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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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**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**IntroductionReview of Definitions and Properties**• Filtering: • Wiener filter:**Balakrishnan’s Sampling Theorem**[Balakrishnan 1957]**Hybrid Wiener Filter**[Huck et. al. 85], [Matthews 00], [Glasbey 01], [Ramani et al 05]**Hybrid Wiener Filter Image scaling**Original Image Bicubic Interpolation Hybrid Wiener**Hybrid Wiener Filter Re-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**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 Approach Non-Stationarity**[Michaeli & Eldar 08]**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

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