Linear Methods in Deconvolution - Research Overview | The University of Tennessee

1. Deconvolution Summer 2008 Research Presentation, The University of Tennessee, Knoxville. by Muharrem Mercimek

2. Linear Methods • Tikhonov Regularization • Where H is a Toeplitz matrix and different from the Fourier transform of function h. The only difference from the Least Squares Method is the regularization parameter . .

3. Linear Methods • The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of h and x can be formulated as: Toeplitz matrix representation gives the opportunity to use numerical approaches

4. Linear Methods In order to analyze we can use SVD of are the jth column vector of the matrices of U are the jth column vector of the matrices V is the jth singular value ofΣ Tikhonov regularization with SVD is a number less than equal to the rank of H

5. Linear Methods 2. Least Squares • Is the basic form can help us to approximate true function with normal equation which is Least Squares solution with SVD

6. Linear Methods 3. Total variation method • Most of the regularization methods expects smooth and continuous information from the data to be reconstructed. • Total variation is independent of this assumption and it preserves the edge information in the reconstructed data. in which  is regularization parameter, are linear first order difference operators at pixel I along horizontal and vertical directions respectively.

7. Iterative Methods • where is a constant. correction term is used to adjust the kth estimate of f.  can be arrange the iterations. 4. Van-Cittert 5. Constrained Iterative : The non-negativity constraint is added.

8. Iterative Methods • Relaxation function is used to put natural corrections during iterative updates. • The upper magnitude limit is taken as the upper bound of the data c • Lower limit is taken 0. 6. Relaxation based iterative or Jannson’s Method

9. Iterative Methods • if the blurred function includes noise the noise strongly deteriorates the quality of the approximation. It is always advantageous pre-filtering the blurred image before iteration starts. 7. Gold’s Ratio • Based on Bayesian theorem of the data. Pre-filtering before applying deconvolution

10. 1D Deconvolution experiments • The data and PSF Functions are created synthetically.

11. Tikhonov regularization Noisy data

12. Tikhonov regularization Noiseless data

13. TV regularization Noisy data

14. TV regularization Noiseless data

15. Van-Cittert Noisy data

16. Van-Cittert with pre-filtering Noisy data with pre-filtering

17. Constrained Iterative Method • Non-negativity constraint is used Noisy data

18. Constrained Iterative Method with pre-filtering Noisy data

19. Jannson’s iterative method Noisy data

20. Jannson’s iterative method with pre-filtering Noisy data

21. 2D Deconvolution experiments a) True image b) PSF c) Observed image d) Pre-filtered Observed Image %5 random noise g’

22. TV Regularization a) TV Approximation b) MSE of a)

23. Tikhonov Regularization a) Tikhonov Approximation b) MSE of a)

24. Van-cittert Method a) Direct Approximation b) MSE of a) c) Approximation with pre filtering d) MSE of c)

25. Truncation Method a) Direct Approximation b) MSE of a) c) Approximation with pre filtering d) MSE of c)

26. Jansson's iterative method a) Direct Approximation b) MSE of a) c) Approximation with pre filtering d) MSE of c)

27. Comments • 1D 2D Deconvolution algorithms are added. • Using Toeplitz matrix makes the problem to handle easier in 1-D. • 2D convolution process is computationally expensive, when using TV and Tikhonov with naïve numerical algorithms, such as finding SVD and calculating the inverse of the functions.

28. Future experiments • More methods • A new idea is missing towards publication. • 3D deconvolution methodology is missing (if I want to get closer to my other research topic).