1. IPIM, IST, José Bioucas, 2007 1 Convolution Operators Spectral Representation Bandlimited Signals/Systems Inverse Operator Null and Range Spaces Sampling, DFT and FFT Tikhonov Regularization/Wiener Filtering

2. IPIM, IST, José Bioucas, 2007 2 Convolution Operators Definition: Spectral representation of a convolution operator: FT

3. IPIM, IST, José Bioucas, 2007 3 A is linear and bounded A is bounded: Let is continuous Adjoint of a convolution operator Properties

4. IPIM, IST, José Bioucas, 2007 4 Adjoint of convolution operator (cont.) since Inverse of a convolution operator or has isolated zeros as is not bounded is defined only if

5. IPIM, IST, José Bioucas, 2007 5 Bandlimited convolution operators/systems is bandlimited with band B, i.e., are orthogonal

6. IPIM, IST, José Bioucas, 2007 6 Convolution of Bandlimited 2D Signals Approximate using periodic sequences

7. IPIM, IST, José Bioucas, 2007 7 From Continuous to Discrete Representation Assume that Let is N-periodic sequences such that Discrete Fourier Transform (DFT)

8. IPIM, IST, José Bioucas, 2007 8 Fast Fourier Transform (FFT) Efficient algorithm to compute When N is a power of 2

9. IPIM, IST, José Bioucas, 2007 9 Vector Space Perspective Let vectors defined in Euclidian vector space with inner product Parseval generalized equality Basis

10. IPIM, IST, José Bioucas, 2007 10 2D Periodic Convolution 2D N-periodic signals (images) Periodic convolution DFT of a convolution Hadamard product

11. IPIM, IST, José Bioucas, 2007 11 Spectral Representation of 2D Periodic Signals Can be represented as a block cyclic matrix Spectral Representation of A eingenvalues of A

12. IPIM, IST, José Bioucas, 2007 12 Adjoint operator Operator

13. IPIM, IST, José Bioucas, 2007 13 Inverse operator Let

14. IPIM, IST, José Bioucas, 2007 14 Deconvolution Examples Imaging Systems Linear Imaging System System noise + Poisson noise Impulsive Response function or Point spread function (PSF) Invariant systems Is the transfer function (TF)

15. IPIM, IST, José Bioucas, 2007 15 Example 1: Linear Motion Blur lens plane Let a(t)=ct for, then

16. IPIM, IST, José Bioucas, 2007 16 Example 1: Linear Motion Blur

17. IPIM, IST, José Bioucas, 2007 17 Example 1: Linear Motion Blur

18. IPIM, IST, José Bioucas, 2007 18 Example 2: Out of Focus Blur lens plane Circle of confusion COC Geometrical optics zeros

19. IPIM, IST, José Bioucas, 2007 19 Deconvolution of Linear Motion Blur Let and

20. IPIM, IST, José Bioucas, 2007 20 Deconvolution of Linear Motion Blur

21. IPIM, IST, José Bioucas, 2007 21 Deconvolution of Linear Motion Blur (TFD) ISNR

22. IPIM, IST, José Bioucas, 2007 22 Deconvolution of Linear Motion Blur (Tikhonov regularization) Assuming that D is cyclic convolution operator Wiener filter Regularization filter

23. IPIM, IST, José Bioucas, 2007 23 Deconvolution of Linear Motion Blur (Tikhonov regularization) Regularization filter Effect of the regularization filter is a frequency selective threshold

24. IPIM, IST, José Bioucas, 2007 24 Deconvolution of Linear Motion Blur ISNR

25. IPIM, IST, José Bioucas, 2007 25 Deconvolution of Linear Motion Blur (Total Variation ) Iterative Denoising algorithm where solves the denoising optimization problem

26. IPIM, IST, José Bioucas, 2007 26 Deconvolution of Linear Motion Blur TFD Tikhonov (D=I) TV