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Shapelets - Caltech Astronomy - California Institute of Technology

1. 29 2 4 3 The decomp structure 22a FRE Etats es 31 2 5 Analysing all objects in an image 31 2 5 1 The shex pro routine 226 tc 2 0 2 1 Be ado TU ae lu A ne 32 2 5 2 The shapecat catalogue 44 234 4454444440 844 5 35 2 5 3 Keep only good objects ar mes Kal bale At 36 3 PSF correction 37 3 1 How to correct for the PSE na a Pele Se Sale 37 3 2 Correcting for the PSF with shapelets 2 2 22 2222 38 3 2 1 Convolution formalism 2 2 3234 2 2a dent ba 38 3 2 2 Deconvolution formalism and method 40 3 2 3 Deconvolution in practice with shapelets 41 3 2 4 Recompose an object deconvolved from the PSF 42 3 3 Is the PSF correction reliable 24 2 64 qu want an Es 42 3 3 1 The two point correlation function of the PSF 42 3 3 2 Looking at objects years aaa pts Bik eee 44 3 3 3 Limits of deconvolution diagnosis 44 4 Interfacing shapelets 45 4 1 The input output programs how to read and write data 45 4 1 1 Read in a fits image aoaaa o 45 4 1 2 Read in a SExtractor catalogue 46 4 1 3 Write a decomp or a shapecat catalogue 47 4 1 4 Read in ashapecat catalogue 47 4 1 5 Obtain a decomp structure from a shapecat catalogue 48 4 2 Plone POUT INES re an wig wae ee de GT ie 48 4 2 1 Draw an image shapelets plot image 48 4 2 2 Statistics of an
2. shapelets_plot_pstamp pstamp options with the arguments e pstamp is the postage stamp to be plotted e OFRAME allows to plot pstamp labeled with the object frame coordinates otherwise it is labeled with the pstamp frame coordi nates e MASK allows to plot contours of masked out regions e NOISE allows to plot the noise e TITLE title is the title of the plot e CRAN cran is the range for color default value is min image max image e CSIZE csize is the size of the color bar e POS pos is identical to an IDL p region variable and allows to specify the position of the plot on a multiplot image e INVERSE allows one to invert the color scale This plotting routine is automatically used when using the PLOTIT keyword in the shapelets_sexcat2pstamp routine Figure 2 1 shows the output of the routine 4 2 PLOTTING ROUTINES ol 4 2 4 Plot the focus route shapelets_plot_focus Looking at the route taken through possible and Nmax during the focus process to obtain shapelets decomposition is important to check how well the focus worked and to which extend it is reliable The shapelets_plot_focus routine achieves this task called as shapelets_plot_focus focus pstamp options with the following arguments focus is the focus structure to be examined e pstamp is its corresponding postage stamp e DECOMP decomp is the corresponding final decomp structure e RECOMP recomp is th
3. shapelets_wl_pipeline As automatically as possible do everything needed to create a PSF corrected shapelet catalogue from an image to measure weak lensing this is is not exactly the pipeline described in Volume II shex Having located objects within an image using SExtractor this routine now decomposes all of the objects into shapelets It uses the shapelets focus suite of routines to optimise the nmax beta and centroid parameters All of the decompositions are written to disc in a shapelets catalgoue This can be later read to memory using shapelets_read_sexcat pro A 8 Plotting routines plotting extract_pixsc extract the value of PIXEL_SCALE from SExtractor configuration files plt_gals plot various figures relative to the galaxies in a catalog plt_gals_shcat plot various figures relative to the galaxies in a shapelet catalog shcat plt_objs plot the results of the focussing of shapelets on an object The history of the search of optimal shapelet parameters is also shown plt_scat_mom plot statistics for the scat_mom catalog In particular compare derived moments from the image to the input catalog plt_sgsep plot the various figures relative to the star galaxy separation for a cat alog of objetcs plt_shear plot shear estimator statistics plt_shear_scat plot shear statistics derived from sextractor catalog This is par ticularly useful if the sextractor catalog is is scat_in input catalog of a s
4. basis functions and output the coefficients in an IDL structure shapelets_hermite Compute the polynomial coefficients for H x the 1D Her mite polynomial of order n shapelets_make_ls_matrix Create an array with which to compute the least squares linear algebra fit of shapelets basis functions to data Can also use this array to perform the overlap integrals calculation shapelets_make_nvec Set up look up tables to specify which number in a vector of shapelet coefficients corresponds to which n and na or which n and nj shapelets make xarr Make arrays containing the values of the x x1 and y x2 coordinates of a grid This is convenient for evaluating 2D functions e g shapelet basis functions on a grid A 3 DECOMPOSITION AND FOCUS DECOMP 61 shapelets_phi Compute dimensionless Cartesian shapelet basis functions amp x in 1D or 2D based on Hermite polynomials Dimensionful basis functions can be calculated with shapelets_phi n x1 beta x2 beta beta shapelets_recomp Compute the recomposed image corresponding to a set of shapelet coefficients calculated using shapelets_decomp pro The input decomp structure also contains a few meta parameters including the shapelet scale size beta and whether or not the basis functions should be integrated within pixels or merely evaluated at the centre of each pixel A 3 1 Focus routines focus shapelets focus Find the optimal 5 Nmax and centroid for a shape
5. rms is strictly positive If NOISE_MAP is not set we do a local estimation of the noise we compute the noise without taking the SExtractor noise estimation into account Here one can choose between two different methods to compute the noise If VARY LOCAL_NOISE_CALC is set we iterate on the estimation of the mean the mode and the rms of the background until the variation of the rms between two iterations is less than 1 When this is the case we create the noise map by giving each of its pixels the value 1 noise_local 5 2 2 back_rms_local Note that this noise map is an inverse variance of the background map 2see section 4 1 3the mode is computed as being mode 2 5 median 1 5 mean 2 2 ISOLATE THE OBJECT TO DECOMPOSE 21 If VARY_LOCAL_NOISE_CALC is not set we first extract an area larger than the postage stamp in order to have more background pixels which improves the statistics and thus the estimation We then grow the region of interest to mask nearby pixels The mean and rms of the background are finally computed if there are enough background pixels no fatal default and not too much zero pixels in the postage stamp using the sky pro routine which computes the sky level in an image and is available in the astrolib package 17 Here again we create the noise map by giving each of its pixels the value 1 noise local 5 2 3 back_rms_local In equations 2 2 and 2 3 the notat
6. An introduction to shapelets based weak lensing image processing Volume I Shapelets Shapelets package version 2 2 Manual for users Version 2 0 Joel Berg Richard Massey amp Alexandre R fr gier il An introduction to shapelets based weak lensing image processing Volume I Shapelets JOEL BERGE RICHARD MASSEY ALEXANDRE REFREGIER 2006 OO Joel Berg CEA Saclay Service d Astrophysique Orme des Merisiers Bat 709 91191 Gif sur Yvette FRANCE e mail joel bergeQ cea fr OO Richard Massey California Institute of Technology MC 105 24 1200 East California Boulevard Pasadena CA 91125 USA e mail rjm astro caltech edu OO Alexandre Refregier CEA Saclay Service d Astrophysique Orme des Merisiers Bat 709 91191 Gif sur Yvette FRANCE e mail refregier cea fr iv Foreword This manuscript is an updated version of the first Introduction to shapelets based weak lensing image processing For convenience the original manual has been split into two volumes The current Volume I deals with and only with shapelets formalism and the publicly available shapelets software the upcoming Volume II deals with the whole method based upon shapelets that we use in weak lensing data analysis particularly PSF modelling This volume can still be considered as twofold a brief overview of the shapelets formalism gives their most useful features then most of the shapelets software s
7. Nmax Mmax be maX Nmax Mmax 3 11 Using these rules we can either convolve or deconvolve every function in an analytic way In astronomy we are interested in deconvolving the PSF from the image thus the convolution formalism will be mostly used for deconvolution 3 2 2 Deconvolution formalism and method As evoked above deconvolution is needed to clear smeared galaxies described by the function h in the above notations from the PSF described by the function g In other words it is aimed at recovering the coefficients fm from the measured coefficients hy of galaxies and the interpolated coefficients g of the PSF at the position of each galaxy For this purpose it is convenient to rewrite equation 3 1 as he Pi 3 12 where Pam gt Cnmig is the PSF matrix As briefly noted in chapter 1 section 1 5 and well explained in 27 this matrix is invertible under some hypothesis There fore equation 3 12 can be directly inverted to obtain the coefficients of unsmeared galaxies a Phn 3 13 This method is intuitive and gives an estimate of the low order coefficients of unconvolved galaxy coefficients But it is not satisfactory because it restricts the deconvolution to low order coefficients whereas we need to know the biggest amount information available to eventually analyze unsmeared coefficients and because the PSF matrix may be not inversible Consequently as already said in section 3 1 we pr
8. S faB p 1 3 where the shapelet coefficient is given by pe de f z Bu e 8 1 4 The series converges quickly if the object is sufficiently localised and if 5 and the origin x 0 are close enough of the size and of the center of the object In such a case a decomposition onto the first few basis functions suffices to catch the shape information of the object of interest and its decomposition into shapelets can be truncated to some maximum order of decomposition denoted Nmax hereafter That is Nmax f z gt FaBn amp 8 1 5 is a good description of the object The functions n and B have many interesting and useful properties we refer the reader to 26 for details Let us just note their invariance up to a rescaling under Fourier transform amp n k i amp n k amp n being the Fourier transform of A very useful property about the integration of B must be pointed out the shapelets basis functions obey the integral property 1 2 CARTESIAN SHAPELETS 7 Figure 1 1 a First few one dimensional basis functions n x b First few two dimen sional basis functions On n The figure is from 26 J i dzB 2 8 2 Ve c2 1 6 co where C2 is the binomial coefficient note that this is true only for even n One can notice an analogy between shapelets and the quantum harmonic oscil lator shapelets basis functions are QHO s eigenstates Hence we can transpose quantum mechani
9. shapelets_plot_image The first compulsory task an image processer must be able to achieve is to plot the image he she wants to analyze This can be done through the shapelets_plot_image routine it is aimed at drawing whatever 2 dimensional pixelated array including astronomical images It is simply called as shapelets_plot_image image options using the arguments and options e image is an image structure as given by shapelets_read_image or a 2D array e FRAME frame is the frame to give to the image note that it can be independent of the image if it is set as FRAME it is the real frame of the image e COLBAR or CBAR allow to draw a color bar above the image e CRANGE crange is the range for color default value is min image max image e CLOG allows to draw a logarithmic color scale e CSIZE csize is the size of the color bar e TITLE title is the title of the plot 4 2 PLOTTING ROUTINES 49 Q 2 Figure 4 1 A very simple array as plotted by shapelets_plot_image 10 13 20 e XTITLE xtitle YTITLE ytitle is the title of the x y coordi nates of the image e CTITLE ctitle is the title of the color bar e INVERSE allows to inverse the color coding e SCALABLE allows to use scalable pixels e NOERASE allows to keep everything was previously drawn in the plotting window otherwise everything is erased before plotting the image e POS pos is identical to an IDL p region v
10. 1 b that as n n na increases with fixed B the cartesian Bn basis functions acquire both a larger extend and smaller scale oscillations The same trend is visible for polar Xn m basis functions for increasing n m as shown by figure 1 3 That implies that a decomposition into shapelets done with certain and Nmax will describe only objects of size within a certain range One can show that the features described by such a decomposition are of size ranging between 26 Omin x B Nmax 1 2 and Onax x BUVimax 1 2 1 28 This can be applied to choose the optimum 9 and nmax for a shapelets decom position as will be described in Volume II 1 5 Convolution and deconvolution Let f and g be two functions and A their convolution product a 8 and y are their respective characteristic scales The convolution product reads lee l defa x g x 1 29 One can link the coefficients of the shapelets decomposition of the three functions thanks to a convolution tensor Cm function of the characteristic scales a 3 and y hn gt gt Camifmg 1 30 m l It can be shown that the tensor Chm depends on the shapelets basis functions By Bm and B So the shapelets formalism allows one to analytically evaluate the convolution product of two functions by decomposing them into shapelets As a consequence it is possible to implement a PSF deconvolution method en tirely based upon shapelets formalism Assume that one knows
11. 4 1 1 Read in a fits image The first step for a shapelets analysis is to read the fits file containing the image data and create an image structure with which shapelets can deal This work is done by shapelets_read_image which also reads in the mask image composed of 0 and 1 whether or not a mask exists or not It can also read in the segmentation map created by SExtractor if requested and read in the SExtractor s noise map or estimate the noise by itself shapelets_read_image image file name options It uses the arguments and options e image is the output image structure e FULL_PATH full_path is the full path of filename e FITS_READ is to be set to use the fits_read pro procedure instead of the readfits pro one for reading the fits file e PIXEL SCALE pixel scale is the scale of pixel in arcsec the de fault value is the one stored in the header of the fits file if available e UNITS units is the units of the image the default value is the one stored in the header of the fits file if available 45 46 CHAPTER 4 INTERFACING SHAPELETS e PHOTO_ZP photo_zp is the photometric zero point for the image the default value is the one stored in the header of the fits file if available e EXPOSURE_TIME exposure_time is the exposure time the de fault value is the one stored in the header of the fits file if available e NO_MASK allows not to read in the mask image to save memory and or CPU e
