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 ( N   ) Oracles … they’re never around when you need them. Plethora of methods If you have a hammer, each signal looks like a …

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!

Theorem:* For each a > 0 there is an algorithm operating in C (a ) n 2 log(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

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

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