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Two Tantalizing Concepts. Randomness “Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin” – Von Neumann There is no such thing as a random number; there are only methods (e.g., arithmetic vs. quantum) to produce random numbers Redundancy

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two tantalizing concepts
Two Tantalizing Concepts
  • Randomness
    • “Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin” – Von Neumann
    • There is no such thing as a random number; there are only methods (e.g., arithmetic vs. quantum) to produce random numbers
  • Redundancy
    • Narrow sense, in information theory
    • Broad sense, in science and engineering
pseudorandom number generator
Pseudorandom Number Generator
  • Complexity-based pseudorandomness is a deep concept in theoretical CS (with wide applications in scientific computing such as Monte Carlo methods)
  • Engineers generate pseudorandom numbers by arithmetic algorithms such as linear congruential generators and linear feedback shift registers (LFSR)

A maximal LFSR produces an

m-sequence (i.e. cycles through

all possible 2n − 1 states within

the shift register except the state

where all bits are zero), unless it

contains all zeros, in which case

it will never change.

slide3
AWGN
  • How is AWGN generated under MATLAB?
  • Yes, by “randn” function but how does it work?
  • RANDN('seed') in MATLAB4 vs. RANDN('state',J) in MATLAB5
  • Subtle implication into denoising experiments
randomness in nature
Randomness in Nature

Star Constellation

Waitomo Glow-worm Caves on Lake Roturura

slide5

Random pin-dropping

Homogeneous Poisson

redundancy
Redundancy
  • What is redundancy in Shannon’s mind?
    • In source coding, redundancy refers to the gap between the source entropy and the actual bit rate (so we want to eliminate redundancy).
    • In channel coding, redundancy refers to the “extra bits” used for error correction (so we add redundancy in a controlled fashion).
  • Divide-and-conquer: an engineering solution
redundancy in nature
Redundancy in Nature
  • Redundancy in language
    • Why are human languages redundant? How is it possible for young kids to learn speaking a language?
  • Redundancy in natural world
    • Why are natural images compressible? How does human vision systems work?
  • Redundancy in genetics
    • Why are Chromosomes in pairs?
redundancy in sp
Redundancy in SP
  • Sampling – an artificial tool which we have not yet understood well
  • No signal from the natural world is band-limited
    • Shannon/Nyquist’s sampling theorem never holds in practice
    • “Truth is much too complicated to allow anything but approximations.”
  • Uniform sampling- a “sin” operator?
two contrasting views
Two Contrasting Views
  • Redundancy is bad
    • Since most natural signals are still so compressible, we should acquire much fewer samples (joint design of sampling and coding)
    • The emerging “compressive sensing” paradigm
  • Redundancy is good
    • Redundancy is essential to human intelligence especially the redundancy exploitation hypothesis advocated by H. Barlow
    • To solve high-level computer vision problems, get low-level (image sampling, feature extraction) done right first.