color star tracking ii: matching - spie...color star tracking ii: matching john enright geoffrey r....

Color star tracking II: matching

John EnrightGeoffrey R. McVittie

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Color star tracking II: matching

John EnrightGeoffrey R. McVittieRyerson University350 Victoria StreetToronto, CanadaE-mail: [email protected]

Abstract. A novel matching algorithm is presented that can identify starsusing raw images of the sky obtained from a CMOS color filter arraydetector. The algorithm combines geometric information with amplituderatios calculated from the red, green, and blue color color channels.Conventional algorithms that match stars based solely on inter-star geom-etry (and sometimes relative brightness), typically require three or morestars for a confident star match. In contrast, the presented algorithmsare able to find matches with only two imaged stars in most regions ofthe sky. The necessary catalog preparation and a simple star-pair match-ing algorithm based on combined color intensity ratios and the angularspacing are discussed. Results from a large set of simulation trials andinitial results from sensor field testing are presented. © The Authors.Published by SPIE under a Creative Commons Attribution 3.0 Unported License.Distribution or reproduction of this work in whole or in part requires full attribution of the originalpublication, including its DOI. [DOI: 10.1117/1.OE.52.1.014406]

Subject terms: star tracker; star matching; attitude determination.

Paper 120540 received Apr. 12, 2012; revised manuscript received Nov. 30, 2012;accepted for publication Dec. 17, 2012; published online Jan. 9, 2013.

1 IntroductionA measure of robustness for autonomous star trackers is theability to obtain an attitude estimate without a priori attitudeinformation, while viewing an arbitrary part of the sky.Performance is usually limited by regions of the sky with lowstar densities. When solving the “lost-in-space” (LIS) prob-lem, imaged stars must be associated with records in the startracker’s on-board catalog. Although some star-matchingroutines can operate with only two stars in the field of view(FOV) once an initial solution has been found, almost all LISmatching algorithms require three visible stars for an initialfit. Unfortunately, the geometry of many pairs and triplets ofstars are ambiguous and unique attitude solutions are notalways assured with the minimum number of imaged stars.The most common remedy for this ambiguity is an opticaldesign that guarantees detection of more than the minimumnumber of stars. Designers can meet this requirement eitherby enlarging the FOV, or observing dimmer stars. Our re-search considers a different approach to resolving ambiguousstar patterns: that of using coarse measurements of stellarspectra as well as inter-star angles.

Awide variety of star identification techniques have beenproposed in literature including triangle, pole-star, grid, andadaptive techniques. A survey by Spratling and Mortari1

provides a good overview of the current state of the field.Matching routines rely on geometric arrangements betweenstars (e.g., arc lengths, dihedral angles, etc.). Some ap-proaches (e.g., Servidia et al.),2 use visual magnitude infor-mation to help resolves ambiguity between similar geometricpatterns. Others, such as Rufino and Accardo3 argue explic-itly against the use of absolute magnitude because accuratephotometry is difficult with the optics and detectors found intypical star trackers.

Although spectroscopy is common practice in astronomy,prior to our current work, there seems to be no discussion ofusing star spectral classification to assist in star tracking.Adopting color detectors will impact many of the steps inthe star tracker image processing sequence. This paper

examines how color information can be used to assist thestar matching process; in our companion paper,4 we developtechniques to reproducibly measure position and color of thestars in a raw detector image captured by a color CMOSdetector.

We begin our analysis by summarizing our recent devel-opment of a nano-satellite star tracker. This section details areference hardware and software implementation that estab-lishes the context for the subsequent analysis. In Sec. 3 weintroduce the basic mechanisms for matching observed starsagainst cataloged data. We present both a basic matchingstrategy and an extension that allows us to match an entirescene’s worth of stars. We then discuss the challenges build-ing catalogs suitable for color matching. Section 4 examinesa number of practical and theoretical issues including: sourc-ing spectral information; encoding and storing the catalogbased on a instrument’s optical properties; and evaluating thetheoretical performance of a star tracker design. Finally, wevalidate the performance of our matching algorithms usingsimulations and night-sky tests.