12. NO SEGMENTATION allows not to read in the SExtractor s seg mentation map e NO_NOISE allows not to read in the SExtractot s noise map e ESTIMATE_NOISE allows to estimate the noise in the image e NOISE_LEVEL noise_level is an optional output containing an estimate of the rms of the background noise e N_GROW n grow is the number of times SExtractor image is grown to mask objects during noise estimation e SKY_SUBTRACT allows to subtract the sky level e SILENT makes the routine operate silently 4 1 2 Read in a SExtractor catalogue SExtractor outputs a catalogue containing astrometric and photometric parameters of detected stars and galaxies It is described in Volume II The eventual shapelets analysis relies on this catalogue The interface with it is done by shapelets_read_sexcat shapelets_read_sexcat sexcat file name options The routine outputs sercat a SExtractor catalogue structure The options are e FULLPATH if set the routine accepts the file name as is can be absolute or relative The default behaviour is to prepend a path name from shapelets_paths pro e ASCII if set the routine assumes the catalogue is in ASCII for mat e COMMENT comment is an optionally output comment on each field it gives the significance of each returned SExtractor s param eter e UNIT unit is the unit of each returned SExtractor s parameter 4 1 THE INPUT OUTPUT PROGRAMS HOW TO READ AND
13. WRITE DATA 47 e XBUGFIXED seemingly a bug often occurs in SExtractor the x coordinate are increased by 1 The default behaviour of the routine is to take this bug into account y subtracting 1 to SExtractor s x coordinates XBUGFIXED must be set if the bug does not occur e SILENT if set the routine operates silently 4 1 3 Write a decomp or a shapecat catalogue We saw that shapelets_decomp and shex output shapelets structures To achieve this they use the shapelets_write routine aimed at writing shapelet coefficients catalogues to disk As it is just a blind routine it is pointless to describe it in details here 4 1 4 Read in a shapecat catalogue Once a shapelets analysis has been made on an image and a shapecat catalogue written to disk containing shapelets parameters of each object in the image one must be able to read it so as to use it The shapelets_read_shapecat is aimed at doing so Futhermore it computes the moments of each object if requested The user can call it as shapelets_read_shapecat shapecat file name options with the following arguments and options e shapecat is the output shapecat catalogue e CARTESIAN converts all objects to cartesian coefficients e POLAR converts all objects to polar coefficients e N MAX n_ max if it set all objects are truncated at this nmax i e only coefficients corresponding to lower Nmax are considered e MOMENTS allows to compute obj
14. achieved for some reason These objects are thus not reliable for futher analysis and must be rejected The eas iest way to do that is to use the trim_failures option in shex otherwise one can still eliminate them after the shapecat catalogue has been created using the shapelets_split pro routine This routine splits any kind of catalogue so as to keep only a selection of objects this selection being entered as an argument of the routine Chapter 3 PSF correction One of the predominant systematics in weak lensing measurements is brought by the Point Spread Function PSF due typically to the atmosphere and to the response of the telescope Keeping in mind that cosmic shear effects are of order 1 2 and that the PSF can be of order 6 7 it becomes obvious that correcting for it is crucial A prior to the correction is to model the variations of the PSF across the image i e on each galaxy as shown in Volume II It then becomes possible to correct for the PSF 3 1 How to correct for the PSF Since the PSF is just a convolved response to the image a simple deconvolution suffices to get rid of its effects Theoritically the problem is very easy but it is less evident from a numerical point of view Indeed as seen earlier chapter 1 section 1 5 one has to invert a matrix to compute a deconvolution this is real not only for the shapelets case but also for every method implying a deconvolution and the problem becomes numer
15. arguments here just let us say that they are sensibly identical to those of shapelets_focus see below Its code is sufficiently commented so as one can easily use it if needed 2 3 3 Focus Nmax The shapelets_focus_nmax pro allows one to find the best maximum order of decom position Nmax for the shapelets decomposition assuming that the scale parameter B is known It looks for the best Nmax by exploring a range of possible Nmax un til the criterion x 1 is reached e is a tolerance value it must negligible compared to 1 The range of checked Nmax depending on the size of the postage stamp we can compute it and restrict the search for Nmax at this range Then we try several allowed nmax starting with the lowest and incrementing it of 2 for each step The x parameter is computed by the shapelets_decomp pro routine and compared to 1 Comparing the two neighbors Nmax we can decide which one gives the best result If they are the same we define the good Nmax as the mean of the two possible Nmax Iterating within the range of allowed nmax the routine finally outputs the focussed Nmax stored in a focus structure The history of the search is also kept in memory Note that this focus structure is the same as the one output by shapelets_focus_beta pro but updated with the focussed nmax As in practice shapelets focus nmax pro is never directly called by the user it is not worth detailing its arguments here just let us say that th
16. chapter 4 describes how we do visualization when dealing with shapelets We present a whole weak lensing measurement method based upon the shapelets formalism from detection of galaxies in an image to the creation of a mass map in Volume II INTRODUCTION Chapter 1 Shapelets formalism For all this chapter we refer the reader to Refregier 2003 26 Refregier amp Bacon 2003 27 Massey amp Refregier 2006 20 Massey Refregier amp Bacon 2004 22 Shapelets are new mathematical entities 26 27 22 aimed at image analysis They represent a complete orthogonal set of basis functions onto one can efficiently decompose every object such as galaxies for example Although the shapelets formalism because of its richness goes beyond the the aim of this manual it is necessary to get an overview of its basics This is the intend of the present section 1 1 Generalities People who work on image processing may be familiar with wavelets appeared in the mid 1980s Since they allows one to decompose an object or an image on several scales they are a performant tool to prospect all scales on an image But they fail in describing the real shape of an object Instead shapelets do not decompose objects on different scales but on different elementary shapes More precisely the shapelets decomposition is a linear decom position into a series of localised basis functions with different shapes the shapelets Two different types of s
17. comp structure shapelets_centroid Computes the centroid from a linear summation of shapelet coefficients read in from a Cartesian decomp structure shapelets_concentration Returns concentration morphology index of a Carte sian decomp structure shapelets_ellipticity Calculates the ellipticity of a shapelet model shapelets_flux Returns total flux of a Cartesian decomp or shapecat structure This does the same as shapelets_moments pro but in function form 70 APPENDIX A ROUTINES OF THE SHAPELETS PACKAGE shapelets_image_moments Compute the zeroth flux and first centroid mo ment of an image shapelets_moments Compute the zeroth flux and first centroid moment for a basis decomposition Also computes the characteristic order n and scale parameter Br shapelets_profile Compute the azymuthally averaged profile f r of an object from its shapelet coefficients shapelets_quadrupole Calculates the unweighted quadrupole moments of a shapelet model shapelets_rsquared Computes the R size measure from a linear summation of shapelet coefficients read in from a Cartesian decomp structure Bibliography 10 11 12 13 14 15 16 17 18 19 20 21 Bacon D Refregier A amp Ellis R 2000 MNRAS 318 625 Bartelmann M amp Schneider P Physics Report 340 2001 291 472 Bekenstein J 2004 Physics ReviewD 70h3509B Bernstein G amp Jarvis M 2002 AJ 123 583 Bertin E SExtractor User s
18. gravitational potential consequently given a mass greater than the one astronomers see dark matter paradigm or given a different gravitational potential than the one General Relativity predicts for the mass we see the effects on local space time will be the same Hence weak lensing must be detectable in both theories It could even maybe allow astronomers to discriminate against concurrent theories if someday one detects a dark clump i e a halo only composed of dark matter with no luminous counterpart then dark matter would become more strongly accepted Or if weak lensing properties its power spectrum for example could be different on both theories one could expect to use it in order to confirm a 2 INTRODUCTION theory In the framework of a theory of gravitation weak lensing is due to the effect of a gravitational potential on its neighborhood Indeed according to the duality between matter energy and space pointed out by Einstein a gravitational potential curves space time Consequently geodesics are no longer straight lines but they are curved by gravitational wells glued to space time hence as all kind of matter travelling through space follow geodesics this stands for photons their trip is not straight but perturbed by all the gravitational wells they encounter from their emission to their detection by an observer Each geodesic is unique and so two light rays emitted by two different parts of a same ga
19. is set all objects will be decom posed using this scale in that case there is no focus on e CHISQ TARGET chisq_target is the ideal value of x for the residual image The default value is 1 0 e CHISQ TOLERANCE chisq_ tolerance is the acceptable accuracy for x default value is 1 0 e CHISQ FLATNESS chisq flatness is the minimum difference in x between two decompositions with nmax differing by two to trigger the flatness constraint in shapelets_focus_nmax e THETA MIN GEOM theta min geom is the minimum scale on which it is possible for the image to contain data default value is 0 2 pixels e FULL_FOCUS if it set all attempts made at decomposition dur ing iteration are recorded e GAUSSIAN_RECENTRING makes the basis functions recentered on the previous decomposition s centre of light e N_MIN n_min is the minimum value allowed for nmax for decom position default value is 2 e N_MAX n_max is the maximum value allowed for nmax for de composition default value is 20 e NOISE_MAP if it is set the noise map supplied in the postage stamp is used otherwise the noise is determined locally e SEG MAP if it is set the segmentation map is determined locally otherwise the segmentation map supplied in the image structure is used 34 CHAPTER 2 SHAPELETS DECOMPOSITION TOO_BIG too_big is the maximum radius of postage stamp default value is 100 pixels Objects that are bigger than
20. m both even or odd are allowed Tr Z eue Cr u Im 2 yim Be e 28 imd 1 20 The real and imaginary parts of the first few polar shapelets basis functions are shown on Figure 1 3 The basis functions with m 0 are wholly real As polar shapelets basis functions are complex number the shapelets coefficients fnm are complex number Their moduli shown on Figure 1 4 determine the strength and their phase the orientation of a component Like in the case of cartesian shapelets most of the shape information is caught by the first few shapelets basis functions This allows us to truncate the decomposition 1 19 to some maximum order of decomposition Nmax such as Nmax n EOSS 2 a 078 1 21 n 0 m n 1 3 POLAR SHAPELETS 11 Figure 1 3 First few polar shapelets basis functions Xn m Moreover as clearly visible on Figure 1 3 basis functions of which n m verify the condition n m cste all have the same scale of oscillation i e they describe same sized details of the object Instead for some fixed n the scale of oscillation of basis functions decreases with increasing m It then seems natural not to use a simple Nmax truncation as in Equation 1 21 but to use a diamond criterium n m lt Nmax for truncating the shapelets decomposition The shapelets coefficients considered in such a decomposition Nmax Fr 0 y gt nom 0 p 1 22 n 0 m lt nmax n are encl
21. model is not a real test on deconvolution It just allows to diagnose the starting point of the deconvolution that is the goodness of the model of the PSF Furthermore plotting objects after deconvolution is not a really accurate test as we do not know their shape without PSF that is precisely what we aim at obtaining in fine here plotting unconvolved stars seems the best way to proceed they must be Dirac functions Reconvolving objects already deconvolved from the PSF and comparing them to observed objects may be a very good way to look at the precision of the deconvolution unfortunately a same problem could occur both in deconvolution and in reconvolution and vanish when reconvolving a previously deconvolved object As a result even this way to proceed does not seem perfect Knowing the difficulty of these steps deconvolving and diagnosing deconvolution we use some tests which are not as accurate as we would wish but which allow to discard very bad deconvolution However so far they have proved to be precise enough for our purposes Chapter 4 Interfacing shapelets 4 1 The input output programs how to read and write data Most shapelets routines need to read information stored in some catalogues or struc tures and to write results to the disk We have already met such input output pro grams e g shapelets_read_image without explaining what they actually do and how they do it This is the goal of this section
22. polar coefficients this is the default option if POLAR is set e ARGUMENT or PHASE allow to plot the phases of polar coeffi cients COMPOSITE_MA not used TITLE title is the title of the plot e XTITLE xtitle YTITLE ytitle is the title of the x y coordi nates of the image FRAME frame is the frame to give to the image 94 CHAPTER 4 INTERFACING SHAPELETS 0 10 20 30 40 50 Figure 4 4 A typical output of shapelets plot decomp the recomposition of the galaxy shown in Fig 2 1 after being decomposed into shapelets e CROSSHAIRS allows to overlay crosshairs on the centre of the basis functions e INVERSE allows one to invert the color scale The default plot is the recomposed object options allow to plot the shapelets coefficients of its decomposition Note that most of the idl plotting keywords are allowed by shapelets_plot_decomp Figure 4 4 shows a typical output of shapelets_plot_decomp the galaxy of Fig 2 1 has been decomposed into shapelets then recomposed and its recomposition plotted by the shapelets_plot_decomp routine 4 2 6 Gather all plots together shapelets_plot For convenience a routine is able to use each one of the previous one to plot whatever the user wants It is called by shex when the PLOTIT keyword is set and allows to check all useful information on an object profile noise the focus of its shapelets parameters and its decomposition This routine is sha
