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The first parallax was measured by Bessel in for the fast moving binary star 61 Cygni Fricke, This confirmed that stars were at a finite distance from us, revealing the scale of the Universe Perryman, Few years later the parallaxes of Vega and Alpha-Centauri were measured, with a uncertainties of the order of 60— mas. Very slowly the number of known parallaxes increased. In Schlesinger listed 1, parallaxes using a new method, based on the use of photographic plates.

In The Yale Catalog included less than objects with a mean error of 10—20 mas van Altena et al. More details can be found in the review by Perryman A gigantic leap took place with the adoption of space techniques, in the ESA Hipparcos mission first and then in Gaia. Since then, astrometry importance has grown enormously, holding the key not only to understand our Galaxy, but also to make an incredible breakthrough in many field of astrophysics. This review focuses on the recent advances on global astrometry and on the future developments and challenges.

In section 2 we summarize the Hipparcos mission; in section 3 we present the performances and the recent exciting results of the Gaia mission; in section 4 we discuss the future directions of global astrometry, their expected science outcome and the main challenges they have to face; in section 5 we draw a few concluding remarks. Earth atmosphere provides a clear limitation to the accuracy of the parallax measurements. The obvious next step was then to go to space making also use of very stable instruments. The idea was first proposed in the sixties, by Pierre Lacroute Perryman, He proposed an instrument able to observe in two separate directions at the same time.

These two lines sight were then combined on the same focal plane, providing wide-angle measurements over the entire sky. Hipparcos was then proposed in and was the first astrometric space mission. Hipparcos provided the positions, parallaxes, and annual proper motions for about , stars with an unprecedented accuracy of 0. The first Catalog was produced in including astrometry and photometry ESA, A new reduction with improved accuracy for the bright stars was published by van Leeuwen Hipparcos improved the astrometry accuracy by a factor of fifty over its predecessors Perryman et al.

One of the main outcome of the Hipparcos mission was the definition of a precise celestial reference frame in the optical, i. This reference frame is the optical counterpart of the radio reference frame. These data were the reference for astrometry for about 20 years Perryman, This mission had an enormous impact on the astrophysics.

A detailed review can be found in Perryman Among the many papers dealing with the properties of the Galactic populations in the solar vicinity, we quote Reddy et al. Hipparcos astrometry has improved the census of nearby young stellar groups and related star-forming regions de Zeeuw et al. It lead to a more precise definition of the evolution of the stars Pietrinferni et al.

Following the success of Hipparcos, more ambitious astrometric space missions were proposed. Most of these proposals were finally not accepted besides their enormous scientific potential. Finally, the European Space Agency Gaia mission was the only one that was approved. Gaia was originally designed as an astrometric optical interferometer with baseline of a few meters contained in a single payload and operated in a continuously scanning mode.

Another main improvement over Hipparcos was the possibility of radial velocities and spectrophotometric measurements by the on-board instrumentation Lindegren and Perryman, a. The initial optical design was revised and became very similar to Hipparcos, but with enhanced capabilities. The satellite was launched in , with 5 year nominal mission prevision. However, an extension to 10 years of the operations has already been proposed to ESA.

At the time of writing, the approval is still pending. The technical description of the spacecraft, of the payload and a summary of the scientific performances are presented in Gaia Collaboration et al. The final one of the nominal mission is planned after a post-operation processing of all the available data at the end of and would include astrometry, photometry, radial velocities and information about a large variety of objects, including stars, quasars, galaxies, variables, binaries, solar system objects, exoplanets.

Gaia mission will revolutionize astronomy in the coming decades. Gaia science case was first presented in de Boer et al. We quote among others Perryman et al. Here we briefly summarize a few key aspects. One of the main scientific targets of Gaia, if not the principle target, is the Milky Way.

Our Galaxy is an impressive laboratory that we can use to reach a detailed understanding of how stars and galaxies form and how they evolve. In turn, stars host planets and our planetary system provides pointers to extra-solar planets. Linking our local environment, the solar vicinity, the disk, the halo, to our Galaxy as a whole, is the key to understand the evolution of the galaxies. Cosmological models predict the formation of large structures through the merging of smaller sub-units.

Deciphering the assembly history of our Galaxy is possible through a detailed mapping of the structure, dynamics, chemical composition, and age distribution of its stellar constituents.

Plate Solving Examples

Gaia is the cutting edge of Galactic astronomy and will have a major impact across all areas of astronomy and astrophysics, and at all scales. Gaia will allow to study the Galactic structure tracing the disk, the halo and spiral structure, the dark matter content through the detection of halo streams.