2 Reference DesignThis paper represents a companion study to the recent devel-opment of a very small star tracker, the S3S ST-16. Acollaborative development effort between Sinclair Inter-planetary (SI), the University of Toronto Space FlightLaboratory (UofT-SFL), and the Ryerson University SpaceAvionics and Instrumentation Laboratory (SAIL), the ST-16(Fig. 1) is designed around a commercial CMOS detectorand a simple lens assembly. This design is described atlength in Enright et al.5 The hardware and processing archi-tecture of the ST-16 have served as a starting point for thiscurrent study. Our performance targets and assumptions con-cerning accuracy, FOV size, resolution, sensitivity, updaterate are derived from the ST-16 design. We recognize thatthis is not an optimal design for a color star tracker, but itis a very easily realized one, for the detector in the ST-16has both monochrome and color variants.

Optical Engineering 014406-1 January 2013/Vol. 52(1)

Optical Engineering 52(1), 014406 (January 2013)


http://dx.doi.org/10.1117/1.OE.52.1.014406

http://dx.doi.org/10.1117/1.OE.52.1.014406

http://dx.doi.org/10.1117/1.OE.52.1.014406

http://dx.doi.org/10.1117/1.OE.52.1.014406

http://dx.doi.org/10.1117/1.OE.52.1.014406

http://dx.doi.org/10.1117/1.OE.52.1.014406

2.1 Optics and Performance

The ST-16 detector is the monochrome version of the AptinaMT9P031. The MT9P031 CMOS detector has a resolutionof 2592 × 1944 pixels and a 2.2 μm pixel pitch. The mono-chrome version is equivalent to the color version of the detec-tor, but lacks the color filter array (CFA) that producesdistinct output color channels. The sensor’s F1.2, 12 mm(dia.) lens assembly gives the ST-16 a threshold detectionmagnitude of Mthresh ¼ 5.75, a 7.5 half-angle FOV, θFOV,and nominal exposure of 100 ms. The design provides atleast three visible stars in view in more than 99.9% of thepossible sensor orientations.

The CFA version of the detector allows distinction be-tween sources of different colors. The CFA consists of small,absorptive filters in front of each pixel in a repeating 2 × 2pattern. These filters absorb some of the incident light, so theoverall sensitivity to dim sources is lower for the color detec-tor than for the monochrome version. An illustration of thetypical differences in quantum efficiency between color andmonochrome detectors is shown in Fig. 2. We note that themonochrome curve is adapted from the specifications forthe MT9V022 detector6 (the monochrome response of theMT9P031 is very similar to the MT9V022, but does notappear on the public datasheet). Although there are somedifferences, based on the spectrum of the source stars, thecolor detector generates about one third of the integratedresponse of the monochrome (i.e., ∼1.25 visual magnitudes).

2.2 Baseline Processing Model

Several processing steps are necessary to convert raw imagesinto attitude estimates (see Fig. 3). In a color star tracker, araw sky image from a CFA detector contains the intensityreadings from the red, green, and blue pixels. The initialimage processing routines in the star tracker must analyzethis intensity map and identify likely stars. These coarsestar locations must then be refined to give precise estimatesof centroid position and color. A camera model rectifies thecentroid locations and predicts corresponding direction vec-tors for each star. Matching routines identify the stars bycomparing the available data to on-board catalog of knownstar patterns. A final attitude estimate follows directly fromthe matched star information. Except for the presence ofcolor information, this processing sequence is typical ofmost modern star trackers.

Color representation is an important consideration; in ourcompanion paper,4 we found that integrated intensity ratioswere an effective representation of star color. Our detectionroutines use the raw CFA output to estimate ratios between

Fig. 1 The ST-16 star tracker prototype, with penny for scale.

400 500 600 700 800 900 10000

10

20

30

40

50

60

Wavelength (nm)

Qua

ntum

Effi

cien

cy (

%)

MonoRedGreenBlue

Fig. 2 Comparison of monochrome and color quantum efficiencies(from Refs. 6 and 7).

ColourRatios

iigr gbΛ ,Nearest-Pixel Sub-Pixel

CoarseDetection

(Th h ldi )

FineDetection

(C t id Fit)

CameraModel

RawImage ,ˆ ,

iDs

Star

Centroids Centroids

Star Matching(Thresholding) (Centroid Fit) Star

Vectors

nm ii nm ii

Λ

Fig. 3 Color star tracker image processing.


Enright and McVittie: Color star tracking II: matching


the blue-green, Λbg, and red-green, Λrg, channels, togetherwith the centroid position, (mc; nc). These algorithms usea nonlinear least-squares cost function and Levenberg-Marquardt minimization. The typical root mean squared(RMS) color noise reported in the study was less than0.1, and often about 0.05. The centroid position error wasequivalent to about 0.2 pixels. These metrics will influencethe effectiveness of our matching algorithms.

