1. Cheblet bases for galaxy modeling: a new tool for surveys Yolanda Jiménez Teja Txitxo Benítez Lozano Instituto de Astrofísica de Andalucía (CSIC) Departamento de Astronomía Extragaláctica E-mail: yojite@iaa.es

2. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Visualization  Chebyshev-Fourier basis  Non-vanishing wings Practical implementation  From the continuous to the discrete domain  Choice of the scale size  Elliptical and irregular galaxies  Spiral galaxies Examples: coefficients and partial reconstruction Practical applications Mathematical background Examples Applications Visualization  Choice of the number of coefficients Motivation

3. 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

4. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Examples Visualization Motivation Chebyshev polynomial: Chebyshev rational function: Applications Conclusions

5. Cheblet bases for galaxy modeling: a new tool for surveys The Cheblet polar basis is separable in r and θ: Chebyshev rational functions in rFourier series in θ, with Outline Mathematical background Examples Visualization Motivation Applications Conclusions

6. Cheblet bases for galaxy modeling: a new tool for surveys A smooth function f can be decomposed into where These coefficients show an algebraic decay rate: where p is related to the smoothness of the function f. Outline Mathematical background Practical implementation Examples Visualization It is a basis of the Hilbert space, with Motivation Applications Conclusions

7. 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

8. 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

9. 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: SèrsicExponentialGaussian Moffat-Lorentzian Cheblet basis Shapelets Outline Mathematical background Examples Visualization C-F basis Wings Motivation Applications Conclusions

10. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Examples Visualization Motivation Applications De Vaucouleurs Sersic Conclusions

11. Cheblet bases for galaxy modeling: a new tool for surveys G S C

12. Outline Mathematical background Examples Visualization Motivation Applications Radial flux Radial profile Conclusions

13. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Examples Visualization Motivation + = = Applications PSF deconvolution Conclusions

14. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Examples Visualization Motivation Applications Conclusions

15. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Examples Visualization Motivation Applications Conclusions

16. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Examples Visualization Motivation Applications Conclusions

17. 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

18. Cheblet bases for galaxy modeling: a new tool for surveys ABELL1703 (F850) (arXiv:1004.4660)

19. Cheblet bases for galaxy modeling: a new tool for surveys ABELL1703 (F435) (arXiv:1004.4660)

20. Cheblet bases for galaxy modeling: a new tool for surveys ABELL1703 (F435) (arXiv:1004.4660)

21. 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 4 th step for each filter. Clusters pipeline Applications Conclusions

22. Cheblet bases for galaxy modeling: a new tool for surveys Outline Mathematical background Practical implementation Examples Visualization Motivation Applications Object shape measurement Flux: Centroid: Rms radius: Ellipticity: If we define then some morphological parameters can be calculated by means of the C-F coefficients: Conclusions

23. Cheblet bases for galaxy modeling: a new tool for surveys Not only galaxies but also arcs: Original Model Residual

24. 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.

25. Cheblet bases for galaxy modeling: a new tool for surveys C-F basis gets a efficient recover of the total light flux of the galaxies, due to the non-vanishing wings of the basis functions. C-F basis properly models objects with elliptical, irregular and spiral shape. C-F basis is highly compact, in the sense that not too many coefficients are needed to efficiently recompose the total light flux of an object and its morphology. The only input the user must introduce is the number of coefficients to be evaluated; the other parameters (the number of sources present in the image, the centre, the scale size,…) are automatically calculated by the algorithm. The software is able to analyze images with neighbouring objects, whenever they can be detected by SExtractor. Outline Mathematical background Practical implementation Examples Conclusions Visualization Motivation