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Instructor: Yonina Eldar Teaching Assistant: Tomer Michaeli Spring 2009 Modern Sampling Methods

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2 Sampling: “Analog Girl in a Digital World…” Judy Gorman 99 Digital worldAnalog world Signal processing Denoising Image analysis … Reconstruction D2A Sampling A2D (Interpolation)

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3 Applications Sampling Rate Conversion Common audio standards: 8 KHz (VOIP, wireless microphone, …) KHz (MPEG audio, …) 16 KHz (VOIP, …) KHz (MPEG audio, …) 32 KHz (miniDV, DVCAM, DAT, NICAM, …) 44.1 KHz (CD, MP3, …) 48 KHz (DVD, DAT, …) …

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4 Lens distortion correction Image scaling Applications Image Transformations

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5 Applications CT Scans

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6 Applications Spatial Superresolution

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7 Applications Temporal Superresolution

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9 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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10 Cauchy (1841): Whittaker (1915) - Shannon (1948): A. J. Jerri, “The Shannon sampling theorem - its various extensions and applications: A tutorial review”, Proc. IEEE, pp , Nov Bandlimited Sampling Theorems

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11 Limitations of Shannon’s Theorem Input bandlimited Impractical reconstruction (sinc) Ideal sampling Towards more robust DSPs: General inputs Nonideal sampling: general pre-filters, nonlinear distortions Simple interpolation kernels

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12 Generalized anti- aliasing filter Sampling Process Linear Distortion Sampling functions Electrical circuit Local averaging

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13 Replace Fourier analysis by functional analysis, Hilbert space algebra, and convex optimization Original + Initial guess Reconstructed signal Sampling Process Nonlinear Distortion Nonlinear distortion Linear distortion

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14 Employ estimation techniques Sampling Process Noise

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15 Signal Priors x(t) bandlimited x(t) piece-wise linear Different priors lead to different reconstructions

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16 Shift invariant subspace: General subspace in a Hilbert space Signal Priors Subspace Priors Common in communication: pulse amplitude modulation (PAM) Bandlimited Spline spaces

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17 Two key ideas in bandlimited sampling: Avoid aliasing Fourier domain analysis Beyond Bandlimited 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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18 Signal Priors Smoothness Priors

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19 Signal Priors Stochastic Priors Original ImageBicubic InterpolationMatern Interpolation

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20 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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21 Reconstruction Constraints Unconstrained Schemes SamplingReconstruction

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22 Reconstruction Constraints Predefined Kernel SamplingReconstructionPredefined Minimax methods Consistency requirement

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23 Reconstruction Constraints Dense Grid InterpolationPredefined (e.g. linear interpolation) To improve performance: Increase reconstruction rate

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24 Reconstruction Constraints Dense Grid Interpolation Bicubic InterpolationSecond Order Approximation to Matern Interpolation with K=2 Optimal Dense Grid Matern Interpolation with K=2

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25 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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