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Sampling Random Signals

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Sampling Random Signals

Introduction

Types of Priors

- Subspace priors:

- Smoothness priors:

- Stochastic priors:

Introduction

Motivation for Stochastic Modeling

- Understanding of artifacts via stationarity analysis
- New scheme for constrained reconstruction
- Error analysis

Introduction

Review of Definitions and Properties

- Filtering:

- Wiener filter:

[Balakrishnan 1957]

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

Hybrid Wiener Filter

Hybrid Wiener FilterImage scaling

Original Image

Bicubic Interpolation

Hybrid Wiener

Hybrid Wiener FilterRe-sampling

- Drawbacks:
- May be hard to implement
- No explicit expression in the time domain

Re-sampling:

Constrained Reconstruction Kernel

Predefined interpolation filter:

The correction filter depends on t !

Non-Stationary Reconstruction

?

Stationary

Non-Stationary Reconstruction

Stationary Signal

Reconstructed Signal

Non-Stationary Reconstruction

Non-Stationary Reconstruction

Artifacts

Original image

Interpolation with rect

Interpolation with sinc

Non-Stationary Reconstruction

Artifacts

Nearest Neighbor

Original Image

Bicubic

Sinc

Constrained Reconstruction Kernel

Predefined interpolation filter:

Solution:

1.

2.

Constrained Reconstruction Kernel

Dense Interpolation Grid

Dense grid approximation of the optimal filter:

Our Approach

Optimal dense grid interpolation:

Our Approach

Motivation

Our ApproachNon-Stationarity

[Michaeli & Eldar 08]

SimulationsSynthetic Data

SimulationsSynthetic Data

SimulationsSynthetic Data

First Order Approximation

- Ttriangular kernel
- Interpolation grid:
- Scaling factor:

Optimal Dense Grid Reconstruction

- Ttriangular kernel
- Interpolation grid:
- Scaling factor:

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

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