Our proposed color-based matching routines (and themonochrome matching routines on the ST-16) rely on know-ing the direction vectors, s in the detector frame of reference.Each ST-16 star tracker must be calibrated to account formanufacturing tolerances and optical aberrations such asdistortion. We use a 17 parameter model to relate centroidposition on the detector array, i.e., (m0; n0), to the star vectordirection, sD. Our calibration procedure and model is basedon the scheme proposed by Heikkila and Silven.8 We employa gimballed platform, a fibre-illuminated pinhole, and aninverted telescope to produce a point source at infinity. Aftercalibration, the typical residual centroid error is 0.2 pixels(RMS). This includes the effects of temporal noise and cal-ibration error. Presently, we assume that the calculation of sDdoes not rely on the star color; if necessary, additions to themodel could correct for chromatic aberration.

3 Color-Based MatchingThe challenge of solving the lost-in-space problem for startrackers rests with the matching of the observed image starsto those in the sensor’s on-board catalog. Whereas otherprocessing steps are deterministic in time and memory, thematching process involves search operations. Thus a viablealgorithm must be both accurate (i.e., it identifies the correctstars), and computationally efficient (the search is boundedin time).

The star matching routines must associate the observablestar characteristics (i.e., mc, nc, Λrg, and Λbg), with knownstars in the on-board catalog. This is essentially a mappingproblem: given a set of observed stars A and a catalog of starsC, we need to identify the relations that map Ak → Cj (in ournotation, we use a subscript index when we need to refer toa particular star from one of these sets). Generally, the morestars that can be matched, the better the resulting attitude fix.However, not all detections made by the sensor are actually

stars—planets, other satellites, or even noise may cause falsedetections—so we may not be able to find a match for all thestars in A.

We first develop the basic kernel matching of a single pairof image stars against the on-board catalog. Not only doesthis represent the fundamental matching operation, but italso provides analysis tools that can be used to predict theperformance of the sensor for a given optical design. Wethen extend the pairwise matching algorithm to efficientlymatch more than two stars. We conclude this section witha discussion of how these algorithms can be implementedefficiently.

3.1 Matching Star Pairs

The basic matching algorithm relies on measurements ob-tained from a pair of image stars. Each measurement set con-sists of five quantities: four color ratios (two for each star)and the angular separation between the pair. In monochromestar trackers, star-pairs alone are too ambiguous enough forreliable matching, but the color data adds sufficient informa-tion for a unique match in many cases.

For a particular image, we select two stars Ak; Al ∈ A andcompute the tuple TAkl

:

TAkl≡ fΛrgk Λbgk φkl Λrgl Λbgl g; (1)

where φkl is computed from the two unit direction vectors:

φkl ¼ arccosðsTk slÞ: (2)

For notational simplicity, we have dropped the frame ofreference subscript from these vectors. Our catalog containssimilar tuples derived from known star locations and spectralproperties. We can compare our observations to the catalogby computing the Mahalanobis distance between each pair oftuples.* In general, this has the form:

Δ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðTA − TCÞTΣðTA − TCÞ

q; (3)

where Σ is the measurement covariance matrix of the ele-ments of TA. If Σ is diagonal (i.e., errors are uncorrelated),then we can write the distance between TAkl

and TCijas:

ΔðTAkl; TCij

Þ ≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Λrgk − Λrgi

σrg

�2

þ�Λrgl − Λrgj

σrg

�2

þ�φkl − φij

σφ

�2

þ�Λbgk − Λbgi

σbg

�2

þ�Λbgl − Λbgj

σbg

�2

s. (4)

The k and l designations in the observed pairs are arbi-trary, so we must compute both ΔðTAkl

; TCijÞ and ΔðTAlk

;TCij

Þ and take the smaller of the two distances. The maxi-mum likelihood match between TAkj

and the cataloged pairsis the TCij

that minimizes Δ. Sorting C in order of increasingφ can make this search efficient.

3.2 Star Pair Ambiguity

We expect that the catalog star tuple TCijthat minimizes Δ

will be the correct match to a pair of observed stars. Thisassumption holds provided that centroid and color noise

does not move the image tuple TAijaway from the correct

match and closer to another catalog pair. Examining the dis-tribution of Δ values calculated between catalog tuplesallows us to assess the likelihood of measurement noisecausing incorrect matches. We outline the basic frameworkfor this analysis here and further develop the technique inSec. 4. This analysis allows us to predict theoretical sensorperformance based on catalog properties and a few opticaldesign parameters.

