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Sampling Random Signals

Sampling Random Signals

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Sampling Random Signals

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Presentation Transcript

  1. Sampling Random Signals

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

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

  4. Introduction Review of Definitions and Properties

  5. IntroductionReview of Definitions and Properties • Filtering: • Wiener filter:

  6. Balakrishnan’s Sampling Theorem [Balakrishnan 1957]

  7. Hybrid Wiener Filter

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

  9. Hybrid Wiener Filter

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

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

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

  13. Non-Stationary Reconstruction ? Stationary

  14. Non-Stationary Reconstruction Stationary Signal Reconstructed Signal

  15. Non-Stationary Reconstruction

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

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

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

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

  20. Our Approach Optimal dense grid interpolation:

  21. Our Approach Motivation

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

  23. Simulations Synthetic Data

  24. Simulations Synthetic Data

  25. Simulations Synthetic Data

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

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

  28. 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

  29. 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