23. routine called shapelets_geometric_constraints computes 60min given by the user or set to 0 2 pixels by default and Omax from the size of the postage stamp We then define a minimum and a maximum as being Sin practice we always use Nmax 2 as a starting point for focus 24 CHAPTER 2 SHAPELETS DECOMPOSITION Dee x Bmax Una Amin gt min On V nmax 1 bass 2 5 and set 3 to be min max Amin FWHM 2 86 Bmax where FWHM is the FWHM of the object given by SExtractor The guessed centre of the basis functions x9 is set to be the center of the postage stamp Those two operations are made by the routine shapelets_guess_nmaxbeta the arguments of which we do not describe here 2 3 2 Focus and xy The shapelets_focus_beta pro routine allows one to find the best scale parameter and centroid for the shapelets decomposition assuming that the maximum order of decomposition Nmax is known It uses 3 and x as guessed by shapelets_guess_ nmaxbeta unless starting 6 and nmax are given by the user as starting points then it iterates on them with Amoeba till x is minimised keeping all steps in a history structure so as one can check later how the focus has gone It decomposes the object with these best parameters and outputs a focus structure containing notably B o n_max not focussed yet x and the history structure As in practice shapelets_focus_beta pro is never directly called by the user it is not worth detailing its
24. successive iterations can have with very similar x in this case x is flat and it represents a problem in the focus process flag 3 e In general y should be monotonic in Nmax that is when increasing nmax the model should become better and x lower for some objects it is not the case and x becomes worse then flag 5 e amoeba can be unable to converge flag 6 or converge towards a bad x flag 7 Nmax Can reach the maximum Nmax allowed for focus in that case flag 8 e The focus on the center of the basis function can make it wander towards the edge of the postage stamp hence basis functions are pushed off the postage stamp flag 9 e The centre of basis functions can be pushed off the postage stamp or the focus process can simply crash flag 10 Flags and their meaning are listed in the right column of table 2 1 Eventually in all shapelets decomposition all objects with focus flag greater than 8 will be rejected 2 3 6 The focus structure The focus structure is output by shapelets_focus it contains all useful characteristics of the focus Among them 1 the focussed 3 xp and Nmax 2 the firstly guessed o and Nmax 10see chapter 3 section 3 2 28 CHAPTER 2 SHAPELETS DECOMPOSITION postage stamp focus 0 OK OK 1 Nearby object Bounced off geometrical constraints Severe overlapping with nearby Entered the region where
25. the 2 object least squares fitting matrix may be singular 3 Object is near a saturated pixel Converged by flatness limit 4 Object is near a masked region Not used 5 Object is near the edge of the image x is not monotonic in nmax 6 Object is itself masked out Amoeba dithered about and not converged to target y 7 Object has 0 FWHM Focus iteration did not converge to target x 8 Too few background pixels around object Maximum nmax reached 9 Object entirely overlapped by neighbors Centroid wandered pushing basis functions off the edge of the pstamp 10 Routine sexcat2pstamp crashed Fatal crash during focus routine Table 2 1 Meaning of postage stamp and focus flags focus flags with a imply that shapelets may incompletely represent the object 3 the minimised x 4 the history of focus 5 the flag giving which errors might have occurred during the focus process We have just seen that the focus procedure is intimately linked to the decompo sition one The next section describes how to decompose an object into shapelets 2 4 Decompose an object into shapelets Once an object has been isolated in a postage stamp and its GB x and Nmax pa rameters focussed one can decompose it into shapelets This section is aimed at introducing the method of decomposition into shapelets and presenting the routine dedicated to this task 2 4 1 How to decompose into shapelets The decomposition into shapelets is nothing m
26. 3 effect it is compulsory to correct for the PSF very well For the moment this remains the main systematic in weak lensing analyses even if new methods allow to correct for it in satisfactory ways Other systematics exist such as intrinsic correlations between close pairs of galaxies whose physical shapes depend on the density field they inhabit or intrinsic ellipticity shear correlation between distant galaxies but in the same line of sight one being physically sheared and the other one lensed by the same potential Those used to be assumed to be negligeable but this assumption seems to be wrong and one should be very cautious when measuring a cosmic shear effect see 9 11 12 19 for recent references Nowadays several methods are implemented to measure well cosmic shear 4 7 10 14 15 20 26 27 28 29 Weak lensing being a shearing of the image of galaxies it is natural to work on shapes of galaxies Hence a method completely based upon shapes should be ideal to deal with weak lensing This is the original idea of the shapelets formalism which relies on the decomposition of the image of galaxies into elementary shapes Our intend here is to introduce the shapelets formalism and method The first chapter gives some useful features of the shapelets formalism Chapter 2 presents the method and the IDL pipeline for decomposing a galaxy into shapelets version 2 2 21 chapter 3 explains how to correct for a PSF Finally
27. C7 cond This function computes the condition number of an N by N array congrid Shrink or expand the size of an array by an arbitrary amount convolve Convolution of an image with a Point Spread Function PSF daycnv Converts Julian dates to Gregorian calendar dates delvarx Delete variables for memory management can call from routines determ This function computes the determinant of an N by N array factorial This function computes the factorial N as the double precision product N N 1 N 2 8 aR A fdecomp Routine to decompose a file name for any operating system fits_close Close a FITS data file 64 APPENDIX A ROUTINES OF THE SHAPELETS PACKAGE fits_help To print a summary of the primary data units and extensions in a FITS file fits_info Provide information about the contents of a FITS file fits_open Opens a FITS Flexible Image Transport System data file fits_ read To read a FITS file fits_write To write a FITS primary data unit or extension fxaddpar Add or modify a parameter in a FITS header array fxmove Skip a specified number of extensions in a FITS file fxpar Obtain the value of a parameter in a FITS header fxposit Return the unit number of a FITS file positioned at specified extension gamma Computes the Gamma function gauss2dfit Fit a2 dimensional elliptical gaussian equation to rectilinearly gridded data get_date Return the current UTC date in CCYY MM DD f
28. Carry out a Principal Components Analysis Karhunen Loeve Transform plothist Plot the histogram of an array with the corresponding abcissa plt_ellipse plot one or several ellipse centered at x y and with major and minor axes a and b respectively and with position angle pa plt_rectangle Draws a rectangle on the current output device randome Generates a random number or nxmxl array of from a rescaled Epanech nikov kernel K x 0 75 1 x for x lt 1 K x 0 elsewhere readcol Read a free format ASCII file with columns of data into IDL vectors remchar Remove all appearances of character char from string st repchr Replace all occurrences of one character with another in a text string reverse Reverse the order of rows or columns in an array or vector sign Compute the sign x of a variable x with sign x 0 1 1 for x 0 0 j0 respectively The variable x can have any dimension sky Determine the sky level in an image using the the procedure MMM spec_dir Complete a file specification by appending the default disk or directory stddev This function computes the stddev of an N element vector strnumber Function to determine if a string is a valid numeric value sxaddpar Add or modify a parameter in a FITS header array sxdelpar Procedure to delete a keyword parameter s from a FITS header 66 APPENDIX A ROUTINES OF THE SHAPELETS PACKAGE sxpar Obtain the value of a parameter in a FITS he
29. EFFICIENTS if it is set the Cartesian shapelet coefficients are plotted POLAR polar if it set one aspect of the polar shapelet coefh cients is plotted depending on the set options 1 Magnitude default or MODULUS option Relative phase ARGUMENT or PHASE option Real part REAL option Imaginary part IMAGINARY option Composite half and half Duce Nr IN ERRORS errors plots on noise depending on the value this key word is given 1 Plot S N of object or coefficients 2 Plots absolute errors NRAN nran sets the n to be considered TOP top if set keep the top largest coefficients NOOVER allows not to oversample the basis functions CRAN cran is the range for color default value is min image max image CLOG allows to draw a logarithmic color scale 4 2 PLOTTING ROUTINES 53 Profile re A E a tt a ak et a 0 Rn 10 00 E I r 0 01 Figure 4 3 The profile of a galaxy The dashed horizontal line represents the noise rms Features of the profile which are in the noise under this line were ignored when fitting the galaxy e CBAR allows to draw a color bar e ISOTROPIC allows to draw a frame for which x and y scale are identical e REAL for polar shapelets the real part of the coefficients is plotted e IMAGINARY for polar shapelets the imaginary part of the co efficients is plotted e COMPOSITE RI not used e MODULUS allows to plot the modulus of
30. FILENAME_OUTPUT filename_output is the name of the out put file containing the shape catalogue by default it is filename e FULL PATH full path optional gives the full path of data files If it is not set the path is assumed to be given by shapelets_paths e INDEX index allows to decompose only the objects given in index must be a scalar or an array e RESTART shape catalogues are periodically written to disk so as to avoid re decomposing thousands of objects in case shex crashed late in its run in that case setting RESTART allows shex to start from the last written to disk decomposition see Volume II 15see chapter 3 2 5 ANALYSING ALL OBJECTS IN AN IMAGE 33 e SAVE_EVERY save_every defines the frequency at which decom position are periodically saved to disk default is every 2000 objects e IMAGE image is an image structure of file name given by shapelets_ read_image if it is not set it is automatically loaded by shex and it can be optionally output if none is entered as input e SEXCAT sexcat is a SExtractor catalogue of file name given by shapelets_read_sexcat if it is not set it is automatically loaded by shex and it can be optionally output if none is entered as input e SHAPECAT shapecat if it is set the shape catalogue is output under this name e BETA_TOLERANCE beta_tolerance is the accuracy with which B is obtained The default value is 107 e FIXED_BETA fixed_beta if it
31. Manual Bertin E amp Arnouts S 1996 A amp AS 117 393 Bridle S Gull S Bardeau S amp Kneib J P 2001 in Scientific NW ed Proceedings of the Yale Cosmology Workshop The Shapes of Galaxies and their Dark Halos Fort B amp Mellier Y 1994 Astron Astrophys Rev 5 239 Heavens A Refregier A amp Heymans C 2000 MNRAS 319 649 Heymans C et al 2006 MNRAS in press astro ph 0506112 Heymans C et al 2006 astro ph 0604001 Hirata C amp Seljak U 2004 Physical ReviewD 70 063526 Holwerda B SExtractor for Dummies available at http www int stsci edu holwerda se html Kaiser N 2000 ApJ 537 555 Kaiser N Squires G amp Broadhurst T 1995 ApJ 449 460 Kaiser N Wilson G amp Luppino GA 2000 Landsman W http idlastro gsfc nasa gov Lupton R 1993 Statistics in theory and practice Princeton University Press Mandelbaum R et al 2006 MNRAS 367 611 Massey R amp Refregier A 2005 MNRAS 363 197 Massey R amp Refregier A Shapelets Web Page http www astro caltech edu rjm shapelets 71 12 BIBLIOGRAPHY 22 Massey R Refregier A amp Bacon D 2004 astro ph 0408458 To be published in 23 24 25 26 27 28 29 30 31 32 33 Impact of Gravitational Lensing on Cosmology IAU Symposium 225 eds Mellier amp Meylan Mellier Y 1999 Annu Rev Astron Astrophys 37 127 Press W Flannery B Teukolsky S amp Vetterling W 2002 Num
32. SKY sky to be set when subtracting the background to be put to 1 to fit it with a constant value around the object 2 to fit it with a plane around the object NONI if set shapelets coefficients with n 1 are forced to be Zero POLAR to be set to use polar shapelets by default cartesian shapelets are used DIAMOND to be set to use the diamond truncation scheme for decomposition e VERBOSE if set the routine operates noisily e SILENT if set the routine operates silently Here decomp is the output decomp structure given by shapelets_decomp pro see section 2 4 for details 8see chapter 3 section 3 2 see chapter 1 section 1 3 2 3 FIND THE OPTIMAL PARAMETERS FOR THE DECOMPOSITION 27 2 3 5 Possible problems and the focus flag Problems can occur both during the focus of and Nmax They are identified with a flag which consists in a number between 0 and 10 Note that the bigger the flag the worse the error If several errors have occurred just the worst is written in the flag Here we list the possible problems together with their representative flag e 3 can bounce off geometrical constraint i e become less than the minimum allowed scale or greater than the maximum one in that case flag 1 e While deconvolving a PSF the least square fitting matrix can become sin gular if the convolved Nmax is lower than the nmax used for focus in that case flag 2 e Two
33. THE SHAPELETS PACKAGE shapelets_polar_reduce Removes the duplicated degenerate coefficients when cartesian shapelets have been converted into polar form and leaves only the minimum number of independent parameters A 4 Inputs Outputs io shapelets_create_decomp Initiliase a brand new decomp structure Works like an IDL_DEFINE procedure but for an anonymous structure which we need because they will shrink or expand to accomodate varying amounts of data shapelets_create_focus Initiliase a brand new decomp structure Works like an IDL_DEFINE procedure but for an anonymous structure which we need because they will shrink or expand to accomodate varying amounts of data shapelets_read_image Read in fits image plus segmentation and inverse vari ance pixel map if available it looks for the same filename but with _objects and _weight added before the extension shapelets_read_psf Creates a shapelet decomp structure representing a PSF this PSF will eventually be the same across one image shapelets_read_sexcat Reads in a SExtractor catalogue shapelets_read_shapecat Reads in a shapelet coefficient catalogue of objects created by shex pro then stores them in a shapecat IDL structure This may then be converted into decomp structures using shapelets_shapecat2decomp pro shapelets_sexcat2pstamp Extract a postage stamp of image data around an object detected by SExtractor data Uses the fact that the inverse variance