*McLachlan9 provides a good introduction for those unfamiliar with thistype of classifier.




The smaller the Mahalanobis distance between two cata-log pairs, the more likely they are to be mistaken for oneanother. Below a certain distance threshold we consider thetwo tuples ambiguous. Using Eq. (4) to compute Δ betweentwo catalog pairs gives:

ΔCðp; qÞ ≡ ΔðTCp; TCq

Þ. (5)

Here we have replaced the star indices i, j from each tuplewith a single catalog index, p or q. Assuming normal errordistributions, we can use linear discriminant analysis theoryto compute the ideal error probability. Adapting the resultfrom McLachlan et al.,9 this becomes:

Perror ¼1

2

�1þ erf

�−Δ2

ffiffiffi2

p��

. (6)

Equivalently, we can solve for the critical separation Δ⋆,for a specified success rate Pcorrect ¼ 1 − Perror:

Δ⋆ ¼ −2ffiffiffi2

perf−1ð1 − 2PcorrectÞ. (7)

Star pairs with a Mahalanobis distance less than Δ⋆ areambiguous. The results in the remainder of this paper usePcorrect ¼ 0.99 to establish a distance threshold. The corre-sponding critical distance is Δ⋆ ¼ 4.65.

Understanding how ambiguous pairs affect system perfor-mance depends on our multistar matching strategy and thecontents of the whole star-pair catalog. These concepts aredeveloped in Secs. 3.3 and 4, respectively.

3.3 Matching Whole Scenes

Our basic matching algorithm gives the best catalog match toa particular pair of image stars. Generally, we can improvethe accuracy of our attitude estimate if we can find consistentmatches to any additional stars in the sensor image. Thisadditional accuracy comes at the cost of increased computa-tional complexity. Most matching algorithms using pairs ortriplets scale with OðN2

AÞ or OðN3AÞ, where NA is the number

of stars (some classes of algorithms such as Grid Algorithmscan moderate this).10,11 For star-rich areas of the sky, we needan efficient approach to matching.

Our strategy for efficiently matching multiple star scenesis based on the random sample consensus (RANSAC)approach12—a popular approach in computer vision. We usea greedy heuristic to quickly assemble a self-consistent setof matches, concentrating on finding ‘inliers’ instead ofrejecting outliers. This algorithm works as follows (see Fig. 4for a pictorial representation of this process):

• From the initial set of imaged stars A [Fig. 4(a)], weselect two stars, Ak and Al [Fig. 4(b)], form the pair-tuple TAkl

and find the best pairwise match betweenTAkl

and the star catalog. The stars Ak and Al formthe basis for a candidate consensus set (or conset), S[Fig. 4(c)].

• For each star Ai; r ≠ k; l we select one star Aj from S(i.e., Aj ∈ S), and form the pair-tuple TAij

[Fig. 4(d)].We then find the best pairwise match between TAij

andthe star catalog, with the additional constraint that themapping j → p is already known. If we can find such a

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

falsestar

1

4

2

3

1

4

2

3

1→p

4→q

2

3

1→p

4→q

2

3

1→p

4→q

2

3

1→p

4→q

2→r

3

1→p

4→q

2→r

1→p

4→q

2→r

1→p

4→q

2→r

Fig. 4 Graphical illustration of consensus matching.




match where Δ < Δ⋆, then we tentatively accept themapping i → r between Ai and the catalog star Cr. Ifwe cannot find an acceptable match for Ai we move onto the next star in A.

• If we have a tentative match for Ai, we need to check toensure that the match is consistent with the other starsin the consensus set S [Fig. 4(e)]. Although we couldevaluate the consistency using the full color informa-tion, it is faster to perform this check based on geom-etry alone. Thus, for each As ∈ S, where we have thetentative mapping i → r and confident mapping s → k,we must have:

jδφj ≡ jφAis− φCrk

j < δφ⋆; (8)

where the maximum acceptable distance error can bederived from the critical Mahalanobis distance, assum-ing the color errors are zero:

δφ⋆ ¼ σφΔ⋆∕2. (9)

The factor of two is needed because Δ⋆ was defined interms of the separation between two catalog entries.

• If the tentative match, i → r, satisfies the consis-tency check outlined in Step-3, then we add Ai to S[Fig. 4(f)]. A false star in the scene, may return avalid pairwise match [Fig. 4(g)], but is very unlikelyto pass the geometric consistency checks [Fig. 4(h)].If the tentative match does not meet the consistencycriteria, Ai is not added to S.

