An unbiased estimator for the ellipticity of an object in a noisy image is
given in terms of the image moments. Three assumptions are made: i) the pixel
noise is normally distributed, although with arbitrary covariance matrix, ii)
the image moments are taken about a fixed centre, and iii) the point-spread
function is known. The relevant combinations of image moments are then jointly
normal and their covariance matrix can be computed. A particular estimator for
the ratio of the means of jointly normal variates is constructed and used to
provide the unbiased estimator for the ellipticity. Furthermore, an unbiased
estimate of the covariance of the new estimator is also given.Comment: 4 pages, accepted by MNRASL; v2 contains explicit covariance matrix
  for moment

Tessore, Nicolas

English

arXiv

An unbiased estimator for the ellipticity of an object in a noisy image is given in terms of the image moments. Three assumptions are made: i) the pixel noise is normally distributed, although with arbitrary covariance matrix, ii) the image moments are taken about a fixed centre, and iii) the point-spread function is known. The relevant combinations of image moments are then jointly normal and their covariance matrix can be computed. A particular estimator for the ratio of the means of jointly normal variates is constructed and used to provide the unbiased estimator for the ellipticity. Furthermore, an unbiased estimate of the covariance of the new estimator is also given

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The University of Manchester ResearchAn unbiased estimator for the ellipticity from imagemomentsDOI:10.1093/mnrasl/slx100Document VersionFinal published versionLink to publication record in Manchester Research ExplorerCitation for published version (APA):Tessore, N. (2017). An unbiased estimator for the ellipticity from image moments. Royal Astronomical Society.Monthly Notices. Letters , 471(1), L57-L60. https://doi.org/10.1093/mnrasl/slx100Published in:Royal Astronomical Society. Monthly Notices. LettersCiting this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact openresearch@manchester.ac.uk providing relevant details, sowe can investigate your claim.Download date:31. Dec. 2025MNRAS 471, L57–L60 (2017) doi:10.1093/mnrasl/slx100Advance Access publication 2017 July 14An unbiased estimator for the ellipticity from image momentsNicolas Tessore‹Jodrell Bank Centre for Astrophysics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, UKAccepted 2017 June 14. Received 2017 May 31; in original form 2017 May 7ABSTRACTAn unbiased estimator for the ellipticity of an object in a noisy image is given in terms ofthe image moments. Three assumptions are made: (i) the pixel noise is normally distributed,although with arbitrary covariance matrix; (ii) the image moments are taken about a fixedcentre; and (iii) the point spread function is known. The relevant combinations of imagemoments are then jointly normal and their covariance matrix can be computed. A particularestimator for the ratio of the means of jointly normal variates is constructed and used to providethe unbiased estimator for the ellipticity. Furthermore, an unbiased estimate of the covarianceof the new estimator is also given.Key words: gravitational lensing: weak – methods: statistical – techniques: image processing.1 IN T RO D U C T I O NA number of applications in astronomy require the measurement ofshapes of objects from observed images. A commonly used shapedescriptor is the ellipticity χ , which is defined in terms of the centralimage moments mpq as the complex numberχ = χ1 + i χ2 = m20 − m02 + 2 i m11m20 + m02 . (1)For example, in weak gravitational lensing, the gravitational fielddistorts the observed shapes of background galaxies, and this shearcan be detected in ellipticity measurements. This is possible becausethe observed ellipticity of a source affected by gravitational lensingis (see e.g. Bartelmann & Schneider 2001)χ = χs + 2 g + g2χ∗s1 + |g|2 + 2[gχ∗s ], (2)where χ s is the intrinsic ellipticity of the source, and g = g1 + i g2is the so-called reduced shear of the gravitational lens. In the limitof weak lensing, this can be approximated to linear order asχ ≈ χs + 2 g. (3)Averaging over an ensemble of randomly oriented backgroundsources, i.e. 〈χ s〉 = 0, the weak lensing equation (3) yields〈χ〉 = 2〈g〉. (4)The observed ellipticities are thus a direct estimator for the shearg from gravitational lensing. Similar ideas are used in Cosmology(for a recent review, see Kilbinger 2015). Here, cosmic shear fromthe large-scale structure of the universe imprints a specific signatureon to the ellipticity two-point correlation functions,ξij (r) = 〈χi(x) χj ( y)〉|x− y|=r, (5) E-mail: nicolas.tessore@manchester.ac.ukwhere the average is taken over pairs of sources with the givenseparation r on the sky. Note that both the one-point function (4)and the two-point function (5) depend on the mean ellipticity overa potentially large sample of sources.In practice, an estimator χ̂ is used to measure the ellipticities ofobserved sources. In order to not introduce systematic errors intoapplications such as the above, the ellipticity estimator χ̂ must beunbiased, i.e. E[χ̂] = χ . One of the biggest problems for estimatorsthat directly work on the data is the noise bias (Refregier et al.2012), arising from pixel noise in the observations. For example,the standard approach to moment-based shape measurement is toobtain estimates m̂pq of the second-order moments from the (noisy)data and use (1) directly as an ellipticity estimate,ê = m̂20 − m̂02 + 2 i m̂11m̂20 + m̂02 . (6)The statistical properties of this estimator have been studied byMelchior & Viola (2012) and Viola, Kitching & Joachimi (2014),who assumed that the image moments are jointly normal with somegiven variance and correlation. The estimator (6) then follows thedistribution of Marsaglia (1965, 2006) for the ratio of jointly normalvariates. None of the moments of this distribution exist, and evenfor a finite sample, small values in the denominator can quicklycause significant biases and large variances. The estimator (6) isthus generally poorly behaved, unless the signal-to-noise ratio ofthe data is very high.Here, a new unbiased estimator for the ellipticity χ from thesecond-order image moments is proposed. First, it is shown thatfor normally distributed pixel noise with known covariance matrixand a fixed centre, the relevant combinations of image moments areindeed jointly normal and that their covariance matrix can easilybe computed. In the appendix, an unbiased estimator for the ratioof the means of jointly normal variates is constructed, which cansubsequently be applied to the image moments. This produces theC© 2017 The AuthorPublished by Oxford University Press on behalf of the Royal Astronomical SocietyDownloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018L58 N. Tessoreellipticity estimate, as well as unbiased estimates of its variance andcovariance.2 A N U N B I A S E D ES T I M ATO R F O R TH EELLIPTICITYIt is assumed that the data are a random vector d = (d1, d2, . . .) ofpixels following a multivariate normal distribution centred on theunknown true signal μ,d ∼ N (μ,), (7)where  is the covariance matrix for the noise, which is assumedto be known but not restricted to a particular diagonal shape. Theobserved signal usually involves a point spread function (PSF), andit is further assumed that this effect can be approximated as a linearconvolution of the true signal and the discretized PSF. In this case,definition (1) can be extended to obtain the true ellipticity of theobject before convolution,χ = m20 − m02 − m00 (π20 − π02) + 2 i m11 − 2 i m00 π11m20 + m02 − m00 (π20 + π02) , (8)from the central moments πpq of the (normalized) PSF. Fixing acentre (x̄, ȳ), the relevant combinations of moments to compute theellipticity from data d are thusu =∑iαidi , αi = wi((xi − x̄)2 − (yi − ȳ)2 − π20 + π02),v =∑iβidi , βi = wi(2 (xi − x̄) (yi − ȳ) − 2 π11),s =∑iγidi , γi = wi((xi − x̄)2 + (yi − ȳ)2 − π20 − π02),(9)where wi is the window function of the observation. To obtain thetrue ellipticity estimate of the signal, and for the PSF correction (8)to remain valid, the window function must be unity over the supportof the signal.1Due to the linearity in the pixel values di, the vector (u, v, s) canbe written in matrix form,(u, v, s) = M d, (10)where the three rows αi, β i, γ i of matrix M are defined by (9). Therandom vector (u, v, s) is hence normally distributed,(u, v, s) ∼ N (μuvs , uvs), (11)with unknown mean μuvs = (μu, μv, μs) = Mμ and known 3 × 3covariance matrix uvs = M MT with entries	uu =∑i,jαi αj 	ij , 	uv = 	vu =∑i,jαi βj 	ij ,	vv =∑i,jβi βj 	ij , 	us = 	su =∑i,jαi γj 	ij ,	ss =∑i,jγi γj 	ij , 	vs = 	sv =∑i,jβi γj 	ij , (12)where 	ij are the entries of the covariance matrix  of the pixelnoise. Hence, the covariance matrix uvs of the moments can becomputed if the pixel noise statistics are known.1 This is in contrast to the weight functions that are sometimes used inmoment-based methods to reduce the influence of noise far from the centre.The true ellipticity (8) of the signal can be written in terms of themean values of the variates u, v and s defined in (9) asχ = χ1 + i χ2 = μu + i μvμs. (13)The problem is to find an unbiased estimate of χ1 and χ2 from theobserved values of u, v and s. In Appendix A, an unbiased estimatoris given for the ratio of the means of two jointly normal randomvariables. It can be applied directly to the ellipticity (13). First, twonew variates p and q are introduced,p = u − a s, a = 	us/	ss,q = v − b s, b = 	vs/	ss, (14)where a, b are constants. This definition corresponds to (A2) inthe univariate case, and therefore both (p, s) and (q, s) are pairs ofindependent normal variates. The desired estimator for the ellipticityis thenχ̂1 = a + p ĝ(s), χ̂2 = b + q ĝ(s). (15)Because the mean μs is always positive for a realistic signal, thefunction ĝ(s) is given by (A3). From the expectationE[χ̂1] = a + E[p]E[ĝ(s)] = a + (μu − a μs)/μs = χ1,E[χ̂2] = b + E[q]E[ĝ(s)] = b + (μv − b μs)/μs = χ2, (16)it follows that χ̂ = χ̂1 + i χ̂2 is indeed an unbiased estimator for thetrue ellipticity χ of the signal.In addition, an unbiased estimate of the variance of χ̂1 and χ̂2 isprovided by (A9),V̂ar[χ̂1] = p2(ĝ(s))2 − ((p2 − 	uu)/	ss + a2)(1 − sĝ(s)),V̂ar[χ̂2] = q2(ĝ(s))2 − ((q2 − 	vv)/	ss + b2)(1 − sĝ(s)). (17)Similarly, there is an unbiased estimator for the covariance,Ĉov[χ̂1, χ̂2] = pq(ĝ(s))2 − ((pq − 	uv)/	ss + ab)(1 − sĝ(s)).(18)It follows that the individual estimates of the ellipticity componentsare in general not independent. However, for realistic pixel noise,window functions and PSFs, the correlations between u, v and s,and hence χ̂1 and χ̂2, can become very small.To demonstrate that the proposed estimator is in fact unbiasedunder the given assumptions of i) normal pixel noise with knowncovariance, ii) a fixed centre for the image moments, and iii) the dis-crete convolution with a known PSF, a Monte Carlo simulation wasperformed using mock observations of an astronomical object. Theimages are postage stamps of 49 × 49 pixels, containing a centredsource that is truncated near the image borders. A circular apertureis used as window function. The source is elliptical and followsthe light profile of de Vaucouleurs (1948). The ellipse containinghalf the total light has a 10 pixel semi-major axis and ellipticity asspecified. Where indicated, the signal is convolved with a GaussianPSF with 5 pixel FWHM. The pixel noise is uncorrelated with unitvariance. The normalization N of the object (i.e. the total number ofcounts) varies to show the effect of the signal-to-noise ratio on thevariance of the estimator.2 The signal ellipticity χ is computed from2 To compare the results to a given data set, it is then possible to scale thedata so that the noise has unit variance, and compare the number of counts.For example, the control-ground-constant data set of the GREAT3 challenge(Mandelbaum et al. 2014) has mostly N = 500–1000.MNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018An unbiased estimator for the ellipticity L59Table 1. Monte Carlo results for the unbiased ellipticity estimator.χ1 χ2 PSF counts 〈χ̂〉 〈χ̂2〉 Var[χ̂1]1/2 Var[χ̂2]1/2 〈V̂ar[χ̂1]〉1/2 〈V̂ar[χ̂2]〉1/20.1107 0.0000 no 500 0.1113 (08) 0.0006 (11) 0.762 1.112 0.762 1.1111000 0.1105 (02) − 0.0002 (02) 0.216 0.214 0.216 0.2142000 0.1108 (01) 0.0000 (01) 0.104 0.103 0.104 0.103yes 500 0.1115 (07) 0.0006 (06) 0.717 0.598 0.717 0.5991000 0.1111 (02) − 0.0001 (02) 0.216 0.214 0.215 0.2142000 0.1108 (01) 0.0000 (01) 0.104 0.103 0.104 0.1030.1820 −0.1842 no 500 0.1820 (07) − 0.1840 (09) 0.704 0.853 0.704 0.8531000 0.1817 (02) − 0.1842 (02) 0.225 0.226 0.225 0.2262000 0.1822 (01) − 0.1842 (01) 0.108 0.108 0.108 0.108yes 500 0.1802 (07) − 0.1840 (06) 0.658 0.616 0.658 0.6161000 0.1819 (02) − 0.1845 (02) 0.223 0.224 0.224 0.2252000 0.1819 (01) − 0.1841 (01) 0.108 0.108 0.108 0.1080.0000 0.5492 no 500 − 0.0005 (08) 0.5489 (15) 0.793 1.451 0.794 1.4501000 − 0.0001 (02) 0.5494 (03) 0.248 0.323 0.248 0.3222000 − 0.0002 (01) 0.5493 (01) 0.117 0.149 0.117 0.149yes 500 0.0001 (07) 0.5473 (10) 0.720 0.957 0.720 0.9601000 0.0002 (02) 0.5488 (03) 0.244 0.315 0.244 0.3162000 0.0001 (01) 0.5491 (01) 0.116 0.147 0.116 0.147the image before the PSF is applied and noise is added. The meanof the estimator χ̂ is computed from 106 realisations of noise forthe same signal. Also computed are the square root of the samplevariance Var[χ̂] and the mean of the estimated variance V̂ar[χ̂], re-spectively. The results shown in Table 1 indicate that the estimatorperforms as expected.3 