modern sampling methods summary of subspace priors spring, 2009
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Modern Sampling Methods
Summary of Subspace Priors
Spring, 2009
[ ]c n[ ]c n[ ]c n
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Outline
Bandlimited Ideal point-wise Ideal interpolation
Subspace priors Smoothness priors Sparsity priors
Linear Sampling Nonlinear distortions
Minimax approach with simple kernels Dense grid recovery
SignalModel
Sampling Reconstruction
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Back to Shannon
Any bandlimited signal is spanned by the sinc function:
The functions are orthonormal
The dual is again so
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Shift Invariant SpacesA subspace that can be expressed as shifts of :
In general is not equal to samples of
Examples: Bandlimited functions
Spline spaces central B-spline
and
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Fourier Transforms All manipulations in SI spaces can be carried out in Fourier
domain!
Continuous time FT:
Discrete time FT: -
periodic
DTFT of sampled sequence :
If is used to create :
Riesz basis condition for :
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Correlation Sequences
Samples can be written as
In the Fourier domain:
The set is orthonormal if
In the Fourier domain
samples
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Generalized anti-aliasing filter
Non-Pointwise Linear Sampling
Sampling
functions
Electrical Electrical circuitcircuit
Local Local averagingaveraging
In the sequel:
Sampling space:
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Outline
Bandlimited Ideal point-wise Ideal interpolation
Subspace priors Smoothness priors Sparsity priors
Non-linear distortions Minimax approach with simple kernels
SignalModel
Sampling Reconstruction
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Perfect Reconstruction
Key observation:
Given which signals can be perfectly reconstructed?
Same samples
Thus, for perfect reconstruction is possible by:
Bandlimited sampling (Shannon theory) is a special case !
sampling space
1
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Some Math
The dual basis is defined by
If then where
In the Fourier domain
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What if lies in a subspace where is generated by ?
If then PR impossible sinceIf then PR possible
Mismatched Sampling
Perfect Reconstruction in a Subspace
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Some Math
Sampling:
After correlation filter: we get back
From we can reconstruct
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Geometric InterpretationWhen and is general:
When and is general:
In both cases we have projections onto the signal space !
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Example: Pointwise Sampling
corresponding to
Input signal not necessarily bandlimited
Recovery possible as long as or
Nonbandlimited functions can be recovered from pointwise samples!
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Example: Bandlimited sampling
Can be recovered even though it is not bandlimited?
YES !1 .Compute convolutional inverse of
2. Convolve the samples with3. Reconstruct with
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Outline
Bandlimited Ideal point-wise Ideal interpolation
Subspace priors Smoothness priors Sparsity priors
Non-linear distortions Minimax approach with simple kernels
SignalModel
Sampling Reconstruction
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Nonlinear Sampling
Saturation in CCD sensors
Dynamic range correction
Optical devices
High power amplifiers
MemorylessNonlinear distortion
Not a subspace !
T. G. Dvorkind, Y. C. Eldar and E. Matusiak, "Nonlinear and Non-Ideal Sampling: Theory and Methods", IEEE Trans. on Signal Processing, vol. 56, no. 12, pp. 5874-5890, Dec. 2008.
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Perfect Reconstruction
Setting: is invertible with bounded derivative lies in a subspace Uniqueness same as in linear case!
Proof: Based on extended frame perturbation theory and geometrical ideas
(Dvorkind, Eldar, Matusiak 07) (Dvorkind, Eldar, Matusiak 07)
Theorem (uniqueness):
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Main idea:1. Minimize error in samples where 2. From uniqueness if
Perfect reconstruction global minimum of
Difficulties:1. Nonlinear, nonconvex problem2. Defined over an infinite space
(Dvorkind, Eldar, Matusiak 07) (Dvorkind, Eldar, Matusiak 07)
Theorem:
Only have to trap a stationary point!
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Algorithm converges to true input !
1. Initial guess
2. Linearization: Replace by its derivative around
3. Solve linear problem and update solution
Algorithm: Linearization
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Example I
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Simulation
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Example IIOptical sampling system:
optical modulator
ADC
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SimulationInitialization with
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Summary: Subspace Priors
Perfect Recovery In A Subspace
General input signals (not necessarily BL)
General samples (anti-aliasing filters), nonlinear samples
Results hold also for nonuniform sampling and more
general spaces
Being bandlimited is not important for recovery
Y. C. Eldar and T. Michaeli, "Beyond Bandlimited Sampling", IEEE Signal Proc. Magazine, vol. 26, no. 3, pp. 48-68, May 2009.