• Once all of the stars in A have been evaluated [Fig. 4(i)],we consider the number of stars we have added to S,i.e., NS. If the number of matched stars exceeds aspecified fraction of the total number of imaged stars,i.e., NS ≥ ηNA, this indicates that we were able toassemble a good consensus set from the initial matchedpair. In this case, the algorithm can terminate early.Conversely, a small consensus set suggests a poor ini-tial match. In our example, this might happen if thefalse star was included in the initial pair of stars. In thiscase, we select a new initial pair and rebuild the conset.The overall matching algorithm terminates when wehave found a good consensus set, or after a specifiednumber of matching attempts has completed.

Like many RANSAC-derived heuristics, good matchingperformance requires some algorithm tuning. We havefound that a matching threshold fraction in the range 0.6 ≤η ≤ 0.7 combines good tolerance to false stars (i.e., low η),with good rejection of bad matches. We must also set anupper bound on the number of RANSAC iterations weallow; for our trials, we have used 3 · NA.

The matching algorithm keeps track of the largest consetfound throughout all of its iterations. Upon termination,we can classify the results as one of three cases: a high-confidence match (conset size exceeds the η, threshold), alow-confidence match (conset is smaller than the η thresh-old), or no match at all (no acceptable starting pairs found).In the first and last cases our next actions are well defined.In the first, we proceed with the processing sequence andcalculate the sensor attitude; in the last case, we report amatching failure and abort. The remaining case—the low-confidence match—may require extra logic to categorize.

In a flight sensor, we may want to formalize more elaboratedecision rules, but for this study, we merely log the resultand complete the attitude calculation.

4 Star Catalog DesignOur matching algorithms rely on comparing the star trackerobservations to catalog values. Thus, we must pay particularattention to the source material, content, and organizationof C. Much of the recent innovation in star tracker processinghas involved different approaches to catalog generation.Calculations that can be performed off-line (areas, distances,etc.) reduce the computational (and often power), require-ments on the star tracker processor. However, precomputinggeometric metrics often increases the need for storage: acatalog of a few thousand stars routinely produces tens ofthousands of star-pairs, and possibly millions of star-triples.

Color star catalogs share many of the same characteristicsas monochrome catalogs, but present a number of uniquechallenges. In this section, we detail the content and organi-zation of our baseline star catalog. We consider both conven-tional requirements for star geometry as well as the extrapreparation required to tabulate color information.

4.1 Preliminary Catalog Processing

The source catalog for this research is the 5th RevisedEdition of the Yale Bright Star (YBS) catalog.13 We denotethis catalog by Y. We subscript this symbol, by the catalogindex, i.e., Yi to refer to a particular star. The on-board cata-log is denoted by C. Working from the source catalog Y, weselect a subset of stars for use in C. We select stars for Cbased on the following criteria:

• Brightness. Our primary criterion for inclusion in theon-board catalog is visual magnitude Mv (i.e., bright-ness). Because the larger values ofMv indicate dimmerstars we select those stars with Mv ≤ Mthresh. Mthresh

represents the brightness of the dimmest stars that aparticular star tracker can reliably detect under nominaloperating conditions.

• Double stars. Binary stars or stars whose apparent sep-arations are too small to be distinctly resolved by thesensor optics are treated as a single star. The brighterstar in each case is included in C.

• Transient Objects. A number of the entries in the YBSare attributed to transient novae. They have been re-tained in Y to preserve the numbering scheme, but areunnecessary when compiling C.

Each star in the source catalog, Y, has a variety of generalastronomical data associated with it. Not all of this informa-tion is necessary in the on-line catalog. For convenience westore some basic information about each star(i.e., its Henry-Draper identifier and visual magnitude), but the star’s posi-tion is most important.

Most astronomical catalogs tabulate star position in termsof right ascension α, and declination, δ. High precision appli-cations may also include proper motion information (weneglect the effect of star motion for this study). To avoid theneed for unnecessary trigonometric calculations, it is con-venient to store a star’s position as a unit vector, expressedin the Earth Centred Inertial (ECI) coordinate system, sI1 .




4.2 Cataloging Color Data

Assembling an on-board catalog requires compilation ofcolor information for each star in C. We must consider whatcolor information will be stored; where it will come from;and whether any off-line processing is necessary to tailor ourdata sources to the specific performance of our sensor.

