On nonstationary preconditioned iterative regularization methods for image deblurring

Transcription

1 On nonstationary preconditioned iterative regularization methods for image deblurring Alessandro Buccini joint work with Prof. Marco Donatelli University of Insubria Department of Science and High Technology Cetraro Summer School Cetraro, Italy June 22-26, 2015

2 Outline 1 Image Deblur 2 Tikhonov Regularization 3 4 Conclusions and Future Work Alessandro Buccini (University of Insubria) Methods for image deblurring 1 / 27

3 Image Deblur Outline 1 Image Deblur 2 Tikhonov Regularization 3 4 Conclusions and Future Work Alessandro Buccini (University of Insubria) Methods for image deblurring 2 / 27

4 Image Deblur Image Deblur The image deblur problem is a deconvolution problem. g(x,y) = K(s x,t y)f(s,t)dsdt = K f, (1) K is called PSF (Point Spread Function). Alessandro Buccini (University of Insubria) Methods for image deblurring 3 / 27

5 Image Deblur Image Deblur The image deblur problem is a deconvolution problem. g(x,y) = K(s x,t y)f(s,t)dsdt = K f, (1) K is called PSF (Point Spread Function). Discretizing (1) we obtain a linear system Ax = b. (2) Alessandro Buccini (University of Insubria) Methods for image deblurring 3 / 27

6 Image Deblur Image Deblur The image deblur problem is a deconvolution problem. g(x,y) = K(s x,t y)f(s,t)dsdt = K f, (1) K is called PSF (Point Spread Function). Discretizing (1) we obtain a linear system It is impossible to avoid noise so Ax = b. (2) b = b true +η. Alessandro Buccini (University of Insubria) Methods for image deblurring 3 / 27

7 Image Deblur Image Deblur The image deblur problem is a deconvolution problem. g(x,y) = K(s x,t y)f(s,t)dsdt = K f, (1) K is called PSF (Point Spread Function). Discretizing (1) we obtain a linear system It is impossible to avoid noise so Ax = b. (2) b = b true +η. We can not directly solve this linear system for x because A is severely ill-conditioned, we have to use regularization methods. Alessandro Buccini (University of Insubria) Methods for image deblurring 3 / 27

8 Image Deblur In practical application we only have finite data, so a problem arises near the boundary. In particular we need to make assumptions on what is outside the field of view. Alessandro Buccini (University of Insubria) Methods for image deblurring 4 / 27

9 Image Deblur (a) Zero (b) Periodic (c) Reflexive (d) Antireflexive Reflexive: [Ng et al., SISC1999]; Antireflexive: [S. Serra-Capizzano, SISC2003]. Alessandro Buccini (University of Insubria) Methods for image deblurring 5 / 27

10 Tikhonov Regularization Outline 1 Image Deblur 2 Tikhonov Regularization 3 4 Conclusions and Future Work Alessandro Buccini (University of Insubria) Methods for image deblurring 6 / 27

11 Tikhonov Regularization Tikhonov Regularization The general Tikhonov regularization has the form with N(L) N(A) = {0}. x α = argmin x Ax b 2 2 +α Lx 2 2, (3) Alessandro Buccini (University of Insubria) Methods for image deblurring 7 / 27

12 Tikhonov Regularization Tikhonov Regularization The general Tikhonov regularization has the form with N(L) N(A) = {0}. The solution of (3) is x α = argmin x Ax b 2 2 +α Lx 2 2, (3) x α = (A A+αL L) 1 A b. Alessandro Buccini (University of Insubria) Methods for image deblurring 7 / 27

13 Tikhonov Regularization Tikhonov Regularization The general Tikhonov regularization has the form with N(L) N(A) = {0}. The solution of (3) is x α = argmin x Ax b 2 2 +α Lx 2 2, (3) x α = (A A+αL L) 1 A b. By applying a refinement technique we obtain the Iterated Tikhonov method x k+1 = x k +(A A+αL L) 1 A (b Ax k ), which can be seen as a preconditioned Landweber. Alessandro Buccini (University of Insubria) Methods for image deblurring 7 / 27

