Healpixels and Wavelets

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Healpixels and Wavelets

• Toby Burnett and Bruce LesnickUniversity of
  Washington

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Problem with binning for images and source
analysis

• Angular resolution varies dramatically with
  energy
• expect 1/E from multiple scattering
• measure E-0.8
• Images dont show localization without removing
  low energies, increasing resolution
• Full information not used in point source
  searches

gt3 decades of energy gt2 decades in resolution!
Note 68 containment is 3?
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HEALpix to the rescue

• Equal-area sky pixelization, invented by/for WMAP
• We use the Hierarchial aspect

Note Npix 12?4level
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From the basic 12 to 48 pixels (level 0 to
1)(with nested indexing)
Nesting is crucial to our variable-pixel scheme
divide by 4 to get mother pixel
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Application to GLAST define a pixelized photon
data base

• Take advantage of Hierarchical property, easy to
  correlate index for contained pixels.
• Create pixels in a sparse structure according to
  8 bins in photon energy, sorted according to
  position.
• Make selecting subset according to outer pixel
  level easy for projection integrals
• Numerous low energy photons are effectively
  binned
• Rare high energy photons occupy single pixels
• Simplifies database indexing

Correlated bins in energy and position
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Apply it to the 56-day simulated data set
(checkout3)
Low levels saturated, many photons/pixel. High
levels single photons (diffuse) multiple
photons (point sources)
1.7M photons w/ Egt100 MeV, front (thin)
section ?300 K pixels. Sparse structure, loads
quickly from ROOT file
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Image generation define a density function

• High energy photons are more localized we
  express this by defining photons/area
• Easily determined from the data base and the
  Healpix code.

3C273 density vs. all photons above 100 Mev
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The whole sky
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Density at the galactic center the simulated
EGRET source
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Seans Wavelets
The Mexican hat

• A projection function
• constant or linear background cancels, so it is a
  measure of the signal strength in presence of
  background
• Adjust width to optimize detection for our psf
  function width depends on photon energy
• Evaluate using the pixelized photon data base
• Assign width corresponding to average photon
  energy in the bin

• Use to measure
• Spectrum, measured value for each energy/healpix
  level
• position maximize value

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Localization by maximizing the wavelet

• Galactic center
• generated at ltcelestial_dir ra"266.51"
  dec"-28.86"/gt
• fit (266.50, -28.88) deviation 134 arc sec.
• But expected error, predicted from curvature of
  the function, is 17 arc sec.
• The background degrades the precision check a
  high-latitude source, like 3C273
• generated at ltcelestial_dir ra"187.25"
  dec"2.17"/gt
• fit (187.251, 2.16998), deviation 4 arc sec,
  predicted 2.
• Vela

ltsource name" F_PSRB0833m45 " gt     ltspectrum
escale"MeV"gt ltSpectrumClass name"PulsarSpectrum"
params"F_B0833m45, 128.83, -45.18, 1.0e5,
3.0e8, 1, 61243, 3, 1e6, 8e6, -1.62,
1.7"/gt      ltuse_spectrum frame"galaxy"/gt     lt/s
pectrumgt  lt/sourcegt
3C273 density
Fit (128.83, -45.18) Deviation 1 arc sec (13
predicted projection error)
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A look at many sources

• We get fits to all but one of the blazars. Fits
  to all 100 in a few min. (Starting at know
  positions)
• About ½ within 100 arc sec, almost independent of
  flux, a surprise

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In progress source detection

• Select candidates based on multiple photons in
  high-level pixels
• Apply wavelet fit to verify a likely source