34. ader tag_exist To test whether a tag name exists in a structure valid_num Check if a string is a valid number representation writefits Write IDL array and header variables to a disk FITS file xgen Generate a vector containing equally spaced numbers between x1 and x2 This is typically used to generate an x axis vector in a plot zparcheck Routine to check user parameters to a procedure A 6 Shapelets algebra operations shapelets_add Superimposes two images stored as shapelet decompositions append a shapelet decomposition to a shapelet catalogue or concatenates two catalogues shapelets_circularise Circularises objects by setting to zero all of their polar shapelet coefficients where m is nonzero When averaged around all angles the negative parts of these basis states cancel out the positive parts shapelets_convolution_matrix Create an array of which unconvolved basis func tions with scale alpha combine to create a basis function convolved with a decomp structure PSF and scale gamma shapelets_convolve Convolves one image with another e g a PSF shapelets_detect_diamond Detect if a shapelet structure has been created using the diamond shape or not shapelets_dilate Dilates changes the size of an object There are four methods available to perform this operation 1 Simply rescale beta the shapelet size parameter ladde 2 To first order in Cartesian shapelets using a amp a r operato
35. ariable and allows to specify the position of the plot on a multiplot image e ISOTROPIC allows to force scales on the x and y axis to be iden tical note that it is not stable yet Figure 4 1 shows a simple array as plotted by shapelets_plot_image The same routine is involved when plotting postage stamps like figure 2 1 4 2 2 Statistics of an image shapelets_plot_image_statitics Once one possesses an image after drawing and visualizing it one can be interested in its statistics distribution of the intensity in pixels mean and rms of the image statistics of the background 50 CHAPTER 4 INTERFACING SHAPELETS This can be done using shapelets_plot_image_statitics image where image is an image structure The two routines presented above are very general and could be adapted to all kind of astronomical image processing even not using the shapelets formalism But some other routines are typical of the shapelets processing they are presented below 4 2 3 Draw a postage stamp shapelets_plot_pstamp When isolating an object in a postage stamp see section 2 2 one must be able to plot it so as to see what it looks like and to check whether or not its decomposition must be problematic In particular some bad objects can make the whole process crash in such a case being able to draw the incriminated object is of primordial importance This task is done by plotting the object s postage stamp using
36. can originate for instance using X ray wavelengths or the Sunyaev Zel dovich effect That is why one can define weak lensing as a unique probe to detect dark matter halos This dark matter has been expected to exist since the mid 20th century when astronomers noticed a discrepancy between virial and luminosity masses of clusters of galaxies Later they observed that the rotation curves of galaxies were flat and could not be explained through the luminous mass only dark matter halo was added around all galaxies Nowadays some new theories are said to be able to explain these discrepancies without invoking dark matter but just a modified potential The MOND paradigm 3 deals with a modified newtonian dynamics which may act almost the same way as the relativistic potential of General Relativity but without dark matter Despite some troubles in this new theory it is said to predict between others the rotation curves of galaxies the Tully Fisher law for galaxies and weak lensing Intuitively thinking that General Relativity may be wrong for great masses potential and or great spatial scales is more elegant than introducing a strange dark matter never seen in more than twenty years even though supersymmetry theories could account for it Although this question is still wide open it is not our intend here to take part in this debate Whatever dark matter exists or not weak lensing is expected to occur because it depends on the
37. cs formalism to deal with shapelets In particular we introduce the creation and annihilation operators defined respectively as 1 1 ip ip 1 7 V2 2 1 i t 2 is the momentum operator and where x is the position operator and p 7 t is the hermitic conjugate The and f operators act on the basis functions as APn Vnbn_1 On Vn lp 1 8 With these operators we define the numeration operator N ata which has the useful property Non non 1 9 and which we will use hereafter to compute the moments of an object 8 CHAPTER 1 SHAPELETS FORMALISM 1 2 2 Two dimensional cartesian shapelets The dimensionless two dimensional shapelets are the product of the one dimensional ones On X Pn 1 On 22 1 10 where x 21 25 and n n na The first few basis functions n are shown on figure 1 1 b The dimensioned basis functions are defined in an equivalent way By x 8 5 1 11 They are still orthonormal Decomposing an object into 2 dimensional shapelets is done in the same way than into 1 dimensional shapelets fx Dee fa Balx 3 fa f drf x Ba x 8 1 12 where fn fn fno Similarly to a one dimensional shapelet decomposition most of the shape in formation of a two dimensional object is contained in the first few 2 dimensional shapelet basis functions Hence one can restrict the decomposition to orders less than a ce
38. e maximum order of convolved object if it is not set it is computed using the above empirical rule for deconvolution equation 3 11 42 CHAPTER 3 PSF CORRECTION In practice we could measure the scale and maximum order of the observed convolved galaxy and set them as arguments of shapelets_convolution_matrix to gether with the guessed one for the model of the unsmeared galaxy This would probably be the most efficient way to obtain an accurate model but it would need a first iteration of shex on the image without deconvolution and then a second one with deconvolution Since computer time would become prohibitive we prefer to use the default gamma and n_max_gamma given by the above rules for deconvolution As long as simulations give good results with this method we trust and use it NB Note that doing so we never measure the shapelets scale and maximum order of decomposition and we never compute shapelets coefficients of the smeared galaxy Once we are in possession of the convolution matrix we can compute the con volved basis functions using a least squares fitting then the unconvolved ones and finally we can compute the unconvolved shapelets coefficients To conclude at the end of these operations we have computed a model of the unsmeared galaxy In other words we have deconvolved the PSF from the observed image We will eventually use this shape information to measure cosmic shear Nevertheless we must be sure t
39. e object recomposed after shapelets decom position e PSF psf is the PSF to be convolved to the image if it exists e IMAGE_IN image_in is a second input image to plot e g noise free simulated image e SCAT_IN scat_in is the SExtractor catalogue of the image it al lows to plot the ellipse whoch encloses the object e NOISE_MAP noise_map if it is set background statistics are computed from the estimation made during the postage stamp ex traction otherwise they are locally computed The focus route as plotted by shapelets_plot_focus is shown in figure 4 2 This figure is to be looked at in a similar way to figure 2 2 The profile of the object is also plotted as shown by figure 4 3 4 2 5 Draw the model of the object shapelets_plot_decomp Last step of a shapelets decomposition the decomposed object can be recomposed for example to be compared to the observed object Plotting it can be done using shapelets_plot_decomp decomp recomp options with the following arguments e decomp is the final decomp structure e recomp is the output object recomposed after shapelets decompo sition CHAPTER 4 INTERFACING SHAPELETS Focus history sol T B 3 64 nmex 12 x 1 08564 snr 42 5839 25 pstamp Severe overlap E focus Flatness H Fi LA 20 rN H H N F 15 Le DE FF 5 0 ee ese eS Er ET EEE i Sr a 0 5 10 15 Tmax Figure 4 2 A focus route CARTESIAN or CO
40. e postage stamp and the centroid of the object if necessary which will be the center of the basis functions Then it effec tively decomposes the object into shapelets see section 2 4 1 recomposes it still convolved with the PSF even though PSF deconvolution is performed when decom posing computes the covariance matrix of the shapelets coefficients and x the difference between the original and reconstructed image and writes the shapelets coefficients in a decomp structure together with other useful quantities The user does not have to call the routine himself because it is directly used by the shapelets_focus pro function which outputs the decomp structure see below decomp shapelets_decomp f beta n_maz options e fis the 2D function to decompose into shapelets Practically this is the intensity of the postage stamp containing the object to de compose pstamp image in the command line pstamp a postage stamp given by shapelets_pstamp is passed as argument The rou tine then extracts and decomposes only pstamp image i e the value of the pixels in the postage stamp e beta is the focussed scale parameter 11To compute x we use the noise calculated when creating the postage stamp by shapelets_ sexcat2pstamp this noise is entered as an optional input in shapelets_decomp 30 CHAPTER 2 SHAPELETS DECOMPOSITION n_max is the focussed maximal order for decomposition CENTER center is the ce
41. ect f is the un sheared profile of the object f is its sheared profile The operator depends on the creation and annihilation operators and it permits to define the shear matrix see 27 for details ae d rBa x S Bm x 1 36 It is then possible thanks to some properties of the shear in the shapelets space to obtain an unbiased estimator for each component of the shear see 27 for details N _ Jas lt fa gt ET ny No pairs Yin Sinm lt fm gt 1 N2 P 1 37 Jon Be my ng odd Yan Sonm lt fm gt a The following chapter describes the publicly available IDL package we imple mented to use shapelets for image processing 16 CHAPTER 1 SHAPELETS FORMALISM Chapter 2 Shapelets decomposition The shapelets package is the publicly available IDL software which contains the shapelets formalism i e which deals with decomposition into shapelets and all the shapelets algebra convolution deconvolution rotation dilatation etc This section is aimed at introducing the most useful routines for the shear analysis of astronomical images Due to the package s richness it cannot be an exhaustive explanation of all routines gathered therein Nevertheless each one is sufficiently explained in its header so as the user can easily understand the way it works The purpose of each routine can also be found in appendix A 2 1 Detect objects to decompose A prior to a shapelets based weak lensin
42. ects moments unweighted quadrupoles size and flux e FULL_PATH full_path is the full path of file name e PARITY is not available yet e N MOMENTS n_ moments is the Nmax till which moments are com puted e SILENT is to be set to operate silently e VERBOSE is to be set to operate noisily e DESCRIPTION description is an optional output which contains information about the catalogue 48 CHAPTER 4 INTERFACING SHAPELETS 4 1 5 Obtain a decomp structure from a shapecat catalogue After all objects have been decomposed into shapelets by shex one could want to look at a given object As the shapecat catalogue contains information for all objects and a decomp catalogue contains information for one given object one must be able to extract a decomp structure from a shapecat catalogue This is done using the shapelets shapecat2decomp routine Being able to deal with input output procedures the user must now be able to draw some plots for example to look for possible problems during the decomposition process Plotting procedures are the topic of the next section 4 2 Plotting routines When processing an image one needs one to be able to visualize it and to draw several plots linked to the image analysis That is why some important plotting rou tines are included in the shapelets package which are often called automatically by numerous analysis routines if requested by the PLOTIT keyword when it exists 4 2 1 Draw an image
43. eded for the treat ment of neighbours which pollute the postage stamp if neigh bour 0 default the errors will be infinite i e the fit is uncon strained if neighbour 1 neighbour pixels are set to the back ground level e N_PIXELS n_pixels output is the size of the postage stamp e PLOTIT if set the postage stamp is ploted to the screen e SILENT if set the routine acts silently In practice shapelets_sexcat2pstamp guesses the size and the coordinates edges of the postage stamp from the position and the FWHM of the object given by SEx tractor It then checks if there are no problems e g the object can be near an edge of the image or can be masked out before extracting the desired region pstamp and creating segmentation and noise maps Finally neighbors are removed from the postage stamp so as the desired object can be alone and all useful quantities are stored into the pstamp structure 2 2 2 Creation of a segmentation map The segmentation map shows the position and shape of each detected object rep resented by its ID SExtractor can create one that can be read later Thus if the SEG_MAP keyword is not set shapelets_sexcat2pstamp creates the segmentation 20 CHAPTER 2 SHAPELETS DECOMPOSITION map of the postage stamp only by extracting the region of the postage stamp off the SExtractor segmentation map of the whole image Otherwise if the SEG MAP is set shapelets sexcat2pstamp creates a b
44. efer to convolve the PSF to an unconvolved model than deconvolving it to a convolved model That is while decomposing a smeared galaxy into shapelets assuming that the PSF coefficients on this galaxy are known we directly search a model of the un smeared galaxy We convolve this model to the PSF and tune it by focusing its scale and its maximum order of decomposition until the convolution product model PSF reproduces the observed smeared galaxy Thus we do not proceed to a direct deconvolution and never invert a matrix all we do is a convolution The next section describes how the shapelets package deals with this process 3 2 CORRECTING FOR THE PSF WITH SHAPELETS 41 3 2 3 Deconvolution in practice with shapelets The PSF deconvolution occurs in the whole shapelets decomposition process Thus it is not a well defined routine which achieves it but it is a sparse process in which each routine implicated in the decomposition takes part We do not describe this process again and refer the reader to chapter 2 for a precise view of it Here we just focus on how the PSF deconvolution works through it It begins at the very begining of the shapelets decomposition when we try to guess starting scale 3 and maximum order Nmax We search them using the usual geometric constraints and transform which as this stage is the scale of the smeared galaxy convolved with the PSF inverting the above empirical rule for convolu
