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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)