14 Outline 1 Image Deblur 2 Tikhonov Regularization 3 4 Conclusions and Future Work Alessandro Buccini (University of Insubria) Methods for image deblurring 8 / 27

15 General Idea Iterated Tikhonov works on the error equation Ae k r k, (4) where e k = x x k and r k = b Ax k. Alessandro Buccini (University of Insubria) Methods for image deblurring 9 / 27

16 General Idea Iterated Tikhonov works on the error equation where e k = x x k and r k = b Ax k. Let C be such that for some 0 < ρ < 1 2. Ae k r k, (4) (C A)z 2 ρ Az 2, z, (5) Alessandro Buccini (University of Insubria) Methods for image deblurring 9 / 27

17 General Idea Iterated Tikhonov works on the error equation where e k = x x k and r k = b Ax k. Let C be such that for some 0 < ρ < 1 2. We approximate equation (4) with Ae k r k, (4) (C A)z 2 ρ Az 2, z, (5) Ce k r k. Alessandro Buccini (University of Insubria) Methods for image deblurring 9 / 27

18 AIT Algorithm (AIT, [M. Donatelli and M. Hanke, IP2013]) Let x 0 be fixed and set r 0 = b Ax 0. Choose τ = 1+2ρ 1 2ρ with ρ as in (5), and fix q [2ρ,1]. { While r k 2 > τ η 2, let τ k = r k 2 / η 2 and q k = max q,2ρ+ 1+ρ τ k }, compute h k = C (CC +α k I) 1 r k, where α k is such that and update r k Ch k 2 = q k r k 2, x k+1 = x k +h k, r k+1 = b Ax k+1. Alessandro Buccini (University of Insubria) Methods for image deblurring 10 / 27

19 AIT Remark In AIT, L = I. Alessandro Buccini (University of Insubria) Methods for image deblurring 11 / 27

20 AIT Remark In AIT, L = I. Proposition While r k 2 > τ η 2 the norm of the reconstruction error e k = x x k decreases monotonically. Alessandro Buccini (University of Insubria) Methods for image deblurring 11 / 27

21 AIT Remark In AIT, L = I. Proposition While r k 2 > τ η 2 the norm of the reconstruction error e k = x x k decreases monotonically. Theorem Let δ b δ a function such that δ b δ b 2 true < δ. Let C,τ, and q as in AIT and denote with x δ the approximation obtained with AIT with right hand side b δ. For δ 0 we have that x δ x 0, which is the nearest solution to x 0 of the system. Alessandro Buccini (University of Insubria) Methods for image deblurring 11 / 27

22 ARIT Algorithm (ARIT, [A. B. and M. Donatelli, submitted 2015]) Let x 0 be fixed and set r 0 = b Ax 0. Choose τ = 1+2ρ 1 2ρ with ρ as in (5), and fix q [2ρ,1]. { While r k 2 > τ η 2, let τ k = r k 2 / η 2 and q k = max q,2ρ+ 1+ρ τ k }, compute h k = C (CC +α k LL ) 1 r k, where α k is such that and update r k Ch k 2 = q k r k 2, x k+1 = x k +h k, r k+1 = b Ax k+1. Alessandro Buccini (University of Insubria) Methods for image deblurring 12 / 27

23 ARIT Remark For ARIT to work we need to make the same assumptions of AIT on C and A and the following N(A) N(L) = {0}; C N(L) = A N(L) ; L and C are diagonalized by the same unitary transformation. Alessandro Buccini (University of Insubria) Methods for image deblurring 13 / 27

24 ARIT Remark For ARIT to work we need to make the same assumptions of AIT on C and A and the following N(A) N(L) = {0}; C N(L) = A N(L) ; L and C are diagonalized by the same unitary transformation. Proposition While r k 2 > τ η 2 we have L 2 Le k L 2 Le k+1 0. Alessandro Buccini (University of Insubria) Methods for image deblurring 13 / 27

25 ARIT Remark For ARIT to work we need to make the same assumptions of AIT on C and A and the following N(A) N(L) = {0}; C N(L) = A N(L) ; L and C are diagonalized by the same unitary transformation. Proposition While r k 2 > τ η 2 we have L 2 Le k L 2 Le k+1 0. Theorem As δ 0 we have that x δ x 0, with the same notation of AIT. Alessandro Buccini (University of Insubria) Methods for image deblurring 13 / 27

