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Image restoration from noisy incomplete frequency data by alternative iteration scheme Jijun Liu School of Mathematics Southeast University, Nanjing, P.R.China The Hong Kong University of Science and Technology, May 20-24, 2019 This is a joint work with Ph.D Xiaoman Liu J.J. Liu (SEU) image restoration by AIS 1 / 63

Outline 1 Introduction 2 The multi-penalty regularization modeling 3 The alternative iteration scheme 4 Convergence property of iteration process 5 Numerical experiments 6 Summary J.J. Liu (SEU) image restoration by AIS 2 / 63

Introduction Outline 1 Introduction 2 The multi-penalty regularization modeling 3 The alternative iteration scheme 4 Convergence property of iteration process 5 Numerical experiments 6 Summary J.J. Liu (SEU) image restoration by AIS 3 / 63

Introduction Introduction (a) exact image (b) Gaussian noise (c) 50% random noise (d) exact color image (e) salt pepper noise (f) speckle noise Figure 1: Exact image and noisy images with different noise. J.J. Liu (SEU) image restoration by AIS 4 / 63

Introduction Introduction J.J. Liu (SEU) image restoration by AIS 5 / 63

Introduction Introduction 1 Signal penalty regularization ROF model; Compressive sensing (CS); ... l 2 ; Total variation (TV), Total generalized variation (TGV) [1]; l q := � l 0 → l 1 , or � f � q i | f i | q (0 < q < 1) [2]; l 0 → low rank matrix, like truncated norm l 1 − 2 [3]: denoted as l t, 1 − 2 for sparse (vector) recovery and (matrix) rank minimization, � � � � x � t, 1 − 2 := | x i | − x 2 i , i/ i/ ∈ Γ x ,t ∈ Γ x ,t for any i / ∈ Γ x ,t and j ∈ Γ x ,t , | x i | ≤ | x j | . 2 Multi-penalty regularization [1] Bredies K, Kunisch K, Pock T. SIAM J. Imaging Sci. , 2010. [2] Chartrand R. IEEE Signal Process. Lett. , 2007. [3] Ma T H, Lou Y F, Huang T Z. SIAM J. Imaging Sci. , 2017. J.J. Liu (SEU) image restoration by AIS 6 / 63

The multi-penalty regularization modeling Outline 1 Introduction 2 The multi-penalty regularization modeling 3 The alternative iteration scheme 4 Convergence property of iteration process 5 Numerical experiments 6 Summary J.J. Liu (SEU) image restoration by AIS 7 / 63

The multi-penalty regularization modeling Notation and Symbols f : image vector for an N × N two-dimensional image f := ( f m,n ) where m, n = 1 , · · · , N . F : Fourier transform. J.J. Liu (SEU) image restoration by AIS 8 / 63

The multi-penalty regularization modeling Notation and Symbols f : image vector for an N × N two-dimensional image f := ( f m,n ) where m, n = 1 , · · · , N . F : Fourier transform. F : Fourier transform matrix, F m,n = e − i 2 π N mn . F : Two-dimensional discrete Fourier transform (DFT) matrix, f := vect [ F T fF ] = ( F ⊗ F ) f := Ff , ˆ where ⊗ is the tensor product of two matrices. J.J. Liu (SEU) image restoration by AIS 8 / 63

The multi-penalty regularization modeling Sampling matrix P : the N × N matrix generating from the identity matrix I by setting its N − M rows as null vectors, i.e., P = diag( p 11 , p 22 , · · · , p NN ) , p ii ∈ { 0 , 1 } . P ˆ f : only take M ( ≤ N ) rows of ˆ f , vect [ P ˆ f ] = ( I ⊗ P ) vect [ ˆ f ] = ( I ⊗ P ) ˆ f := Pˆ f . Remark vect [ P ˆ f ] = ( I ⊗ P ) vect [ ˆ f ] = ( I ⊗ P ) ˆ f := P r ˆ f . vect [ ˆ fP ] = ( P ⊗ I ) vect [ ˆ f ] = ( P ⊗ I ) ˆ f := P c ˆ f . J.J. Liu (SEU) image restoration by AIS 9 / 63

The multi-penalty regularization modeling Sampling operator P ∗ : random band sampling (RBS), i.e., takes both M r -row sampling and M c -column sampling together � ˆ P ∗ : ˆ f → P r ˆ f + ˆ fP c − P r ˆ f fP c . R center : the efficient elements among all the sampling elements, R center := m r × m c . M r M c R total : the sampling ratio of P ∗ , R total := N × M r + N × M c − M r × M c , N 2 which m r rows and m c columns located in the center area. J.J. Liu (SEU) image restoration by AIS 10 / 63

The multi-penalty regularization modeling General multi-penalty model The reconstruction of f with the sparsity requirement under the basis { ψ m,n : m, n = 1 , · · · , N } can be modeled by   {� ˜ f � l 0 : �PF Ψ[ ˜ g δ � F ≤ δ } , min f ] − P ˆ ˜ (1) f  f := Ψ[ ˜ f ] , J α ( f ) := 1 g δ � 2 F + α 1 � Ψ − 1 [ f ] � l 1 + α 2 | f | TV . 2 �PF f − P ˆ (2) With Charbonnier approximation, the model can be rewritten as � 1 � g δ � 2 F + α 1 � Ψ − 1 [ f ] � l 1 ,φ C min 2 �PF f − P ˆ β + α 2 | f | TV,φ C . (3) f β g δ : the incomplete noisy frequency data; P ˆ α 1 , α 2 : regularizing parameters, α := ( α 1 , α 2 ) > 0. J.J. Liu (SEU) image restoration by AIS 11 / 63