The difficulty of establishing accurate distances over cosmologically significant scales has made the determination of the expansion rate at the current epoch H 0 extremely challenging.

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Deriving the extra-galactic distance scale is a crucial problem of modern astrophysics. Gaia will provide the calibration through direct measurement of parallaxes for the local distance indicators Cepheids, RR-Lyrae. Gaia will unravel the star formation history of our Galaxy and the kinematics of nearby galaxies. Extra-solar planet detection through astrometry and photometric transit will be possible down to Jupiter-mass objects. In addition fundamental physics tests relativistic parameters and stellar physics improvements will be under reach.

The challenge will be to built theories and models to reproduce the exquisite details in which our Milky Way is being depicted Binney, The first Gaia data release is based on the data collected during the first 14 months of the nominal mission. Gaia DR1 properties and limitations are described in Gaia Collaboration et al. Gaia DR1 supplies the astronomical community with positions, G-band photometry for slightly more than 1 billion objects, and a number of RR-Lyrae and Cepheids light curves.

For a subset of objects, about two millions, the TGAS sample positions, parallaxes and proper motions are derived using as prior the positions of the Hipparcos and Tycho 2 Catalogs. It should be pointed out that DR1 parallaxes and proper motions are independent from Hipparcos and Tycho-2 data. More information can be found in Lindegren et al. The typical uncertainties are of the order of about 0. This first data release is already a substantial improvement over the Hipparcos astrometry. However, the quality of the data were limited by the available sky coverage, the still immature calibrations for instance preliminary modeling of dependence in color and time of the point spread function of the Gaia telescopes, and of the satellite attitude.

A systematic term on parallax zero point is still present, i. Due to the combined effect of the filtering, sky coverage and crowding, the completeness of the Gaia Catalog is a complex function of the position in the sky, of the star density, and of the magnitude. It is also very limited in dense areas on the sky the Bulge, the globular cluster centers, the low latitude disk Arenou et al.

Independent verification of Gaia DR1 astrometric quality was performed by the scientific community with a variety of methods. One of the first results of the Gaia DR1 was the solution of the so-called Pleiades distance controversy. The fact that Hipparcos distance to the Pleiades is significantly different from any distance derived with other methods Melis et al. Now, the new Gaia parallax reconciles Pleiades distance with literature determinations Gaia Collaboration et al. Exciting results on Galactic structure and kinematics were derived combining Gaia astrometry with other photometric or spectroscopic surveys.

We quote, among others, Iorio et al. Hunt et al. Massari et al. Cantat-Gaudin et al.

Applied Optics

Ward and Kruijssen analyzing the kinematics of young OB associations, come to the conclusion that not all the stars formed in clusters. TGAS data support the idea that star formation has a hierarchical nature. In this view, large-scale associations can form in-situ following the fractal structure of the molecular cloud. The second data release Gaia DR2 is now approaching. Based on 22 months of data, a full astrometric solution is derived, without imposing any prior. Positions, parallaxes, and proper motions will be available for more than 1.

The uncertainties on the parallaxes reach 0. This is the largest data base of objects having six-parameter solution. More than 1. In addition Gaia DR2 will contain epoch astrometry for more than 13, known asteroids, light curves for more than , variables. This represents undoubtedly an unprecedented data base for astronomers.

Completeness has improved over Gaia DR1. The milli-mag level magnitudes will allow a comparison with stellar models. Star clusters having secure membership derived from DR2 astrometry will provide a test-bed for stellar models allowing further refinements and calibration of effects such as the overshoot and mixing in general in the stellar cores, the rotation, the magnetic fields. This should ultimately lead to a significant improvement in the reliability of those models and their application in various fields, such as providing age estimates for field stars, and stellar population templates for extra-galactic studies.

Cluster formation and disruption in different Galactic environments will be within the reach of DR2 data, putting additional constraints on the relation between clusters and field population. Galactic kinematics, halo substructures and Galaxy modeling will benefit from this enormous data-base. Gaia DR2 will clarify how streaming motions affect the Galactic disc, allowing a better definition of the processes involved in galaxy formation and evolution. A recent work has pointed out that the distance uncertainties can create artificial wave-like patterns Carrillo et al.

