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Comments on Rebecca Willett’s paper “Multiscale Analysis of Photon-Limited Astronomical Images”

Comments on Rebecca Willett’s paper “Multiscale Analysis of Photon-Limited Astronomical Images” By Jeff Scargle Space Science Division NASA Ames Research Center. Good lesson: Try your algorithm out on pure noise! Cautions for the astronomer: Performance = rate of asymptotic convergence

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Comments on Rebecca Willett’s paper “Multiscale Analysis of Photon-Limited Astronomical Images”

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  1. Comments on Rebecca Willett’s paper “Multiscale Analysis of Photon-Limited Astronomical Images” By Jeff Scargle Space Science Division NASA Ames Research Center

  2. Good lesson: Try your algorithm out on pure noise! • Cautions for the astronomer: • Performance = rate of asymptotic convergence • ( N  ) • Oracles … they’re never around when you need them. • Plethora of methods • If you have a hammer, each signal looks like a …

  3. Good lesson: Try your algorithm out on pure noise! • Cautions for the astronomer: • Performance = rate of asymptotic convergence • ( N  ) • Oracles … they’re never around when you need them. • Plethora of methods • If you have a hammer, each signal looks like a … hammer!

  4. Theorem:* For each a > 0 there is an algorithm operating in C (a ) n 2log(n ) flops which is asymptotically powerful for detecting signals with amplitudes A n = 2 (1 + a ) log n (against unit variance i.i.d Gaussian noise). The asymptotic behavior of wavelet coefficients in equation (2) of Rebecca’s paper is related to this somwhat magical result. * “Near-Optimal Detection of Geometric Objects by Fast Multiscale Methods,” Ery Arias-Castro, David Donoho, Xiaoming Huo, August 18, 2003

  5. Trade-off: • Try to detect a signal with known properties • (linear transforms, matched filtering, etc.) • Vs. • Try to find what, if any, signal is present: • Representation in complete basis • Representation in overcomplete bases • Generic, non-parametric representation

  6. 2 gray levels 4 gray levels 8 gray levels 16 gray levels 32 gray levels 64 gray levels

  7. 8 16 32 40

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