Image processing challenges in weak gravitational lensing
IIMAGE PROCESSING CHALLENGES IN WEAK GRAVITATIONAL LENSING
Adam Amara
Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland
ABSTRACT
The field of weak gravitational lensing, which measuresthe basic properties of the Universe by studying the way thatlight from distant galaxies is perturbed as it travels towards us,is a very active field in astronomy. This short article presentsa broad overview of the field, including some of the impor-tant questions that cosmologists are trying to address, such asunderstanding the nature of dark energy and dark matter. Todo this, there is an increasing feeling within the weak lens-ing community that other disciplines, such as computer sci-ence, machine learning, signal processing and image process-ing, have the expertise that would bring enormous advantageif channelled into lensing studies. To illustrate this point, thearticle below outlines some of the key steps in a weak lens-ing analysis chain. The challenges are distinct at each step,but each could benefit from ideas developed in the signal pro-cessing domain. This article also gives a brief overview ofcurrent and planned lensing experiments that will soon bringabout an influx of data sets that are substantially larger thanthose analysed to date. It is, therefore, inevitable that cur-rent techniques are likely to be insufficient, thus leading to anexciting era where new methods will become crucial for thecontinued success of the field.
1. INTRODUCTION
Understanding the Dark Universe [1] has become one of themost pressing issues in physics and cosmology today. Fun-damental topics, such as the nature and composition of darkmatter and dark energy, are being addressed by building onthe remarkable period of growth that we have seen over re-cent decades in cosmology. The experimental evidence forthese components of the Universe is strong, though our the-oretical understanding of them is still not clear. By studyingthe Universe out to a distance (away from us) that correspondsto a redshift of 2, which is the target for a number of currentand future experiments, we can build a detailed understandingof the development of the Universe over two-thirds of its cos-mic history. This, in combination with what we have alreadylearnt from the CMB, will very likely lead to new discoveriesin fundamental physics.Due to the complex physical processes involved in the lateUniverse, we need to bring together multiple probes. The fourmain techniques used in cosmology are: (i) Distribution of Galaxies [2]; (ii) Supernovae [3]; (iii) Galaxy Clusters [4];and (iv) Gravitational Lensing [5]. Each method has poten-tial strengths and drawbacks. For instance, the simplicity ofthe supernovae analyses, measuring the distance-redshift re-lation, has led to early success. However, dark energy has asecond observable signature in that it affects the way that cos-mic structure grows. Supernovae observations are not ableto measure this effect. Instead, the number of galaxy clus-ters is sensitive to this. The drawback with cluster studiesis that the statistic relies on the mass of the clusters, whichis very difficult to determine observationally. The statisticsof the distribution of galaxies and weak gravitational lensingare seen as the methods with the most potential in measur-ing the effects of dark energy. However, the primary diffi-culty for the method that uses galaxy positions is that the re-lation between the galaxies and the underlying dark matter isnot direct. Therefore, assumptions would need to be made.For weak gravitational lensing, the main challenge is that themeasurements are very difficult to acquire and analyse. Thisarticle will focus on the method of gravitational lensing.
2. WEAK GRAVITATIONAL LENSING
As light travels towards us from distant galaxies, it is bentand perturbed by intervening matter along the line of sight.This leads to what is known as gravitational lensing. In weaklensing, the lensing effects of large-scale structure lead toa correlated distortion pattern being imprinted onto galaxyimages. This effect is subtle and hard to detect due to theweak signal. The difficulty for lensing experiments is alsocompounded by the fact that the galaxies we wish to measureare small and faint due to their distance from us. A numberof ground-based facilities have been custom designed andbuilt with weak lensing in mind, such as the Dark EnergySurvey (DES), PanSTARRS and the Large Synoptic SurveyTelescope (LSST), which are all likely to make significantprogress on what is possible today with the worlds largestlensing survey CFTHLS. Nonetheless, these surveys arelikely to continue to have problems dealing with the intrinsicdifficulties of observing through the Earth’s atmosphere. Forthis reason, there are a number of planned experiments thatwill take data from above Earth. As was the case for CMBstudies, this can be done with a combination of spacecraftand balloon observatories. These include HALO, Euclid and a r X i v : . [ a s t r o - ph . C O ] S e p FIRST, which will greatly extend on what we have today– COSMOS, a 2 square degree survey imaged with the Hub-ble Space Telescope (HST). Table 1 shows some of the keyproperties of these experiments.
