Post on 04-Jan-2016
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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
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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