45. erical Recipes in C C Cambridge University Press Refregier A 2008 Annu Rev Astron Astrophys 41 645 Refregier A 2003 MNRAS 338 35 Refregier A amp Bacon D 2003 MNRAS 338 48 Rhodes J Refregier A amp Groth E 2000 ApJ 536 79 Smith D Bernstein G Fisher P amp Jarvis M 2001 ApJ 551 643 Soucail G Fort B Mellier Y amp Picat J B 1987a Astron Astrophys 172 L14 Soucail G Mellier Y Fort B et al 1987b Astron Astrophys 184 L7 Subaru Collaboration 2004 http www subarutelescope org van Waerbeke L Mellier Y Pell R et al 2000 Astron Astrophys 393 369 x 29 PSF matrix 13 40 x 22 minimization 22 quadrupoles 9 background scale noise 20 33 of decomposition 13 39 focus shapelets 3 24 centroid 9 22 guess shapelets 8 23 focus 24 shapelets 8 6 22 guess 23 segmentation map 19 33 convolution tensor 13 38 sexcat 33 46 SExtractor decomp read outputs 46 plot 51 sex pro 67 structure 26 31 47 48 shapecat 33 35 47 Beus shapelets decomp pro 29 60 Ase 27 32 shapelets focus pro 25 61 shapelets_paths pro 59 shapelets_read_image pro 45 62 shapelets_read_sexcat pro 46 62 plot history 51 structure 27 image shapelets_read_shapecat pro 47 62 plot 48 shapelets_sexcat2pstamp pro 18 62 read a FITS 45 shapelets shapecat2decomp pro 48 62 statistics 49 shapelets_split pro 36 67 structure 33 45 shex pro 32 68 size 10 order o
46. ey are sensibly 6see section 2 4 2 3 FIND THE OPTIMAL PARAMETERS FOR THE DECOMPOSITION 25 identical to those of shapelets_focus see below Its code is sufficiently commented so as one can easily use it if needed 2 3 4 The shapelets_focus routine In practice the scale parameter the centroid and the maximum order of decom position are focussed with the shapelets_focus pro routine Of course it relies on shapelets_focus_beta and shapelets_focus_nmax After guessing the starting 5 and Nmax it passes them to shapelets_focus_beta which searches for the best and zo for the guessed Nmax then computes the best Nmax with these 3 and x Some itera tions are made in which those two routines run one after each other so as the three searched parameters are focussed the best way possible While performing the focus of 3 o and Nmax the routine decomposes the object into shapelets and outputs a decomp structure calling shapelets_decomp pro A focus structure is also output see below Call shapelets_focus pro as decomp shapelets_focus pstamp options using the arguments e pstamp the input postage stamp containing the object for which GB o and Nmax must be focussed e FOCUS focus the output focus structure see below for details e RECOMP recomp the optional output recomposed image see section 2 4 for details e BETA_GUESS beta_guess the starting point for 8 if not set the rout
47. f decomposition Nmax 22 focus 24 guess 23 plt_stars pro 68 postage stamp 18 21 25 flag 22 32 plot 50 PSF correction for 37 correlation function 42 73
48. g measurement is the detection of all galaxies to be decomposed This is done with SExtractor 5 6 13 and is described in Volume I Once all objects have been detected we perform a shapelets decomposition on each object idividually one after the other 2 2 Isolate the object to decompose The first step in a shapelets decomposition is to isolate the object from the SEx tractor catalogue containing the all detected objects The goal is to cut the wanted object and its near neighborhood out of the image in other words one wants to extract a postage stamp of the image illustrated by figure 2 1 This is done thanks to a function called shapelets_sexcat2pstamp pro which we present in the next subsection 17 18 CHAPTER 2 SHAPELETS DECOMPOSITION 30 Postage stamp 31043 1940 1950 1960 1970 1980 1990 x pixels Figure 2 1 A postage stamp isolating a galaxy 2 2 1 The shapelets_sexcat2pstamp routine As its name suggests this routine extracts postage stamps of the SExtractor cata logue In order to create a postage stamp named pstamp call it as pstamp shapelets_sexcat2pstamp file_name image sexcat id options using the arguments e image is an image structure given by shapelets_read_image e sexcat is the SExtractor catalogue out of which one wants to isolate pstamp id is the identity number of the object to place in the postage stamp NOISE_MAP if set a noise map from image
49. hape parameters associated with shapelets Pol shapelete ss ses tes ae ee e 2S Bee ES 1 3 1 1 3 2 1 3 3 B rmals n 2 2 at see a ee Ae ee Shape parameters sgt LM dd cove aoe Os deen as we ad See Conversion polar cartesian shapelets Scales of decomposition a ar Ein onde doe OR Sew A Baw te Convolution and deconvolution PIgelizalion re zur Coe BS a Ne deal ok kit A shear estimator for cartesian shapelets 2 Shapelets decomposition 2 1 Detect objects to decompose 2 2 Isolate the object to decompose 2 3 2 2 1 page 2 2 3 2 2 4 The shapelets_sexcat2pstamp routine Creation of a segmentation map u 24 i e es ak Gee es Noise estimation and creation of anoisemap The pstamp structure The fagin Shem af que hes hte et hk Saad ee ee ee Find the optimal parameters for the decomposition 2 3 1 2 3 2 2 3 3 2 3 4 2 3 9 Guess starting 8 o and Nmax e es sn orne oies Focus and to ss areas FOCUS RAR ee ee N ee The shapelets focus routine Possible problems and the focus flag vi viii CONTENTS 2 3 6 The focus structure SE Len ee BE GL Dale 27 2 4 Decompose an object into shapelets 28 2 4 1 How to decompose into shapelets 28 2 4 2 In practice shapelets decomp pro
50. hapelets exist which are very similar The cartesian shapelets are weighted Hermite polynomials which correspond to perturbations about a cir cular gaussian and in their asymptotic form to the Edgeworth expansion in several dimensions They are also the eigenstates of the 2 dimensional Quantum Harmonic Oscillator QHO and they allow to use the powerful formalism developed in quan tum mechanics The polar shapelets are weighted Laguerre polynomials and are closely linked to the cartesian ones Both kinds of shapelets have several interesting mathematical properties some of which we will deal with hereafter 5 6 CHAPTER 1 SHAPELETS FORMALISM 1 2 Cartesian shapelets 1 2 1 One dimensional cartesian shapelets To describe a 1 dimensional object we define the dimensionless basis functions n x 2 Van PH x e 72 1 1 in which n N and H is an nth order Hermite polynomial They are orthogonal i e Aes drPn t Pm T mn Where fmn is the Kronecker symbol They can be seen as perturbations around the gaussian do The first few basis functions are shown on figure 1 1 a In practice we rather use the dimensional basis functions By x 3 Bol 1 2 where 8 is a characteristic size close to that of the object to decompose The basis functions B also are orthogonal They create a complete set of basis functions for C and integrable functions so as the profile f x of an object can be decomposed on these f x
51. hat the deconvolution process went good we use some diagnosis that we present in the next section 3 2 4 Recompose an object deconvolved from the PSF We just saw how to deconvolve the PSF from a given object but we did not evoke how to look at this corrected object This can be done using the shapelets_recomp pro routine which recomposes an object after it has been decomposed into shapelets Particularly if one introduces the PSF in this routine it recomposes the object re convolved from the PSF otherwise the unconvolved object is recomposed 3 3 Is the PSF correction reliable 3 3 1 The two point correlation function of the PSF The two point correlation function of the PSF ellipticity is defined as the average over all couples of stars of the product of the ellipticity of both members of each couple It depends on the separation r of the stars Co lt 1 0 e1 0 r gt lt E 2 0 E2 0 r gt 3 14 If the ellipticity of two stars is positively negatively correlated then their corre lation function will be positive negative If no correlation exists then the correla tion function should be zero A two point correlation function can also be measured for the size of the PSF 3 3 IS THE PSF CORRECTION RELIABLE 43 Figure 3 1 Two point correlation functions for the PSF ellipticity left panels and PSF size right panels In both cases top panels compare the raw correlation function dia monds and the cor
52. ical analysis of an image requires dealing with thousands of objects e g Subaru images contain approximately 70000 utilizable objects It would be inhuman to decompose them one by one creating thousands of postage stamps and then calling each time shapelets_focus let us still emphasize on the fact that the user does not have to call shapelets_decomp the routine is included in and automatically called by shapelets_focus pro One program built upon those two former routines deals with all objects We present it in the next section 2 5 Analysing all objects in an image So as to perform a shapelets analysis of an image the user ideally has to run only one program which is in charge of decomposing each object detected by SExtractor into shapelets As it is not limited in terms of number of possible decompositions just the memory let available by idl and the upper limit of long type variable can limit it one can easily treat a physical current image For example a Subaru image covers 0 25 deg and contains approximately 70000 galaxies this is easy to treat them all 32 CHAPTER 2 SHAPELETS DECOMPOSITION 2 5 1 The shex pro routine A routine called shex pro is the orchestra leader of all the shapelets analysis we ex clude from this term the processes aimed at modeling the PSF and at estimating the shear in the case of cosmic shear measurements It reads in the SExtrac tor catalogue and extracts and decomposes each object i
53. ically ill as soon as the matrix is degenerate or not inversible So as to avoid this problem some methods have been developped to correct the PSF without a direct deconvolution See for example the RRG Rhodes Refregier amp Groth 28 method Nevertheless it must confessed that a method using a direct deconvolution is more elegant and accurate provided that it can deal with numerical problems Two kinds of method can be thought of to deconvolve the PSF Both rely on reproducing data i e observed objects by modeling either 1 the smeared object i e the object one sees through the telescope or 2 the unsmeared object i e the object as it would be seen without a PSF In the case 1 the PSF is deconvolved afterwards in case 2 the PSF is convolved to the model and the model tuned so as the convolved model reproduces data Case 2 has the advantage that it does not compute a deconvolution but a convolution and hence does not need to invert a matrix as already mentionned shapelets use this method to correct for the PSF 37 38 CHAPTER 3 PSF CORRECTION 3 2 Correcting for the PSF with shapelets 3 2 1 Convolution formalism The convolution formalism for shapelets has already been briefly introduced in chap ter 1 section 1 5 We develop it here and give some useful formulae needed to understand the shapelets deconvolution scheme section 3 2 3 They can all been found in 27 together with their justification Le
54. image shapelets_plot_image_statitics 49 4 2 3 Draw a postage stamp shapelets_plot_pstamp 50 4 2 4 Plot the focus route shapelets plot focus 51 4 2 5 Draw the model of the object shapelets_plot_decomp 51 4 2 6 Gather all plots together shapelets plot 2 2 2 2 54 43 Other routines of the shapelets package 55 Conclusion 57 A Routines of the shapelets package 59 A 1 General routines a re pnd ess Be ee Bah koe 59 A 2 Alias routines alias 2 oaoa Bere ee 59 A 3 Decomposition and focus decomp 60 CONTENTS A 3 1 Focus routines focus A 3 2 Polar shapelets decomposition polar A 4 Inputs Outputs io A 4 1 Deal with ascii catalogues asciicatalogues A 5 Library of useful routines library A 6 Shapelets algebra operations A 7 Pipeline pipeline A 8 Plotting routines plotting A 9 Objects properties properties Bibliography Index 61 61 62 63 63 66 67 68 69 71 73 CONTENTS Introduction Because it relies only on General Relativity and not on the physics of halos weak lensing represents an observable directly related to the mass or to the potential of halos and therefore it allows to have recourse to scaling laws that we do not necessarily perfectly know from which degeneracies in the detection of clusters of galaxies
55. imula tion plt_stars plot various figures relative to the stars in a catalog A 9 OBJECTS PROPERTIES PROPERTIES 69 shapelets_plot Generic plotting routine Acts as a wrapper for many more spe cific plotting utilities First decides what the structure type is rather as you would expect from object oriented code then calls the relevant plotting sub routine shapelets_plot_basis Makes a plot of all the polar shapelet basis functions out tO Nmax shapelets_plot_chisq_grid plot x on the 3 Nmax grid shapelets_plot_colourbar draw an annotated color bar at the top of the plotting window The window parameters are then set to the remainder of the window shapelets_plot_decomp Repixellate a shapelet model decomp structure and display it shapelets_plot_focus Plot the route taken through possible Nmax and 3 values to obtain a final decomposition shapelets_plot_image Draws a 2D pixellated image shapelets_plot_image_statitics Plot statistics about an image shapelets_plot_pstamp plot the image and sextractor parameters for an object shapelets_plot_sexcat Displays an image if input and overlays the positions of all objects found on it by SExtractor If no image structure is given it just plots the positions of objects shapelets_plot_shapecat plot statistics relative to shapelet catalog A 9 Objects properties properties shapelets_asymmetry Returns asymmetry morphology index of a Cartesian de
56. in and Omax equation 1 28 If this is not the case we search other better 3 o and Nmax Figure 2 2 shows this process 3 and Nmax are varied along the solid black line labeled Algorithm until the reduced x shown in color is 1 e During the path of the algorithm and Nmax must be such as they lie between the dashed lines labeled Omin and Omax 4in practice focus and decomposition are simultaneous 2 3 FIND THE OPTIMAL PARAMETERS FOR THE DECOMPOSITION 23 logio Xr N max Figure 2 2 Focus of 3 and Nmax They are varied along the solid black line labeled Algorithm until the reduced x shown in color is 1 no The Algorithm line must lie between the dashed lines labeled Omin and Omax The minimisation of x is done using the Numerical Recipes routine Amoeba 24 re written in idl langage Practically this focus work is achieved in two times Firstly we focus x and 3 and secondly Nmax is fixed The former task is done through the shapelets_focus_beta pro routine the latter by shapelets_focus_nmax pro But first of all we must guess with which 6 center and Nmax we shall begin the focus process 2 3 1 Guess starting 0 to and Nmax As we saw in chapter 1 section 1 2 2 shapelets scale 8 and maximum order of decomposition Nmax can represent objects with sizes between certain Omin and Omax given by equation 1 28 Consequently we choose starting 3 and nmax using these rules A
