Lecture 2: Applied Harmonic Analysis and Compressed Sensing Gitta Kutyniok (Technische Universit¨ at Berlin) Winter School on “Compressed Sensing”, TU Berlin December 3–5, 2015 Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 1 / 30
Challenge in Data Analysis Recovery of missing data is a common problem in many applications. In imaging science, this is termed the inpainting problem. Examples include... Removal of scratches on old photos. Removal of overlaid text or graphics. Missing sensors for measuring seismic data. Filling-in missing blocks in unreliably transmitted images. ... Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 2 / 30
Inpainting, I (Source: Elad, Starck, Querre, and Donoho; 2005) Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 3 / 30
Inpainting, II (Source: Hennenfent and Herrmann; 2008) Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 4 / 30
Compressed Sensing Approach to Inpainting Main Philosophy: Original image is sparsified by a particular representation system. Optimize the sparsity of the reconstructed image. (Incomplete) List of Contributors: Donoho, Elad, Querre, and Starck (2005) Cai, Chan, Dong, Fadili, Gribonval, Hennenfent, Herrmann, Li, Setzer, Shen, Steidl, Ye, Xu,... Another Path are Variational Approaches: Propagate information from the boundaries & guarantee smoothness. Contributors: Bornemann, Chan, Esedoglu, Kang, Osher, Sapiro, Setzer, Shen, Steidl, Vese, ... Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 5 / 30
What is the Relation of Compressed Sensing and Inpainting? Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 6 / 30
‘Mathematical Model’ Model for missing data: Given a signal x = x K + x M ∈ H K ⊕ H M . Recover x , if only x K is known. Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 7 / 30
‘Mathematical Model’ Model for missing data: Given a signal x = x K + x M ∈ H K ⊕ H M . Recover x , if only x K is known. Isn’t this impossible? x M lives in an orthogonal space. Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 7 / 30
‘Mathematical Model’ Model for missing data: Given a signal x = x K + x M ∈ H K ⊕ H M . Recover x , if only x K is known. Isn’t this impossible? x M lives in an orthogonal space. But we have additional information: x and x K are morphologically different. Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 7 / 30
Underdetermined Problem! Main Idea: Let x ∈ H be a signal. Φ be an ONB, and set x = Φ c . H = H K ⊕ H M with orthogonal projections P K and P M . Task: Given y = P K x ∈ H K , solve y = P K ˜ x for ˜ x ∈ H . Problem: This is an underdetermined problem! Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 8 / 30
Compressed Sensing enters the Picture! Solve y = Ax , A = ( a i ) i ∈ R n × m with n << m , if x is sparse, i.e., � x � 0 = # { i : x i � = 0 } is small. A is incoherent, e.g., max i � = j |� a i , a j �| is small. Then x can be recovered from y by Convex optimization: min ˜ x � ˜ x � 1 s.t. y = A ˜ x . Greedy algorithms: maximize |� y , a i �| etc. Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 9 / 30
Compressed Sensing enters the Picture! Solve y = Ax , A = ( a i ) i ∈ R n × m with n << m , if x is sparse, i.e., � x � 0 = # { i : x i � = 0 } is small. A is incoherent, e.g., max i � = j |� a i , a j �| is small. Then x can be recovered from y by Convex optimization: min ˜ x � ˜ x � 1 s.t. y = A ˜ x . Greedy algorithms: maximize |� y , a i �| etc. Theorem (Bruckstein, Elad; 2002)(Donoho, Elad; 2003) Let A ∈ R n × m with normalized columns, n << m , and let x satisfy � x � 0 < 1 1 + µ ( A ) − 1 � � , 2 with coherence µ ( A ) = max i � = j |� a i , a j �| . For y = Ax , we then have x = argmin ˜ x � ˜ x � 1 subject to y = A ˜ x . Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 9 / 30
Birth of Inpainting via Compressed Sensing Situation: Let x ∈ H be a signal. Φ be an ONB such that c with x = Φ c is sparse. H = H K ⊕ H M with orthogonal projections P K and P M . ℓ 1 Minimization Problem (Elad, Starck, Querre, Donoho; 2005): c = argmin ˜ ˆ c � ˜ c � 1 s.t. P K x = P K Φ˜ c x = Φˆ ˆ c � Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 10 / 30