The multi-penalty regularization modeling General multi-penalty model Theorem 2.1 For given sampling operator P and α = { α 1 , α 2 } > 0 , β ≥ 0 , there exists a local minimizer f ∗ ,δ α,β (maybe not unique) to the optimization problem (3) in R N × N . Moreover, f ∗ ,δ α,β has the following error estimates:  F ≤ δ 2 + 2( α 1 + α 2 ) N 2 √ β �PF f ∗ ,δ g δ � 2 α,β − P ˆ     + 2 α 1 � Ψ − 1 [ f † ] � l 1 + 2 α 2 | f † | TV ,   α 1 ) N 2 √ β � Ψ − 1 [ f ∗ ,δ δ 2 2 α 1 + α 2 α 1 | f † | TV + (1 + α 2 α,β ] � l 1 ≤ (4)   + � Ψ − 1 [ f † ] � l 1 ,    α 2 ) N 2 √ β + | f † | TV  δ 2 | f ∗ ,δ 2 α 2 + α 1 α 2 � Ψ − 1 [ f † ] � l 1 + (1 + α 1 α,β | TV ≤ where f † ∈ R N × N is the grey matrix for exact image. · Liu X M, Liu J J. On image restoration from random sampling noisy frequency data with regularization. Inverse Problems in Science and Engineering , 2018. J.J. Liu (SEU) image restoration by AIS 12 / 63

The multi-penalty regularization modeling General multi-penalty model (4a) the data-fitting error: when α 1 = α 2 = δ 2 and small constant β > 0, the optimal order is O ( δ 2 ); (4b) the sparsity estimate: depends on the ratio α 2 α 1 ; (4c) the smoothness estimate: depends on the ratio α 1 α 2 . We can take α 1 = δ, α 2 = δ 2 , β = δ 2 such that �PF f ∗ ,δ g δ � 2 F ≤ Cδ 2 , α,β − P ˆ 0 ≤ � Ψ − 1 ◦ f ∗ ,δ α,β � l 1 − � Ψ − 1 ◦ f † � l 1 ≤ Cδ, α,β | TV − | f † | TV ≤ C 1 0 ≤ | f ∗ ,δ δ . J.J. Liu (SEU) image restoration by AIS 13 / 63

The multi-penalty regularization modeling Simplify multi-penalty model � 1 � g δ � 2 F + α 1 � Ψ − 1 [ f ] � l 1 + α 2 | f | TV min 2 �PF f − P ˆ (2) f Difficulties: 1) l 1 penalty term: with complex sparsity framework. 2) huge computational works. 3) non-smooth, non-convex. J.J. Liu (SEU) image restoration by AIS 14 / 63

The multi-penalty regularization modeling Simplify multi-penalty model � 1 � g δ � 2 F + α 1 � Ψ − 1 [ f ] � l 1 + α 2 | f | TV min 2 �PF f − P ˆ (2) f Difficulties: 1) l 1 penalty term: with complex sparsity framework. 2) huge computational works. 3) non-smooth, non-convex. The sparse basis Ψ (like Danbechies) ⇒ boundary operator C 1 � vect [ f ] � l 1 ≤ � Ψ − 1 [ f ] � l 1 ≤ C 2 � vect [ f ] � l 1 . � 1 � g δ � 2 (2) ⇒ min 2 �PF f − P ˆ F + α 1 � vect [ f ] � l 1 + α 2 | f | TV (5) f J.J. Liu (SEU) image restoration by AIS 14 / 63

The alternative iteration scheme Outline 1 Introduction 2 The multi-penalty regularization modeling 3 The alternative iteration scheme 4 Convergence property of iteration process 5 Numerical experiments 6 Summary J.J. Liu (SEU) image restoration by AIS 15 / 63

The alternative iteration scheme Reformulation of the simplify model Outline 1 Introduction 2 The multi-penalty regularization modeling 3 The alternative iteration scheme Reformulation of the simplify model Fast algorithm based on alternative iteration 4 Convergence property of iteration process 5 Numerical experiments 6 Summary J.J. Liu (SEU) image restoration by AIS 16 / 63

The alternative iteration scheme Reformulation of the simplify model Simplify model � g δ � α,ν ( f ) := 1 2 � � J Z � PF f − P ˆ F + α 1 � vect [ f ] � l 1 ,φ Z ν + α 2 | f | TV (6) � 2 for ( Z, ν ) = ( C, β ) or ( Z, ν ) = ( H, ǫ ). N N � � φ C φ H � vect [ f ] � l 1 ,φ C β = β ( f m,n ) , � vect [ f ] � l 1 ,φ H ǫ = ǫ ( f m,n ) . m,n =1 m,n =1 � s 2 � 2 ǫ , | s | ≤ ǫ, s 2 + β, φ C φ H β ( s ) := ǫ ( s ) := | s | − ǫ 2 , | s | > ǫ. J.J. Liu (SEU) image restoration by AIS 17 / 63

The alternative iteration scheme Reformulation of the simplify model Simplify model 3 3 f1=|x| f1 f2=sqrt(x 2 +beta) f2 2 2 f3=Huber function f3 f(x) 1 df 1 0 0 -1 -1 -2 -1 0 1 2 -2 -1 0 1 2 x x (a) Three functions (b) The derivatives of three functions Figure 2: The absolute value function comparing with Charbonnier and Huber function, β = ǫ = 0 . 1. J.J. Liu (SEU) image restoration by AIS 18 / 63