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Sampling and Interpolation

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Presentation on theme: "Sampling and Interpolation"— Presentation transcript:

1 Sampling and Interpolation
Martin Vetterli, 2, 9 and with Benjamin Bejar Haro, Mihailo Kolundzija and R.Parhizkar

2 Outline 1. Introduction The world is analog but computation is digital! 2. Sampling and Interpolation Finite dimensional vectors Sequences Functions Periodic functions 3. Sampling beyond the classical model Finite rate of innovation sampling Plenacoustic sampling Plenoptic sampling Sampling along trajectories 4. Conclusions

3 The Situation The world is analog Computation is digital
How to go between these representations? Ex: Audio, sensing, imaging, simulation, communications Sometimes: Analog to digital: Estimation Digital to analog: Synthesis Processing Analog World

4 From Analog to Digital

5 Dual situation Information is digital Communication is analog
How to go between these representations? Ex: Digital communications Analog to digital: ADC, error correction, estimation, Digital to analog: DAC, modulation Analog channel Digital Information

6 The sampling question:
Given a class of objects, like a class of functions (e.g. bandlimited, SISS) And given a sampling device, as usual to acquire the real world Smoothing kernel or low pass filter Regular, uniform sampling Obvious question: When is there a countable representation? When does a minimum number of samples uniquely specify the function? sampling kernel

7 Kernel and Sampling Rate
ESO, Chile 1. About the observation kernel: Given by nature Diffusion equation, Green function Ex: sensor networks Given by the set up Designed by somebody else (Hubble telescope) Given by design Pick the best kernel Ex: engineered systems, but constraints 2. About the sampling rate: Given by problem Ex: sensor networks Usually, as low as possible (UWB)

8 Are these Real Problems?
Google Street view as a popular example How many images per second to reconstruct the real world What resolution images to give a precise view of the world

9 Plenoptic Sampling Computer vision and graphics problem: Epipolar geometry Points map to lines Approximate sampling theorem Light fields

10 When there are problems….
Rolex Learning Center at EPFL SANAA Architects ( Kazuyo Sejima, Ryue Nishizawa)

11 Are these Real Problems?
Google maps as another popular example How to register images What resolution images to give an adequate view of the world

12 Are these Real Problems?
Sensor networks as another relevant example How many sensors How to reconstruct

13 Diffusion Equation: Real World Problem
Point source, know location, unknown time-varying intensity Observation by multiple, possibly mobile sensors

14 Classic Sampling Case [WNKWRGSS, 1915-1949]
If a function x(t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/(2W) seconds apart. [if approx. T long, W wide, 2TW numbers specify the function] It is a representation theorem: sinc(t-n) is an orthobasis for BL[-,] x(t)  BL[-,] can be written as Linear subspace, Shift-invariant Notes: Slow convergence of the sinc series Shannon-Bandwidth and information rate Bandlimited is sufficient, not necessary! (e.g. bandpass sampling) Many variations (irregular, multichannel, SISS, etc) x(t) = \sum_{n=-\infty}^{\infty}x[n]\mathrm{sinc} \bigg( \frac{t-nT_s}{T_s}\bigg)

15 Shannon’s Theorem… a Bit of History
Whittaker Kotelnikov Raabe Shannon 1935 1928 1946 1949 1915 1933 1938 1948 Nyquist Someya Whittaker Gabor

16 Classic Case: Subspaces
Shannon bandlimited case or 1/T degrees of freedom per unit time But: a single discontinuity, and no more sampling theorem… Are there other signals with finite number of degrees of freedom per unit of time that allow exact sampling results? Latex: x(t) = \sum_{n\in Z}x(nT)\mathrm{sinc}(t/T-n)

17 Examples of non-bandlimited signals

18 Goal of this set of lectures
1. Develop a thorough understanding of sampling and interpolation Hilbert space setting All cases: RN, l2, L2, periodic Shannon sampling as a particular (but very important) case Shift invarint subspace as the general case 2. Sampling theory beyond Shannon Discuss extensions Sparse sampling 3. Applications Glimpse at a few applications

19 Recap

20 Recap

21 Recap

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