Laboratorio di Problemi Inversi Esercitazione 4: metodi Bayesiani e importance sampling

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1 Laboratorio di Problemi Inversi Esercitazione 4: metodi Bayesiani e importance sampling Luca Calatroni Dipartimento di Matematica, Universitá degli studi di Genova May 19, Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

2 Outline 1 Recap on the theory 2 MATLAB implementation Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

3 Bayesian method for inverse problems I Again, the inverse problem y = Ax + n, n N (0, σ 2 ), but we consider this from a different perspective. A different point of view deterministic approaches: we look for a reconstruction of measurement y; Bayes approaches: we aim to determine the probability of reconstructing x given y (posterior probability). Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

4 Bayesian methods for inverse problems II We have: p(y x) (probability of observing y if x is given, but x is unknown). We seek: p(x y). Bayes theorem p(x y) = p(y x)p(x) p(y) Ingredients: p(y x): likelihood; p(x): encodes a-priori information on the desired reconstruction (prior). p(y): often interpreted as a normalisation constant. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

5 Bayesian methods for inverse problems II We have: p(y x) (probability of observing y if x is given, but x is unknown). We seek: p(x y). Bayes theorem p(x y) = p(y x)p(x) p(y) Ingredients: p(y x): likelihood; p(x): encodes a-priori information on the desired reconstruction (prior). p(y): often interpreted as a normalisation constant. The unknown can be estimated using Maximum A Posteriori (MAP) estimate: x MAP = argmax p(x y) Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

6 Bayesian methods for inverse problems II We have: p(y x) (probability of observing y if x is given, but x is unknown). We seek: p(x y). Bayes theorem p(x y) = p(y x)p(x) p(y) Ingredients: p(y x): likelihood; p(x): encodes a-priori information on the desired reconstruction (prior). p(y): often interpreted as a normalisation constant. The unknown can be estimated using Maximum A Posteriori (MAP) estimate: Bayesian inverse problem solution x MAP = argmax p(x y) The solution of the inverse problem is the posterior itself. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

7 Work plan Often, it is difficult to determine an analytic expression of the posterior. Objectives We want to implement two basic Monte Carlo methods to estimate the posterior distribution solution of a very simplified deblurring/denoising problem. We will consider: Importance Sampling; Metropolis-Hastings. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

8 Outline 1 Recap on the theory 2 MATLAB implementation Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

9 Building the parametric image Test image: construct the image of a blurred circle with given: - centre coordinates and radius; - background/foreground values; - Gaussian blur with fixed blur variance. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

10 Building the parametric image Test image: construct the image of a blurred circle with given: - centre coordinates and radius; - background/foreground values; - Gaussian blur with fixed blur variance. Parameters: x C = y C = 32, r = 15, bg val = 200, fg val = 125, σ 2 PSF = 2. Write a MATLAB function create circle that creates a circle depending on the input parameters above. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

11 Building the parametric image Test image: construct the image of a blurred circle with given: - centre coordinates and radius; - background/foreground values; - Gaussian blur with fixed blur variance. Parameters: x C = y C = 32, r = 15, bg val = 200, fg val = 125, σ 2 PSF = 2. Write a MATLAB function create circle that creates a circle depending on the input parameters above. For blurring: use fft2, ifft2, fftshift. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

12 Building the parametric image Test image: construct the image of a blurred circle with given: - centre coordinates and radius; - background/foreground values; - Gaussian blur with fixed blur variance. Parameters: x C = y C = 32, r = 15, bg val = 200, fg val = 125, σ 2 PSF = 2. Write a MATLAB function create circle that creates a circle depending on the input parameters above. For blurring: use fft2, ifft2, fftshift. Add Gaussian noise with noise level σ noise = 12. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

13 Writing Importance Sampling Fix N = samples. Unknowns: x C, y C, r, bg val, fg val, σ 2 PSF Idea In each sample, draw a random value for each parameter from a known probability distribution (we fix uniform distribution), build the corresponding blurred image and construct a weight encoding how likely that combination of parameters is. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

14 Writing Importance Sampling Fix N = samples. Unknowns: x C, y C, r, bg val, fg val, σ 2 PSF Idea In each sample, draw a random value for each parameter from a known probability distribution (we fix uniform distribution), build the corresponding blurred image and construct a weight encoding how likely that combination of parameters is. - For every sample, for each parameter involved: Use MATLAB command rand to generate a random number in the interval [0, 1] with uniform probability distribution. Scale the generate number within the desired interval; For simplicity, use ceil to take the biggest integer to the found value. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

15 Writing Importance Sampling Fix N = samples. Unknowns: x C, y C, r, bg val, fg val, σ 2 PSF Idea In each sample, draw a random value for each parameter from a known probability distribution (we fix uniform distribution), build the corresponding blurred image and construct a weight encoding how likely that combination of parameters is. - For every sample, for each parameter involved: Use MATLAB command rand to generate a random number in the interval [0, 1] with uniform probability distribution. Scale the generate number within the desired interval; For simplicity, use ceil to take the biggest integer to the found value. - Construct for each sample the corresponding PSF and use it to construct a blurred version of the estimated image. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

16 Importance weights - For each sample, construct the weight depending on the density we want to estimate and the importance density η as follows: w i = p(x i y) η(x i ) = p(y x i)p(x i ) η(x i )p(y) = p(y x i ) p(y) p(y x i ) assuming that p(x i ) = η(x i ) (importance distribution = prior) and up to multiplicative constants. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