In astronomy, color is an ambiguous term related to anobject’s emission spectrum. In addition to standard identi-fiers and celestial coordinates, conventional catalogs, likethe YBS catalog, often include several quantities relatedto the spectrum of the star. These include: spectral type,color index, and effective temperature. Other collated data-sets contain detailed spectra for individual stars.

Our discussion of color-based star detection relies on afamiliarity with spectral type classification schemes. For sim-plicity we use the popular Morgan-Keenan (and Kelmann)(MK) system.14–16 Formally, cataloged spectral types arecomposite classifications based mainly on star temperature,but also on specific features (e.g., emission or absorptionlines, etc) of the complete spectrum.

Although spectral type is a multiattribute categorization,the notion of color index is much closer to a directly mea-surable quantity. Classical color indices were introduced byJohnson and Morgan and indicate magnitude difference inthe integrated response across standardized spectral bands.14

Unfortunately, the color filters on our detector do not havethe same response as the official Johnson and Morgan pass-bands, making direct conversion between the two difficult.Instead, we calculate our own version of these color indicesfor each star in the star tracker catalog. Given a star’s spec-trum, IðλÞ, we can calculate the red-green and blue-greencolor indices:

Λrg ≡

R λ2λ1χrIðλÞdλR λ2

λ1χgIðλÞdλ

; (10)

Λbg ≡

R λ2λ1χbIðλÞdλR λ2

λ1χgIðλÞdλ

. (11)

The χr;g;b capture the detector quantum efficiency for eachof the color channels.

Choosing a source of stellar spectra for our star trackercatalog presented its own challenges. Complete spectra isunnecessary in the on-line catalog; we need only store theexpected color ratios. Three approaches seemed feasible:(a) continuous black-body curves based on the stars’ effec-tive temperatures; (b) reference solar spectra based on spec-tral type; and (c) tabulated spectra for each individual star.The reference spectra used in this study are taken from thePickles spectral library;17 individual spectra are taken fromthe recent Indo-US Library of Coud Feed Stellar Spectra,18

and to a lesser extent from an earlier catalog by Burnashev.19

Temperature estimates are based on data in the Indo-US cata-log. A comparison of these approaches is shown in plots forγ-Ori (Fig. 5) and α-Ori (Fig. 6). We used the Pickles B2IIspectrum for γ-Ori (there is no B2III spectrum provided), anda black-body temperature of 22,570 K. For α-Ori we use theM2I spectrum and a temperature of 3540 K.

We decided to use the Pickles catalog as our color data-base, eliminating the alternatives based on the following

rationale. First, although the black-body curve is a goodmatch to the hot spectrum of γ-Ori, it is a very poor matchto the much cooler α-Ori. Because effective temperature isbased on total integrated flux, not on a match to the spectra,black-body distributions introduced inaccuracies in the cal-culated color ratios [i.e., from Eqs. (10) and (11)]. Second,even though the Indo-US catalog is considered to have broadspectral coverage (346-946 nm) it is not quite sufficientto cover the full sensitive range of our CMOS detector.Improved color ratios could be calculated by merging partialspectral catalogs for all members of C—Pickles describessuch strategies in context of building his reference spectra—but this process would be laborious. Furthermore, exceptfor small local features, the Pickles and Indo-US curves arein reasonable agreement. The integration in the evaluation ofthe Λ ratios will minimize the effect of local differences. Inthe balance, the Pickles’ data was a pragmatic compromise.

Having selected the Pickles catalog, P, as our source ofcolor information, we must still associate each element ofC with one of the reference spectra. Assigning reference

400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Wavelength (nm)

Nor

mal

ized

Flu

x

B2II Reference Spectra (Pickles)γ−Ori (IndoUS)Black−body

Fig. 5 Comparison of Spectra for γ-Ori (Bellatrix). Flux normalized to550 nm.

400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

Wavelength (nm)

Nor

mal

ized

Flu

x

M2I Reference Spectra (Pickles)α−Ori (IndoUS)Black−body

Fig. 6 Comparison of Spectra for α-Ori (Betelgeuse). Flux normalizedto 550 nm.




spectra to the stars selected from the YBS is not alwaysstraightforward. The spectral types given in the latter catalogdo not always have exact matches in the former. Rules forselecting the closest match depend of the stellar classifica-tion: for hot stars, the class and subclass dominate; for coolerstars, luminosity can be quite important as well. We havedeveloped a set of heuristic rules for this process, but furthercross-checking with the available spectra in the Indo-UScatalog is necessary. Figure 7 shows the predicted color ratioresponse across different spectral types for our baselinedetector. The hook in the bottom right of the figure is causedby the second passband in the blue and green curves around800 nm (see Fig. 2).

