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Modern Sampling Methods
049033
Instructor: Yonina Eldar
Teaching Assistant: Tomer Michaeli
Spring 2009
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Sampling: “Analog Girl in a Digital World…”
Judy Gorman 99
Analog world
Digital world
Sampling
A2D
Signal processing
Denoising
Image analysis …
Reconstruction
D2A
(Interpolation)
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Applications
Sampling 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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Applications
Image Transformations
Lens distortion correction
Image scaling
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Applications
CT Scans
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Applications
Spatial Superresolution
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Applications
Temporal Superresolution
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Applications
Temporal 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 domain
Sampling can be viewed in a broader sense of projection onto any
subspace or union of subspaces
Can 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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Bandlimited Sampling Theorems
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.
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Limitations of Shannon’s Theorem
Ideal
sampling

Input
bandlimited
Impractical
reconstruction (sinc)
Towards more robust DSPs:
General inputs
Nonideal sampling: general pre-filters, nonlinear distortions
Simple interpolation kernels
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Sampling Process
Linear Distortion
Sampling
functions
Generalized antialiasing filter
Local averaging
Electrical circuit
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Sampling Process
Nonlinear Distortion
Original + Initial guess
Nonlinear
distortion
Linear
distortion
Reconstructed signal
Replace Fourier analysis by functional analysis, Hilbert space
algebra, and convex optimization
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Sampling Process
Noise
Employ estimation techniques
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Signal Priors
bandlimited
x(t) piece-wise
linear
Different priors lead to different reconstructions
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Signal Priors
Subspace Priors
X(f )
x(t )
Bandlimited
x(t )
Spline spaces
Shift invariant subspace:
Common in communication: pulse amplitude modulation (PAM)
General subspace in a Hilbert space
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Beyond Bandlimited
Two key ideas in bandlimited sampling:
Avoid aliasing
Fourier domain analysis
Misleading concepts!
Suppose that
with
Signal is clearly not bandlimited
Aliasing in frequency and time
Perfect reconstruction possible from samples

Aliasing is not the issue …
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Signal Priors
Smoothness Priors
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Signal Priors
Stochastic Priors
Original Image
Bicubic Interpolation
Matern Interpolation
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Signal Priors
Sparsity Priors
Wavelet transform of images is commonly sparse
STFT transform of speech signals is commonly sparse
Fourier transform of radio signals is commonly sparse
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Reconstruction Constraints
Unconstrained Schemes

Sampling
Reconstruction
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Reconstruction Constraints
Predefined Kernel
Predefined

Sampling
Reconstruction
Minimax methods
Consistency requirement
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Reconstruction Constraints
Dense Grid Interpolation
To improve performance: Increase reconstruction rate
Predefined
(e.g. linear interpolation)

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Reconstruction Constraints
Dense 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)
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