All Figures

$\begin{figure} \par\includegraphics[width=14.2cm,clip]{4794f1.eps} \end{figure}$ Figure 1: The information used in the ellipticity determination, for shapelet expansion to order N=8. On the left, the Cartesian shapelet coefficients that are fitted to describe a source and its associated PSF. On the right, the same information has been rearranged into a polar shapelet expansion (the two may be transformed into one another by appropriate mixing of the terms at order $N\equiv a+b$ ). The heavy line indicates the polar shapelets from which the ellipticity is estimated by a fit to Eq. (16). Note the safety margin at order N-1 and N, and the fact that only azimuthal orders between -2 and +2 are fitted.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4794f2.eps} \end{figure}$ Figure 2: Empirical calibration of the post-linear correction to the measurement of e. Top left: fractional error on derived 1st-order ellipticity e for a range of different scale parameters and Sersic indices. Top right: fractional error of the corrected ellipticity. Bottom left: the coverage of the c₂/c₀, c₄/c₀plane by the models. The four groups of points correspond, from top to bottom, to Sersic index 4, 3, 2 and 1. The horizontal spread is mostly a consequence of using different scale radii $\beta$ . Bottom Right: residuals on 1st-order ellipticity vs. the linear combination of c'/c₀=(0.085c₂+0.63c₄)/c₀ for input ellipticity 0.4, showing that this parameter drives the scatter.
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In the text

$\begin{figure} \par\includegraphics[width=6cm,clip]{4794f3.eps} \end{figure}$ Figure 3: Different shapelet fits to a Moffat PSF $(1+\alpha r^2)^{-2}$ with FWHM of 12 pixels. Top panel: the solid line shows a cross-section of an 8th-order fit, using the dispersion of the best-fit Gaussian as scale radius. The dotted line shows the corresponding fit for a scale radius that is a factor of 1.3 times larger. The lower panel shows the PSF multiplied by radius³, in order to accentuate the residuals in the wings. Dots are the actual (monte-carlo realized) pixelated PSFs used in the simulations of Sect. 4. Note how the increased scale radius makes for a much better fit in the outer regions, without a serious degradation in the core of the PSF.
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In the text

$\begin{figure} \par\includegraphics[width=6.5cm,clip]{4794f4.eps} \end{figure}$ Figure 4: Illustration of the effect of the choice of scale radius on the ellipticity measurement. Each curve shows, for a different galaxy profile, how the derived e depends on the choice of scale radius (expressed as a multiple of the dispersion of the best-fitting round Gaussian). Each model galaxy had an effective radius of 4 pixels and ellipticity 0.3, and was convolved with a PSF of Moffat index 2 and FWHM 8 pixels. The $\beta$ for convolved source and PSF are both scaled by the same factor.
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In the text

$\begin{figure} \par\includegraphics[width=14cm,clip]{4794f5.eps} \end{figure}$ Figure 5: Fractional error in recovering a 10% shear, using round PSFs. Each panel represents a different Moffat PSF; the rightmost panels are very nearly Gaussian. The simulated galaxies have effective radii of 4 pixels. Top row: 8th-order shapelets; bottom row: 12th-order shapelets.
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In the text

$\begin{figure} \par\includegraphics[width=14cm,clip]{4794f6.eps}\end{figure}$ Figure 6: Residual shear after correction for an elliptical PSF (the same PSFs as Fig. 5, sheared by 10%).
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In the text

$\begin{figure} \par\includegraphics[width=13.8cm,clip]{4794f7.eps}\end{figure}$ Figure 7: Residual shear after correction for a lopsided PSF (the same PSFs as Fig. 5, but a third of their flux is displaced by (2/3) FWHM).
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In the text