Table 1 . Overview summary of weak lensing experiments.The details should be seen as a rough guide since, for in-stance, the survey area can depend on one’s definition of use-ful sky for extra-galactic observations.Name Survey Area Facility Type Time Scale[Sq. deg.]COSMOS 2 Space CurrentCFHTLS 50 ( →
3. OVERVIEW OF WEAK LENSING DATA3.1. Overview
Weak lensing data consist of deep, wide field images of thesky. Figure 1 shows an example of one such image. This is asmall portion, roughly 1% of the COSMOS field [6]. Theseimages contain a large number of galaxies. As a rough guide,the number density of galaxies in most surveys is between 10to 40 galaxies per square arc-minute, and the number densityof stars is typically around one star per square arc-minute.Most weak lensing experiments use or plan to take images inthe visible (roughly 400 - 900 nm). The main reasons for thisare twofold. The first is that the transmission of the Earth’satmosphere is high in this range. Second, the detectors forthis range (CCD’s or Charge Coupled Devices) have beenused extensively, and their performances and shortcomingsare well-studied [7]. However, missions, such as WFIRST,are exploring the possibility of capturing images in the Near-Infrared (NIR), which would correspond roughly to the rangeof 1 to 2 microns. The resolution of weak lensing imagesis sub arc-second, so the planned, future all-sky surveys willproduce mosaic images with more than pixels, whichwill contain billions of galaxies and several hundred millionstars. This needs to be reduced to test our current model of theUniverse, the ΛCDM model [8], which can be well describedby only seven parameters.
We now briefly discuss the processes that galaxy and star im-ages go through in order to illustrate where and how the un-
Fig. 1 . An example of the type of images used in weak lensingstudies. This is a small section of the COSMOS field. Theimage contains galaxies (two examples have been highlightedin the yellow inset) and stars (red insets).derlying cosmological information is encoded. This forwardprocess is described in detail in the GREAT08 Handbook [9].This is a challenge set up by the weak lensing communitythrough the PASCAL network to engage with the machinelearning community. Figure 2 shows how the intrinsic im-ages of galaxies are modified as their light travels towards us.From top to bottom, the steps are:1. Gravitational Lensing: Extended objects such as galax-ies have their images distorted by gravitational lensing.The simplest image distortion is a shear which can bedescribed by a simple distortion matrix (see equation1). Intrinsic star images are effectively delta functions,so they do not respond to a shear and are, therefore, notsensitive to weak gravitational lensing.2. ‘Blurring’ by the Point Spread Function (PSF): Boththe atmosphere and our telescopes cause the images ofobjects to become blurred. This effect is a convolution,where the convolution kernel is known as the PSF.3. Pixelisation and Noise: The light from the images fallsonto our detectors and is recorded. This process leavesus with a noisy, pixelated reproduction.In lensing, we begin at the bottom and systematicallywork our way back to recover the original lensing signal. Wecan do this because star images suffer from the same contam-inating effects as galaxies, but, crucially, they are not lensed. ensing
PSF ConvolutionPixel + Noise PSF ConvolutionPixel + Noise
Fig. 2 . Illustration of the steps that affect the images of galax-ies and stars as their light travels towards us. From the top, wesee: (i) the intrinsic images of the objects; (ii) the post-lensingimages (note that only the galaxy images experience a sheardue to gravitational lensing); (iii) the images after a blurringdue to the PSF of the atmosphere and/or the instrument; and(iv) the light falls onto a detector, which results in noisy andpixelated images.The lensing shear effect can be described by a distortionmatrix such that (cid:18) x y (cid:19) = (cid:18) − g − g − g g (cid:19) (cid:18) x y (cid:19) , (1)where the coordinates ( x y ) are for the original image and( x y ) are for the lensed image. The elements g and g arethe two components of shear. This shear comes directly froma weighted integral of the mass along the line of sight . One ofthe powerful things about gravitational lensing is that it doesnot matter what form the mass is in. It is equally sensitiveto both normal matter, as well as the otherwise invisible darkmatter. Lensing observables can be derived from the ’lensing potential’, ψ ,which can be calculated from the Poison Equation ∇ ψ = 2 κ , where κ is the convergence and is a weighted integral of mass along the line of sight.In terms of image distortion, convergence causes a change in image size. Im-age shear also can be calculated from the second derivatives of the lensingpotential: g = ( ∂ ψ/∂x − ∂ ψ/∂y ) and g = ∂ ψ/∂x∂y . Each galaxy gives us a noisy measure of the lensing