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Modern Sampling Methods 049033

Modern Sampling Methods 049033. Instructor: Yonina Eldar Teaching Assistant: Tomer Michaeli Spring 2009. Sampling: “Analog Girl in a Digital World…” Judy Gorman 99. Analog world. Digital world. Sampling A2D. Signal processing Denoising Image analysis …. Reconstruction D2A.

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Modern Sampling Methods 049033

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  1. Modern Sampling Methods 049033 Instructor: Yonina Eldar Teaching Assistant: Tomer Michaeli Spring 2009

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

  3. 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, …) • …

  4. ApplicationsImage Transformations • Lens distortion correction • Image scaling

  5. ApplicationsCT Scans

  6. Applications Spatial Superresolution

  7. Applications Temporal Superresolution

  8. Applications Temporal Superresolution

  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 …)

  10. 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.

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

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

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

  14. Employ estimation techniques Sampling Process Noise

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

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

  17. 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 …

  18. Signal Priors Smoothness Priors

  19. Signal Priors Stochastic Priors Original Image Bicubic Interpolation Matern Interpolation

  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

  21. Reconstruction Constraints Unconstrained Schemes Sampling Reconstruction

  22. Reconstruction Constraints Predefined Kernel Predefined Sampling Reconstruction • Minimax methods • Consistency requirement

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

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

  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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