Sampling Random Signals

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

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

4 Introduction Review of Definitions and Properties

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

6 Balakrishnan’s Sampling Theorem [Balakrishnan 1957]

7 Hybrid Wiener Filter

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

9 Hybrid Wiener Filter

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

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

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

13 Stationary ? Non-Stationary Reconstruction

14 Non-Stationary Reconstruction Stationary Signal Reconstructed Signal

15 Non-Stationary Reconstruction

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

17 BicubicSinc Nearest Neighbor Original Image Non-Stationary Reconstruction Artifacts

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

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

20 Optimal dense grid interpolation: Our Approach

21 Our Approach Motivation

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

23 Simulations Synthetic Data

24 Simulations Synthetic Data

25 Simulations Synthetic Data

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

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

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 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