4/27/16 Basic Problem Making the Sky Searchable: • I show you a picture of the night sky. Fast Geometric Hashing for Automated Astrometry Sam Roweis, Dustin Lang & Keir Mierle University of Toronto David Hogg & Michael Blanton • You tell me where on the sky it came from. New York University http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu Rules of the game Rules of the game • We start with a catalogue of stars in the sky, • We start with a catalogue of stars in the sky, and from it build an index which is used to and from it build an index which is used to assist us in locating ( ‘ solving ’ ) new test assist us in locating ( ‘ solving ’ ) new test images. images. • We can spend as much time as we want building the index but ? solving should be fast. • Challenges: 1) The sky is big. 2) Both catalogues and pictures are noisy. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 1
4/27/16 Distractors and Dropouts You try • Bad news: Query images may contain some extra stars that are not in your index catalogue, and some catalogue stars may be missing from the image. • These “ distractors ” & “ dropouts ” mean that naïve matching techniques will not Find this “ field ” on this “ sky ” . work. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu You try You try Hint #1: Missing stars. Hint #1: Missing stars. Hint #2: Extra stars. Find this “ field ” on this Find this “ field ” on this “ sky ” . “ sky ” . http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 2
4/27/16 You try Robust Matching • We need to do some sort of robust matching of the test image to any proposed location on the sky. • Intuitively, we need to ask: “ Is there an alignment of the test image and the catalogue so that (almost * ) every catalogue star in the field of view of the test image lies (almost * ) exactly on top of Find this “ field ” on this an observed star? ” “ sky ” . [*The details depend on the rate of distractors/dropouts. ] http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu Solving the search problem (Inverted) Index of Features • Even if we can succeed in • To solve this problem, we will employ finding a good robust matching the classic idea of an “ inverted index ” . algorithm, there is still a huge • We define a set of “ features ” for any search problem. particular view of the sky (image). • Which proposed location • Then we make an (inverted) index, should we match to? ? telling us which views on the sky exhibit certain (combinations of) • Exhaustive search? feature values. too expensive! • This is like the question: Which web pages contain the words “ machine learning ” ? TM The Sky is Big http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 3
4/27/16 Matching a test image Caching Computation • When we see a new test image, • The idea of an inverted index is that is we compute which features are pushes the computation from search time present, and use our inverted back to index construction time. index to look up which possible • We actually do perform an exhaustive views from the catalogue also have those feature values. search of sorts, but it happens during the building of the inverted index and not at • Each feature generates a candidate list in this way, search time, so queries can still be fast. and by intersecting the lists • There are millions of patches of the scale we can zero in on the true of a test image on the sky (plus rotation), matching view. so we need to extract about 30 bits. The features in our inverted index act as “ hash codes ” for locations on the sky. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu Quads as Robust Features Robust Features for Geometric Hashing • We encode the relative positions • In simple search domains like The features we of nearby quadruples of stars text, the inverted index idea use are the (ABCD) using a coordinate can be applied directly. B relative positions of system defined by the most • However, in our star matching nearby quadruples widely separated pair (AB). C task, the features we chose of stars. • Within this coordinate system, D must be invariant to scale, the positions of the remaining rotation and translation. two stars form a 4-dimensional A code for the shape of the quad. • They must also be robust to • Swapping AB or CD does not small positional noise. change the shape but it does • Finally, there is the additional “ reflect ” the code, so there is problem of distractor & some degeneracy. dropout stars. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 4
4/27/16 Quads as Robust Features Catalogues: USNO-B 1.0 + TYCHO-2 • This geometric hash code is • USNO-B is an all-sky invariant to scale, translation catalogue compiled and rotation. B from scans of old • It also has the property that if Schmidt plates. C stars are uniformly distributed D Contains about 10 9 in space, codes are uniformly distributed in 4D. objects, both stars A • We compute codes for most and galaxies. nearby quadruples of stars, but • TYCHO-2 is a tiny not all; we require C&D to lie in subset of 2.5M the unit circle with diameter AB. brightest stars. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu Making a uniform catalogue Building the index • Starting with USNO+ • Start with the catalogue; build a TYCHO we “ cut ” to get kdtree on the 3D object positions. a spatially uniform set • Place a fine healpix grid on the of the ~150M brightest sky. Within each pixel, identify a stars & galaxies. valid quad whose size is near the • We do this by laying target scale for the index. down a fine “ healpix ” • Compute 4D codes for those grid and taking the quads; enter them into another brightest K unique kdtree remembering their original objects in each pixel. locations. This is the index. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 5
4/27/16 A Typical Final Index Solving a new test image • Identify objects (stars+galaxies) in the image • 144M stars bitmap and create a list of their 2D positions. (6 quads/star) • Cycle through all possible valid * quads (brightest • 205M quads first) and compute their corresponding codes. (4-5 arcmin) • Look up the codes in the code KD-tree to find matches within some tolerance; this stage incurs • 12 healpixes some false positive and false negative matches. Codes • Each code match returns a candidate position & in rotation on the sky. As soon as 2 quads agree 4D on a candidate, we proceed to verify that candidate against all objects in the image. Quads on the sky http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 6
4/27/16 A Real Example from SDSS A Real Example from SDSS Query image An all-sky catalogue. Query image Zoomed in by a (after object detection). (after object detection). factor of ~ 1 million. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu A Real Example from SDSS A Real Example from SDSS Query image The objects in our index. All the quads in our index which (after object detection). are present in the query image. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 7
4/27/16 A Real Example from SDSS A Real Example from SDSS The query image scaled, translated & rotated A single quad which we happened to try. as specified by the quad. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu A Real Example from SDSS A Real Example from SDSS The proposed match, on The verified answer, overlaid The proposed match, on which we run verification. on the original catalogue. which we run verification. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 8
4/27/16 Final Verification Preliminary Results: SDSS • After hash code • The Sloan Digital Sky matching, we are left with Survey (SDSS) is an a list of candidate views all-sky, multi-band that >1 codes agree on. survey which includes targeted spectroscopy • If this list is empty, the of interesting objects. search has failed. • The telescope is • If this list is non-empty, located at Apache we do a slower positional Point Observatory. verification on each candidate to see if it • Fields are 14x9arcmin really is the correct corresponding to position in the catalogue. 2048x1361 pixels. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu Preliminary Results: SDSS Preliminary Results: GALEX • 336,554 fields • GALEX is a space-based telescope, seeing only in science grade+ the ultraviolet. • 0 false positives • It was launched in April • 99.84% solved 2003 by Caltech&NASA 530 unsolved and is just about finished collecting data now. • 99.27% solve w/ • It takes huge (80 arcmin) 60 brightest objs circular fields with 5arcsec Assume known pixel scale resolution and spectra (for speedup of solving only.) of all objects. http://astrometry.net roweis@cs.toronto.edu http://astrometry.net roweis@cs.toronto.edu 9
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