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Cheblet bases for galaxy modeling: a new tool for surveys
Cheblet bases for galaxy modeling: a new tool for surveys
Yolanda Jiménez TejaTxitxo Benítez Lozano
Instituto de Astrofísica de Andalucía (CSIC)Departamento de Astronomía Extragaláctica
E-mail: [email protected]
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Examples
Visualization
Motivation
Applications
• ALHAMBRA: http://arxiv.org/abs/0806.3021
• PAU: http://fr.arxiv.org/abs/0807.0535
• CLASH
Large photometric redshift surveys
• Photometry measurements
• Morphological features
• Linear
• Flexible
• Capable of modelling both the bulge and the disk of extended galaxies.
Galaxies decomposition method
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Examples
Visualization
Motivation
rnrTn arccoscos Chebyshev polynomial:
Lr
LrnLrTLn arccoscos;Chebyshev rational
function:
Applications
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
The Cheblet polar basis is separable in r and θ:
Chebyshev rational functions in r
Lr
Lrn
Lr
LrTrTL nn arccoscos 111
Fourier series in θ
2ine
21
2
12121;;,
nn
innnnnn eLrTL
CLr
, with
0if,2
0if,1
2
2
n
nC
Outline
Mathematical background
Examples
Visualization
Motivation
Applications
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
Cheblet polar basis functions
Outline
Mathematical background
Examples
Visualization
C-F basis
Wings
Motivation
Applications
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
Cheblet polar basis functions
Outline
Mathematical background
Examples
Visualization
C-F basis
Wings
Motivation
Applications
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
The wings of the basis functions tend to vanish, so the light flux is bounded by the basis:
Sèrsic Exponential Gaussian Moffat-Lorentzian
1,6 21 nn 4,10 21 nn2,6 21 nn0,2 21 nn
0,0 mn 1,1 mn 4,4 mn 2,6 mn
Cheblet basis
Shapelets
Outline
Mathematical background
Examples
Visualization
C-F basis
Wings
Motivation
Applications
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Examples
Visualization
Motivation
Applications
De Vaucouleurs
Sersic
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
-0.09 -0.03 0.03 0.09 0.15
-0.32 -0.16 0.00 0.16
-0.18 -0.06 0.06 0.18 0.30
-0.04 0.00 0.04 0.08
-0.18 -0.06 0.06 0.18 0.30
-0.075 -0.025 0.025 0.075 0.125
G
S
C
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Examples
Visualization
Motivation
Applications
Radial flux
Radial profile
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Examples
Visualization
Motivation
Applications
Clusters processing
• Method 1
One-by-one processing of the objects, taking different frames.
• Method 2
Simultaneous processing of the objects, centering a grid in each object.
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
ABELL1703 (F850) (arXiv:1004.4660)
Cheblet bases for galaxy modeling: a new tool for surveys
ABELL1703 (F435) (arXiv:1004.4660)
Cheblet bases for galaxy modeling: a new tool for surveys
ABELL1703 (F435) (arXiv:1004.4660)
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Examples
Visualization
Motivation
1) Settling the grid and evaluation of the Cheblet basis with some certain n1_max and n2_max.
2) Ortonormalization of the basis functions: Modified Gram-Schmidt Method SLOWEST STEP!
3) Storing the orthonormalized basis.
4) Calculation of the optimal number of coefficients n1 and n2: Chi2 minimization.
5) Extraction of the corresponding basis functions for those optimal n1 and n2.
6) Evaluation of the Cheblet coefficients.
7) Building the model with these coefficients.
8) Repeat the process from 4th step for each filter.
Clusters pipeline
Applications
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Practical implementation
Examples
Visualization
Motivation
Applications
Object shape measurement
€
F = 2π fn1 ,0I1n1
n1 = 0
+∞
∑• Flux:
€
xc + i yc =2π
Ffn1 ,1I2
n1
n1 = 0
+∞
∑• Centroid: €
R2 =2π
Ffn1 ,0I3
n1
n1 = 0
+∞
∑• Rms radius:
€
ε =fn1 ,−2I3
n1
n1 = 0
+∞
∑
fn1 ,0I3n1
n1 = 0
+∞
∑• Ellipticity:
€
Ipn1 =
2n1
j
⎛
⎝ ⎜
⎞
⎠ ⎟ −1( )
jL− j / 2 R
p+ j / 2+1
2p+ j + 2Re e in1π / 2in1 + j
2F1 n1, 2p+ j + 2, 2p+ j + 3;−i R
L
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥j= 0
n1
∑ , if n1 > 0
R p+1
p+1, if n1 = 0
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
If we define
then some morphological parameters can be calculated by means of the C-F coefficients:
Conclusions
Cheblet bases for galaxy modeling: a new tool for surveys
Not only galaxies but also arcs:
Original
Model
Residual
Cheblet bases for galaxy modeling: a new tool for surveys
Outline
Mathematical background
Examples
Visualization
Motivation
• Cheblet bases allow us to efficiently reproduce the morphology of the galaxies and measure their photometry.
Conclusions
• Cheblet bases have proved to be a highly reliable method to analyze galaxy images, with better results than GALFIT and shapelet techniques.
Applications
• PSF deconvolution is easily implemented due to the bases linearity.
• Different morphological parameters can be directly inferred from Cheblet coefficients, with great accuracy.
• Not only single image processing is possible, but also cluster images, just overlapping grids with origin on the different object centers.