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.

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Sampling Random Signals

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Sampling random signals

Sampling Random Signals


Sampling random signals

Introduction

Types of Priors

  • Subspace priors:

  • Smoothness priors:

  • Stochastic priors:


Sampling random signals

Introduction

Motivation for Stochastic Modeling

  • Understanding of artifacts via stationarity analysis

  • New scheme for constrained reconstruction

  • Error analysis


Sampling random signals

Introduction

Review of Definitions and Properties


Introduction review of definitions and properties

IntroductionReview of Definitions and Properties

  • Filtering:

  • Wiener filter:


Balakrishnan s sampling theorem

Balakrishnan’s Sampling Theorem

[Balakrishnan 1957]


Hybrid wiener filter

Hybrid Wiener Filter


Hybrid wiener filter1

Hybrid Wiener Filter

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


Sampling random signals

Hybrid Wiener Filter


Sampling random signals

Hybrid Wiener FilterImage scaling

Original Image

Bicubic Interpolation

Hybrid Wiener


Sampling random signals

Hybrid Wiener FilterRe-sampling

  • Drawbacks:

  • May be hard to implement

  • No explicit expression in the time domain

Re-sampling:


Sampling random signals

Constrained Reconstruction Kernel

Predefined interpolation filter:

The correction filter depends on t !


Sampling random signals

Non-Stationary Reconstruction

?

Stationary


Sampling random signals

Non-Stationary Reconstruction

Stationary Signal

Reconstructed Signal


Sampling random signals

Non-Stationary Reconstruction


Sampling random signals

Non-Stationary Reconstruction

Artifacts

Original image

Interpolation with rect

Interpolation with sinc


Sampling random signals

Non-Stationary Reconstruction

Artifacts

Nearest Neighbor

Original Image

Bicubic

Sinc


Sampling random signals

Constrained Reconstruction Kernel

Predefined interpolation filter:

Solution:

1.

2.


Sampling random signals

Constrained Reconstruction Kernel

Dense Interpolation Grid

Dense grid approximation of the optimal filter:


Sampling random signals

Our Approach

Optimal dense grid interpolation:


Sampling random signals

Our Approach

Motivation


Sampling random signals

Our ApproachNon-Stationarity

[Michaeli & Eldar 08]


Sampling random signals

SimulationsSynthetic Data


Sampling random signals

SimulationsSynthetic Data


Sampling random signals

SimulationsSynthetic Data


Sampling random signals

First Order Approximation

  • Ttriangular kernel

  • Interpolation grid:

  • Scaling factor:


Sampling random signals

Optimal Dense Grid Reconstruction

  • Ttriangular kernel

  • Interpolation grid:

  • Scaling factor:


Sampling random signals

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


Sampling random signals

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


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