Modern Sampling Methods Summary of Subspace Priors Spring, 2009

2 Outline BandlimitedIdeal point-wiseIdeal interpolation Subspace priors Smoothness priors Sparsity priors Linear Sampling Nonlinear distortions Minimax approach with simple kernels Dense grid recovery Signal Model Sampling Reconstruction

3 Back to Shannon Any bandlimited signal is spanned by the sinc function: The functions are orthonormal The dual is again so

4 Shift Invariant Spaces A 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

5 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 :

6 Correlation Sequences Samples can be written as In the Fourier domain: The set is orthonormal if In the Fourier domain samples

7 Generalized anti- aliasing filter Non-Pointwise Linear Sampling Sampling functions Electrical circuit Local averaging In the sequel: Sampling space:

8 Outline BandlimitedIdeal point-wiseIdeal interpolation Subspace priors Smoothness priors Sparsity priors  Non-linear distortions  Minimax approach with simple kernels Signal Model Sampling Reconstruction

9 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 Knowing is equivalent to knowing (* if is a Riesz basis or frame) 1

10 Some Math The dual basis is defined by If then where In the Fourier domain

11 What if lies in a subspace where is generated by ? If then PR impossible since If then PR possible Mismatched Sampling Perfect Reconstruction in a Subspace

12 Some Math Sampling: After correlation filter: we get back From we can reconstruct

13 Geometric Interpretation When and is general: In both cases we have projections onto the signal space !

14 Example: Pointwise Sampling corresponding to Input signal not necessarily bandlimited Recovery possible as long as or Nonbandlimited functions can be recovered from pointwise samples!

15 Example: Bandlimited sampling Can be recovered even though it is not bandlimited? YES ! 1. Compute convolutional inverse of 2. Convolve the samples with 3. Reconstruct with

16 Outline BandlimitedIdeal point-wiseIdeal interpolation Subspace priors Smoothness priors Sparsity priors Non-linear distortions  Minimax approach with simple kernels Signal Model Sampling Reconstruction

17 Nonlinear Sampling Saturation in CCD sensors Dynamic range correction Optical devices High power amplifiers Memoryless Nonlinear 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 , Dec Nonlinear and Non-Ideal Sampling: Theory and Methods

18 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) Theorem (uniqueness):

19 Main idea: 1. Minimize error in samples where 2. From uniqueness if Perfect reconstruction global minimum of Difficulties: 1. Nonlinear, nonconvex problem 2. Defined over an infinite space ( Dvorkind, Eldar, Matusiak 07) Theorem : Only have to trap a stationary point!

20 Algorithm converges to true input ! 1.Initial guess 2.Linearization: Replace by its derivative around 3.Solve linear problem and update solution Algorithm: Linearization

21 Example I

22 Simulation

23 Example II Optical sampling system: optical modulator ADC

24 Simulation First iteration: Third iteration: Initialization with

25 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 , May 2009.