Laboratorio di Problemi Inversi Esercitazione 3: regolarizzazione iterativa, metodo di Landweber
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1 Laboratorio di Problemi Inversi Esercitazione 3: regolarizzazione iterativa, metodo di Landweber Luca Calatroni Dipartimento di Matematica, Universitá degli studi di Genova May 18, Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
2 Outline 1 Recap on the theory 2 Landweber regularisation implementation Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
3 Surprisingly: the inverse problem We are always considering the following linear, ill-posed, denoising/deblurring inverse problem: y = Bx + n and try to solve it in a clever way (i.e. avoiding the ill-posedness). Recap So far, we have seen some standard methods to solve it (both from the point of view of minimisation/spectral filtering approaches): TSVD: truncate the bad components; Tikhonov: regularise the bad components depending on λ (choice of lambda: generalised cross-validation,seen!).... then, invert. Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
4 Iterative regularization: idea Another approach consists in solving the least square problem: 1 min x 2 Bx y 2. Taking the gradient we have that in correspondence of the minimum x there holds: B T (B x y) = 0 Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
5 Iterative regularization: idea Another approach consists in solving the least square problem: 1 min x 2 Bx y 2. Taking the gradient we have that in correspondence of the minimum x there holds: B T (B x y) = 0 We can now either use fixed-point iterations or used gradient descent methods to write down a time-stepping: Landweber iterations For every k compute x by iterating for an appropriate choice of τ. x k+1 = x k τb T (Bx k y) Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
6 Outline 1 Recap on the theory 2 Landweber regularisation implementation Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
7 Work plan Objectives We want to implement the Landweber iterative regularisation technique to solve the deblurring/denoising problem: easy! Furthermore, we want to understand: its convergence properties, also when noise is in the image; its approximation properties; its computational limits. Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
8 Let s start to... As usual, load the Shepp-Logan phantom; Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
9 Let s start to... As usual, load the Shepp-Logan phantom; Create Gaussian PSF, with σ 2 = 5; Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
10 Let s start to... As usual, load the Shepp-Logan phantom; Create Gaussian PSF, with σ 2 = 5; Use Fourier to blur the images (recall: fft2, ifft2, fftshift). Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
11 Let s start to... As usual, load the Shepp-Logan phantom; Create Gaussian PSF, with σ 2 = 5; Use Fourier to blur the images (recall: fft2, ifft2, fftshift). Assign to sigma the level of Gaussian noise you want to add (take it small) and add noise to the blurred image. Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
12 Let s start to... As usual, load the Shepp-Logan phantom; Create Gaussian PSF, with σ 2 = 5; Use Fourier to blur the images (recall: fft2, ifft2, fftshift). Assign to sigma the level of Gaussian noise you want to add (take it small) and add noise to the blurred image. Show the result. Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
13 How to choose the Landweber parameter? Without optimising, the rule of thumbs corresponds to choose: To Do s: set-up τ appropriately; 0 < τ < 2 F(PSF ) 2 = 2 B 2 set-up max it as the maximum number of iterations (use 2000). Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
14 How to choose the Landweber parameter? Without optimising, the rule of thumbs corresponds to choose: To Do s: set-up τ appropriately; 0 < τ < 2 F(PSF ) 2 = 2 B 2 set-up max it as the maximum number of iterations (use 2000). OK, now let us create a MATLAB function having as input whatever is needed + the original image (not realistic!) to compute residual and as output the denoised/deblurred image + the vector of residuals. [deb,res]=landweber deb(...,phantom) Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
15 How to do it? Essential steps Initialise: vector of relative error and array of the iterates. Iteratively (till maximum of iterations) apply Landweber. Recall How did we compute Bx using the PSF? What about B T z? Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
16 How to do it? Essential steps Initialise: vector of relative error and array of the iterates. Iteratively (till maximum of iterations) apply Landweber. Recall How did we compute Bx using the PSF? What about B T z? Use: fftshift, fft2, ifft2. To get B T from B using the PSF, flip upside-down and left-to-right the PSF. commands: fliplr, flipud Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
