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.