C O N C L U S I O N A N D D I S C U S S I O NThe unbiased estimator (15) provides a new method for elliptic-ity measurement from noisy images. It’s simple and analytic formallows quick implementation and fast evaluation, and statistics forthe results can be obtained directly with unbiased estimates of thevariance (17) and covariance (18).Using an unbiased estimator for the ellipticity of the signal μeliminates the influence of noise from a subsequent analysis ofthe results (the so-called ‘noise bias’ in weak lensing, Refregieret al. 2012). However, depending on the application, other kinds ofbiases might exist even for a noise-free image. For example, due tothe discretization of the image, the signal ellipticity can differ fromthe intrinsic ellipticity of the observed object. This ‘pixelation bias’(Simon & Schneider 2016) remains an issue in applications suchas weak lensing, where the relevant effects must be measured fromthe intrinsic ellipticity of the objects.Furthermore, a fixed centre (x̄, ȳ) for the moments has beenassumed throughout. For a correct ellipticity estimate, this must bethe centroid of the signal, which is usually estimated from the dataitself. Centroid errors (Melchior & Viola 2012) might thereforeultimately bias the ellipticity estimator or increase its variance,although this currently does not seem to be a significant effect.In practice, additional biases might arise when the assumed re-quirements for the window function and PSF are not fulfilled by thedata. The estimator should therefore always be carefully tested forthe application at hand.Lastly, the ellipticity estimate might be improved by suitablefiltering of the observed image. A linear filter with matrix A canbe applied to the image before estimating the ellipticity, since thetransformed pixels remain multivariate normal with mean μ′ = Aμand covariance matrix ′ = A AT. Examples of viable filters arenearest neighbour or bilinear interpolation, as well as convolution.A combination of these filters could be used to perform PSF decon-volution on the observed image, as an alternative to the algebraicPSF correction (8).AC K N OW L E D G E M E N T SNT would like to thank S. Bridle for encouragement and manyconversations about shape measurement. The author acknowledgessupport from the European Research Council in the form of a Con-solidator Grant with number 681431.R E F E R E N C E SBartelmann M., Schneider P., 2001, Phys. Rep., 340, 291de Vaucouleurs G., 1948, Ann. Astrophys., 11, 247Kilbinger M., 2015, Rep. Prog. Phys., 78, 086901Mandelbaum R. et al., 2014, ApJS, 212, 5Marsaglia G., 1965, J. Am. Stat. Assoc., 60, 193Marsaglia G., 2006, J. Stat. Softw., 16, 1Melchior P., Viola M., 2012, MNRAS, 424, 2757Refregier A., Kacprzak T., Amara A., Bridle S., Rowe B., 2012, MNRAS,425, 1951Simon P., Schneider P., 2016, A&A, preprint (arXiv:1609.07937)Viola M., Kitching T. D., Joachimi B., 2014, MNRAS, 439, 1909Voinov V. G., 1985, Sankhyā Ser. B, 47, 354APPENDI X A : A RATI O ESTI MATOR FO RN O R M A L VA R I AT E SLet x and y be jointly normal variates with unknown means μx andμy, and known variances σ 2x and σ2y and correlation ρ. The goalhere is to find an unbiased estimate of the ratio r of their means,r = μx/μy. (A1)Under the additional assumption that the sign of μy is known, anunbiased estimator for r can be found in two short steps.First, the transformation of Marsaglia (1965, 2006) is used toconstruct a new variate,w = x − c y, c = ρ σx/σy. (A2)The constant c is chosen so that E[wy] = E[w]E[y]. It is clear that wis normal, and that variates w and y are jointly normal, uncorrelatedMNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018L60 N. Tessoreand thus independent. Note that c = 0 and w = x for independent xand y.Secondly, Voinov (1985) derived an unbiased estimator for theinverse mean of the normal variate y, i.e. a function ĝ(y) withE[ĝ(y)] = 1/μy . For the relevant case of an unknown but positivemean μy > 0, this function is given byĝ(y) =√π√2 σyexp(y22 σ 2y)erfc(y√2 σy). (A3)It is then straightforward to construct an estimator for the ratio(A1),r̂ = c + w ĝ(y). (A4)Since w and y are independent, the expectation isE[r̂] = c + E[w]E[ĝ(y)] = c + (μx − c μy)/μy = μx/μy, (A5)which shows that r̂ is in fact an unbiased estimator for r.The variance of the ratio estimator r̂ is formally given byVar[r̂] = (E[w]2 + Var[w])Var[ĝ(y)] + Var[w] E[ĝ(y)]2. (A6)As pointed out by Voinov (1985), the variance Var[ĝ(y)] does notexist for function (A3) due to a divergence at infinity. The confidenceinterval h with probability p,Pr(|ĝ(y) − 1/μy | < h) = p, (A7)however, is well-defined, and the variance of ĝ(y) remains finite inapplications where infinite values of y are not observed. In this case,an unbiased estimatorV̂ar[ĝ(y)] = (ĝ(y))2 − (1 − yĝ(y))/σ 2y (A8)exists and, together with Voinov’s estimator for 1/μ2y , yieldsV̂ar[r̂] = w2 (ĝ(y))2 − ((w2 − σ 2x ) /σ 2y + c2) (1 − yĝ(y)) (A9)as an unbiased estimate of the variance of the estimator r̂ .When y is significantly larger than its standard deviation, e.g.y/σ y > 10, the function ĝ(y) given by (A3) is susceptible to numer-ical overflow. However, in this regime, it is also very well approxi-mated by its series expansion,ĝ(y) ≈ 1/y − σ 2y /y3 + 3σ 4y /y5, y/σy > 10. (A10)For even larger values y  σ y, this approaches 1/y, as expected.This paper has been typeset from a TEX/LATEX file prepared by the author.MNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018

An unbiased estimator for the ellipticity from image moments

The University of Manchester ResearchAn unbiased estimator for the ellipticity from imagemomentsDOI:10.1093/mnrasl/slx100Document VersionFinal published versionLink to publication record in Manchester Research ExplorerCitation for published version (APA):Tessore, N. (2017). An unbiased estimator for the ellipticity from image moments. Royal Astronomical Society.Monthly Notices. Letters , 471(1), L57-L60. https://doi.org/10.1093/mnrasl/slx100Published in:Royal Astronomical Society. Monthly Notices. LettersCiting this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:12. Nov. 2022MNRAS 471, L57–L60 (2017) doi:10.1093/mnrasl/slx100Advance Access publication 2017 July 14An unbiased estimator for the ellipticity from image momentsNicolas Tessore‹Jodrell Bank Centre for Astrophysics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, UKAccepted 2017 June 14. Received 2017 May 31; in original form 2017 May 7ABSTRACTAn unbiased estimator for the ellipticity of an object in a noisy image is given in terms ofthe image moments. Three assumptions are made: (i) the pixel noise is normally distributed,although with arbitrary covariance matrix; (ii) the image moments are taken about a fixedcentre; and (iii) the point spread function is known. The relevant combinations of imagemoments are then jointly