Comments on Rebecca Willett

1
Comments on Rebecca Willetts 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 theyre 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 theyre 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 gt 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 Rebeccas 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
(No Transcript)
6
(No Transcript)
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

8
(No Transcript)
9
2 gray levels 4 gray levels 8 gray levels 16
gray levels 32 gray levels 64 gray levels
10
8 16 32
40
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)