26 APIT Algorithm (APIT, [A. B. and M. Donatelli, submitted 2015]) Let x 0 be fixed and set r 0 = b Ax 0. Choose τ = 1+2ρ 1 2ρ with ρ as in (5), and fix q [2ρ,1]. Let Ω be a closed and convex set such that x true Ω and P Ω the metric projection over it. { While r k 2 > τ η 2, let τ k = r k 2 / η 2 and q k = max q,2ρ+ 1+ρ τ k }, compute h k = C (CC +α k I) 1 r k, where α k is such that r k Ch k 2 = q k r k 2, and update x k+1 = P Ω (x k +h k ), r k+1 = b Ax k+1. Alessandro Buccini (University of Insubria) Methods for image deblurring 14 / 27

27 APIT Remark In APIT, L = I. Alessandro Buccini (University of Insubria) Methods for image deblurring 15 / 27

28 APIT Remark In APIT, L = I. Proposition While r k 2 > τ η 2 the norm of the reconstruction error e k = x x k decreases monotonically. Alessandro Buccini (University of Insubria) Methods for image deblurring 15 / 27

29 APIT Remark In APIT, L = I. Proposition While r k 2 > τ η 2 the norm of the reconstruction error e k = x x k decreases monotonically. Theorem As δ 0 we have that x δ x 0, with the same notation of AIT. Alessandro Buccini (University of Insubria) Methods for image deblurring 15 / 27

30 APRIT Algorithm (APRIT, [A. B. and M. Donatelli, submitted 2015]) Let x 0 be fixed and set r 0 = b Ax 0. Choose τ = 1+2ρ 1 2ρ with ρ as in (5), and fix q [2ρ,1]. Let Ω be a closed and convex set such that x true Ω and P Ω the metric projection over it. { While r k 2 > τ η 2, let τ k = r k 2 / η 2 and q k = max q,2ρ+ 1+ρ τ k }, compute h k = C (CC +α k LL ) 1 r k, where α k is such that r k Ch k 2 = q k r k 2, and update x k+1 = P Ω (x k +h k ), r k+1 = b Ax k+1. Alessandro Buccini (University of Insubria) Methods for image deblurring 16 / 27

31 Numerical Experiments Numerical Experiments A is the blurring matrix with the desired boundary conditions; Alessandro Buccini (University of Insubria) Methods for image deblurring 17 / 27

32 Numerical Experiments Numerical Experiments A is the blurring matrix with the desired boundary conditions; C is the blurring matrix with the same PSF of A but with periodic boundary conditions; Alessandro Buccini (University of Insubria) Methods for image deblurring 17 / 27

33 Numerical Experiments Numerical Experiments A is the blurring matrix with the desired boundary conditions; C is the blurring matrix with the same PSF of A but with periodic boundary conditions; L = L 1 I +I L 1, where L 1 is defined as L 1 = ; Alessandro Buccini (University of Insubria) Methods for image deblurring 17 / 27

34 Numerical Experiments Numerical Experiments A is the blurring matrix with the desired boundary conditions; C is the blurring matrix with the same PSF of A but with periodic boundary conditions; L = L 1 I +I L 1, where L 1 is defined as L 1 = ; Ω = {x R M : i = 1,...,M, x i 0}; Alessandro Buccini (University of Insubria) Methods for image deblurring 17 / 27

35 Numerical Experiments Numerical Experiments A is the blurring matrix with the desired boundary conditions; C is the blurring matrix with the same PSF of A but with periodic boundary conditions; L = L 1 I +I L 1, where L 1 is defined as L 1 = ; Ω = {x R M : i = 1,...,M, x i 0}; ρ = 10 3, q = 0.7; Alessandro Buccini (University of Insubria) Methods for image deblurring 17 / 27

36 Numerical Experiments Numerical Experiments A is the blurring matrix with the desired boundary conditions; C is the blurring matrix with the same PSF of A but with periodic boundary conditions; L = L 1 I +I L 1, where L 1 is defined as L 1 = ; Ω = {x R M : i = 1,...,M, x i 0}; ρ = 10 3, q = 0.7; Note that N(L) N(A) = {0}. Alessandro Buccini (University of Insubria) Methods for image deblurring 17 / 27