normal and their covariance matrix can be computed. A particularestimator for the ratio of the means of jointly normal variates is constructed and used to providethe unbiased estimator for the ellipticity. Furthermore, an unbiased estimate of the covarianceof the new estimator is also given.Key words: gravitational lensing: weak – methods: statistical – techniques: image processing.1 IN T RO D U C T I O NA number of applications in astronomy require the measurement ofshapes of objects from observed images. A commonly used shapedescriptor is the ellipticity χ , which is defined in terms of the centralimage moments mpq as the complex numberχ = χ1 + i χ2 = m20 − m02 + 2 i m11m20 + m02 . (1)For example, in weak gravitational lensing, the gravitational fielddistorts the observed shapes of background galaxies, and this shearcan be detected in ellipticity measurements. This is possible becausethe observed ellipticity of a source affected by gravitational lensingis (see e.g. Bartelmann & Schneider 2001)χ = χs + 2 g + g2χ∗s1 + |g|2 + 2[gχ∗s ], (2)where χ s is the intrinsic ellipticity of the source, and g = g1 + i g2is the so-called reduced shear of the gravitational lens. In the limitof weak lensing, this can be approximated to linear order asχ ≈ χs + 2 g. (3)Averaging over an ensemble of randomly oriented backgroundsources, i.e. 〈χ s〉 = 0, the weak lensing equation (3) yields〈χ〉 = 2〈g〉. (4)The observed ellipticities are thus a direct estimator for the shearg from gravitational lensing. Similar ideas are used in Cosmology(for a recent review, see Kilbinger 2015). Here, cosmic shear fromthe large-scale structure of the universe imprints a specific signatureon to the ellipticity two-point correlation functions,ξij (r) = 〈χi(x) χj ( y)〉|x− y|=r, (5) E-mail: nicolas.tessore@manchester.ac.ukwhere the average is taken over pairs of sources with the givenseparation r on the sky. Note that both the one-point function (4)and the two-point function (5) depend on the mean ellipticity overa potentially large sample of sources.In practice, an estimator χ̂ is used to measure the ellipticities ofobserved sources. In order to not introduce systematic errors intoapplications such as the above, the ellipticity estimator χ̂ must beunbiased, i.e. E[χ̂] = χ . One of the biggest problems for estimatorsthat directly work on the data is the noise bias (Refregier et al.2012), arising from pixel noise in the observations. For example,the standard approach to moment-based shape measurement is toobtain estimates m̂pq of the second-order moments from the (noisy)data and use (1) directly as an ellipticity estimate,ê = m̂20 − m̂02 + 2 i m̂11m̂20 + m̂02 . (6)The statistical properties of this estimator have been studied byMelchior & Viola (2012) and Viola, Kitching & Joachimi (2014),who assumed that the image moments are jointly normal with somegiven variance and correlation. The estimator (6) then follows thedistribution of Marsaglia (1965, 2006) for the ratio of jointly normalvariates. None of the moments of this distribution exist, and evenfor a finite sample, small values in the denominator can quicklycause significant biases and large variances. The estimator (6) isthus generally poorly behaved, unless the signal-to-noise ratio ofthe data is very high.Here, a new unbiased estimator for the ellipticity χ from thesecond-order image moments is proposed. First, it is shown thatfor normally distributed pixel noise with known covariance matrixand a fixed centre, the relevant combinations of image moments areindeed jointly normal and that their covariance matrix can easilybe computed. In the appendix, an unbiased estimator for the ratioof the means of jointly normal variates is constructed, which cansubsequently be applied to the image moments. This produces theC© 2017 The AuthorPublished by Oxford University Press on behalf of the Royal Astronomical SocietyDownloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018L58 N. Tessoreellipticity estimate, as well as unbiased estimates of its variance andcovariance.2 A N U N B I A S E D ES T I M ATO R F O R TH EELLIPTICITYIt is assumed that the data are a random vector d = (d1, d2, . . .) ofpixels following a multivariate normal distribution centred on theunknown true signal μ,d ∼ N (μ,), (7)where  is the covariance matrix for the noise, which is assumedto be known but not restricted to a particular diagonal shape. Theobserved signal usually involves a point spread function (PSF), andit is further assumed that this effect can be approximated as a linearconvolution of the true signal and the discretized PSF. In this case,definition (1) can be extended to obtain the true ellipticity of theobject before convolution,χ = m20 − m02 − m00 (π20 − π02) + 2 i m11 − 2 i m00 π11m20 + m02 − m00 (π20 + π02) , (8)from the central moments πpq of the (normalized) PSF. Fixing acentre (x̄, ȳ), the relevant combinations of moments to compute theellipticity from data d are thusu =∑iαidi , αi = wi((xi − x̄)2 − (yi − ȳ)2 − π20 + π02),v =∑iβidi , βi = wi(2 (xi − x̄) (yi − ȳ) − 2 π11),s =∑iγidi , γi = wi((xi − x̄)2 + (yi − ȳ)2 − π20 − π02),(9)where wi is the window function of the observation. To obtain thetrue ellipticity estimate of the signal, and for the PSF correction (8)to remain valid, the window function must be unity over the supportof the signal.1Due to the linearity in the pixel values di, the vector (u, v, s) canbe written in matrix form,(u, v, s) = M d, (10)where the three rows αi, β i, γ i of matrix M are defined by (9). Therandom vector (u, v, s) is hence normally distributed,(u, v, s) ∼ N (μuvs , uvs), (11)with unknown mean μuvs = (μu, μv, μs) = Mμ and known 3 × 3covariance matrix uvs = M MT with entries	uu =∑i,jαi αj 	ij , 	uv = 	vu =∑i,jαi βj 	ij ,	vv =∑i,jβi βj 	ij , 	us = 	su =∑i,jαi γj 	ij ,	ss =∑i,jγi γj 	ij , 	vs = 	sv =∑i,jβi γj 	ij , (12)where 	ij are the entries of the covariance matrix  of the pixelnoise. Hence, the covariance matrix uvs of the moments can becomputed if the pixel noise statistics are known.1 This is in contrast to the weight functions that are sometimes used inmoment-based methods to reduce the influence of noise far from the centre.The true ellipticity (8) of the signal can be written in terms of themean values of the variates u, v and s defined in (9) asχ = χ1 + i χ2 = μu + i μvμs. (13)The problem is to find an unbiased estimate of χ1 and χ2 from theobserved values of u, v and s. In Appendix A, an unbiased estimatoris given for the ratio of the means of two jointly normal randomvariables. It can be applied directly to the ellipticity (13). First, twonew variates p and q are introduced,p = u − a s, a = 	us/	ss,q = v − b s, b = 	vs/	ss, (14)where a, b are constants. This definition corresponds to (A2) inthe univariate case, and therefore both (p, s) and (q, s) are pairs ofindependent normal variates. The desired estimator for the ellipticityis thenχ̂1 = a + p ĝ(s), χ̂2 = b + q ĝ(s). (15)Because the mean μs is always positive for a realistic signal, thefunction ĝ(s) is given by (A3). From the expectationE[χ̂1] = a + E[p]E[ĝ(s)] = a + (μu − a μs)/μs = χ1,E[χ̂2] = b + E[q]E[ĝ(s)] = b + (μv − b μs)/μs = χ2, (16)it follows that χ̂ = χ̂1 + i χ̂2 is indeed an unbiased estimator for thetrue ellipticity χ of the signal.In addition, an unbiased estimate of the variance of χ̂1 and χ̂2 isprovided by (A9),V̂ar[χ̂1] = p2(ĝ(s))2 − ((p2 − 	uu)/	ss + a2)(1 − sĝ(s)),V̂ar[χ̂2] = q2(ĝ(s))2 − ((q2 − 	vv)/	ss + b2)(1 − sĝ(s)). (17)Similarly, there is an unbiased estimator for the covariance,Ĉov[χ̂1, χ̂2] = pq(ĝ(s))2 − ((pq − 	uv)/	ss + ab)(1 − sĝ(s)).