GLAST LAT Project PHYSTAT 05, Oxford, 12-15 September 2005 1 Application of a multidimensional...

GLAST LAT Project

PHYSTAT 05, Oxford, 12-15 September 2005 1

Application of a multidimensional Application of a multidimensional wavelet denoising algorithm for the wavelet denoising algorithm for the

detection and characterization of detection and characterization of astrophysical sources of astrophysical sources of -rays-rays

S. W. DigelStanford Linear Accelerator CenterB. ZhangQuantitative Image Analysis Unit, Institut PasteurJ. ChiangJoint Center for Astrophysics/Physics Department, University of Maryland, Baltimore CountyJ. M. FadiliGREYC CNRS UMR 6072, Image Processing GroupJ.-L. StarckService d'Astrophysique CEA/Saclay

Gamma-ray Large Gamma-ray Large Area Space Area Space TelescopeTelescope

GLAST LAT Project


OutlineOutline

• What is the application?– GLAST Large Area Telescope observations of the high-

energy gamma-ray sky– Characteristics of the sources and of the data

• Denoising via wavelet domain hypothesis testing

GLAST LAT Project


Sp

ect

rum

Ast

ro

GLAST Large Area Telescope (LAT)GLAST Large Area Telescope (LAT)

e+ e–

Calorimeter

Tracker

ACD

1.8 m

3000 kg

• Within its first few weeks, the LAT will double the number of celestial gamma rays ever detected

• 5-year design life, goal of 10 years

YearsYears Ang. Res.Ang. Res.(100 MeV)(100 MeV)

Ang. Res. Ang. Res. (10 GeV)(10 GeV)

Eng. Rng. Eng. Rng. (GeV)(GeV)

AAeffeff Ω Ω

(cm(cm22 sr) sr)# # raysrays

EGRETEGRET 1991–001991–00 5.8°5.8° 0.5°0.5° 0.03–100.03–10 750750 1.4 × 101.4 × 1066

AGILEAGILE 2006–2006– 4.7°4.7° 0.2°0.2° 0.03–500.03–50 1,5001,500 4 × 104 × 1066/yr/yr

AMSAMS 2008+?–2008+?– –– 0.1°0.1° 0.3–3000.3–300 1,6001,600 7 × 107 × 1055/yr/yr

GLAST LATGLAST LAT 2007–2007– 3.53.5°° 0.1°0.1° 0.02–3000.02–300 25,00025,000 1 × 101 × 1088/yr/yr

GLAST LAT Project


Some relevant detailsSome relevant details

• Response functions– Angular resolution and collecting

area are great for a -ray telescope, but poor for an astronomical instrument

• Observing strategy– FOV is huge (>2 sr), and the earth is

a bright background– For various reasons GLAST will be

in low earth orbit – so scan and rock while pointing away from the earth

• Astronomical analyses are underpinned by event reconstruction and classification, which we will take for granted here as being done optimally well

GLAST LAT Project


Gamma-ray skyGamma-ray sky

• High-energy gamma rays are produced by interactions of high-energy particles (surprise), accelerated in a remarkable variety of ways– Blazars, Pulsars, Plerions, SNR, OB star

associations, microquasars, Gamma-Ray Bursts are known or plausible classes of sources

• The universe is basically transparent to gamma rays in the range up to ~10s of GeV

• EGRET detected gamma rays from sources as close as the moon and as distant as redshifts of a few

• Unlike cosmic rays, gamma rays point back to their sources

Crab pulsar & nebula (CXC)

M87 jet (STScI)

GLAST LAT Project


Gamma-ray sky (2)Gamma-ray sky (2)

• Cosmic-ray interactions with the interstellar medium and interstellar radiation field make the Milky Way itself a bright, structured celestial foreground

• The gamma-ray sky is far from quiescent; blazars are intrinsically remarkably variable – Many EGRET blazars were detected only once, while flaring

EGRET (>100 MeV)

GLAST LAT Project


More on gamma-ray skyMore on gamma-ray sky

• Celestial fluxes are low & constraints (financial and physical) severely limit the collecting area and angular resolution– Gamma-ray astronomy is quite expensive – GLAST should

be a relative bargain in this field at ~50 cents per celestial gamma ray

• Source detection and characterization will require careful analysis

>1

GeV

in V

irgo

clus

ter

LAT simulationEGRET data

GLAST LAT Project


The ProblemThe Problem

• We’d like to detect all of the celestial sources in the data, and to recognize when a blazar [or member of another other class of source] are flaring (and quite likely being detected for the first time)– Some of this is real time searching (keeping up with ~6

downlinks and 8 sky coverages per day) – although the very brightest flares worth alerting other astronomers about are easiest to detect

– Some is wanting to be careful not to miss a blazar that turns on for a week or two

• We do not know in advance where the sources are that we will detect– For blazars projections are that we will detect several

thousand, i.e., a few times more than are known at other wavelengths

GLAST LAT Project


More on the problemMore on the problem

• In high-energy gamma-ray astronomy, the standard analysis technique has been model fitting, likelihood analysis to detect and characterize sources– Searching for new sources with likelihood analysis is slow,

brute force– Also does not extrapolate well to finding extended sources

• The utility of a good, ideally nonparametric method for initially detecting and characterizing gamma-ray sources in LAT data is clear

• Within the LAT collaboration, several algorithms have been investigated at some level– Optimal filtering, aperture photometry, and wavelet-based

(followed by source detection step)

GLAST LAT Project


Wavelet domain hypothesis testingWavelet domain hypothesis testing

• From Zhang, Fadili, & Starck et al. (2005)• For this analysis, the data are treated as 2D+1D, maps of

photon counts in planes with different energy ranges• Photon counts v(x, y, E), and typically 0-few counts per

pixel/bin• Indexing the pixels/bins (I the index set)

• vi’s are iid Poisson is the underlying image (intensity × exposure).