17 How to do it? Essential steps Initialise: vector of relative error and array of the iterates. Iteratively (till maximum of iterations) apply Landweber. Recall How did we compute Bx using the PSF? What about B T z? Use: fftshift, fft2, ifft2. To get B T from B using the PSF, flip upside-down and left-to-right the PSF. commands: fliplr, flipud Write the Landweber iteration and show the result every 50 iterations, storing also the relative error: err(k) = x k x or 2 F x or 2 F Compute the deblurred/denoised solution as the one having minimum relative error output. Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
18 Remark Matrices VS. vectors Note that even though we are thinking at the problem from a matrix - vector perspective, we are still using matrices... the properties of the linear operators involved are the one we are exploiting to solve the problem efficiently! Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
19 Comments on the results 1 Show the denoised/deblurred result when a small choice of the noise level is made. What do you observe? How fast is the method? Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
20 Comments on the results 1 Show the denoised/deblurred result when a small choice of the noise level is made. What do you observe? How fast is the method? 2 What if the noise level is larger? What do you observe on the error plot? How does this reflect on iterates? Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
21 Comments on the results 1 Show the denoised/deblurred result when a small choice of the noise level is made. What do you observe? How fast is the method? 2 What if the noise level is larger? What do you observe on the error plot? How does this reflect on iterates? Semi-convergence There is a k after which error amplifies and noise oscillations become more and more visible. Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
22 Comments on the results 1 Show the denoised/deblurred result when a small choice of the noise level is made. What do you observe? How fast is the method? 2 What if the noise level is larger? What do you observe on the error plot? How does this reflect on iterates? Semi-convergence There is a k after which error amplifies and noise oscillations become more and more visible. 3 Use scatter to find the minimum of the error plot. Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
23 Part II: approximating property of Landweber No inversion! By construction, the Landweber method does not require any inversion step along the iterates: that s good! We can re write: x k+1 = (I τb T B)x k + τb T y =... = (I τb T B) k+1 x 0 + τ k (I τb T B) i B T y i=0 Now, if I τb T B 1 the first term goes to zero as k. We can then study the behaviour of the second term by looking at the polynomial: S k (t) = τ k (1 τt) i = i=0 1 (1 τt)k+1 t Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
24 No inversion! The following convergence property holds true: S k (t) 1 t if 1 λt < 1, for k. The inversion step is represented by the ratio 1/t which would imply for Landweber iterates: x k x = (B T B) 1 B T y, but such inversion is avoided through the analytical identity above, provided the terms in the series satisfy 1 τt < 1 Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
25 Regularisation properties 1 Define the appropriate interval such that: 0 < 1 τt < 1 (1) (suggestion: take the first extreme to be a bit bigger than 0) Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
26 Regularisation properties 1 Define the appropriate interval such that: 0 < 1 τt < 1 (1) (suggestion: take the first extreme to be a bit bigger than 0) 2 Define a MATLAB handle computing the partial sums S k (t) = depending on the parameter t and k. 1 (1 τt)k+1 t 3 Show on the same figure, the plots of S k for different values of k and compare with the graph of 1/t (representing the inversion we want to avoid). What do you observe? Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
27 Regularisation properties 1 Define the appropriate interval such that: 0 < 1 τt < 1 (1) (suggestion: take the first extreme to be a bit bigger than 0) 2 Define a MATLAB handle computing the partial sums S k (t) = depending on the parameter t and k. 1 (1 τt)k+1 t 3 Show on the same figure, the plots of S k for different values of k and compare with the graph of 1/t (representing the inversion we want to avoid). What do you observe? 4 Now increase the interval in (1) and do the same. What do you observe? Is the approximating property holding for every t R? Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
28 End of Part I: deterministic approaches See you tomorrow with Figure: Thomas Bayes Luca Calatroni (DIMA, Unige) Esercitazione 3, Lab. Prob. Inv. May 18, / 15
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