1. Sampling Random Signals

2. 2 Introduction Types of Priors Subspace priors: Smoothness priors: Stochastic priors:

3. 3 Introduction Motivation for Stochastic Modeling Understanding of artifacts via stationarity analysis New scheme for constrained reconstruction Error analysis

4. 4 Introduction Review of Definitions and Properties

5. 5 Introduction Review of Definitions and Properties Filtering: Wiener filter:

6. 6 Balakrishnan’s Sampling Theorem [Balakrishnan 1957]

7. 7 Hybrid Wiener Filter

8. 8 [Huck et. al. 85], [Matthews 00], [Glasbey 01], [Ramani et al 05]

9. 9 Hybrid Wiener Filter

10. 10 Hybrid Wiener Filter Image scaling Bicubic Interpolation Original Image Hybrid Wiener

11. 11 Hybrid Wiener Filter Re-sampling Drawbacks: May be hard to implement No explicit expression in the time domain Re-sampling:

12. 12 Predefined interpolation filter: Constrained Reconstruction Kernel The correction filter depends on t !

13. 13 Stationary ? Non-Stationary Reconstruction

14. 14 Non-Stationary Reconstruction Stationary Signal Reconstructed Signal

15. 15 Non-Stationary Reconstruction

16. 16 Non-Stationary Reconstruction Artifacts Original image Interpolation with rect Interpolation with sinc

17. 17 BicubicSinc Nearest Neighbor Original Image Non-Stationary Reconstruction Artifacts

18. 18 Predefined interpolation filter: Constrained Reconstruction Kernel Solution:1.2.

19. 19 Constrained Reconstruction Kernel Dense Interpolation Grid Dense grid approximation of the optimal filter:

20. 20 Optimal dense grid interpolation: Our Approach

21. 21 Our Approach Motivation

22. 22 Our Approach Non-Stationarity [Michaeli & Eldar 08]

23. 23 Simulations Synthetic Data

24. 24 Simulations Synthetic Data

25. 25 Simulations Synthetic Data

26. 26 First Order Approximation Ttriangular kernel Interpolation grid: Scaling factor:

27. 27 Optimal Dense Grid Reconstruction Ttriangular kernel Interpolation grid: Scaling factor:

28. 28 Error Analysis Average MSE of dense grid system with predefined kernel Average MSE of standard system (K=1) with predefined kernel For K=1: optimal sampling filter for predefined interpolation kernel

29. 29 Average MSE of the hybrid Wiener filter Necessary & Sufficient conditions for linear perfect recovery Necessary & Sufficient condition for our scheme to be optimal Theoretical Analysis