37 Numerical Experiments (a) (b) (c) Figure: Barbara Test case: (a) Barbara Test image, (b) Diagonal motion PSF, (c) Blurred image, η 2 = 0.03 b true 2. Since the image is generic we use the antireflexive boundary conditions. Alessandro Buccini (University of Insubria) Methods for image deblurring 18 / 27

38 Numerical Experiments AIT APIT ARIT APRIT Figure: Barbara Test Case: RRE History Alessandro Buccini (University of Insubria) Methods for image deblurring 19 / 27

39 Numerical Experiments Method RRE Computational Time AIT sec. ARIT sec. APIT sec. APRIT sec. Hybrid (Optimal: ) sec. TwIST sec. FlexiAT sec. RRAT sec. NN-ReStart-GAT sec. Table: Barbara: Relative errors comparisons Hybrid: [J. Chung, J. Nagy and, D. O Leary, ETNA2008]; TwIST: [J. Bioucas-Dias and M. Figueiredo, IEEE2007]; FlexiAT: [S. Gazzola and J. Nagy, SISC2014]; RRAT: [A. Neuman, L. Reichel and, H. Sadok, NA2012]; NN-ReStart-GAT: [S. Gazzola and J. Nagy, SISC2014]. Alessandro Buccini (University of Insubria) Methods for image deblurring 20 / 27

40 Numerical Experiments (a) (b) (c) (d) Figure: Barbara Reconstructions: (a) ARIT, (b) APIT, (c) Hybrid, (d) TwIST. Alessandro Buccini (University of Insubria) Methods for image deblurring 21 / 27

41 Numerical Experiments (a) (b) (c) Figure: Satellite Test case: (a) Test image, (b) Astronomic PSF, (c) Blurred image, η 2 = 0.01 b true 2. Since the image is all black around the boundaries we use the zero boundary conditions. Alessandro Buccini (University of Insubria) Methods for image deblurring 22 / 27

42 Numerical Experiments AIT APIT ARIT APRIT Figure: Satellite Test Case: RRE History Alessandro Buccini (University of Insubria) Methods for image deblurring 23 / 27

43 Numerical Experiments Method RRE Computational Time AIT sec. ARIT sec. APIT sec. APRIT sec. Hybrid sec. TwIST sec. FlexiAT sec. RRAT sec. NN-ReStart-GAT sec. Table: Satellite: Relative errors comparisons Alessandro Buccini (University of Insubria) Methods for image deblurring 24 / 27

44 Numerical Experiments (a) (b) (c) (d) Figure: Satellite Reconstructions: (a) APIT, (b) APRIT, (c) RRAT, (d) NN-Restart-GAT. Alessandro Buccini (University of Insubria) Methods for image deblurring 25 / 27

45 Conclusions and Future Work Outline 1 Image Deblur 2 Tikhonov Regularization 3 4 Conclusions and Future Work Alessandro Buccini (University of Insubria) Methods for image deblurring 26 / 27

46 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

47 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

48 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Capable of dealing with high level of noise; Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

49 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Capable of dealing with high level of noise; Any boundary condition can be easily implemented. Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

50 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Capable of dealing with high level of noise; Any boundary condition can be easily implemented. Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

51 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Capable of dealing with high level of noise; Any boundary condition can be easily implemented. Possible future improvements Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

52 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Capable of dealing with high level of noise; Any boundary condition can be easily implemented. Possible future improvements Non spatially invariant blur; Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

53 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Capable of dealing with high level of noise; Any boundary condition can be easily implemented. Possible future improvements Non spatially invariant blur; More advanced regularization term (p norms, TV,...); Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

54 Conclusions and Future Work Conclusions and Future Work We proposed four methods with good properties Almost parameter free; Capable of dealing with high level of noise; Any boundary condition can be easily implemented. Possible future improvements Non spatially invariant blur; More advanced regularization term (p norms, TV,...); Blind deconvolution. Alessandro Buccini (University of Insubria) Methods for image deblurring 27 / 27

55 Thank you for your attention.