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
- Slides: 29