Gaia DR2 can probe unexplored regions of the Galaxy, allowing us to put significant constraints on the geometry of the Galactic potential Posti et al. Gaia DR2 and the increasingly richer following releases will undoubtedly provide answers, but may also point to unexpected problems, and to completely new directions in which to take astrophysics. The decision of undertaking astrometry from space done with Hipparcos made a true revolution in comparison with ground-based Catalogs.

Then Gaia constitutes a jump of two orders of magnitudes in accuracy with respect to Hipparcos.

While still awaiting the final Gaia Catalog, the debate in the scientific community is on the future directions of astrometry. In the near future several space missions and facilities will become active. One of the main goals of LSST is to obtain high-quality astrometry, i. Predicted parallax errors of 0. ESA Euclid space mission will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky.

Clearly the main driver for Euclid is the nature of dark energy, but it goes without saying that Euclid science will cover a vast range of topics, from cosmology to galaxy evolution to planetary research. In this global framework, future directions of global astrometry should provide complementary information.

Several options have been envisaged:. This review focuses on global astrometry. However, before discussing its prospects in the following sections, we would like to spend a few words on the possibilities offered by relative astrometry. The advantage of this approach is that high relative position accuracy can be achieved on small angular scale, allowing the study of higher order positional effects modifying relative positions, as in the case of binary stars or extra-solar planets. High precision sub-microarcsec relative astrometry can address many science cases, from the exoplanet detection to the probing of the dark matter.

The Hubble Space Telescope has proven to be able to yield sub-milli-arc-second accuracy in differential astrometry see for instance Libralato et al. The new generation space telescopes such as WFIRST and JWST can in principle go much further, provided that instrumental issues and systematics such as stability, geometric distortions in the field are understood and accounted for.

Its science case well illustrates the possibilities of sub-micro-arcsec relative astrometry. Theia goal is to perform relative astrometry at 0. Theia can probe the dark matter distribution in dwarf spheroidals, the outer shape of the Milky Way dark matter DM halo, and the power spectrum of density perturbations. Dwarf galaxies are dark matter dominated. The shape of the dark matter cores either large or cuspy depends on a number of different processes star formation, self-interaction , Read et al.

Concerning exoplanets, differential astrometry provides estimates of the mass and three-dimensional orbital parameters which are fundamental to models of planetary evolution, bio-signature identification, and habitability. Theia precision can easily be reached with a 0. Relative astrometry can be a very realistic option for future astrometric developments, although then existing or planned facilities can provide a way through. It will bring astrometry from the local environment to the cosmological scale. While the science case for this is exciting, this option has to face major technological challenges.

Thus satellites with precision formation flying will be mandatory. Thermo-mechanical stability of the entire spacecraft, attitude control, and knowledge of the barycentric velocity of the spacecraft down to the required level will be really far from trivial and would imply significant technological development Brown, The whole data analysis will be extremely challenging.

Indeed the simple treatment of the time dependence of source coordinates now applied to the interpretation of the Gaia data will not be sufficient and new modeling paradigms need to be explored. The effects of the sources of astrometric jitter star spots, faculae, or micro-lensing in crowded regions on the interpretation of image locations in the data stream will be relevant and requires further research.

In addition we are still lacking of suitable modeling to correct astrometric measurements for relativistic effects at the nano-arcsec level Klioner, This requires a refinement of the currently employed models and also a substantially improved knowledge of the solar system e. Here we focus on two other science cases that will open a new window, i.

One of most exciting applications of the astrometry stays in a new field of study, real-time cosmology Brown, The main point is that while most cosmological observations have a geometric nature see for instance the use of standard candles to measure distances and the expansion of the Universe , in real time cosmology the measurements involve dynamics, and are independent from cosmological models. This was already noticed in de Boer et al. The red-shift is the first well known real time observable. The transverse motion of external galaxies and quasars is the analogous in the transverse direction.


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The astrometry of quasars is affected by several effects, first of all the intrinsic proper motion due to relativistic jets and the secular aberration drift due to the acceleration of the Sun around the Galactic center. The secular aberration drift cannot be distinguished from other effects, like cosmic parallaxes. However, if we assume we can model it as a dipole, we get an amplitude of the order of 4. Then, correlated proper motions of quasars can be used to probe cosmological models, for instance obtaining geometrical distances independent of the cosmological distance ladders, testing the Universe isotropic expansion, measuring the collapse of large scale structures, and, finally providing estimates of the upper limit of the energy flux of the stochastic gravitational wave background.