Birth of Inpainting via Compressed Sensing Situation: Let x ∈ H be a signal. Φ be an ONB such that c with x = Φ c is sparse. H = H K ⊕ H M with orthogonal projections P K and P M . ℓ 1 Minimization Problem (Elad, Starck, Querre, Donoho; 2005): c = argmin ˜ ˆ c � ˜ c � 1 s.t. P K x = P K Φ˜ c x = Φˆ ˆ c � Theorem (Donoho, Elad; 2003) If � c � 0 < 1 2 (1 + µ ( P K Φ) − 1 ) ( µ ( A ) = max i � = j |� a i , a j �| ), then x = Φˆ c . Morphological Difference: µ ( P K Φ) = µ ([ Φ | P K Φ ]) Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 10 / 30
Let’s go back to Imaging Science... Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 11 / 30
Back to Imaging Science... Main steps: Original image governed by curvilinear structures: ? Choose suitable representation system which optimally sparsely approximates curvelike structures � Shearlets. Minimize the ℓ 1 norm of the coefficients. The original image will be recovered as the sparsest possible continuation of the known part. Question: What is a good model for an image? Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 12 / 30
Fitting Model for Images Definition (Donoho; 2001): The set of cartoon-like images E 2 ( R 2 ) is defined by E 2 ( R 2 ) = { f ∈ L 2 ( R 2 ) : f = f 0 + f 1 · χ B } , where B ⊂ [0 , 1] 2 with ∂ B a closed C 2 -curve, f 0 , f 1 ∈ C 2 0 ([0 , 1] 2 ). Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 13 / 30
Fitting Model for Images Definition (Donoho; 2001): The set of cartoon-like images E 2 ( R 2 ) is defined by E 2 ( R 2 ) = { f ∈ L 2 ( R 2 ) : f = f 0 + f 1 · χ B } , where B ⊂ [0 , 1] 2 with ∂ B a closed C 2 -curve, f 0 , f 1 ∈ C 2 0 ([0 , 1] 2 ). Theorem (Donoho; 2001): Let ( ψ λ ) λ ⊆ L 2 ( R 2 ). Allowing only polynomial depth search, the optimal asymptotic approximation error of f ∈ E 2 ( R 2 ) is � � f − f N � 2 2 ≍ N − 2 , N → ∞ , where f N = c λ ψ λ . λ ∈ I N Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 13 / 30
(Cone-adapted) Discrete Shearlet Systems Definition (K, Labate; 2006): The (cone-adapted) discrete shearlet system SH ( c ; φ, ψ, ˜ ψ ), c > 0, generated by φ ∈ L 2 ( R 2 ) and ψ, ˜ ψ ∈ L 2 ( R 2 ) is the union of { φ ( · − m ) : m ∈ Z 2 } , { 2 3 j / 4 ψ ( S k A 2 j · − m ) : j ≥ 0 , | k | ≤ ⌈ 2 j / 2 ⌉ , m ∈ Z 2 } , { 2 3 j / 4 ˜ ψ ( ˜ S k ˜ A 2 j · − m ) : j ≥ 0 , | k | ≤ ⌈ 2 j / 2 ⌉ , m ∈ Z 2 } . Theorem (K, Lim; 2011): ψ , ˆ Let φ, ψ, ˜ ψ ∈ L 2 ( R 2 ) be compactly supported, and let ˆ ˜ ψ satisfy certain decay condition. Then SH ( φ, ψ, ˜ ψ ) provides an optimally sparse approximation of f ∈ E 2 ( R 2 ), i.e., 2 ≤ C · N − 2 · (log N ) 3 , � f − f N � 2 N → ∞ . Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 14 / 30
Analysis of Shearlet-Based Inpainting... Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 15 / 30
Numerical Results of Inpainting, I (Source: Lim; 2014) Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 16 / 30
Numerical Results of Inpainting, II Undersampled seismic data Reconstructed image (Source: K, Lim; 2012) Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 17 / 30
Numerical Results for Inpainting, III Original Noisy Version (80% missing) Curvelets (29.95dB, 182.15sec) Shearlets (31.04dB, 85.18sec) (Source: Lim; 2012) Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 18 / 30
Microlocal Model Masked Seismic image: Curvilinear Singularity: � ρ C = w ( t ) δ τ ( t ) dt , − ρ τ : [ − 1 , 1] → R 2 a C 2 -curve, ρ < 1, and w : [ − ρ, ρ ] → R + 0 ‘bump’. Mask: M h = { ( x 1 , x 2 ) ∈ R 2 : | x 1 | ≤ h } , h > 0 . Observed Signal: f = 1 R 2 \ M h · C . Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 19 / 30
Scale-Dependent Decomposition Observed Object: f = 1 R 2 \ M h · C . Subband Decomposition: C �→ C j = C ⋆ F j . Question: Shall we consider 1 R 2 \ M h · C j ? Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 20 / 30
Size of the Mask Intuition: Inpainting at scale j : Inpainting seems possible provided h = o (2 − j / 2 ) as j → ∞ . Asymptotic analysis: Consider h = h j . Set f j = 1 R 2 \ M hj · C j . Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 21 / 30
Size of the Mask Intuition: Inpainting at scale j : Inpainting seems possible provided h = o (2 − j / 2 ) as j → ∞ . Asymptotic analysis: Consider h = h j . Set f j = 1 R 2 \ M hj · C j . Shearlet-Inpainting: S j � ( � ˜ S j , σ η � ) η � 1 s.t. f j = 1 R 2 \ M hj · ˜ S j = argmin ˜ S j Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 21 / 30
Asymptotic Inpainting Theorem (King, K, Zhuang; 2014) For h j = o (2 − j / 2 ) as j → ∞ , � S j − C j � 2 → 0 , j → ∞ . �C j � 2 Gitta Kutyniok (TU Berlin) Lecture 2 Winter School 2015 22 / 30
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