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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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Presentation Transcript
slide2

Introduction

Types of Priors

  • Subspace priors:
  • Smoothness priors:
  • Stochastic priors:
slide3

Introduction

Motivation for Stochastic Modeling

  • Understanding of artifacts via stationarity analysis
  • New scheme for constrained reconstruction
  • Error analysis
slide4

Introduction

Review of Definitions and Properties

hybrid wiener filter1
Hybrid Wiener Filter

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

slide10

Hybrid Wiener Filter Image scaling

Original Image

Bicubic Interpolation

Hybrid Wiener

slide11

Hybrid Wiener Filter Re-sampling

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

Re-sampling:

slide12

Constrained Reconstruction Kernel

Predefined interpolation filter:

The correction filter depends on t !

slide14

Non-Stationary Reconstruction

Stationary Signal

Reconstructed Signal

slide16

Non-Stationary Reconstruction

Artifacts

Original image

Interpolation with rect

Interpolation with sinc

slide17

Non-Stationary Reconstruction

Artifacts

Nearest Neighbor

Original Image

Bicubic

Sinc

slide18

Constrained Reconstruction Kernel

Predefined interpolation filter:

Solution:

1.

2.

slide19

Constrained Reconstruction Kernel

Dense Interpolation Grid

Dense grid approximation of the optimal filter:

slide20

Our Approach

Optimal dense grid interpolation:

slide21

Our Approach

Motivation

slide22

Our Approach Non-Stationarity

[Michaeli & Eldar 08]

slide26

First Order Approximation

  • Ttriangular kernel
  • Interpolation grid:
  • Scaling factor:
slide27

Optimal Dense Grid Reconstruction

  • Ttriangular kernel
  • Interpolation grid:
  • Scaling factor:
slide28

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
slide29

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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