Instructor: Yonina EldarTeaching Assistant: Tomer

Michaeli

Spring 2009

Modern Sampling Methods

049033

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Sampling: “Analog Girl in a Digital World…” Judy Gorman 99

Digital worldAnalog world

Signal processingDenoisingImage analysis …

ReconstructionD2A

SamplingA2D

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(Interpolation)

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ApplicationsSampling Rate Conversion

Common audio standards: 8 KHz (VOIP, wireless microphone, …) 11.025 KHz (MPEG audio, …) 16 KHz (VOIP, …) 22.05 KHz (MPEG audio, …) 32 KHz (miniDV, DVCAM, DAT, NICAM, …) 44.1 KHz (CD, MP3, …) 48 KHz (DVD, DAT, …) …

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Lens distortion correction

Image scaling

ApplicationsImage Transformations

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ApplicationsCT Scans

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ApplicationsSpatial Superresolution

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ApplicationsTemporal Superresolution

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ApplicationsTemporal Superresolution

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Our Point-Of-View

The field of sampling was traditionally associated with methods implemented either in the frequency domain, or in the time domainSampling can be viewed in a broader sense of projection onto any subspace or union of subspacesCan choose the subspaces to yield interesting new possibilities (below Nyquist sampling of sparse signals, pointwise samples of non bandlimited signals, perfect compensation of nonlinear effects …)

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Cauchy (1841):

Whittaker (1915) - Shannon (1948):

A. J. Jerri, “The Shannon sampling theorem - its various extensions and applications: A tutorial review”, Proc. IEEE, pp. 1565-1595, Nov. 1977.

Bandlimited Sampling Theorems

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Limitations of Shannon’s Theorem

Input Input bandlimitbandlimit

eded

Impractical Impractical reconstruction reconstruction

(sinc)(sinc)

Ideal Ideal samplisampli

ngng

Towards more robust DSPs:General inputsNonideal sampling: general pre-filters, nonlinear distortionsSimple interpolation kernels

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Generalized anti-aliasing filter

Sampling ProcessLinear Distortion

Sampling

functionsElectrical Electrical

circuitcircuitLocal Local

averagingaveraging

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Replace Fourier analysis by functional analysis, Hilbert space algebra, and convex optimization

Original + Initial guess

Reconstructed signal

Sampling ProcessNonlinear Distortion

Nonlinear distortion

Linear distortion

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Employ estimation techniques

Sampling ProcessNoise

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Signal Priors

x(t) bandlimitedx(t) piece-wise linear

Different priors lead to different reconstructions

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Shift invariant subspace:

General subspace in a Hilbert space

Signal PriorsSubspace Priors

Common in communication: pulse amplitude modulation (PAM)

Bandlimited Spline spaces

( )X f ( )x t( )x t

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Two key ideas in bandlimited sampling:Avoid aliasingFourier domain analysis

Beyond Bandlimited

Misleading concepts!

Suppose that with Signal is clearly not bandlimitedAliasing in frequency and timePerfect reconstruction possible from samples

Aliasing is not the issue

…

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Signal PriorsSmoothness Priors

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Signal PriorsStochastic Priors

Original Image Bicubic Interpolation Matern Interpolation

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Signal PriorsSparsity Priors

Wavelet transform of images is commonly sparseSTFT transform of speech signals is commonly sparseFourier transform of radio signals is commonly sparse

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Reconstruction ConstraintsUnconstrained Schemes

Sampling Reconstruction

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Reconstruction ConstraintsPredefined Kernel

Sampling Reconstruction

PredefinPredefineded

Minimax methodsConsistency requirement

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Reconstruction ConstraintsDense Grid Interpolation

PredefinedPredefined

(e.g. linear (e.g. linear interpolation)interpolation)

To improve performance: Increase reconstruction rate

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Reconstruction ConstraintsDense Grid Interpolation

Bicubic Interpolation Second Order Approximation to

Matern Interpolation with K=2

Optimal Dense Grid Matern Interpolation

with K=2

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Course Outline(Subject to change without further notice)

Motivating introduction after which you will all want to take this course (1 lesson)Crash course on linear algebra (basically no prior knowledge is assumed but strong interest in algebra is highly recommended) (~3 lessons)Subspace sampling (sampling of nonbandlimited signals, interpolation methods) (~2 lessons)Nonlinear sampling (~1 lesson)Minimax recovery techniques (~1 lesson)Constrained reconstruction: minimax and consistent methods (~2 lessons)Sampling sparse signals (1 lesson)Sampling random signals (1 lesson)