Expectation Maximization Deconvolution Algorithm

Transcription

1 Epectation Maimization Deconvolution Algorithm Miaomiao ZHANG March 30, 2011 Abstract In this paper, we use a general mathematical and eperimental methodology to analyze image deconvolution. The main procedure is to use an eample image convolving it with a know Gaussian point spread function and then develop algorithms to recover the image. Observe the deconvolution process by adding Gaussian and Poisson noise at different signal to noise ratios. In addition, we will describe the effect of the width of the Gaussian which is used to blur the image. The core algorithms in this paper is the iterative E-M algorithm as well as the non-iterative least squares deconvolution algorithm. 1 Introduction In this paper, we will discuss the Epectation Maimization (EM) algorithm which was proposed by Dempster, Laird and Rubin. It can deal with problems which involve incomplete data, or miture estimation. Generally, in statistics, it aims to find maimum likelihood or maimum a posteriori (MAP) estimates of parameters in the model, where the model depends on unobserved data sets. EM algorithm has two iterative steps: E step is to estimate the epectation; M step is to maimize the likelihood. We will introduce this in the net section in detail. In addition, here we will also do the eperiment on non-iterative methods which takes advantage of the Fourier transformation. It would be pretty faster than EM algorithm. 1

2 2 Methods The practical problem we solve here is the linear imaging system which can be epressed as: D(y) = P (y )λ()d (1) Where λ() is the object space, which stands for the intensity of an image. D(y) is the detector space, which is the observed data. P (y ) is the point spread function which is the probability depends on the distance between y and. And it describes how much of the original information from λ() arrives the detector space D(y). Actually, if the P (y ) is shift invariant, here we can consider it as the same as P (y ). Thus, we can get: D(y) = P (y )λ()d (2) From the equation above, It is obviously to see that equation (2) is equivalent to the image convolution. However, there always eists noise in D(y) which is generated by the imaging system. Therefore, it leads to the uncertainty of the detected data in detector space. In this case, we cannot just solve the linear system which was introduced above, so we need to maimize the likelihood of D(y) according to the distribution of the noise. Net, we will discuss two kinds of important noise distribution, one is Gaussian distribution, and the other one is the Poisson distribution. 2.1 Gaussian Case Generally, the Gaussian distribution is: p() = 1 2πδ ep ( µ)2 2δ 2 (3) Where µ is the mean and δ 2 is the variance of the distribution. In Gaussian case, we assume that: µ(d(y)) = P (y )λ()dp (D(y) µ(d(y))) = 1 ( ep (D(y) µ(d(y))) 2 2δ 2 ) (4) 2πδ Where D(y) is the Gaussian distribution with the mean µ(d(y)) and covariance δ 2. P (D(y) µ(d(y))) is the joint probability of D(y). It is easy to notice that to maimize the likelihood is to minimize the linear equation (D(y) µ(d(y))) 2. 2

3 2.2 Poisson Case The poisson distribution can be written as: P (k µ) = µk e µ Where µ is the number of occurrence, k is the integer of epected number of occurrence. When µ is large enough, the distribution of poisson will be close to the gaussian model. As the same inference as we did in gaussian model, the joint probability for poisson distribution is: µ(d(y)) = k! P (y )λ()dp (D(y) µ(d(y))) = µ(d(y)) D(y) e µ(d(y)) ) (6) D(y)! In order to reduce the compleity of computation, we take the log operation of the likelihood, then we can obtain the de-convoluted λ() through maimize the equation as follows: ma λ() 0 [D(y) log y 3 Implementation P (y )λ()d 3.1 Implementation of Gaussian Case (5) P (y )λ()d log(d(y)!)] (7) In this eperiment, by taking advantage of the Prseval s equality and Fourier transformation, we can obtain: (D(y) λ()p (y )) 2 = (F (D(y) λ P )) 2 dω (8) = (D(ω) λ(ω) P (ω)) 2 dω (9) Where F is the Fourier transformation operator, is the symbol for image convolution. To solve this equation, we can get: λ(ω) = D(ω) P (ω) (10) 3

4 Then, we should get the output image back to image space from the Fourier space, thus we have: λ() = F 1 (λ(ω)) (11) Note: For equation (11), it is necessary to pad P (y ) to have the same size as the input image before doing the Fourier transformation, one trick here is to pad all the other elements as zeros. 3.2 Implementation of Poisson Case By solving Equation 7 above, we can get the iteration formulation for each piel i in an image for λ()as : λ k+1 ( i ) = λ k ( i ) j D(y j )P (y j i ) i λk ( i )P (y j i ) (12) Where j is each piel in the detector space. 3.3 Regularization Term In this paper, we choose the Gradient Descent method to eliminate the noise. Then for the poisson noise, after added by the regularization term, the formulation is: ma I( i ) Z( i ) log(i( i )) i i I( i ) + 1 τ Where, Z is any data, I is the mean of poisson distribution. I( i ) 2 (13) i 4 Results and Discussion In this section, we will show different results of the methods which we discussed above and give the discussion of why the results look like this. How the algorithms make effects on the same input image. If we change the parameters in a different value, how will the algorithm work, whether it is bad or good. Basically, the steps for our eperiments are: step 1: We blur the original image with different width. step 2: Apply different implementation algorithms above to get the result. 4

5 Figure 1: Deconvolution results with Gaussian model by different width of σ : Top: original image; Middle: left with the σ = 0.5, right with the σ = 1; bottom: left with σ = 2, right with σ = 3. 5

6 Figure 2: Deconvolution results with poisson model by different width of σ : Top: left is original image, middle is blurred image with σ = 1, right is deconvolution result; Middle: left is original image, middle is blurred image with σ = 5, right is deconvolution result; bottom: left is original image, middle is blurred image with σ = 8, right is deconvolution result. All the iterations in these tested images are 50. 6

7 4.1 Discussion From the results above, in figure.1, the noise has different levels. It is clearly that when the value of σ increases, the deconvolution results would get worse. This is because when the P (ω) is near zero, the λ(ω) will be very large, thus the result will become worse. In addition, there is no big difference if we add the regularization term to this case. However, the convergence by adding this term would be better. In figure.2, we test the algorithm on Poisson distribution noise, it is more stable than the Gaussian model when comparing the results. When we add the regularization term, it seems that it also does not help too much. 5 Conclusion In a word, from the whole eperiment, we find that the Poisson model is much more stable than the Gaussian model no matter by adding the noise or increasing the width of σ. What s more, for the regularization term, it does not always help, in other words, it depends on which information we use, or else the result would get even worse. 6 Reference All the reference materials come from: [1] M.Bertero and P.Boccacci. Image Deconvolution. Springer Netherlands, [2] Sean Borman. The epectation maimization algorithm: A short tutorial. [3] Class Notes, Sarang Joshi, Mathematics of Imaging. 7