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

Astronomical Images”

By Jeff ScargleSpace 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).

* “Near-Optimal Detection of Geometric Objects by Fast Multiscale Methods,” Ery Arias-Castro, David Donoho, Xiaoming Huo, August 18, 2003

The asymptotic behavior of wavelet coefficients in equation (2) of Rebecca’s paper is related to this somwhat magical result.

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

2 gray levels 4 gray levels 8 gray levels16 gray levels 32 gray levels 64 gray levels

8 16 32 40