Sampling Random Signals

Sampling Random Signals Sampling Random Signals

[ ]c n[ ]c n[ ]c n

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IntroductionTypes of Priors

Subspace priors:

Smoothness priors:

Stochastic priors:

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IntroductionMotivation for Stochastic

Modeling

Understanding of artifacts via stationarity analysis

New scheme for constrained reconstruction

Error analysis

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IntroductionReview of Definitions and

Properties

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IntroductionReview of Definitions and

Properties

Filtering:

Wiener filter:

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Balakrishnan’s Sampling Theorem

[Balakrishnan 1957]

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Hybrid Wiener Filter

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Hybrid Wiener Filter

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

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Hybrid Wiener Filter

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Hybrid Wiener FilterImage scaling

Bicubic Bicubic InterpolationInterpolation

Original ImageOriginal Image Hybrid WienerHybrid Wiener

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Hybrid Wiener FilterRe-sampling

Drawbacks:Drawbacks:

•May be hard to implement

•No explicit expression in the time domain

Re-sampling:Re-sampling:

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Predefined interpolation Predefined interpolation filter:filter:

Constrained Reconstruction Kernel

The correction filter depends on t !

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StationaryStationary

??

Non-Stationary Reconstruction

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Non-Stationary Reconstruction

Stationary Stationary SignalSignal

Reconstructed Reconstructed SignalSignal

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Non-Stationary Reconstruction

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Non-Stationary ReconstructionArtifacts

Original image

Interpolation with rect

Interpolation with sinc

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BicubicBicubic SincSincNearest Nearest

NeighborNeighborOriginal ImageOriginal Image

Non-Stationary ReconstructionArtifacts

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Predefined interpolation Predefined interpolation filter:filter:

Constrained Reconstruction Kernel

Solution:Solution:

1.1.

2.2.

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Constrained Reconstruction Kernel

Dense Interpolation GridDense grid approximation of the Dense grid approximation of the optimal filter:optimal filter:

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Optimal dense grid interpolation:Optimal dense grid interpolation:

Our Approach

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optspan w t n nKspan w t

1K

2K

3K

Our ApproachMotivation

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Our ApproachNon-Stationarity

[Michaeli & Eldar 08]

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SimulationsSynthetic Data

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SimulationsSynthetic Data

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SimulationsSynthetic Data

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First Order ApproximationFirst Order Approximation

•Ttriangular kernel

•Interpolation grid:

•Scaling factor:

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Optimal Dense Grid Optimal Dense Grid ReconstructionReconstruction

•Ttriangular kernel

•Interpolation grid:

•Scaling factor:

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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

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• Average MSE of the hybrid Wiener filter

• Necessary & Sufficient conditions for linear perfect recovery

• Necessary & Sufficient condition for our scheme to be optimal

Theoretical Analysis