57. ine guesses itself e BETA TOLERANCE beta tolerance the fractional tolerance for finding 8 The default value is 1078 e FIXED BETA fixed_beta fixed value of beta that should be used for decomposition e N MAX GUESS n max guess the starting point for Nmax if not set the routine guesses itself e N MAX RANGE n max range range of allowed and tried values for n_max default is 2 20 e CENTRE_GUESS centre_guess the starting point for the centre of shapelets basis functions Tsee section 2 4 26 CHAPTER 2 SHAPELETS DECOMPOSITION CHISQ_TARGET chisq_target the target value of x to achieve default value is 1 CHISQ TOLERANCE chisq_tolerance the fractional tolerance for y minimisation called in the main text CHISQ_FLATNESS chisq flatness is the minimum difference in x between two decompositions with nmax differing by two to trigger the flatness constraint in shapelets_focus_nmax THETA MIN GEOM theta min geom minimum geometrical scale Omin default value is 0 5 pixels GAUSSIAN_RECENTRING if it is set the basis functions are recentered on the previous decomposition s centre of light MAX_LOOPS max_loops maximum number of alternative runs of shapelets_focus_beta and shapelets_focus_nmax default value is 20 FULL_FOCUS if it is set all attempts made at decomposition during iteration are recorded PSF psf to be set when the focus is done while deconvolving the PSF 8
58. ion If the two point correlation of the stars after correction for the PSF is negligible then one can be really confident on one s model of the PSF and its correction 44 CHAPTER 3 PSF CORRECTION Nevertheless something wrong could have occurred in the deconvolution process The next section describes how comparing some objects at different steps of their treatment can help decide if the deconvolution is reliable 3 3 2 Looking at objects After deconvolution stars should be Dirac function Plotting their unconvolved model can allow one to decide instantly whether the deconvolution seems OK or not an unconvolved stars well peaked on a pixel is the sign that the deconvolution has worked on the opposite an unconvolved star which spreads out on lots of pixels thus not a Dirac function reveals the fact that the deconvolution is likely to have gone wrong One can also reconvolve unconvolved stars with the PSF and compare the results to the original star as seen before deconvolution they should be identical If this is not the case that means that a problem occured during deconvolution The same work can be done on galaxies but seems to be less efficient to check problems 3 3 3 Limits of deconvolution diagnosis Using the two diagnosis described above one can check whether the correction of the PSF has gone well or wrong Nevertheless it must be emphasized that computing the two point correlation function of the PSF and of its
59. ions are the same as in the routine the local stands for local estimation of the noise Note that the noise we compute is gaussian and uncorrelated To be fair this might decrease the efficiency of a shapelets decomposition especially if the intrinsic data noise is correlated and or non gaussian This is an aspect we still have to work on 2 2 4 The pstamp structure The pstamp structure is the output of shapelets_sexcat2pstamp it contains all useful quantities for the postage stamp and the extracted object Among them 1 the image in the postage stamp 2 the object s position in the image coordinates as given by SExtractor 3 the coordinates of the corners and edges of the postage stamp the preliminary square one if it is circular 4 the object s centroid in pstamp coordinates 5 the noise and segmentation maps 6 the estimated noises s mean and rms 7 some SExtractor quantities 8 a flag giving the possible errors 22 CHAPTER 2 SHAPELETS DECOMPOSITION The flag in pstamp This flag is aimed at further detecting possible bad objects It consists in a number between 0 and 10 For example an object can lie near the object of interest flag 1 or the object can be masked out flag 6 or the background could not be estimated flag 8 Possible flags and their meaning are given in the left column of table 2 1 Eventually in all shapelets decomposition all objects with postage stamp flag grea
60. is zero on saturated stars shapelets_shapecat2decomp Extracts one decomp structure object from a shapecat structure catalogue shapelets_structure_type Determine whether a variable is a known shapelets structure in a very robust manner shapelets_update_history Append a string to an object s history record shapelets_write Writes a shapelet coefficient catalogue of objects to disc NB all these routines work with binary catalogue files A 5 LIBRARY OF USEFUL ROUTINES LIBRARY 63 A 4 1 Deal with ascii catalogues asciicatalogues shapelets_ascii2idl shapecat Converts a shapecat stored on disc from a shapelets shape file in ASCII format to a shapelets shapelet file in IDL SAVE for mat shapelets_concatenate_ascii_shapecats Concatenates two shapecat structure catalogues shapelets_idl2ascii shapecats Converts a shapecat stored on disc from a shapelets shapelet file in IDL SAVE format to a shapelets shape file in ASCII for mat shapelets_read_ascii_sexcat Reads in a SExtractor catalogue shapelets_read_ascii_shapecat Reads just the header information in shapelet catalogues shapelets_read_ascii_shapecat_hdf Reads in and concatenates shapelet cata logues containing all of the objects from both Hubble Deep Fields A 5 Library of useful routines library check_fits Check that keywords in a FITS header array match the associated data comb Calculates binomial coefficients
61. is used otherwise the background is locally estimated a noise map is created SEG_MAP if set a locally determined segmentation map is used otherwise the one from SExtractor stored in image is used e SATURATION_LEVEL saturation_level not used Isee section 4 1 1 2 2 ISOLATE THE OBJECT TO DECOMPOSE 19 e SQUARE if set the postage stamp will be squared otherwise it will be circular If SQUARE is not set the routine starts with creating a square postage stamp then masks the corner off so as it becomes a circle e BORDER if set the border around the edges of postage stamp is increased to make pretty plots in papers but it is useless elsewhere e BACK_SIZE back size is the size of postage stamp used for noise estimation default value is 120 pixels e N_GROW n_grow is the number of times SExtractor s segmen tation map is grown to mask objects during noise estimation e VARY_LOCAL_NOISE_CALC allows to estimate the noise by a simple iteration on the statistics of the background e SHOT_NOISE allows to add the photon noise in the noise map e LAZY if it is set then if a deffect is found in the surrounding of the object in such a case the object will eventually be eliminated the noise map is not computed but a fake is created instead e NFWHM nfwhm is the size of the postage stamp in units of the a parameter of SExtractor default 3 e NEIGHBOUR neighbour this keyword is ne
62. laxy will be differently curved i e deviated by a same gravitational well just because they travel on two different geodesics that is what is called differential deviation As a result the image one sees of a distant galaxy will be distorted one does not see the galaxy as it really is but amplified and sheared There is a direct analogy with optical lenses physicists talk about lensing Some cases appear in which the gravitational potential is so large that the image of a distant galaxy is extremely amplified and sheared and appears as an arc this is strong lensing In other cases the distortion is pretty negligible and is rather called weak lensing or cosmic shear and that is what weak lensing observers intend to detect and measure For details about lensing the reader is referred to 2 23 25 Since General Relativity predicts it the idea of both strong and weak gravi tational lensing grew very early in the first half of the 20th century initiated by Zwicky In the 1980s a team first observed strong lensing 30 31 with gravita tional arcs in some clusters Then in the early 1990s weak lensing was detected in clusters of galaxies 8 Since the early 2000s astronomers have had more and more accurate telescopes and methods dedicated to weak lensing and cosmic shear due to large scale structures is currently being detected and measured 1 16 33 If it took so long to detect and to measure cosmic shear that is because i
63. let decompo sition via an iterative search throughout the space spanned by these meta variables shapelets_focus_beta Find the optimal beta and centroid centre_guess for the decomposition of an image by minimising x This is done using a 1 dimensional Amoeba search The maximum shapelet order nmax is assumed to be known shapelets_focus_nmax Find the optimal nmax for the decomposition of an image by exploring different values until x is equal to a set value x The shapelet scale 3 is assumed to be known shapelets_geometric_constraints Determine min and max for an object shapelets_guess_nmaxbeta Guess the optimal 3 and Nmax for an object shapelets_make_chi2_grid Compute the x difference between the input and the reconstructed image on a grid of values of and Nmax shapelets_make_chisq_grid Compute the x difference between the input and the reconstructed image on a grid of values of 8 and Nmax A 3 2 Polar shapelets decomposition polar shapelets_polar_convert Converts Cartesian shapelet coefficients to their equiv alent polar shapelet representation or vice versa shapelets_polar_expand Manufactures and reinserts the degenerate complex parts of polar shapelet coefficients so they can be transformed back into carte sian shapelets shapelets_polar_matrix Calculates a matrix to convert Cartesian shapelet co efficients to their equivalent polar shapelet representation 62 APPENDIX A ROUTINES OF
64. lets are used DIAMOND to be set to use the diamond truncation scheme for decomposition FULL_ERROR is to be set to return the full covariance matrix of the coefficients in error tag name deprecated in current version e SILENT is to be set so as the routine operates silently l2see chapter 3 section 3 2 13 see chapter 1 section 1 6 2 5 ANALYSING ALL OBJECTS IN AN IMAGE 31 e LS is obsolete it used to be set to fit the coefficients with the least squares method now by default instead of Fourier like overlap integrals now used when setting OVERLAP e X0 x0 is obsolete it used to be CENTER We describe the decomp structure hereafter 2 4 3 The decomp structure It gathers all useful parameters of the shapelets decomposition of an object It notably contains 1 beta the scale parameter 3 used for decomposition nmax the maximum order for decomposition Nmax used for decomposition n_coeffs the number of coefficients used in decomposition coeffs and coeffs_error the coefficients of the decomposition and their errors n_pixels the size of the postage stamp expressed in pixels nl and n2 the one dimensional orders of coefficients oy Se SOU En 5 4 D chisq x the difference between the original and recomposed image So far we have shown how to properly decompose an object into shapelets either a star or a galaxy But the goal of a weak lensing or simply an astronom
65. methods and IDL routines are described Of course explaining each routine of the package would have been not only painless but also pointless and only the most important ones have been dealt with In that sense shapelets algebra codes are not dealt with since this manuscript aiming at emphasizing cosmic shear analysis focuses on the shape measurement with shapelets An exhaustive list of the shapelets package s routines is given in an appendix though and some help can be found on the Internet at the below address Thank you Alex and Richard for your support when I was learning how to use shapelets I want to thank Mandeep Gill for comments on the first version of this manuscript and Richard Ellis for providing me computer space at Caltech where I can find a home for this manual All routines of the shapelets package can be downloaded at the URL http www astro caltech edu rjm shapelets code Updated versions of this manuscript can be found at the URL http www astro caltech edu jberge shapelets manual For any comment or suggestion feel free to contact me directly J B May 13 2006 vi FOREWORD Contents Foreword Introduction 1 Shapelets formalism Ls Generalities de de a ea els ave a En ere 1 2 Cartesian shapelets 22 2 zu zu nad Hak ai 1 3 1 4 1 5 1 6 1 7 1 2 1 1 22 1 2 3 One dimensional cartesian shapelets 2 22 2 2 Two dimensional cartesian shapelets S
66. n of the PSF allows to get rid of the PSF in an efficient way and has been well described Finally the plotting and interface machinery of the shapelets package have been presented Having introduced shapelets we are now in a position where we can use them for weak lensing data analyses In Volume II we explain how we proceed from the detection of galaxies on an image to the measure of the shear and the reconstruction of the dark matter distribution In particular we present our shapelets based PSF modelling scheme We also describe the practical use that we make of the shapelets package That is we give a full overview of our whole shapelets based weak lensing image processing The shapelets package can be downloaded on the Internet at http www astro caltech edu rjm shapelets code 57 58 CONCLUSION Appendix A Routines of the shapelets package In this appendix we list all the routines belonging to the shapelets package version 2 18 together with their purpose Each sub section corresponds to a sub directory of the package the name of which is given between brackets A 1 General routines shapelets_paths This stores the locations of data on a locally accesible hard disc Please update this file so that all the strings point to the correct locations shapelets_demo Runs through the main shapelet routines to demonstrate their use and to check that they are installed correctly A 2 Alias routines alias pl
67. nd so on for x The quadrupoles Jj f d xzx z f x F are obtained the same way x and x2 being symmetric the centroid expression makes it obvious that even 3 1 2 1 25 2 n1 n2 2 n 1 1 2 nah 1 2 Ja Joy PAV TBP S t net 06m CE Oa Sum n1 n2 1 16 Computing the other two quadrupoles demands to use the numeration operator It gives even Ja FRE Y ery pon CE CA fame ID n1 n2 and so on for Ja 10 CHAPTER 1 SHAPELETS FORMALISM Finally one can calculate the size of the object even F T 2l TAES n2 2 1 Ny na ca he Case 1 18 n1 n2 1 3 Polar shapelets Polar shapelets have been introduced in 26 and fully described in 20 1 3 1 Formalism They are directly linked to cartesian shapelets and share all their useful properties and a similar gaussian weighting function of scale size 5 But as they are separable in r and 0 polar shapelets coefficients are easier to understand in terms of rotational symmetries and many operations are made simpler and more intuitive A function f r 0 in polar coordinates can be decomposed as a weighted sum of the basis functions Yu m r 9 3 f r 0 an pa Tree ons Wi 0 p fam Jfr Fr 0 Xnm r 9 B rdrd fam n EN n lt m lt n are the polar shapelets coefficients of order n m 1 19 The basis functions Xn m are related to Laguerre polynomials D Iml 2 through note that only the states with n and
