Image deblurring by multigrid methods. Department of Physics and Mathematics University of Insubria
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1 Image deblurring by multigrid methods Marco Donatelli Stefano Serra-Capizzano Department of Physics and Mathematics University of Insubria
2 Outline 1 Restoration of blurred and noisy images The model problem Properties of the PSF Iterative regularization methods 2 Multigrid regularization Multigrid methods Iterative Multigrid regularization Computational Cost 3 Numerical experiments An airplane Direct multigrid regularization 4 Conclusions Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 2 / 27
3 Outline Restoration of blurred and noisy images 1 Restoration of blurred and noisy images The model problem Properties of the PSF Iterative regularization methods 2 Multigrid regularization Multigrid methods Iterative Multigrid regularization Computational Cost 3 Numerical experiments An airplane Direct multigrid regularization 4 Conclusions Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 3 / 27
4 Restoration of blurred and noisy images The model problem Image restoration with Boundary Conditions Using Boundary Conditions (BCs), the restored image f is obtained solving: (in some way...) Af = g + ξ g = blurred image, ξ = noise (random vector), A = two-level matrix depending on the point spread function (PSF) and the BCs. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 4 / 27
5 Restoration of blurred and noisy images The model problem Image restoration with Boundary Conditions Using Boundary Conditions (BCs), the restored image f is obtained solving: (in some way...) Af = g + ξ g = blurred image, ξ = noise (random vector), A = two-level matrix depending on the point spread function (PSF) and the BCs. The PSF is the observation of a single point (e.g., a star in astronomy). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 4 / 27
6 Restoration of blurred and noisy images Coefficient matrix structure The model problem The matrix-vector product computed in O(n 2 log(n)) ops for n n images while the inversion costs O(n 2 log(n)) ops only in the periodic case. BCs Dirichlet periodic Neumann (reflective) anti-reflective A Toeplitz circulant Toeplitz + Hankel Toeplitz + Hankel Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 5 / 27
7 Restoration of blurred and noisy images Coefficient matrix structure The model problem The matrix-vector product computed in O(n 2 log(n)) ops for n n images while the inversion costs O(n 2 log(n)) ops only in the periodic case. BCs Dirichlet periodic Neumann (reflective) anti-reflective A Toeplitz circulant Toeplitz + Hankel Toeplitz + Hankel If the PSF is symmetric with respect to each direction: BCs Neumann (reflective) anti-reflective A DCT III DST I + low-rank Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 5 / 27
8 Restoration of blurred and noisy images Generating function of PSF Properties of the PSF The eigenvalues of A(z) are about a uniform sampling of z. PSF Generating function z(x) Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 6 / 27
9 Restoration of blurred and noisy images Generating function of PSF Properties of the PSF The eigenvalues of A(z) are about a uniform sampling of z. PSF Generating function z(x) The ill-conditioned subspace is mainly constituted by the middle/high frequencies. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 6 / 27
10 Restoration of blurred and noisy images Iterative regularization methods Iterative regularization methods Semi-convergence behavior Some iterative methods (Landweber, CGNE,...) have regularization properties: the restoration error firstly decreases and then increases. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 7 / 27
11 Restoration of blurred and noisy images Iterative regularization methods Iterative regularization methods Semi-convergence behavior Some iterative methods (Landweber, CGNE,...) have regularization properties: the restoration error firstly decreases and then increases. Reason They firstly reduce the algebraic error in the low frequencies (well-conditioned subspace). When they arrive to reduce the algebraic error in the high frequencies then the restoration error increases because of the noise. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 7 / 27
12 Outline Multigrid regularization 1 Restoration of blurred and noisy images The model problem Properties of the PSF Iterative regularization methods 2 Multigrid regularization Multigrid methods Iterative Multigrid regularization Computational Cost 3 Numerical experiments An airplane Direct multigrid regularization 4 Conclusions Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 8 / 27
13 The algorithm Multigrid regularization Multigrid methods The choices 1 We apply only the pre-smoother simply called smoother. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 9 / 27
14 The algorithm Multigrid regularization Multigrid methods The choices 1 We apply only the pre-smoother simply called smoother. 2 Let R i and P i be the restriction and the prolongation operators at the level i, respectively. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 9 / 27
15 Multigrid regularization Multigrid methods The algorithm The choices 1 We apply only the pre-smoother simply called smoother. 2 Let R i and P i be the restriction and the prolongation operators at the level i, respectively. 3 We use the Galerkin approach P i = R T i A i+1 = RA i R T i Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 9 / 27
16 The algorithm Multigrid regularization Multigrid methods The choices 1 We apply only the pre-smoother simply called smoother. 2 Let R i and P i be the restriction and the prolongation operators at the level i, respectively. 3 We use the Galerkin approach P i = R T i A i+1 = RA i R T i 4 Coarser grid of size 8 8 independent of the size of the finer grid. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 9 / 27