To do this one needs to assume that all systematics in higher order harmonics come from the gravitational wave background. The quadrupole component and higher order harmonics should be sensitive to Hubble constant anisotropy or to primordial gravitational waves of long wavelengths Titov et al. Bachchan et al. In the following, we briefly comment on these items. An homogeneous and isotropic metric the Friedmann-Robertson-Walker metric is one of the main assumption behind our standard cosmological models.

However this is still an open issue. A deviation from homogeneity and isotropy is proposed as an alternative view to explain the accelerated expansion of the Universe without a dark energy field. Any anisotropy in the expansion of the Universe would produce a variation in time of the angular separation between two sources, called cosmic parallaxes Quercellini et al.

This effect is a function of the red-shift.

Plate Solving and The PinPoint Astrometric Engine

In an isotropic expansion, the cosmic parallax vanishes, since the angular separations between sources is expected to be constant but for the peculiar motions. Quercellini et al. A positive detection of large-scale cosmic parallax would disprove one of the main hypotheses of modern cosmology, isotropy. Astrometry can measure the collapse of structures. Small separation objects that are in gravitationally interacting systems for instance filaments are expected to present a net decrease in their angular separations while collapsing.

Unbound objects just moving with the Hubble flow will show no net angular separation changes. Most recent determinations using VLBI quasar Catalog, put an upper limit on the rate of convergence of large-scale structure of — This effect depends on the redshift and on the fundamental cosmological parameters. This shift would lead to geometrical constraints on the dark energy with a significant figure of merit Ding and Croft, As we have discussed, all these measurements present considerable technical challenges and require an enormous control of both systematic and statistical errors. With a few exceptions, they are far beyond the possibilities of Gaia Ding and Croft, , and even combining present measurements with a new Gaia mission either in the optical or in the infrared, see below would only provide very limited constraints Hobbs et al.

The detection of gravitational waves by the Advanced LIGO experiment has just opened the way to new explorations Abbott et al. However lower frequencies are inaccessible to ground-based instruments. Further progress is expected from the space-based detector, LISA.

The possibility of using astrometry to detect Gravitational waves was proposed by many authors see for instance Braginsky et al. The passage of a gravitational wave provokes a deflection of the apparent position of the star, i. The astrometric deflection depends on a telescope term and on a star term. This second term will be different from each star and definitely smaller for distant sources, while the telescope term is common. This implies that the motions are coherent across the sky. Thus, repeated astrometric measurements across the whole sky would allow the identification of the deflection pattern due to a gravitational wave passage.

One should distinguish the regime of ultra-low frequency gravitational waves that directly influence apparent proper motions Pyne and Carroll, ; Gwinn et al. These frequencies correspond to the stochastic background produced by the superposition many monochromatic signals, or of primordial origin.

Recent papers have explored the astrometric detection of gravitational waves of higher frequencies in the context of Gaia-like astrometric missions Moore et al. Assuming that all the signals with period less than 2 Gaia rotational periods will not detectable and that the period of the detectable waves should be considerably shorter than the length of the mission, one comes to the conclusion that Gaia is sensitive to wavelengths in the range 6. A detailed practical algorithm to search the gravitational wave signatures in the residuals of Gaia-like astrometric solution is formulated in Klioner In this frequency range Gaia can search for signals of e.

In general, prospects for astrometric detection are not very encouraging Schutz, , but nevertheless gravitational wave sources detectable by Gaia still may exist.

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What are the implications for a future nano-as mission? The astrometric effect due to a stochastic gravitational wave background on a single object proper motion is expected to be less than 0. A future mission at the nano-as level has fair chances to make a detection. Gaia is operating at optical wavelength. The very obscured regions of our Galaxy, i. A natural evolution would be to switch to astrometry in the infrared.

Near-field cosmology is of fundamental importance, since detailed studies of the Galaxy underlie our understanding of universal processes over cosmic time. It has become clear in the recent past that one component of the Milky Way cannot be separated from another Bland-Hawthorn and Gerhard, Disk instability might be responsible for the bar formation see among others Athanassoula et al. The inner Galaxy is still puzzling. The bulge is formed by a mixture of populations having different orbits, but is unclear how many we have Ness et al.

It is still under discussion if a classical bulge formed by early mergers is present or if the old-metal poor population traced by the RR-Lyrae is indeed related to the halo Pietrukowicz et al. Finally we do not know how nuclear disk, inner disk and bar are connected Ness et al. In this context, it is important to have a global view of the kinematics and dynamics of all the Galaxy components. New astrometric missions in the NIR are already under study to cover the gap and allow a better understanding of the our Galaxy as a whole. The goal is to discuss the above issues with an accuracy comparable to Gaia in the obscured regions of the Galaxy.