(18)It follows that the individual estimates of the ellipticity componentsare in general not independent. However, for realistic pixel noise,window functions and PSFs, the correlations between u, v and s,and hence χ̂1 and χ̂2, can become very small.To demonstrate that the proposed estimator is in fact unbiasedunder the given assumptions of i) normal pixel noise with knowncovariance, ii) a fixed centre for the image moments, and iii) the dis-crete convolution with a known PSF, a Monte Carlo simulation wasperformed using mock observations of an astronomical object. Theimages are postage stamps of 49 × 49 pixels, containing a centredsource that is truncated near the image borders. A circular apertureis used as window function. The source is elliptical and followsthe light profile of de Vaucouleurs (1948). The ellipse containinghalf the total light has a 10 pixel semi-major axis and ellipticity asspecified. Where indicated, the signal is convolved with a GaussianPSF with 5 pixel FWHM. The pixel noise is uncorrelated with unitvariance. The normalization N of the object (i.e. the total number ofcounts) varies to show the effect of the signal-to-noise ratio on thevariance of the estimator.2 The signal ellipticity χ is computed from2 To compare the results to a given data set, it is then possible to scale thedata so that the noise has unit variance, and compare the number of counts.For example, the control-ground-constant data set of the GREAT3 challenge(Mandelbaum et al. 2014) has mostly N = 500–1000.MNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018An unbiased estimator for the ellipticity L59Table 1. Monte Carlo results for the unbiased ellipticity estimator.χ1 χ2 PSF counts 〈χ̂〉 〈χ̂2〉 Var[χ̂1]1/2 Var[χ̂2]1/2 〈V̂ar[χ̂1]〉1/2 〈V̂ar[χ̂2]〉1/20.1107 0.0000 no 500 0.1113 (08) 0.0006 (11) 0.762 1.112 0.762 1.1111000 0.1105 (02) − 0.0002 (02) 0.216 0.214 0.216 0.2142000 0.1108 (01) 0.0000 (01) 0.104 0.103 0.104 0.103yes 500 0.1115 (07) 0.0006 (06) 0.717 0.598 0.717 0.5991000 0.1111 (02) − 0.0001 (02) 0.216 0.214 0.215 0.2142000 0.1108 (01) 0.0000 (01) 0.104 0.103 0.104 0.1030.1820 −0.1842 no 500 0.1820 (07) − 0.1840 (09) 0.704 0.853 0.704 0.8531000 0.1817 (02) − 0.1842 (02) 0.225 0.226 0.225 0.2262000 0.1822 (01) − 0.1842 (01) 0.108 0.108 0.108 0.108yes 500 0.1802 (07) − 0.1840 (06) 0.658 0.616 0.658 0.6161000 0.1819 (02) − 0.1845 (02) 0.223 0.224 0.224 0.2252000 0.1819 (01) − 0.1841 (01) 0.108 0.108 0.108 0.1080.0000 0.5492 no 500 − 0.0005 (08) 0.5489 (15) 0.793 1.451 0.794 1.4501000 − 0.0001 (02) 0.5494 (03) 0.248 0.323 0.248 0.3222000 − 0.0002 (01) 0.5493 (01) 0.117 0.149 0.117 0.149yes 500 0.0001 (07) 0.5473 (10) 0.720 0.957 0.720 0.9601000 0.0002 (02) 0.5488 (03) 0.244 0.315 0.244 0.3162000 0.0001 (01) 0.5491 (01) 0.116 0.147 0.116 0.147the image before the PSF is applied and noise is added. The meanof the estimator χ̂ is computed from 106 realisations of noise forthe same signal. Also computed are the square root of the samplevariance Var[χ̂] and the mean of the estimated variance V̂ar[χ̂], re-spectively. The results shown in Table 1 indicate that the estimatorperforms as expected.3 C O N C L U S I O N A N D D I S C U S S I O NThe unbiased estimator (15) provides a new method for elliptic-ity measurement from noisy images. It’s simple and analytic formallows quick implementation and fast evaluation, and statistics forthe results can be obtained directly with unbiased estimates of thevariance (17) and covariance (18).Using an unbiased estimator for the ellipticity of the signal μeliminates the influence of noise from a subsequent analysis ofthe results (the so-called ‘noise bias’ in weak lensing, Refregieret al. 2012). However, depending on the application, other kinds ofbiases might exist even for a noise-free image. For example, due tothe discretization of the image, the signal ellipticity can differ fromthe intrinsic ellipticity of the observed object. This ‘pixelation bias’(Simon & Schneider 2016) remains an issue in applications suchas weak lensing, where the relevant effects must be measured fromthe intrinsic ellipticity of the objects.Furthermore, a fixed centre (x̄, ȳ) for the moments has beenassumed throughout. For a correct ellipticity estimate, this must bethe centroid of the signal, which is usually estimated from the dataitself. Centroid errors (Melchior & Viola 2012) might thereforeultimately bias the ellipticity estimator or increase its variance,although this currently does not seem to be a significant effect.In practice, additional biases might arise when the assumed re-quirements for the window function and PSF are not fulfilled by thedata. The estimator should therefore always be carefully tested forthe application at hand.Lastly, the ellipticity estimate might be improved by suitablefiltering of the observed image. A linear filter with matrix A canbe applied to the image before estimating the ellipticity, since thetransformed pixels remain multivariate normal with mean μ′ = Aμand covariance matrix ′ = A AT. Examples of viable filters arenearest neighbour or bilinear interpolation, as well as convolution.A combination of these filters could be used to perform PSF decon-volution on the observed image, as an alternative to the algebraicPSF correction (8).AC K N OW L E D G E M E N T SNT would like to thank S. Bridle for encouragement and manyconversations about shape measurement. The author acknowledgessupport from the European Research Council in the form of a Con-solidator Grant with number 681431.R E F E R E N C E SBartelmann M., Schneider P., 2001, Phys. Rep., 340, 291de Vaucouleurs G., 1948, Ann. Astrophys., 11, 247Kilbinger M., 2015, Rep. Prog. Phys., 78, 086901Mandelbaum R. et al., 2014, ApJS, 212, 5Marsaglia G., 1965, J. Am. Stat. Assoc., 60, 193Marsaglia G., 2006, J. Stat. Softw., 16, 1Melchior P., Viola M., 2012, MNRAS, 424, 2757Refregier A., Kacprzak T., Amara A., Bridle S., Rowe B., 2012, MNRAS,425, 1951Simon P., Schneider P., 2016, A&A, preprint (arXiv:1609.07937)Viola M., Kitching T. D., Joachimi B., 2014, MNRAS, 439, 1909Voinov V. G., 1985, Sankhyā