17 Importance weights - For each sample, construct the weight depending on the density we want to estimate and the importance density η as follows: w i = p(x i y) η(x i ) = p(y x i)p(x i ) η(x i )p(y) = p(y x i ) p(y) p(y x i ) assuming that p(x i ) = η(x i ) (importance distribution = prior) and up to multiplicative constants. In our case the likelihood is the one corresponding to the noise distribution: p(y x i ) = 1 ( exp y Bx i 2 ) ( F 2πσ 2 2σ 2 p(y x i ) = exp y Bx i 2 ) F 2σ 2 Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

18 Importance weights - For each sample, construct the weight depending on the density we want to estimate and the importance density η as follows: w i = p(x i y) η(x i ) = p(y x i)p(x i ) η(x i )p(y) = p(y x i ) p(y) p(y x i ) assuming that p(x i ) = η(x i ) (importance distribution = prior) and up to multiplicative constants. In our case the likelihood is the one corresponding to the noise distribution: p(y x i ) = 1 ( exp y Bx i 2 ) ( F 2πσ 2 2σ 2 p(y x i ) = exp y Bx i 2 ) F 2σ 2 Numerical problems in MATLAB (numerical zeros)! Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

19 Way around: log-sum-exp formula MATLAB may get 0 s when calculating the weights due to large numerical values at the power of e. When computed for each sample, the weight vector needs to be normalised: W = w wi log(w ) = log(w) log( w i ) (1) in the MC loop, instead of the w i s, compute their log (and avoid the numerical problems); Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

20 Way around: log-sum-exp formula MATLAB may get 0 s when calculating the weights due to large numerical values at the power of e. When computed for each sample, the weight vector needs to be normalised: W = w wi log(w ) = log(w) log( w i ) (1) in the MC loop, instead of the w i s, compute their log (and avoid the numerical problems); use: log( ( ( )) w i ) = max(log(w i )) + log exp log(w i ) max(log(w i ) and calculate log(w ) in (1). Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

21 Way around: log-sum-exp formula MATLAB may get 0 s when calculating the weights due to large numerical values at the power of e. When computed for each sample, the weight vector needs to be normalised: W = w wi log(w ) = log(w) log( w i ) (1) in the MC loop, instead of the w i s, compute their log (and avoid the numerical problems); use: log( ( ( )) w i ) = max(log(w i )) + log exp log(w i ) max(log(w i ) and calculate log(w ) in (1). compute W and find the indices where the weight vector is maximum. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

22 Way around: log-sum-exp formula MATLAB may get 0 s when calculating the weights due to large numerical values at the power of e. When computed for each sample, the weight vector needs to be normalised: W = w wi log(w ) = log(w) log( w i ) (1) in the MC loop, instead of the w i s, compute their log (and avoid the numerical problems); use: log( ( ( )) w i ) = max(log(w i )) + log exp log(w i ) max(log(w i ) and calculate log(w ) in (1). compute W and find the indices where the weight vector is maximum. compute the corresponding set of parameters and use it to plot the reconstructed image. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

23 Weighted histograms We now want to plot the weighted histograms to see how IS works and find their peaks (MAP estimation). Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

24 Weighted histograms We now want to plot the weighted histograms to see how IS works and find their peaks (MAP estimation). on the x-axis: range of values for the parameter to estimate; on the y-axis: accumulators h for each possible value of the parameter. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

25 Problems Narrow distributions The likelihood concentrates values on a very narrow set of values the posterior distribution promotes very few values. Recall: w i = p(x i y) η(x i ) = p(y x i)p(x i ) η(x i )p(y) = p(y x i ) p(y) p(y x i ) Consequences: y Bx i 2 F are very large weights are very small and focus just in correspondence of these values (MATLAB has numerical limitations... ). log-sum exp formula: we can at least save the maximum but no weighted histograms... TASK: how to make the posterior larger? Choose a different importance distribution centred on real values but not uniform... Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

26 Weighted histograms in MATLAB Initialise the range of values and the histograms for each parameter estimated; Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

27 Weighted histograms in MATLAB Initialise the range of values and the histograms for each parameter estimated; For each sample i = 1,..., N: whenever I find an occurrence of that value, increment the corresponding histogram of w i. (think about Hough: here, your accumulator increases of w i ) Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

28 Weighted histograms in MATLAB Initialise the range of values and the histograms for each parameter estimated; For each sample i = 1,..., N: whenever I find an occurrence of that value, increment the corresponding histogram of w i. (think about Hough: here, your accumulator increases of w i ) Example for a generic parameter t: tt=[1:max t]; ht=zeros(size(tt)); for i=1:n ht(t(i))=ht(t(i))+w(i); end Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

29 Weighted histograms in MATLAB Initialise the range of values and the histograms for each parameter estimated; For each sample i = 1,..., N: whenever I find an occurrence of that value, increment the corresponding histogram of w i. (think about Hough: here, your accumulator increases of w i ) Example for a generic parameter t: tt=[1:max t]; ht=zeros(size(tt)); for i=1:n ht(t(i))=ht(t(i))+w(i); end Plot the histograms for every parameter estimated. Comment on the resulting plots and find the peaks Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

30 MAP estimation In this way, we are computing the MAP estimate: MAP estimate max p(x y) = max N w i h(x x i ) i=1 and finding the marginal distributions whose peaks are exactly the parameters maximising the posterior. Luca Calatroni (DIMA, Unige) Esercitazione 4, Lab. Prob. Inv. May 19, / 14

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