Gaia and Gaia-NIR measurements can be combined producing higher accuracy astrometry in the whole Galaxy see following section. The challenge of this astrometric mission is related to the use of infrared detectors in time delay integration TDI mode, for which technology is not yet mature. HgCdTe detectors covering a range between and 2, nm are proposed. The problem is the high read-out noise of these detectors. At the current status, this level can be reached in near-infrared detectors per pixel.

However, this would lead to an extremely high read-out noise when the detectors are used in TDI mode. Alternatively, a limits at 1, nm is a fall back option that would still allow to meet the scientific goals. The scientific objectives are to study the dynamics of the Galactic center and of the bulge, and clarify the nature of the super-massive black hole. To repeat a mission is not usually done, but here it would have the advantage of a very realistic possibility, built on the knowledge of Gaia.

The accuracy on proper motion will be of the order of 1. While Gaia at its 5-years nominal duration provides high precision proper motion mapping of the streams in the halo up to 20 Kpc or 30 Kpc in 10 years , with a new mission stellar streams in the halo out to Kpc can be detected providing fundamental information about galaxy formation process.

Indeed recent studies have pointed out that a transition in halo properties metallicity, density distribution, degree of lumpiness between the inner and outer regions takes place at farther distances. This interface could also be related to the in-situ vs. Young stellar debris, accreted less than 8 Gyr ago are expected to be more easily found at larger distances. High precision kinematics of gaps in the streams can be used to trace the dark matter content of our Galaxy Helmi and Koppelman, Several other science cases can be addressed by these observations going from exoplanets, to brown dwarfs, and binaries.

Internal proper motions of Local Group galaxies can be resolved, allowing determination of the dark matter content. This sample will include SPhs and Ultra-faint galaxies that could be the building blocks of our Galaxy. Ultimately this can help understanding the missing satellite problem and the too big to fail problem. Finally, Gaia optical reference frame will be degraded by about 40 mas over 50 years, as an effect of the cosmic acceleration or galactic aberration acting like a systematic proper motion field for the QSOs.

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With a new mission, the reference frame degradation will be slow down by the same factor affecting the proper motions Mignard et al. We are now in the golden age of space astrometry. Gaia exquisite capabilities have established the European science and industry as leaders in high precision astrometry. Gaia science exploitation is just at the beginning, but will likely continue in the coming decades and will drive the development of next major steps in astrophysics.

Potential key science drivers and development pathways will become clearer after this process and it cannot be excluded that astrophysics might take unexpected directions. However both options will require to overcome enormous technical and scientific challenges. AV is responsible of the conception and design of the work. The manuscripts submitted to Frontiers was not previously published or be under consideration for publication elsewhere, neither in whole nor in part.

The Author agrees to be accountable for the content of the work. Funding for the Gaia Data Processing and Analysis Consortium has been provided by national institutions, in particular the institutions participating to the multi-lateral agreement. Not bad for a single DSLR raw frame and a mm lens! The image covers a large region of the Milky Way around M8 and M20, which includes several solar system objects at the date of observation: Saturn and the asteroids Amalthea, Lilaea, Adorea, and 77 Frigga. I have solved this frame astrometrically with Tycho-2 data up to magnitude Only asteroids Amalthea and Lilaea, with visual magnituides of The image of Saturn is a saturated flat disk, which prevents valid PSF measurements.


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  • Here are the topocentric coordinates of Amalthea measured with the DynamicPSF tool on the plate-solved image:. For a resolution of 7. Also take into account that the image has been solved with Tycho-2 data, since unfortunately, no Vizier server was able to provide Gaia DR2 data for this region.

    With less accurate star positions and proper motions from Tycho-2, the generated astrometric solution is also somewhat less accurate. As in the preceding example, here is a comparison of the topocentric and geocentric annotations. The geocentric distance was 2. With the new version of our ImageSolver script, you may have to change the way you solve your images astrometrically in order to achieve the best possible results. Here are a few guidelines to help you with this task:. One of the main practical problems we currently have to compute accurate astrometric solutions is our dependency on online catalogs.

    The ImageSolver script downloads astrometric data, mainly star coordinates, proper motions and magnitudes, from astronomical catalogs available online through VizieR services. Although a clear benefit of this system is that we don't have to have astronomical catalogs stored as huge local files, this dependency can be problematic, not just because it requires an active Internet connection to work—with the associated connectivity and availability issues—, but chiefly because data availability from VizieR servers has important restrictions.