68. nter of the basis functions It should be the focus one but if it is not set the center of the postage stamp is taken to be the center of the basis functions NAME name is the name of the object By default the object has no name PSF psf is a decomp structure representing the PSF on the object if set the PSF is deconvolved from the object NOISE noise is a noise map i e an inverse variance map of pix els If it is not set the noise is assumed to be constant In practice we use the noise estimated by shapelets_sexcat2pstamp and stored in the pstamp structure RECOMP recomp is the output recomp image still convolved with the PSF to remove this one must use the shapelets_recomp routine 12 OVERSAMPLE oversample forces an oversampling factor to eval uate the basis functions There is no oversampling if over 1 OVERLAP is to be set to fit the coefficients with Fourier like over lap integrals instead of the least square method INTEGRATE integrate serves for pixelizing the shapelets model 13 to use basis functions integrated within pixels instead of pixels central value It is set by default to take center set integrate 2 SKY sky to be set when subtracting the background to be set to 1 to fit it with a constant value around the object 2 to fit it with a plane around the object NONI if set ao and ajo are required to be 0 POLAR to be set to use polar shapelets by default cartesian shape
69. nto shapelets one after each other Therefore the routine is just a loop on objects to decompose Each loop is aimed at decomposing one object by creating a postage stamp shapelets sexcat2pstamp focussing the needed parameters and decomposing the object into shapelets shapelets_focus The routine can decompose objects either with or with out PSF correction using deconvolution 15 Each object is analysed and all those that have a focus flag greater than 8 and a postage stamp flag greater than 3 are skipped but still written in the output catalogue except if explicitly requested that they are not recorded Before de scribing the routine s options two features must be emphasized First weak lensing measurements usually deal with images with thousands of galaxies and shex can run for hours even for days until each galaxy is decomposed since it would be frustrating that an object makes shex crash just before the end say the 65000th object over 67000 in an image and a great waste of time shex is enabled to peri odically record temporary output catalogues and to recover them in case it crashes Second defining the size of postage stamps using SExtractor s parameter can prove wrong and give too small a postage stamp as a consequence shex can be enabled to automatically increase the size of postage stamps shex is simply called as shex file name options using the arguments and options e
70. nts with n 1 are forced to be zero TRUE_IMAGE true_image is an additional input image for plot ting purpose only e g an image in other color SCAT_IN scat_in is a SExtractor catalogue for plotting purpose only PIXEL_SCALE pixel_scale is the scale of pixel in arcsec the de fault one is given in shapelets_read_image pro 2 5 ANALYSING ALL OBJECTS IN AN IMAGE 35 e UNITS units is the units of the image the default one is given in shapelets_read_image pro e PHOTO_ZP photo_zp is the photometric zero point for the image the default one is given in shapelets_read_image pro e EXPOSURE_TIME exposure_time is the exposure time the de fault one is given in shapelets_read_image pro e REDRAW_ADAPTIVE allows shex to increase the size of the postage stamp if the decomposition crashes because the original postage stamp is too small NREDRAWS and DELTA_REDRAW parametrize this option e N_REDRAWS n redraws is the number of times shex is autho rized to increase the postage stamp each time by a fractional factor DELTA_REDRAW the default value is 5 e DELTA_REDRAW delta_redraw is the fractional increase of the postage stamp if an increase is necessary the default value is 0 05 e PLOTIT allows to draw some plots on the screen e SILENT is to be set to operate silently e VERBOSE is to be set to operate noisily shex outputs a catalogue containing all useful quantities about each decomposed object shapelet
71. o missing border due for instance to electronics the integration is given by 20 by ba Lino On On y drdy In no 1 33 2 where the integrals In f On x d x are given by the recurrence relation h ye lerta I V28ldo a Ia 1 34 l n AET By 2 6 a nn n gt 2 Note that missing pixel borders can be dealt with by altering the limits of the integration Finally note that when using polar shapelets which cannot be integrated in rectangular pixels we first convert the model into cartesian shapelets then convert the pixelised cartesian model in polar shapelets As a result the unconvolved analytical shapelets model after being convolved with the PSF has been pixelized It therefore undergoes the same operations as real photons through an observation Thus it can be directly matched to data so as to perform a least square fitting on our model to make it reliable It is then possible to measure the shape of the model and to infer a shear estimator 1 7 A SHEAR ESTIMATOR FOR CARTESIAN SHAPELETS 15 1 7 A shear estimator for cartesian shapelets Shapelets were primarily defined for weak lensing dedicated image analysis As a result they naturally permit to estimate a shear estimator based upon the intro duction of a new quantic operator the shear operator which acts on the profile of the object as shown Fayns 1 35 at the first order in y y being the shear applicated to the obj
72. onents can be obtained by symmetry e g Lozo 2 2b courtesy from 27 Shapelets thus allow to analytically compute convolution products To calculate the convolution product of two functions f and g one just has to decompose them into shapelets compute their convolution tensor Cjmn and apply equation 3 1 The main difficulty in this calcul is to evaluate shapelets scales a 3 and y and maximum order of decomposition Nmax Mmax and Imax Of the two convolved functions f g and of their product of convolution h Contrary to A Nmax and 3 Max which are directly known when decomposing f and g into shapelets y and Imax are unknown But as they are needed to compute h one has to be able to evaluate them Convolution algebra allows one to justify that l Oy min Oain GE OB min 3 7 Oey max Oo max ar Og max where 04 min resp Og max is the minimum resp maximum size described by shapelets decomposition of function f resp g as defined in chapter 1 section 1 2 2 Thus using equations 1 28 we can conclude that y and Imax should be given by V i 1 B2 Mmax 1 A Mmes 1 BANmex 1 4 v ne en _ a Nmax 1 B Mmax 1 2 ee O 1 8 nmax 1 as D re In practice instead of those formulae we use empirical recipes which give better results We evaluate y through the equation 40 CHAPTER 3 PSF CORRECTION 3 10 and we set Imax as being the maximum of
73. ore than fitting a model on data That is we create a shapelets model of a certain galaxy that we fit after possible 2 4 DECOMPOSE AN OBJECT INTO SHAPELETS 29 convolution with a PSF and pixelization to the galaxy we want to model Techni cally two methods can be used the first one uses a least squares fitting method the other one is a normal linear overlap integral method As the default s one is the least squares fitting hereafter we only describe this method The needed least squares matrix which gathers the shapelets basis functions needed for the fit is used as prescribed by 18 to compute the shapelets coefficients of the fitting model of the galaxy That is the galaxy has been decomposed into shapelets 2 4 2 In practice shapelets_decomp pro The shapelets_decomp pro routine is aimed at decomposing all kind of function into shapelets In practice it decomposes the intensity of galaxies or stars It should always use the scale parameter and the maximum order of decomposition as found by the focus process That is why we call this routine only within the shapelets_focus pro one Thus we are certain that we do not incorporate errors by hand The practical decomposition into shapelets is done as follows The function to be decomposed is the intensity of the object i e the value of pixels in the postage stamp which contains the object so the postage stamp is an input of the rou tine The routine computes the size of th
74. ormat for FITS headers gettok Retrieve the first part of the string up to a specified character idl_validname Modify a string if necessary so that it can used as a IDL variable name ieee_to_host Translate an IDL variable from IEEE 754 to host representation inside see if point is inside polygon is_ieee_big Determine if the current machine is use IEEE big endian numbers loadct Load predefined color tables match Routine to match values in two vectors mean This function computes the mean of an N element vector mkhdr Make a minimal primary or IMAGE extension FITS header mmm Estimate the sky background in a stellar contaminated field moment This function computes the mean variance skewness and kurtosis of an N element vector A 5 LIBRARY OF USEFUL ROUTINES LIBRARY 65 mrd_hread Reads a FITS header from an opened disk file or Unix pipe mrd_skip Skip a number of bytes from the current location in a file or a pipe mrd_struct Return a structure as defined in the names and values data mrdfits Read all standard FITS data types into arrays or structures numlines Return the number of lines in a file path_sep Return the proper file path segment separator character for the current operating system This is the character used by the host operating system for delimiting subdirectory names in a path specification Use of this function instead of hardwiring separators makes code more portable pca
75. osed within the diamond symbol depicted by the solid lines on Figure 1 4 Besides its natural definition this truncation schemes allows to further compress the decomposition by getting rid of inappropriate with too much oscillations basis functions 1 3 2 Shape parameters One can use polar shapelets to compute the photometry and astrometry of objects Here we give some formulae Their demonstration can be found in 20 The flux is given by even F f flde BVI Y fao 1 23 A centroid e Ye is given by 12 CHAPTER 1 SHAPELETS FORMALISM T aE Moduli of i coefficients Figure 1 4 Shapelets coefficients of the decomposition of a galaxy 2 odd itis 2 Lt 1 24 The size and the ellipticity are derived from the unweighted quadrupoles mo ments RP nz Sn 1 fno 1 25 n and 3 even e me Zn n 2 fn2 1 26 with e cos 20 isin 20 1 3 3 Conversion polar cartesian shapelets Although polar shapelets are complex functions and cartesian shapelets are real functions it is possible to convert a set of cartesian shapelets coefficients fh no with ni na lt Nmax into polar shapelets coefficients with n lt Nmax with a direct one to one mapping through the relation 20 1 2 9 n 2 m Ing Trini 2 1 rie Dh 1 27 x Xi 0 oe i nr m 2 n tn n dna na 1 4 SCALES OF DECOMPOSITION 13 1 4 Scales of decomposition It must be noted from figure 1
76. pelets_plot structure structure2 options and its arguments are 4 3 OTHER ROUTINES OF THE SHAPELETS PACKAGE 55 e structure is one shapelet structure to deal with pstamp or focus or decomp e structure is another shapelet structure e CARTESIAN COEFFICIENTS if it is set the Cartesian shapelet coefficients are plotted e POLAR_COEFFICIENTS polar_coefficients if it set one aspect of the polar shapelet coefficients is plotted 1 Magnitude default 2 Relative phase 3 Composite half and half e STATISTICS allows to plot the image statistics e All optional inputs and keywords of other plotting routines are al lowed 4 3 Other routines of the shapelets package So far we have presented and described the more useful and visible routines of the shapelets package But other ones exist which are aimed at dealing with shapelets algebra or which are just useful and sometimes used everywhere programs used for some repetitive operations It would be irrelevant and pointless to describe them all here Appendix A lists all the routines of the shapelets package together with their purpose 56 CHAPTER 4 INTERFACING SHAPELETS Conclusion Shapelets are a new generation shear measurement technique that is now reliable for precision cosmology We gave a short overview of their formalism and described the pipeline we developed for dealing with shapelets algebra and decomposition The analytical deconvolutio
77. rand new segmentation map within the postage stamp To achieve this the object s neighbors within the postage stamp are detected besides the object itself their spatial extension determined and their potential overlap with the object to decom pose checked The creation of the segmentation map is finalised by giving to each object s pixels the value of its ID 1 to follow SExtractor convention background pixels which do not belong to any object value is set to 0 2 2 3 Noise estimation and creation of a noise map One crucial task is to estimate the noise which will be needed later for the compu tation and minimization of the least square x during the focus process see section 2 3 Doing so we create a noise map of the postage stamp which is an inverse variance map of the image Two methods can be used either the NOISE_MAP keyword is set or not The first step in both methods is to discriminate background pixels against object pixels using the segmentation map as background pixels have zero segmentation value If NOISE_MAP is set we do an external estimation of the noise we use the noise already estimated when reading the image with shapelets_read_image In the mean time the rms of the noise is computed 1 back_rms_ext a 2 1 J noise_ext where we use the notations used in the routine the ext stands for external estimation of the noise Note that this estimation method can be used only if the
78. relation function of the modeled PSF squares bottom panels com pare the raw correlation function diamonds and the correlation function of the residuals between the raw and the modeled PSF squares As the PSF consists in a pattern stars within a same neighbourhood should be correlated whereas distant stars should be uncorrelated As a result the two point correlation of the PSF i e of the stars before correction for the PSF is expected to be significant for small scales and negligible for large ones On the contrary after correction for the PSF the pattern of the PSF should have been smoothed almost erased Thus the two point correlation function should be negligible at all scales Figure 3 1 shows the behavior of the two point correlation function of a PSF ellipticity left panels and of the same PSF size right panels Left top panel shows the correlation of the raw measured PSF ellipticity diamond symbols together with the correlation of the modeled PSF ellipticity square symbols Left bottom panel shows the correlation of the raw PSF ellipticity diamond symbols together with the correlation of the residuals model raw ellipticity PSF square symbols Right panels show the same quantities but for the PSF size The model of the PSF here was done using a polynomial interpolation of PSF shapelets coefficients Note the slight inaccuracy at scales less than 3 arcmin which is the smoothing scale used for the interpolat