17 Multigrid regularization The Algebraic Multigrid (AMG) Multigrid methods The AMG uses only information on the coefficient matrix. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 10 / 27
18 Multigrid regularization Multigrid methods The Algebraic Multigrid (AMG) The AMG uses only information on the coefficient matrix. Different classic smoothers have similar behavior: in the initial iterations they are not able to reduce effectively the error in the subspace generated by the eigenvectors associated to small eigenvalues (ill-conditioned subspace) Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 10 / 27
19 Multigrid regularization Multigrid methods The Algebraic Multigrid (AMG) The AMG uses only information on the coefficient matrix. Different classic smoothers have similar behavior: in the initial iterations they are not able to reduce effectively the error in the subspace generated by the eigenvectors associated to small eigenvalues (ill-conditioned subspace) To obtain a fast solver, the restriction is chosen in order to project the error equation in such subspace. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 10 / 27
20 Multigrid regularization Multigrid methods The Algebraic Multigrid (AMG) The AMG uses only information on the coefficient matrix. Different classic smoothers have similar behavior: in the initial iterations they are not able to reduce effectively the error in the subspace generated by the eigenvectors associated to small eigenvalues (ill-conditioned subspace) To obtain a fast solver, the restriction is chosen in order to project the error equation in such subspace. For Toeplitz matrices there are two talks: Fischer on Wednesday and Aricò on Friday. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 10 / 27
21 Multigrid regularization Multigrid methods Algebraic analysis of the Geometric Multigrid For elliptic PDEs the ill-conditioned subspace is made by low frequencies (complementary with respect to the gaussian blur). Poisson s problem π 3.5 Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 11 / 27
22 Multigrid regularization Multigrid methods Algebraic analysis of the Geometric Multigrid For elliptic PDEs the ill-conditioned subspace is made by low frequencies (complementary with respect to the gaussian blur). Poisson s problem π 3.5 Projector operator R = Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 11 / 27
23 Multigrid regularization Multigrid for structured matrices Multigrid methods Preserve the structure In order to apply recursively the MGM, it is necessary to keep the same structure at each level (Toeplitz,...). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 12 / 27
24 Multigrid regularization Multigrid for structured matrices Multigrid methods Preserve the structure In order to apply recursively the MGM, it is necessary to keep the same structure at each level (Toeplitz,...). For every structure arising from the proposed BCs, there exist projectors that preserve the same structure. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 12 / 27
25 Multigrid regularization Multigrid for structured matrices Multigrid methods Preserve the structure In order to apply recursively the MGM, it is necessary to keep the same structure at each level (Toeplitz,...). For every structure arising from the proposed BCs, there exist projectors that preserve the same structure. R i = K Ni A Ni (p), where Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 12 / 27
26 Multigrid regularization Multigrid for structured matrices Multigrid methods Preserve the structure In order to apply recursively the MGM, it is necessary to keep the same structure at each level (Toeplitz,...). For every structure arising from the proposed BCs, there exist projectors that preserve the same structure. R i = K Ni A Ni (p), where K Ni R N i 4 N i is the cutting matrix that preserves the structure at the lower level. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 12 / 27
27 Multigrid regularization Multigrid for structured matrices Multigrid methods Preserve the structure In order to apply recursively the MGM, it is necessary to keep the same structure at each level (Toeplitz,...). For every structure arising from the proposed BCs, there exist projectors that preserve the same structure. R i = K Ni A Ni (p), where K Ni R N i 4 N i is the cutting matrix that preserves the structure at the lower level. p(x, y) is the generating function of the projector, which selects the subspace where to project the linear system. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 12 / 27
28 Multigrid regularization Multigrid methods Multigrid, structured matrices, and images The cutting matrix K ni in 1D circulant Toeplitz&DST I DCT III [ ] [ ] [ ] Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 13 / 27
29 Multigrid regularization Multigrid methods Multigrid, structured matrices, and images The cutting matrix K ni in 1D circulant Toeplitz&DST I DCT III [ ] [ ] [ ] Low-pass filter: Low frequencies projection noise reduction 2D p(x, y) = (1 + cos(x))(1 + cos(y)) ց Full weighting ր Bilinear interpolation Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 13 / 27
30 Multigrid regularization Image deblurring and Multigrid Iterative Multigrid regularization In the image deblurring the ill-conditioned subspace is related to high frequencies, while the well-conditioned subspace is generated by low frequencies. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 14 / 27