    These restrictions are particularly problematic for the Gaia DR2 catalog, and very especially for wide and medium fields, where current VizieR services often cannot provide the large amounts of data that we need, forcing us to use reduced limit magnitudes that may degrade the quality of our results. Fortunately, the solution to this problem exists and is obvious: to have the required astronomical catalogs available as local files.

    We are working to generate an optimized version of the Gaia DR2 catalog, which will be released publicly when completed. This is a huge and complex work, but definitely necessary. For extremely wide fields, for example images acquired with focal lengths in the range from 50 to 20 mm, the ImageSolver script still has problems that we must address as soon as possible. Basically, the problem is the long computation times required to identify stars under very large field distortions, typically on the corners of the image.

    The culprit here is the StarAlignment process, which needs a new local distortion modeling algorithm to be more efficient and capable. We have a new algorithm already designed and ready for implementation, time and priorities permitting, which will improve significantly on these extreme cases. For annotation and localization of solar system bodies, currently we are limited to the planets and the most massive asteroids for which we already have JPL ephemeris data.

    It is evident that we should be able to locate and annotate more objects, including comets and much more asteroids. To solve this problem we need an appropriate numerical integration routine. Once this algorithm is implemented as a new tool, the user will have the possibility to locate and plot essentially any moving object for which a valid set of initial conditions is available, such as orbital elements, or position and velocity vectors. As noted in the introduction, to understand the changes we have made to our astrometry tools, an informed user should get a good grasp of the way surface simplification algorithms work in practice.

    A few examples with synthetic data sets can help to achieve this goal. The examples in this section have been produced with the following script executed in the PixInsight JavaScript runtime. With its default configuration, the script generates a sample of , pixel values interpolated from the active image at random pixel coordinates. This dense input surface is simplified using our algorithm with a tolerance of 0. In these renditions, red dots represent surface data points. If you are interested in the technical details of these algorithms, or if you want to get a deeper understanding of the way they can be controlled with our current implementation, we encourage you to experiment by varying working parameters, especially the properties of the SurfaceSimplifier object created in the SurfaceTest function.

    Three-dimensional representation of the bivariate function. Image representation of the test surface. Input set of , pixel samples at random image coordinates. With the script's working parameters 0. Simplification test with a sample of two-dimensional simplex noise. Note that more points are preserved on the regions of higher variation of the sampled function, i. Note: Input sets of random points will be omitted from now on. A posterized image is a particularly interesting test case for surface simplification. The simplified surface concentrates more points on the transitions between regions of constant value, as is necessary to represent jump discontinuities.

    Surface simplification test with a saturated image, where a large region has been truncated to white. The saturated region is flat and hence can be maximally simplified. This section is intended for readers with interest in development topics. Some technical knowledge of these algorithms is always desirable, although not required for their practical application. If you are not interested in technical details about the algorithms we have designed and implemented to achieve important improvements in our astrometry scripts and tools, you can skip this section completely. Given a finite set of three-dimensional points representing sampled values of a real bivariate function.

    Succinctly, the algorithm divides the input point space recursively on the XY plane into rectangular regions using custom quadtree structures. For each region, the algorithm finds the orientation of the dominant plane through principal component analysis. The deviation of function values from the dominant plane is then evaluated for the points in the region, and if the region is considered flat to within the tolerance parameter, its points are replaced with a simplified reduced set of points that tends to preserve the local shape of the original function over the region.

    If the region is tagged as curve, it is further divided using a new quadtree recursion, until no additional simplification can be achieved. The implemented algorithm can be summarized as follows:. Build a bucket point-region quadtree [1][2] structure from the input set. In our current implementation, each quadtree recursion tends to split into four quadrants. This can be controlled by varying the bucket size. Perform a tree traversal. For each tree leaf node:. Let be the least eigenvector of that is, the eigenvector associated with the smallest eigenvalue :.

    From the plane equation we can compute the distance to the dominant plane for each :. In step 2. Intuitively, the convex hull can be seen as the best simplified representation in the sense of local shape preservation: since the region is flat, all of the points inside the convex hull are redundant for function fitting. In the current implementation, the centroid point of the region in step 2. The outlier rejection performed in step 2. Uncertainty originates here mainly from errors in astrometric catalogs, from PSF fitting errors resulting from image noise and spurious data, and from difficulties for star identification on crowded fields.

    Outlier rejection is essential in this case to achieve the robust results that we need.