Sampling Random Signals Sampling Random Signals
[ ]c n[ ]c n[ ]c n
2
IntroductionTypes of Priors
Subspace priors:
Smoothness priors:
Stochastic priors:
3
IntroductionMotivation for Stochastic
Modeling
Understanding of artifacts via stationarity analysis
New scheme for constrained reconstruction
Error analysis
4
IntroductionReview 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 FilterImage scaling
Bicubic Bicubic InterpolationInterpolation
Original ImageOriginal Image Hybrid WienerHybrid Wiener
11
Hybrid Wiener FilterRe-sampling
Drawbacks:Drawbacks:
•May be hard to implement
•No explicit expression in the time domain
Re-sampling:Re-sampling:
12
Predefined interpolation Predefined interpolation filter:filter:
Constrained Reconstruction Kernel
The correction filter depends on t !
13
StationaryStationary
??
Non-Stationary Reconstruction
14
Non-Stationary Reconstruction
Stationary Stationary SignalSignal
Reconstructed Reconstructed SignalSignal
15
Non-Stationary Reconstruction
16
Non-Stationary ReconstructionArtifacts
Original image
Interpolation with rect
Interpolation with sinc
17
BicubicBicubic SincSincNearest Nearest
NeighborNeighborOriginal ImageOriginal Image
Non-Stationary ReconstructionArtifacts
18
Predefined interpolation Predefined interpolation filter:filter:
Constrained Reconstruction Kernel
Solution:Solution:
1.1.
2.2.
19
Constrained Reconstruction Kernel
Dense Interpolation GridDense grid approximation of the Dense grid approximation of the optimal filter:optimal filter:
20
Optimal dense grid interpolation:Optimal dense grid interpolation:
Our Approach
21
optspan w t n nKspan w t
1K
2K
3K
Our ApproachMotivation
22
Our ApproachNon-Stationarity
[Michaeli & Eldar 08]
23
SimulationsSynthetic Data
24
SimulationsSynthetic Data
25
SimulationsSynthetic Data
26
First Order ApproximationFirst Order Approximation
•Ttriangular kernel
•Interpolation grid:
•Scaling factor:
27
Optimal Dense Grid Optimal Dense Grid ReconstructionReconstruction
•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
• 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