X

Y

E

GLAST LAT Project


Requirements of the methodRequirements of the method

• Get an estimate of the intensity using statistical estimation/decision theory in the wavelet domain

• Must be efficient enough in very low-count setting• Regularity preservation (e.g., isotropic structures)• Flux preservation

GLAST LAT Project


Wavelet Poisson intensity estimation: Overview

• Transformation-based methods– Variance stabilizing transforms (e.g. Anscombe, Fisz)

[Donoho 93][Frylewizc 04][Fadili 04]• Classical direct methods

– Wavelet histogram auto-convolution [Bijaoui et al. 01]

– l1 penalized estimator [Sardy et al. 03]

– Adaptive Wiener filter [Nowak et al. 99]– Modulation estimators [Antoniadis et al. 01]

• Bayesian direct methods– Bayesian multiscale models [Kolaczyk 99,04]– Mixture-based prior [Timmerman 99],[Lu 04]

• Hypothesis testing methods[Kolaczyk 99][Zhang et al. 05]

GLAST LAT Project


Formulation of the methodFormulation of the method

• The background intensity, if not known a priori, is assumed to be locally constant (in the wavelet support).

• Can be estimated from the approximation coefficients at a coarser scale.

• Define the binary hypothesis testing for each wavelet coefficient w:

– H0 : w is consistent with the background;

– H1 : w is inconsistent with the background.

• H0 coefficients (consistency with the background with statistical fluctuations) are zeroed

• H1 coefficients (significant change from the background) are retained

• This is a controlled (via the p-value) hard thresholding scheme.

GLAST LAT Project


Hypothesis testing of individual coefficientsHypothesis testing of individual coefficients

• The coefficients are tested separately without correction for multiple comparisons. The probability of false detection is upper bounded by p:

• If the pdf under the null is symmetric around the (unique) mode at 0, the double-sided critical threshold at significance level p is

• For computational tractability, we use the Haar wavelet, thanks to the availability of the closed form for its pdf

• Zhang et al. also show that the same thresholds can be applied to the coefficients of biorthogonal Haar wavelets, giving an excellent regularity preservation.

GLAST LAT Project


Deriving the thresholdsDeriving the thresholds

• The pdf of Haar coefficients follows a Non-Central Chi-Square distribution

• Using the Fisher normal approximation, a threshold can be derived for every Gaussian significance level– It has a higher level of detection power than [Kolaczyk 99]– We have extended it to non-constant, or unknown a priori,

background intensity• This decision threshold is found to be sensitive to low-count

situations

GLAST LAT Project


Relative detection powersRelative detection powers

Zhang et al. (2005)

Kolaczyk

Zhang et al.

1-dim examples

GLAST LAT Project


Extension to the energy dimensionExtension to the energy dimension

• A 2D bi-orthogonal haar (BH) transform is applied spatially to each image at energy band i

• To each BH spatial coefficient, a 1D BH transform is applied along the energy axis

• Apply the hypothesis testing estimator to the 2D+1D BH coeffs

• Apply inverse 1D BH transform along energy axis

• Apply inverse 2D BH spatially to get the final estimated multi-spectral data

• Simulations show that the method offers good flux preservation in various count settings

GLAST LAT Project


Application to simulated LAT dataApplication to simulated LAT data

• Power-law spectra with range of fluxes (factors of sqrt(2) from 1 x 10-8 cm-2 s-1 ) and spectral indexes (1.6, 1.8, 2, 2.2, 2.4), one month scanning sky survey, counts summed over energy

Point sources only Point sources + background After denoising

GLAST LAT Project


More examplesMore examples

• Fluxes as before, hard spectral cutoffs at 500, 1000, 2000, 4000, 8000 MeV

Point sources + background

After denoising

GLAST LAT Project


More examples (2)More examples (2)

• One-month observation at high Galactic latitudes, source flux 5 × 10-8 cm-2 s-1

GLAST LAT Project



• Detecting 5 × 10-8 cm-2 s-1 sources against the Milky Way

Point sources + Galactic + extragalactic diffuse

After denoising After denoising, with model for diffuse emission supplied

Profiles in latitude

GLAST LAT Project



• Recovery of spectra looks possible– A source from the previous simulation

Expected E-1 slope in counts per bin

GLAST LAT Project


SummarySummary

• Motivations for a reliable nonparametric source detection algorithm to apply to GLAST LAT data are clear – Especially for the relatively short time ranges over which

we will want to study sources, the data will be squarely in the low counts regime with widely varying response functions and significant celestial foregrounds

• The hypothesis testing wavelet denoising algorithm of Zhang et al. has firm statistical underpinnings, is demonstrably sensitive relative to other wavelet approaches, and allows spectra to be recovered

GLAST LAT Project PHYSTAT 05, Oxford, 12-15 September 2005 1 Application of a multidimensional...

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