31 Multigrid regularization Image deblurring and Multigrid Iterative Multigrid regularization In the image deblurring the ill-conditioned subspace is related to high frequencies, while the well-conditioned subspace is generated by low frequencies. In order to obtain a fast convergence the algebraic multigrid projects in the high frequencies where the noise lives = noise explosion already at the first iteration (it requires Tikhonov regularization [Donatelli, NLAA in press]). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 14 / 27
32 Multigrid regularization Image deblurring and Multigrid Iterative Multigrid regularization In the image deblurring the ill-conditioned subspace is related to high frequencies, while the well-conditioned subspace is generated by low frequencies. In order to obtain a fast convergence the algebraic multigrid projects in the high frequencies where the noise lives = noise explosion already at the first iteration (it requires Tikhonov regularization [Donatelli, NLAA in press]). In this case the low-pass filter projects in the well-conditioned subspace (low frequencies) = it is slowly convergent but it can be a good iterative regularizer. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 14 / 27
33 Multigrid regularization Iterative multigrid regularization Iterative Multigrid regularization The Multigrid as an iterative regularization method If we have an iterative regularization method we can improve its regularizing properties and/or accelerate its convergence using it as smoother in a Multigrid algorithm. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 15 / 27
34 Multigrid regularization Iterative multigrid regularization Iterative Multigrid regularization The Multigrid as an iterative regularization method If we have an iterative regularization method we can improve its regularizing properties and/or accelerate its convergence using it as smoother in a Multigrid algorithm. Regularization The regularization properties of the smoother are preserved since it is combined with a low-pass filter. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 15 / 27
35 Multigrid regularization Two-Level (TL) regularization Iterative Multigrid regularization Idea: project into the low frequencies and then apply an iterative regularization method. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 16 / 27
36 Multigrid regularization Iterative Multigrid regularization Two-Level (TL) regularization Idea: project into the low frequencies and then apply an iterative regularization method. TL as a specialization of TGM Smoother: iterative regularization method Projector: low-pass filter Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 16 / 27
37 Multigrid regularization Iterative Multigrid regularization Two-Level (TL) regularization Idea: project into the low frequencies and then apply an iterative regularization method. TL as a specialization of TGM Smoother: iterative regularization method Projector: low-pass filter TL Algorithm 1 No smoothing at the finer level 2 At the coarser level to apply one step of the smoother instead of to solve directly the linear system Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 16 / 27
38 Multigrid regularization Iterative Multigrid regularization Multigrid regularization (applying recursively the TL) V-cycle Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 17 / 27
39 Multigrid regularization Iterative Multigrid regularization Multigrid regularization (applying recursively the TL) V-cycle Using a larger number of recursive calls (e.g. W-cycle), the algorithm works more in the well-conditioned subspace, but it is more difficult to define an early stopping criterium. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 17 / 27
40 Computational Cost Multigrid regularization Computational Cost Assumptions: n n images and m m PSFs with m n. Let S(n) be the computational cost of one smoother iteration. The computational cost of one iteration of our multigrid regularization method with γ recursive calls is 1 3 S(n), γ = 1 C(γ,n) S(n), γ = 2 3S(n), γ = 3 C(γ,n) = { O(n 2 log(n)), γ = 4 O(n 2log 4 (γ) ), γ > 4 if m n then S(n) = O(n 2 log(n)). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 18 / 27
41 Outline Numerical experiments 1 Restoration of blurred and noisy images The model problem Properties of the PSF Iterative regularization methods 2 Multigrid regularization Multigrid methods Iterative Multigrid regularization Computational Cost 3 Numerical experiments An airplane Direct multigrid regularization 4 Conclusions Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 19 / 27
42 An airplane Numerical experiments An airplane Periodic BCs Gaussian PSF (A spd) noise = 1% Original Image Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 20 / 27
43 An airplane Numerical experiments An airplane Periodic BCs Gaussian PSF (A spd) noise = 1% Original Image Inner part Observed image Restored with MGM Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 20 / 27
44 Numerical experiments Restoration error: noise = 1% An airplane e j = f f (j) 2 / f 2 restoration error at the j-th iteration. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 21 / 27
45 Numerical experiments Restoration error: noise = 1% An airplane e j = f f (j) 2 / f 2 restoration error at the j-th iteration. Minimum restoration error Method min j) j=1,... arg min j) j=1,... CG Richardson TL(CG) TL(Rich) MGM(Rich, 1) MGM(Rich, 2) CGNE RichNE Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 21 / 27
46 Numerical experiments Restoration error: noise = 1% An airplane e j = f f (j) 2 / f 2 restoration error at the j-th iteration. Relative error vs. number of iterations CG Richardson TL(CG) TL(Richardson) MGM(Ric,1) MGM(Ric,2) Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 21 / 27
47 Noise = 10% Numerical experiments An airplane For CG and Richardson it is better to resort to normal equations. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 22 / 27