79. rs 3 Convert to polar shapelets and use ladder operators but again only to first order Then convert back to Cartesian coefficients 4 An alternative method uses the Cy formalism for convolution Convo lution with a delta function i e a PSF with 3 0 leaves the object un changed but we can now choose a new in shapelets_convolution_matrix pro the notation is to change alpha to gamma This is a general case no longer limited to first order A 7 PIPELINE PIPELINE 67 Option 4 is the default the others are selected using switches shapelets_exponentiate_operations Applies a pure shear to an object using ladder operators in Cartesian shapelet space shapelets_extend_nmax Increases the value of Nmax in a structure by Nincrease DY padding the coefficient arrays with zeros Can also decrease nmax by truncating the coefficient arrays shapelets_flex Applies a flexion the slight bending due to gradients in a shear field to one object in a decomp structure or to all objects in a shapecat catalogue shapelets_keep_it_real Makes sure a polar shapelet model is wholly real by dis carding any imaginary part Has no effect on a Cartesian shapelet model shapelets_recentre Applies a translation to an object to ensure that its centre of light lies exactly on the origin the centre of the basis functions shapelets_reflect Makes a mirror image of an object s by swapping parity shapelets_rotate Rotate
80. rtain maximum order of decomposition so as equation 1 12 becomes n Nn2 lt Nmax fO D AB 1 13 n1 n2 0 Figure 1 2 a shows how a galaxy can be decomposed and reconstructed to different orders of decomposition note that as n increases the reconstruction goes better For big enough n the reconstruction is almost indistinguishabe from the original image Figure 1 2 b shows the shapelets coefficients associated with the same galaxy in other words it shows the previous galaxy in shapelets space 1 2 3 Shape parameters associated with shapelets Shapelets allow an accurate computation of the astrometry the photometry and the shape parameters of the object to decompose for example its moments Using equation 1 6 the total flux F is given by F er YaBB pm 1 14 n1 n2 where uns fn 1 2 CARTESIAN SHAPELETS 9 10 0 0000 0 0005 0 0010 0 0015 0 0020 ESL 0 4 6 8 10 li B original ns5 5 ns10 ns20 as ren Sean RER 0 5 10 15 20 ny _ Shi ape let C oe effic ie nts 1 ey b Figure 1 2 a A galaxy and its reconstruction after a shapelets decomposition at different n b the shapelets coefficients of the same galaxy The figure is from 26 Identically and using creation and annihilation operators it can be shown that the centroid of the object zf f d a f x F is given by odd even 1 27 n2 1 2 Fr SOS n 1 1 29 2 n1 A CE c u 1 15 ni n2 a
81. s an object by converting it s Cartesian shapelet coeffi cients to polar shapelet coefficients then applying the rotation matrix which in this basis is a simple multiplication shapelets_shear Applies a pure shear to one object in a decomp structure or to all objects in a shapecat catalogue shapelets_split Splits a IDL shapecat or sexcat structure catalogue shapelets_subtract Subtracts one image from another when both are stored as a shapelet decomposition shapelets_translate Translates an object to first order using ladder operators some number of pixels A 7 Pipeline pipeline config subdirectory containing SExtractor configuration files sex Runs SExtractor by spawning it as a child process under the shell used to run IDL to locate objects within an image shapelets_interpolate_psf Fourier interpolation of the PSF not dealt with nei ther in Volume I nor in Volume IN 68 APPENDIX A ROUTINES OF THE SHAPELETS PACKAGE shapelets_interpolate_psf_Ismatrix Fourier interpolation of the PSF not dealt with neither in Volume I nor in Volume II shapelets make image mask Create a binary mask of an image that is zero everywhere except for ones around saturated or potentially corrupted regions not used in this manual neither in Volume I nor in Volume ID shapelets_select_stars Interactively select various different object types not used in this manual neither in Volume I nor in Volume I
82. s parameters astrometry photometry The name of this structure is of the form file_name shapecat shapecat being a reference to shapelets The next section presents it 2 5 2 The shapecat catalogue It gathers lots of parameters all of them can be useful later Among them one can notably find 1 2 the number of objects shex tried to decompose n B and Nmax for each object the maximum nmax called maxn_max the position of each object x the number of coefficients n_coeffs the shapelets coefficients and their errors for each decomposed object the postage stamp and focus flags for each object 36 CHAPTER 2 SHAPELETS DECOMPOSITION 8 the reduced x for each object 9 the estimated background level for each object 10 the used CHISQ_TARGET 11 the rms of x 12 the estimated seeing 13 SExtractor parameters the name of which begins by sex for each object 14 the edges of postage stamps 15 if requested the moments of each objects flux centroid size quadrupole ellipticity and their errors This catalogue stored in filename shapecat can be later read using the shapelets_ read_shapecat pro routine and will be used for all shape and cosmic shear analysis 2 5 3 Keep only good objects If shex pro has been used without the trim failures options then the shapecat cat alogue contains all detected objects on the image That is it contains objects for which decomposition into shapelets has not been
83. t f and g be two 2 dimensional functions and h their convolution product Their shapelets coefficients respectively fn gn and hy are linked by the equation hn gt Craft 3 1 m l where the 2 dimensional convolution tensor Cami y a 8 can be written in terms of the one dimensional one Chmi y 8 Caml a b Ca Gr a Ben 7 Q p 3 2 The convolution tensor Chm depends on the scales of f g and h respectively named a 3 and y through the relation Cama eye e e 971 a7 B71 3 3 where B is defined as OO B ay a9 a3 dr Bi x a1 Bm x a2 B x a3 3 4 00 and B x a is the nth order one dimensional shapelets basis function and is given by equation 1 2 3 One can evaluate the integral expression 3 4 giving By B a 2 03 V alte rminillarasas x L v22 v2 v2 1 2 3 3 5 where we define v aj a3 a3 and 1 OO D ee f wer Ha Ele 3 6 VTJ H being an nth order Hermite polynomial By parity Limn 0 if m n lis odd Table 3 1 gives the first few non zero components of this function 3 2 CORRECTING FOR THE PSF WITH SHAPELETS 39 Loo 1 Loo2 2 2c Louw 2cb Loz 4 4b 4c 12b c Lio 4ab 12062 Lois 12bc 12bc Lo04 12 24c 126 Loos 120 360c 360c 120c Table 3 1 First few components of the normalized three product integral Limn b c other comp
84. t_chi2_grid alias for shapelets_plot_chisq_grid plt_colbar alias for shapelets_plot_colourbar plt_decomp alias for shapelets_plot_decomp plt_decomp_polar alias for shapelets_plot_decomp_polar plt_focus alias for shapelets_plot_focus plt_image alias for shapelets_plot_image plt_imstat alias for shapelets_plot_image_statitics plt_obj alias for shapelets_plot plt_polar alias for shapelets_plot_image_polar 59 60 APPENDIX A ROUTINES OF THE SHAPELETS PACKAGE plt_pstamp alias for shapelets_plot_pstamp plt_scat alias for shapelets_plot_sexcat plt_sexcat alias for shapelets_plot_sexcat plt_shcat alias for shapelets_plot_shapecat shapelets_c2p alias for shapelets_polar_convert cartesian to polar shapelets_concatenate_shapecats alias for shapelets_add two shapecats shapelets_p2c alias for shapelets_polar_convert polar to cartesian shapelets_shapecat_c2p alias for shapelets_polar_convert shapelets_shapecat_nmax Alter nmax in a shapelet catalogue obsolete shapelets_split_sexcat alias for shapelets_split shapelets_split_shapecat alias for shapelets_split shapelets_write_shapecat alias for shapelets_write A 3 Decomposition and focus decomp shapelets_chi Compute dimensionless 2D polar shapelet basis functions Xnm Z y based on Laguerre polynomials Dimensional basis functions can be calculated with shapelets_chi nm x1 beta x2 beta beta shapelets_decomp Decompose a function f into
85. ter than 3 will be rejected The pstamp structure is then the seed of all the decomposition process which must begin by finding optimal parameters for the decomposition into shapelets 2 3 Find the optimal parameters for the decom position When an object is isolated in a postage stamp and even before decomposing it into shapelets one must determine the parameters that will define it its scale parameter 8 its centroid x i e the center of the basis functions its maximum order of decomposition Nmax Indeed while 8 and x are well fixed the object shape information is contained within only the first few shapelet coefficients and that allows to truncate the expansion at Nmax A crucial but very delicate part of the shapelets decomposition is to focus these three parameters To achieve this we search the best 3 o and Nmax which minimize the difference between the observed and reconstructed images renormalized with respect to the local noise level 1 E a En X 3 number of pixels 4 o pixels P Doing so one must note that a decomposition into shapelets done with certain B and Nmax will describe objects of size within a certain range eq 1 28 26 Thus we focus 3 o and Nmax by changing them until x is minimized and putting geometrical constraints For instance the size of the eventual reconstructed image must be larger than the seeing of the image and less than the size of the postage stamp and lie between Am
86. the PSF g x con volved with f Writing Pam gt gt Cnmigi the PSF matrix equation 1 30 becomes ea Pile 1 31 It can be shown that to low enough order the PSF matrix is invertible which allows us to deconvolve the PSF and to obtain the wanted coefficients fin Fi Peli 1 32 and so to obtain the deconvolved profile f of the PSF 14 CHAPTER 1 SHAPELETS FORMALISM In practice we do not use the scheme presented above in order to avoid nu merical instabilities while inverting the matrix Pam Instead we create an a priori unconvolved model of galaxies that we convolve Eq 1 30 with a previously mea sured model of the PSF This deconvolution scheme will be further developped in chapter 3 1 6 Pixelization Real data are usually stored into pixels Instead shapelets provide an analytical decomposition of objects Therefore to eventually match the continuous analytical decomposition and the discrete data object we pixelize our analytical shapelets model Although some caution must be taken in pixelizing an analytical model 20 we make use of the fact that cartesian shapelets basis function are separable in x and y and can be analytically integrated within rectangular pixels This is similar to what really happens to photons hitting a CCD which are integrated in pixels Let the pixel in which we want to integrate our model be enclosed in the region x la bil y a gt b2 one can show that if there is n
87. this radius are skipped SG_CUT sg_cut is the threshold for SExtractor star galaxy clas sification If this is positive then only galaxies with lower class are decomposed if this is negative only stars with class greater than sg_cut are decomposed if it is zero or not set everything is decomposed LAZY PSTAMP allows not to decompose objects for which defects have been detected in their postage stamp SQUARE if it set squared postage stamps are used otherwise circular postage stamps are used NEIGHBOUR this keyword codes the treatment of neighbors The default behaviour is to do an unconstraint fit if the keyword is set neighbors pixels are set to the background level and associated errors NFWHM nfwhm is the size of the postage stamp in units of SExtractor semi major axis a TRIM_FAILURES if it is set objects with bad postage stamp or bad focus iterations are discarded and not kept in memory N_OBJECTS n_ objects is the output number of decomposed ob jects during the run of shex and written in the shape catalogue PSF psf is the PSF structure to deconvolve from the object if it is not set the object is decomposed without doing any deconvo lution SKY sky allows one to subtract sky level It can be set to 0 default not to subtract the sky 1 to fit it with a constant value around the object 2 to fit it with a plane around the object NONI if set shapelets coefficie
88. tion equation 3 10 thus the 8 we will use in further steps is the scale of the unsmeared deconvolved from the PSF galaxy We then pass this scale 3 to further routines shapelets_focus pro shapelets_ focus_beta pro shapelets_focus_nmax pro all relying on shapelets_decomp pro The goal of these routines is to search for an accurate model of the unsmeared galaxy that is of the unconvolved galaxy The deconvolution heart lies in shapelets_decomp pro where the model of the unsmeared galaxy is convolved to the PSF and compared to the observed smeared galaxy while focus is done so as we have a model as accurate as possible To achieve that one routine is essential which we describe here shapelets_ convolution_matrix pro As its name suggests it is aimed at computing the convo lution matrix Pam which links the smeared galaxy the unsmeared one and the PSF It is called as Pam shapelets convolution matrix psf alpha n max _alpha gamma n max gamma where e psfis the PSF structure to deconvolve e g given by psf_interpolation e alpha is the scale of the unconvolved object that is of the model we want to create basically it is the 3 given by shapelets_guess_ nmaxbeta e n_maz_alpha is the maximum order of the unconvolved object e gamma is the scale of the convolved observed object if it is not set it is computed using the above rules for deconvolution equation 3 10 e n_max_gamma is th
89. ts effect is rather small because large surveys are needed and because astronomical images are used to be smeared Indeed cosmic shear is particularly a statistical effect it is so small that it is absolutely impossible to detect it on a single galaxy One has to measure the shape of lots of galaxies so as to detect a shear The larger and the deeper the image the better the results Not so long ago no large surveys existed and therefore no cosmic shear could be detected Furthermore the effect of weak lensing being so small astronomers need excellent telescopes They have had some for just a few years Subaru Telescope 32 CHFT and its Megacam camera Keck Telescopes Hubble Space Telescope But the main difficulty is due to the response of the telescope and the camera the Point Spread Function PSF On the one hand a CCD camera smears images the smearing depends on the camera and can be known very accurately and easily corrected for on the other hand the atmosphere troubles images These two effects add to each other creating this so called PSF which is convolved to the real image the one one could see in space INTRODUCTION 3 with perfect camera and telescope It smears images and therefore contaminates weak lensing measurements Consequently if one wants to measure the shear on an image one must correct it for the PSF Knowing furthermore that the PSF has a 8 10 effect when cosmic shear is expected to be a 1

Shapelets - Caltech Astronomy - California Institute of Technology

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