signalat its position. There are a number of error sources, includ-ing ‘photon noise’ and ‘shape noise’. The former is due tothe fact that the majority of the galaxies used in lensing havea very low signal to noise galaxies (down to S/N of roughly10), so that the galaxy is barely above the background noiseof the image. The latter is due to the fact that galaxies’ in-trinsic shapes are not circular. Instead, galaxies are elliptical,with ellipticities of ∼ . , where the change in ellipticity dueto lensing is significantly smaller than this ( ∼ . ). Ourchallenge is to collect the measurements from a large numberof galaxies that are distributed in space in order to map theunderlying dark matter and measure its statistics.Figure 3 shows a simulations of a lensing mass field thatwe would expect in a ΛCDM
Universe. The field size shownhere is comparable to that of the COSMOS field. This isthe noise-free version of the kappa field. The measurements,however, give us a noisy realisation. Optimal method for de-noising such data will depend on how it is to be used. To mea-sure cosmology parameters, the focus is to measure the pow-erspectrum (or two-point correlation function) of the shearfield. This reduces the random noise by averaging over a largenumber of galaxy pairs. Systematic errors that do not averageto zero, therefore, are of particular concern, especially thosethat are more present in low-signal to noise data. Anothernoteworthy source of potential error is the impact of masking.Large sections of the raw images used in lensing studies mustbe masked. For instance, pixels that surround very bright starscannot be used. Since the signal we seek is the spatial corre-lation function of the lensing signal, care must be taken whendealing with these holes in the analysis.
4. MEASURING COSMIC SHEAR
With this brief overview of weak lensing in place, we cannow go through the main steps of a data analysis chain andhighlight the current state of the art.1. Object Detection: The first step in the analysis processis to identify and classify the objects of interest in theimages. Specifically, we are concerned with finding thegalaxies and stars and being able to distinguish clearlybetween them (as illustrated in Figure 1). At present,this step is almost exclusively performed using the rou-tine Sextractor [10].2. Measuring the Point Spread Function (PSF): A key stepin weak lensing is to correct for the adverse effects ofthe observations. We do this by measuring the PSFfrom the stars and then interpolating this to the galaxypositions. Developing new techniques for this PSF in-terpolation process is one of the main objectives of the ig. 3 . Simulation of the lensing mass (convergence) that wewould expect for a survey with a similar size as COSMOSfor a
ΛCDM
Universe. It is the statistical properties of thisfield, such as its two-point correlation function, that allow usto measure cosmological parameters. The example shownhere is for the lensing signal at a redshift of one. The darkregions are under dense, and the coloured peak show wherethe dark matter is concentrated.GREAT10 challenge and an overview of current meth-ods is given in Appendix D of the challenge handbook[11]. This, in my view, is an area that requires specialattention.3. Measuring the Shear Per Galaxy: Some of the greatestdifficulties we face come from the fact that the majorityof the galaxies have a very low signal to noise and thatthe intrinsic shapes of galaxies are complex. A detaileddiscussion of some of these problems is presented inthe GREAT08 results paper [12].4. Measuring the Correlation Function: Using a catalog ofthe weak lensing estimators coming from each galaxy,the two-point correlation of galaxy pairs can be mea-sured. However, there has recently been considerableinterest in the lensing community to find better waysof constructing the correlation functions in the data.For instance, a number of studies have explored fastermethods, such as in-painting [13], for dealing with theholes in the data.All the steps above are challenging. Since weak lensingstudies rely heavily on careful image analysis techniques, im-provements in all areas will be needed for the future surveysoutlined in Table 1. In particular, understanding how to con-struct a detailed model of the PSF, which varies in both spaceand time, will be key.
5. CONCLUSIONS AND FUTURE PROSPECTS
Weak lensing experiments are expected to grow rapidly in thecoming decade, providing us with a wealth of new data thatis several orders of magnitude greater than what is currentlyavailable. These new datasets will require new analysis meth-ods, and image analysis techniques may play critical roles ineach of the main steps of (i) object detection and classifica-tion, (ii) PSF measurements from stars, (iii) lensing measure-ments from galaxies and (iv) constructing statistical measuresof the lensing signal over the sky.
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