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 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

7. 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

9. 2 gray levels 4 gray levels 8 gray levels