4.3 Star Pair Catalog

The Pickles catalog, P, provides a mechanism for computingthe expected color ratios for each star in C. To complete ourcatalog, we compile a secondary table of feasible star tuplesTCij

. Our matching algorithm is based on these pairs of starsand this table provides the basis for finding matches. Theangular separation between two stars, k; l ∈ C, φkl can becomputed from their direction vectors:

φkl ¼ arccosðsTIksIlÞ: (12)

To compile the table of star-pairs, we evaluate all the paircombinations and select those that satisfy

φmin < φkl ≤ 2θFOV: (13)

The upper bound ensures that all catalog star-pairs fitwithin the minor axis of the sensor FOV; the lower boundis chosen so that the star peaks can be resolved separately.Typically φmin ¼ θFOV∕500, an empirically chosen distance,equivalent to about four pixels.

It is unnecessary to store the entire five-element tuple on-line. The color information can be stored in a separate tableindexed by the star identifier or reference spectrum identifier.

When we need to find the best match to a star-pair tuple T,we do not have to calculate the Mahalanobis distances for all

tuples in the catalog. Rather we recognize that each compo-nents ofΔmust lie in a bounded range. Thus, if the measuredimage arc-length is φkl, we need only check those catalogentries whose arc-length, φp, satisfies:

φkl − δφ⋆ ≤ φp ≤ φkl − δφ⋆: (14)

If we sort the catalog entries in order of increasing φ, thenall of the feasible matches will be sequential in the pairs cata-log T. We use a hash-based index scheme to find the startof the appropriate section of the catalog (see Fig. 8). We allo-cate storage for a pairs index Ph; 0 ≤ h < Nh (typically weuse Nh ¼ 1024, but this choice was somewhat arbitrary).For an observed φkl we calculate the separation hash h:

h ¼ bH · φklc; (15)

where the constant scaling factor H is chosen such thatH ¼ Nh∕φmax. The value of Ph is the smallest catalog indexp, where

φp ≥hH

− δφ⋆. (16)

The matching routine then starts at table index Ph andmoves forward sequentially until the catalog value of φexceeds the upper bound of Eq. (14). The hash h correspondsto a range of different φ values, so the initial φPh

mayactually lie outside the lower bound of Eq. (14). In practice,this is a negligible concern because there are typically only ahandful of extra pairs, and these can all be skipped quicklybased on their arc-length.

5 Experimental ValidationWe conducted a number of tests to assess the overall feasibil-ity of the color star matching concept. First, studying thedistribution of ambiguous tuples (Sec. 3.2) in a color starcatalog measures the effectiveness of the Δ-based classifierand allows us to compare the performance of sensor point-designs. Second, we examine the performance of the fullmatching algorithm using a combination of scene simulationand night-sky testing.

5.1 Sensor Design Trade Studies

One of the key performance metrics for a star tracker is itsavailability, A. This quantity represents the fraction of timewhen an image will provide a correct attitude solution. Straylight or occlusion from the sun, Earth or moon, can reducethe effective availability for a given mission, but more

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Red/Green Ratio

Blu

e/G

reen

Rat

io

O5V

B5I

A5V

F5V

G0I

G8IK2I

K7V

Fig. 7 Color intensity ratios as a function of spectral type. The curl inthe lower-right is caused by secondary pass-bands in the array filtersat near infrared wavelengths.

Fig. 8 Pair look-up tables.




fundamental is the instrument’s raw ability to image enoughstars to provide an attitude fix. In this section, we examinethe sensor performance as predicted from the properties ofthe star-pair catalogs (from Sec. 4). We use these results toevaluate the suitability of the baseline ST-16 optical designand identify evolutionary design changes that may be bettersuited for color-matching.

Sensor availability is strongly tied to the dimensions ofthe FOV, θFOV, the threshold detection magnitude, Mthresh,and the star matching scheme. For our color-based matching,availability will be limited by instrument orientations thatmeet one of the following criteria:

• Zero or one visible stars. This is strictly a function ofθFOV and Mthresh.

• Views that only image ambiguous star-pairs. For exam-ple, if two stars are visible, then that pair must beambiguous; if three stars are visible then all three pairscombinations must be ambiguous. Thus, ambiguouspairs are only harmful if no unambiguous pairs areavailable.†

To predict the performance of a particular star trackerdesign, we must consider both of these cases. Although thetwo situations appear to be similar, there is an important dis-tinction between them. In views with an insufficient numberof stars, the matching routines are aware they cannot providean attitude solution. In contrast, when only ambiguous pairsare visible, the algorithms may give thewrong solution. Thus,the second situation is much more serious than the first.