Ser. B, 47, 354APPENDI X A : A RATI O ESTI MATOR FO RN O R M A L VA R I AT E SLet x and y be jointly normal variates with unknown means μx andμy, and known variances σ 2x and σ2y and correlation ρ. The goalhere is to find an unbiased estimate of the ratio r of their means,r = μx/μy. (A1)Under the additional assumption that the sign of μy is known, anunbiased estimator for r can be found in two short steps.First, the transformation of Marsaglia (1965, 2006) is used toconstruct a new variate,w = x − c y, c = ρ σx/σy. (A2)The constant c is chosen so that E[wy] = E[w]E[y]. It is clear that wis normal, and that variates w and y are jointly normal, uncorrelatedMNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018L60 N. Tessoreand thus independent. Note that c = 0 and w = x for independent xand y.Secondly, Voinov (1985) derived an unbiased estimator for theinverse mean of the normal variate y, i.e. a function ĝ(y) withE[ĝ(y)] = 1/μy . For the relevant case of an unknown but positivemean μy > 0, this function is given byĝ(y) =√π√2 σyexp(y22 σ 2y)erfc(y√2 σy). (A3)It is then straightforward to construct an estimator for the ratio(A1),r̂ = c + w ĝ(y). (A4)Since w and y are independent, the expectation isE[r̂] = c + E[w]E[ĝ(y)] = c + (μx − c μy)/μy = μx/μy, (A5)which shows that r̂ is in fact an unbiased estimator for r.The variance of the ratio estimator r̂ is formally given byVar[r̂] = (E[w]2 + Var[w])Var[ĝ(y)] + Var[w] E[ĝ(y)]2. (A6)As pointed out by Voinov (1985), the variance Var[ĝ(y)] does notexist for function (A3) due to a divergence at infinity. The confidenceinterval h with probability p,Pr(|ĝ(y) − 1/μy | < h) = p, (A7)however, is well-defined, and the variance of ĝ(y) remains finite inapplications where infinite values of y are not observed. In this case,an unbiased estimatorV̂ar[ĝ(y)] = (ĝ(y))2 − (1 − yĝ(y))/σ 2y (A8)exists and, together with Voinov’s estimator for 1/μ2y , yieldsV̂ar[r̂] = w2 (ĝ(y))2 − ((w2 − σ 2x ) /σ 2y + c2) (1 − yĝ(y)) (A9)as an unbiased estimate of the variance of the estimator r̂ .When y is significantly larger than its standard deviation, e.g.y/σ y > 10, the function ĝ(y) given by (A3) is susceptible to numer-ical overflow. However, in this regime, it is also very well approxi-mated by its series expansion,ĝ(y) ≈ 1/y − σ 2y /y3 + 3σ 4y /y5, y/σy > 10. (A10)For even larger values y  σ y, this approaches 1/y, as expected.This paper has been typeset from a TEX/LATEX file prepared by the author.MNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018

Nicolas Tessore

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Monthly Notices of the Royal Astronomical Society Letters

The University of Manchester ResearchAn unbiased estimator for the ellipticity from imagemomentsDOI:10.1093/mnrasl/slx100Document VersionFinal published versionLink to publication record in Manchester Research ExplorerCitation for published version (APA):Tessore, N. (2017). An unbiased estimator for the ellipticity from image moments. Royal Astronomical Society.Monthly Notices. Letters , 471(1), L57-L60. https://doi.org/10.1093/mnrasl/slx100Published in:Royal Astronomical Society. Monthly Notices. LettersCiting this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact openresearch@manchester.ac.uk providing relevant details, sowe can investigate your claim.Download date:15. Oct. 2025MNRAS 471, L57–L60 (2017) doi:10.1093/mnrasl/slx100Advance Access publication 2017 July 14An unbiased estimator for the ellipticity from image momentsNicolas Tessore‹Jodrell Bank Centre for Astrophysics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, UKAccepted 2017 June 14. Received 2017 May 31; in original form 2017 May 7ABSTRACTAn unbiased estimator for the ellipticity of an object in a noisy image is given in terms ofthe image moments. Three assumptions are made: (i) the pixel noise is normally distributed,although with arbitrary covariance matrix; (ii) the image moments are taken about a fixedcentre; and (iii) the point spread function is known. The relevant combinations of imagemoments are then jointly normal and their covariance matrix can be computed. A particularestimator for the ratio of the means of jointly normal variates is constructed and used to providethe unbiased estimator for the ellipticity. Furthermore, an unbiased estimate of the covarianceof the new estimator is also given.Key words: gravitational lensing: weak – methods: statistical – techniques: image processing.1 IN T RO D U C T I O NA number of applications in astronomy require the measurement ofshapes of objects from observed images. A commonly used shapedescriptor is the ellipticity χ , which is defined in terms of the centralimage moments mpq as the complex numberχ = χ1 + i χ2 = m20 − m02 + 2 i m11m20 + m02 . (1)For example, in weak gravitational lensing, the gravitational fielddistorts the observed shapes of background galaxies, and this shearcan be detected in ellipticity measurements. This is possible becausethe observed ellipticity of a source affected by gravitational lensingis (see e.g. Bartelmann & Schneider 2001)χ = χs + 2 g + g2χ∗s1 + |g|2 + 2[gχ∗s ], (2)where χ s is the intrinsic ellipticity of the source, and g = g1 + i g2is the so-called reduced shear of the gravitational lens. In the limitof weak lensing, this can be approximated to linear order asχ ≈ χs + 2 g. (3)Averaging over an ensemble of randomly oriented backgroundsources, i.e. 〈χ s〉 = 0, the weak lensing equation (3) yields〈χ〉 = 2〈g〉. (4)The observed ellipticities are thus a direct estimator for the shearg from gravitational lensing. Similar ideas are used in Cosmology(for a recent review, see Kilbinger 2015). Here, cosmic shear fromthe large-scale structure of the universe imprints a specific signatureon to the ellipticity two-point correlation functions,ξij (r) = 〈χi(x) χj ( y)〉|x− y|=r, (5) E-mail: nicolas.tessore@manchester.ac.ukwhere the average is taken over pairs of sources with the givenseparation r on the sky. Note that both the one-point function (4)and the two-point function (5) depend on the mean ellipticity overa potentially large sample of sources.In practice, an estimator χ̂ is used to measure the ellipticities ofobserved sources. In order to not introduce systematic errors intoapplications such as the above, the ellipticity estimator χ̂ must beunbiased, i.e. E[χ̂] = χ . One of the biggest problems for estimatorsthat directly work on the data is the noise bias (Refregier et al.2012), arising from pixel noise in the observations. For example,the standard approach to moment-based shape measurement is toobtain estimates m̂pq of the second-order moments from the (noisy)data and use (1) directly as an ellipticity estimate,ê = m̂20 − m̂02 + 2 i m̂11m̂20 + m̂02 . (6)The statistical properties of this estimator have been studied byMelchior & Viola (2012) and Viola, Kitching & Joachimi (2014),who assumed that the image moments are jointly normal with somegiven variance and correlation. The estimator (6) then follows thedistribution of Marsaglia (1965, 2006) for the ratio of jointly normalvariates. None of the moments of this distribution exist, and evenfor a finite sample, small values in the denominator can quicklycause significant biases and large variances. The estimator (6) isthus generally poorly behaved, unless the signal-to-noise ratio ofthe data is very high.Here, a new unbiased estimator for the ellipticity χ from thesecond-order image moments is proposed. First, it is shown thatfor normally distributed pixel noise with known covariance matrixand a fixed centre, the relevant combinations of image moments areindeed jointly normal and that their covariance matrix can easilybe computed. In the appendix, an unbiased estimator for the ratioof the means of jointly normal variates is constructed, which cansubsequently be applied to the image moments. This produces theC© 2017 The AuthorPublished by Oxford University Press on behalf of the Royal Astronomical SocietyDownloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018L58 N. Tessoreellipticity estimate, as well as unbiased estimates of its variance andcovariance.2 A N U N B I A S E D ES T I M ATO R F O R TH EELLIPTICITYIt is assumed that the data are a random vector d = (d1, d2, . . .) ofpixels following a multivariate normal distribution centred on theunknown true signal μ,d ∼ N (μ,), (7)where  is the covariance matrix for the noise, which is assumedto be known but not restricted to a particular diagonal shape. Theobserved signal usually involves a point spread function (PSF), andit is further assumed that this effect can be approximated as a linearconvolution of the true signal and the discretized PSF. In this case,definition (1) can be extended to obtain the true ellipticity of theobject before convolution,χ = m20 − m02 − m00 (π20 − π02) + 2 i m11 − 2 i m00 π11m20 + m02 − m00 (π20 + π02) , (8)from the central moments πpq of the (normalized) PSF. Fixing acentre (x̄, ȳ), the relevant combinations of moments to compute theellipticity from data d are thusu =∑iαidi , αi = wi((xi − x̄)2 − (yi − ȳ)2 − π20 + π02),v =∑iβidi , βi = wi(2 (xi − x̄) (yi − ȳ) − 2 π11),s =∑iγidi , γi = wi((xi − x̄)2 + (yi − ȳ)2 − π20 − π02),(9)where wi is the window function of the observation. To obtain thetrue ellipticity estimate of the signal, and for the PSF correction (8)to remain valid, the window function must be unity over the supportof the signal.1Due to the linearity in the pixel values di, the vector (u, v, s) canbe written in matrix form,(u, v, s) = M d, (10)where the three rows αi, β i, γ i of matrix M are defined by (9). Therandom vector (u, v, s) is hence normally distributed,(u, v, s) ∼ N (μuvs , uvs), (11)with unknown mean μuvs = (μu, μv, μs) = Mμ and known 3 × 3covariance matrix uvs = M MT with entries	uu =∑i,jαi αj 	ij , 	uv = 	vu =∑i,jαi βj 	ij ,	vv =∑i,jβi βj 	ij , 	us = 	su =∑i,jαi γj 	ij ,	ss =∑i,jγi γj 	ij , 	vs = 	sv =∑i,jβi γj 	ij , (12)where 	ij are the entries of the covariance matrix  of the pixelnoise. Hence, the covariance matrix uvs of the moments can becomputed if the pixel noise statistics are known.1 This is in contrast to the weight functions that are sometimes used inmoment-based methods to reduce the influence of noise far from the centre.The true ellipticity (8) of the signal can be written in terms of themean values of the variates u, v and s defined in (9) asχ = χ1 + i χ2 = μu + i μvμs. (13)The problem is to find an unbiased estimate of χ1 and χ2 from theobserved values of u, v and s. In Appendix A, an unbiased estimatoris given for the ratio of the means of two jointly normal randomvariables. It can be applied directly to the ellipticity (13). First, twonew variates p and q are introduced,p = u − a s, a = 	us/	ss,q = v − b s, b = 	vs/	ss, (14)where a, b are constants. This definition corresponds to (A2) inthe univariate case, and therefore both (p, s) and (q, s) are pairs ofindependent normal variates. The desired estimator for the ellipticityis thenχ̂1 = a + p ĝ(s), χ̂2 = b + q ĝ(s). (15)Because the mean μs is always positive for a realistic signal, thefunction ĝ(s) is given by (A3). From the expectationE[χ̂1] = a + E[p]E[ĝ(s)] = a + (μu − a μs)/μs = χ1,E[χ̂2] = b + E[q]E[ĝ(s)] = b + (μv − b μs)/μs = χ2, (16)it follows that χ̂ = χ̂1 + i χ̂2 is indeed an unbiased estimator for thetrue ellipticity χ of the signal.In addition, an unbiased estimate of the variance of χ̂1 and χ̂2 isprovided by (A9),V̂ar[χ̂1] = p2(ĝ(s))2 − ((p2 − 	uu)/	ss + a2)(1 − sĝ(s)),V̂ar[χ̂2] = q2(ĝ(s))2 − ((q2 − 	vv)/	ss + b2)(1 − sĝ(s)). (17)Similarly, there is an unbiased estimator for the covariance,Ĉov[χ̂1, χ̂2] = pq(ĝ(s))2 − ((pq − 	uv)/	ss + ab)(1 − sĝ(s)).(18)It follows that the individual estimates of the ellipticity componentsare in general not independent. However, for realistic pixel noise,window functions and PSFs, the correlations between u, v and s,and hence χ̂1 and χ̂2, can become very small.To demonstrate that the proposed estimator is in fact unbiasedunder the given assumptions of i) normal pixel noise with knowncovariance, ii) a fixed centre for the image moments, and iii) the dis-crete convolution with a known PSF, a Monte Carlo simulation wasperformed using mock observations of an astronomical object. Theimages are postage stamps of 49 × 49 pixels, containing a centredsource that is truncated near the image borders. A circular apertureis used as window function. The source is elliptical and followsthe light profile of de Vaucouleurs (1948). The ellipse containinghalf the total light has a 10 pixel semi-major axis and ellipticity asspecified. Where indicated, the signal is convolved with a GaussianPSF with 5 pixel FWHM. The pixel noise is uncorrelated with unitvariance. The normalization N of the object (i.e. the total number ofcounts) varies to show the effect of the signal-to-noise ratio on thevariance of the estimator.2 The signal ellipticity χ is computed from2 To compare the results to a given data set, it is then possible to scale thedata so that the noise has unit variance, and compare the number of counts.For example, the control-ground-constant data set of the GREAT3 challenge(Mandelbaum et al. 2014) has mostly N = 500–1000.MNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018An unbiased estimator for the ellipticity L59Table 1. Monte Carlo results for the unbiased ellipticity estimator.