Sampling Random Signals Introduction Types of Priors Subspace

Sampling Random Signals

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

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

Introduction Review of Definitions and Properties 4

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

Balakrishnan’s Sampling Theorem [Balakrishnan 1957] 6

Hybrid Wiener Filter 7

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

Hybrid Wiener Filter 9

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

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

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

Non-Stationary Reconstruction ? Stationary 13

Non-Stationary Reconstruction Stationary Signal Reconstructed Signal 14

Non-Stationary Reconstruction 15

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

Non-Stationary Reconstruction Artifacts Original Image Nearest Neighbor Bicubic Sinc 17

Constrained Reconstruction Kernel Predefined interpolation filter: Solution: 1. 2. 18

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

Our Approach Optimal dense grid interpolation: 20

Our Approach Motivation 21

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

Simulations Synthetic Data 23

Simulations Synthetic Data 24

Simulations Synthetic Data 25

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

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

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 28

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