48 Noise = 10% Numerical experiments An airplane For CG and Richardson it is better to resort to normal equations. Minimum restoration error Method min j) j=1,... arg min j) j=1,... CGNE RichNE TL(CGNE) TL(RichNE) MGM(RichNE,1) MGM(RichNE,2) MGM(Rich,1) MGM(Rich,2) Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 22 / 27
49 Noise = 10% Numerical experiments An airplane For CG and Richardson it is better to resort to normal equations. Relative error vs. number of iterations CGN RichN TL(CGN) TL(RichN) MGM(RichN,1) MGM(RichN,2) Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 22 / 27
50 The γ regularization Numerical experiments Direct multigrid regularization Varying γ, the proposed multigrid is a direct (one step) regularization method with regularization parameter γ. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 23 / 27
51 The γ regularization Numerical experiments Direct multigrid regularization Varying γ, the proposed multigrid is a direct (one step) regularization method with regularization parameter γ. The airplane example with noise = 1% e 1 = restoration error after one iteration of MGM(Rich,γ). The CGNE reaches a minimum error equal to after 2500 iterations! γ e Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 23 / 27
52 Numerical experiments Direct multigrid regularization The γ regularization (the airplane with noise = 1%) MGM(Rich,1) MGM(Rich,2) MGM(Rich,4) MGM(Rich,6) Restoration error norm vs. number of iterations Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 24 / 27
53 Numerical experiments Direct multigrid regularization The γ regularization (the airplane with noise = 1%) MGM(Rich,1) MGM(Rich,2) MGM(Rich,4) MGM(Rich,6) Restoration error norm vs. number of iterations The computational cost increases with γ but not so much (e.g. γ = 8 O(N 1.5 ) where N = n 2 ). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 24 / 27
54 Outline Conclusions 1 Restoration of blurred and noisy images The model problem Properties of the PSF Iterative regularization methods 2 Multigrid regularization Multigrid methods Iterative Multigrid regularization Computational Cost 3 Numerical experiments An airplane Direct multigrid regularization 4 Conclusions Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 25 / 27
55 Conclusions Summarizing... multigrid regularization method It is a general framework which can be used to improve the regularization properties of an iterative regularizing method. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 26 / 27
56 Conclusions Summarizing... multigrid regularization method It is a general framework which can be used to improve the regularization properties of an iterative regularizing method. It leads to a smaller relative error and a flatter error curve with respect to the smoother applied alone. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 26 / 27
57 Conclusions Summarizing... multigrid regularization method It is a general framework which can be used to improve the regularization properties of an iterative regularizing method. It leads to a smaller relative error and a flatter error curve with respect to the smoother applied alone. It is fast and it obtains a good restored image without resorting to normal equations. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 26 / 27
58 Conclusions Summarizing... multigrid regularization method It is a general framework which can be used to improve the regularization properties of an iterative regularizing method. It leads to a smaller relative error and a flatter error curve with respect to the smoother applied alone. It is fast and it obtains a good restored image without resorting to normal equations. It can be combined with other techniques (e.g., nonnegativity constraints) and it can lead to several generalizations (e.g., γ regularization). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 26 / 27
59 Conclusions Summarizing... multigrid regularization method It is a general framework which can be used to improve the regularization properties of an iterative regularizing method. It leads to a smaller relative error and a flatter error curve with respect to the smoother applied alone. It is fast and it obtains a good restored image without resorting to normal equations. It can be combined with other techniques (e.g., nonnegativity constraints) and it can lead to several generalizations (e.g., γ regularization). Reference M. Donatelli and S. Serra Capizzano, On the regularizing power of multigrid-type algorithms, SIAM J. Sci. Comput., in press. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 26 / 27
60 Future work Conclusions Theoretical analysis of the regularization properties. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 27 / 27
61 Future work Conclusions Theoretical analysis of the regularization properties. Applications: strictly nonsymmetric PSFs. Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 27 / 27
62 Future work Conclusions Theoretical analysis of the regularization properties. Applications: strictly nonsymmetric PSFs. Combination with techniques for edge enhancing (Wavelet, Total Variation,...). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 27 / 27
63 Future work Conclusions Theoretical analysis of the regularization properties. Applications: strictly nonsymmetric PSFs. Combination with techniques for edge enhancing (Wavelet, Total Variation,...). Numerics/Simulations: A complete experimentation with all the proposed BCs (multigrid methods already exist for the arising matrices, see Aricò and Donatelli, Num. Math., to appear). Marco Donatelli (University of Insubria) Image deblurring by multigrid methods 27 / 27
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