3D data is essential for Biology, Seismology,... ◮ Transition 2D → 3D is unique. ◮ Anisotropic features occur in 3D for the first time in different dimensions. Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions Goal for Today Goal: Introduce shearlets for 3D data which... ◮ ...are generated by one ‘mother function’, ◮ ...provide optimally sparse approximation of cartoons, ◮ ...allow for compactly supported analyzing elements, ◮ ...are associated with fast decomposition algorithms, ◮ ...treat the continuum and digital ‘world’ uniformly. Remark: Even new in 2D: ◮ Construction of compactly supported shearlets. ◮ Optimal sparse approximation of such shearlets. Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Review: Discrete Shearlets in 2D Compactly Supported Shearlet Systems Shearlet Systems in 3D Applications Optimal Sparse Approximation Conclusions Frame Properties Which is the Correct Scaling...? In harmonic analysis there have been a number of important applications of decompositions based on parabolic scaling 1 fa(x1, x2)= f (a 2 x1, ax2), 2 which leaves invariant the parabola x2 = x1 . History: ◮ 1970’s: Fefferman and later Seeger, Sogge, and Stein studied boundedness of certain operators. ◮ 1998: Hart Smith defined special molecular decompositions of Fourier Integral Operators. Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Review: Discrete Shearlets in 2D Compactly Supported Shearlet Systems Shearlet Systems in 3D Applications Optimal Sparse Approximation Conclusions Frame Properties Which is the Correct Measure for Directions...? Angle versus slope: ◮ It seems that the appropriate measure for directions is the angle (↔ rotation). Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Review: Discrete Shearlets in 2D Compactly Supported Shearlet Systems Shearlet Systems in 3D Applications Optimal Sparse Approximation Conclusions Frame Properties Which is the Correct Measure for Directions...? Angle versus slope: ◮ It seems that the appropriate measure for directions is the angle (↔ rotation). ◮ However, in today’s digital world slope (↔ shearing) is much more appropriate. ◮ Example: Parametrization by slope is more suitable for measurements in seismology. ◮ Supports the treating of the digital setting: 1 k Z2 = Z2 for all k ∈ Z. 0 1 Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Review: Discrete Shearlets in 2D Compactly Supported Shearlet Systems Shearlet Systems in 3D Applications Optimal Sparse Approximation Conclusions Frame Properties Construction of 2D Discrete Shearlets Idea: ◮ Consider (affine) systems: 1/2 2 {| det M| ψ(Mx − m) : M ∈ G ⊆ GLn, m ∈ Z }. ◮ Recall: For wavelets, G = {diag(2j , 2j ) : j ∈ Z}. ◮ Construct G to be a special 2-parameter dilation group by: 2j 0 1 k Aj = and Sk = , j, k ∈ Z. 0 2j/2 0 1 Aj : Parabolic scaling matrix, Sk : Shear matrix. ◮ Dilation matrices: G = {Sk Aj : j, k ∈ Z}. Gitta Kutyniok, Jakob Lemvig, &