$\begin{figure} \par\includegraphics[width=13.8cm,clip]{4794f8.eps}\end{figure}$ Figure 8: The result of Monte-Carlo simulations, in which many noise realizations of Sersic profile galaxies were run through the ellipticity-fitting procedure described in this Paper. Shapelet order N=8 was used throughout. Each plotted dot represents the average ellipticity of 2500 different noise realizations of the same galaxy image. The same data are plotted in both panels, but coded by different model parameters: the Sersic index on the left, and the PSF size on the right. The vertical axis shows the fractional scatter of the measured fluxes, $\sigma (F)/F$ , of the sources, as determined by integrating their shapelet series. No trend of the mean shear with S/N is seen: noise leads to scatter but no bias.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4794f9.eps}\end{figure}$ Figure 9: Comparison between the scatter in ellipticity measurements from sets of 2500 random noise realizations, and the error predicted by propagating the pixel noise through the calculations.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{4794f10.eps} \end{figure}$ Figure 10: Results from the STEP1 simulations, which are based on modeling of the optics of the CFHT. Each plotted point represents the average ellipticity of about 2200 sources in one STEP1 image. Shapelets to order $N_{\rm max}=8$ were used. PSFs 0 to 5 are, respectively, round, with coma, with astigmatism, with defocus, and with 3rd and 4th-order astigmatism, and results for images with applied shears of 0, 0.05 and 0.1 are shown as three clusters of points in each panel. The correction factor $1-\langle {e}^2\rangle$ is about 0.93 in all cases. Results from cluster to cluster are not statistically independent, but within each cluster of points they are.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{4794f11.eps} \end{figure}$ Figure 11: As Fig. 10, but now the median ellipticity is used as shear estimator. The STEP1 galaxy ellipticity distribution is very peaked, which makes the median a very efficient estimator of the center of the distribution.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4794f12a.eps}\par\bigskip \includegraphics[width=8.8cm,clip]{4794f12b.eps}\end{figure}$ Figure 12: The recovered shear for the STEP1 simulations for PSF0 and input shears of 0, 0.05 and 0.1, split up by source brightness ( top) and size ( bottom). In both cases there is a systematic, so far unexplained trend. The upper panel in each plot shows the ellipticity dispersion correction factor derived for and applied to each bin.
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In the text

$\begin{figure} \par\includegraphics[width=14cm,clip]{4794f5.eps} \end{figure}$	Figure 5: Fractional error in recovering a 10% shear, using round PSFs. Each panel represents a different Moffat PSF; the rightmost panels are very nearly Gaussian. The simulated galaxies have effective radii of 4 pixels. Top row: 8th-order shapelets; bottom row: 12th-order shapelets.
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$\begin{figure} \par\includegraphics[width=14cm,clip]{4794f6.eps}\end{figure}$	Figure 6: Residual shear after correction for an elliptical PSF (the same PSFs as Fig. 5, sheared by 10%).
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$\begin{figure} \par\includegraphics[width=13.8cm,clip]{4794f7.eps}\end{figure}$	Figure 7: Residual shear after correction for a lopsided PSF (the same PSFs as Fig. 5, but a third of their flux is displaced by (2/3) FWHM).
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4794f9.eps}\end{figure}$	Figure 9: Comparison between the scatter in ellipticity measurements from sets of 2500 random noise realizations, and the error predicted by propagating the pixel noise through the calculations.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{4794f11.eps} \end{figure}$	Figure 11: As Fig. 10, but now the median ellipticity is used as shear estimator. The STEP1 galaxy ellipticity distribution is very peaked, which makes the median a very efficient estimator of the center of the distribution.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4794f12a.eps}\par\bigskip \includegraphics[width=8.8cm,clip]{4794f12b.eps}\end{figure}$	Figure 12: The recovered shear for the STEP1 simulations for PSF0 and input shears of 0, 0.05 and 0.1, split up by source brightness ( top) and size ( bottom). In both cases there is a systematic, so far unexplained trend. The upper panel in each plot shows the ellipticity dispersion correction factor derived for and applied to each bin.
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