χ1 χ2 PSF counts 〈χ̂〉 〈χ̂2〉 Var[χ̂1]1/2 Var[χ̂2]1/2 〈V̂ar[χ̂1]〉1/2 〈V̂ar[χ̂2]〉1/20.1107 0.0000 no 500 0.1113 (08) 0.0006 (11) 0.762 1.112 0.762 1.1111000 0.1105 (02) − 0.0002 (02) 0.216 0.214 0.216 0.2142000 0.1108 (01) 0.0000 (01) 0.104 0.103 0.104 0.103yes 500 0.1115 (07) 0.0006 (06) 0.717 0.598 0.717 0.5991000 0.1111 (02) − 0.0001 (02) 0.216 0.214 0.215 0.2142000 0.1108 (01) 0.0000 (01) 0.104 0.103 0.104 0.1030.1820 −0.1842 no 500 0.1820 (07) − 0.1840 (09) 0.704 0.853 0.704 0.8531000 0.1817 (02) − 0.1842 (02) 0.225 0.226 0.225 0.2262000 0.1822 (01) − 0.1842 (01) 0.108 0.108 0.108 0.108yes 500 0.1802 (07) − 0.1840 (06) 0.658 0.616 0.658 0.6161000 0.1819 (02) − 0.1845 (02) 0.223 0.224 0.224 0.2252000 0.1819 (01) − 0.1841 (01) 0.108 0.108 0.108 0.1080.0000 0.5492 no 500 − 0.0005 (08) 0.5489 (15) 0.793 1.451 0.794 1.4501000 − 0.0001 (02) 0.5494 (03) 0.248 0.323 0.248 0.3222000 − 0.0002 (01) 0.5493 (01) 0.117 0.149 0.117 0.149yes 500 0.0001 (07) 0.5473 (10) 0.720 0.957 0.720 0.9601000 0.0002 (02) 0.5488 (03) 0.244 0.315 0.244 0.3162000 0.0001 (01) 0.5491 (01) 0.116 0.147 0.116 0.147the image before the PSF is applied and noise is added. The meanof the estimator χ̂ is computed from 106 realisations of noise forthe same signal. Also computed are the square root of the samplevariance Var[χ̂] and the mean of the estimated variance V̂ar[χ̂], re-spectively. The results shown in Table 1 indicate that the estimatorperforms as expected.3 C O N C L U S I O N A N D D I S C U S S I O NThe unbiased estimator (15) provides a new method for elliptic-ity measurement from noisy images. It’s simple and analytic formallows quick implementation and fast evaluation, and statistics forthe results can be obtained directly with unbiased estimates of thevariance (17) and covariance (18).Using an unbiased estimator for the ellipticity of the signal μeliminates the influence of noise from a subsequent analysis ofthe results (the so-called ‘noise bias’ in weak lensing, Refregieret al. 2012). However, depending on the application, other kinds ofbiases might exist even for a noise-free image. For example, due tothe discretization of the image, the signal ellipticity can differ fromthe intrinsic ellipticity of the observed object. This ‘pixelation bias’(Simon & Schneider 2016) remains an issue in applications suchas weak lensing, where the relevant effects must be measured fromthe intrinsic ellipticity of the objects.Furthermore, a fixed centre (x̄, ȳ) for the moments has beenassumed throughout. For a correct ellipticity estimate, this must bethe centroid of the signal, which is usually estimated from the dataitself. Centroid errors (Melchior & Viola 2012) might thereforeultimately bias the ellipticity estimator or increase its variance,although this currently does not seem to be a significant effect.In practice, additional biases might arise when the assumed re-quirements for the window function and PSF are not fulfilled by thedata. The estimator should therefore always be carefully tested forthe application at hand.Lastly, the ellipticity estimate might be improved by suitablefiltering of the observed image. A linear filter with matrix A canbe applied to the image before estimating the ellipticity, since thetransformed pixels remain multivariate normal with mean μ′ = Aμand covariance matrix ′ = A AT. Examples of viable filters arenearest neighbour or bilinear interpolation, as well as convolution.A combination of these filters could be used to perform PSF decon-volution on the observed image, as an alternative to the algebraicPSF correction (8).AC K N OW L E D G E M E N T SNT would like to thank S. Bridle for encouragement and manyconversations about shape measurement. The author acknowledgessupport from the European Research Council in the form of a Con-solidator Grant with number 681431.R E F E R E N C E SBartelmann M., Schneider P., 2001, Phys. Rep., 340, 291de Vaucouleurs G., 1948, Ann. Astrophys., 11, 247Kilbinger M., 2015, Rep. Prog. Phys., 78, 086901Mandelbaum R. et al., 2014, ApJS, 212, 5Marsaglia G., 1965, J. Am. Stat. Assoc., 60, 193Marsaglia G., 2006, J. Stat. Softw., 16, 1Melchior P., Viola M., 2012, MNRAS, 424, 2757Refregier A., Kacprzak T., Amara A., Bridle S., Rowe B., 2012, MNRAS,425, 1951Simon P., Schneider P., 2016, A&A, preprint (arXiv:1609.07937)Viola M., Kitching T. D., Joachimi B., 2014, MNRAS, 439, 1909Voinov V. G., 1985, Sankhyā Ser. B, 47, 354APPENDI X A : A RATI O ESTI MATOR FO RN O R M A L VA R I AT E SLet x and y be jointly normal variates with unknown means μx andμy, and known variances σ 2x and σ2y and correlation ρ. The goalhere is to find an unbiased estimate of the ratio r of their means,r = μx/μy. (A1)Under the additional assumption that the sign of μy is known, anunbiased estimator for r can be found in two short steps.First, the transformation of Marsaglia (1965, 2006) is used toconstruct a new variate,w = x − c y, c = ρ σx/σy. (A2)The constant c is chosen so that E[wy] = E[w]E[y]. It is clear that wis normal, and that variates w and y are jointly normal, uncorrelatedMNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018L60 N. Tessoreand thus independent. Note that c = 0 and w = x for independent xand y.Secondly, Voinov (1985) derived an unbiased estimator for theinverse mean of the normal variate y, i.e. a function ĝ(y) withE[ĝ(y)] = 1/μy . For the relevant case of an unknown but positivemean μy > 0, this function is given byĝ(y) =√π√2 σyexp(y22 σ 2y)erfc(y√2 σy). (A3)It is then straightforward to construct an estimator for the ratio(A1),r̂ = c + w ĝ(y). (A4)Since w and y are independent, the expectation isE[r̂] = c + E[w]E[ĝ(y)] = c + (μx − c μy)/μy = μx/μy, (A5)which shows that r̂ is in fact an unbiased estimator for r.The variance of the ratio estimator r̂ is formally given byVar[r̂] = (E[w]2 + Var[w])Var[ĝ(y)] + Var[w] E[ĝ(y)]2. (A6)As pointed out by Voinov (1985), the variance Var[ĝ(y)] does notexist for function (A3) due to a divergence at infinity. The confidenceinterval h with probability p,Pr(|ĝ(y) − 1/μy | < h) = p, (A7)however, is well-defined, and the variance of ĝ(y) remains finite inapplications where infinite values of y are not observed. In this case,an unbiased estimatorV̂ar[ĝ(y)] = (ĝ(y))2 − (1 − yĝ(y))/σ 2y (A8)exists and, together with Voinov’s estimator for 1/μ2y , yieldsV̂ar[r̂] = w2 (ĝ(y))2 − ((w2 − σ 2x ) /σ 2y + c2) (1 − yĝ(y)) (A9)as an unbiased estimate of the variance of the estimator r̂ .When y is significantly larger than its standard deviation, e.g.y/σ y > 10, the function ĝ(y) given by (A3) is susceptible to numer-ical overflow. However, in this regime, it is also very well approxi-mated by its series expansion,ĝ(y) ≈ 1/y − σ 2y /y3 + 3σ 4y /y5, y/σy > 10. (A10)For even larger values y  σ y, this approaches 1/y, as expected.This paper has been typeset from a TEX/LATEX file prepared by the author.MNRASL 471, L57–L60 (2017)Downloaded from https://academic.oup.com/mnrasl/article-abstract/471/1/L57/3965718by University of Manchester useron 27 July 2018

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An unbiased estimator for the ellipticity from image moments

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