Our analysis is broken down into the following steps:

1. An exhaustive off-line search identifies all ambiguouspairs in the catalog. These results are stored forreference.

2. We discretize the celestial sphere using a generalizedspiral.20 This spiral generates a set of N, uniformly dis-tributed bore-sight vectors, rk. For these trials we useN¼ 105—this puts about 430 sample points in an areasize of the FOV (θFOV ¼ 7.5 deg). The separationbetween points is approximately 0.7.

3. For each bore-sight vector, we calculate the number ofvisible stars (i.e., Mi ≥ Mthresh), that lie closer thanθFOV to rk. If we have fewer than two stars, we recordthis view direction in a list of bad views.

4. If there are two or more stars in the FOV, then wechoose pairs of stars until we find a pair that is unam-biguous. If all the pair combinations are ambiguous,then make note of this view.

The availability, A, is evaluated as a spatial fraction (ratherthan a temporal fraction):

A ¼ Number of Good Views

Total Number of Views. (17)

To make our results more readily comparable from onedesign to another, we tabulate the fraction of bad views,i.e., 1 − A.

These analyses were repeated for a range of different sen-sor designs. The choice of specific cases was motivated bythe characteristics of the ST-16 design (see Sec. 2.1), andthe laboratory estimates of the centroid and color indexnoise (i.e., σpix, σrg, and σbg obtained from McVittie andEnright).4 The integrated difference in quantum efficiencybetween color and monochrome detectors is equivalent toa visual magnitude difference of approximately 1.25. Thus,the design threshold of the ST-16, Mthresh ¼ 5.75, is equiv-alent to a color sensor with Mthresh ¼ 4.50. We also considermore sensitive yet plausible designs. Manual inspection ofST-16 images reveal a number of magnitude 6.25 stars. Sucha sensor would have a corresponding color thresholdMthresh ¼ 5.0.

The centroid position error, expressed in pixels, is relatedto the arc-length error σφ by the following approximation:

σφ ≈ 2γ

fσpix: (18)

The γ∕f factor converts the error from pixels to radiansand the factor of two computes the diagonal error betweentwo measured centroids. Centroid noise captures the perfor-mance of the centroid estimation routines. The two selectedvalues for the represent nearest-pixel accuracy (σpix ¼ 0.4),and a modest degree of subpixel centroid refinement(σpix ¼ 0.2).

The results from these trials can be seen in Table 1. Themost immediate observation is that no view has only ambigu-ous pairs. This suggests that system availability is limitedby the sensitivity of the detector and optical design, not thepair-based matching scheme. For each test case, we repeatedthe analysis assuming that only monochrome informationwas available. Virtually all of the catalog pairs are ambigu-ous when compared solely on their separation. More impor-tantly, these tests indicate a small but significant number ofunresolvable, ambiguous views. Additionally, increasing thecatalog size (e.g., by increasing Mthresh or θFOV), or increas-ing the noise values, will also increase the number ofambiguous star-pairs.

Several secondary observations can be made from thesedata. The first few rows represent color sensors very close tothe reference optical design. Sensor performance for thesetest cases is not terrible, but substantial areas of the sky(16%) are still subject to bad views. The design solutionswith around 2% bad views offer much better availabilitywith optical designs only slightly different from the baselinesensor. In particular, because we have successfully detectedmagnitude 6.25 stars with the monochrome detector on theST-16, the solutions withMthresh ¼ 5 and θFOV ¼ 7.5may beplausible without any hardware changes. Although a badview fraction of 2% is not negligible, that level of perfor-mance may be acceptable for some missions. The bestavailability is encountered with designs a little furtherfrom the reference values. These cases demand an increasein both θFOV and Mthresh. Such changes would require sub-stantial alterations to the current optics (and possibly thedetector, too).

More generally, the relatively low number of ambiguouspairs suggests that our color -based classifier provides gooddiscrimination between catalog tuples. Ambiguity rises withboth larger catalogs and increasing errors, indicating areas ofthe design-space that should be avoided.

†This criterion is actually more conservative than it has to be. A variety ofadditional inferences involving the distribution of ambiguous star-pairscould help resolve many ambiguous observations.




5.2 Simulation Experiments

Simulation of the entire matching process provides comple-mentary results to the static catalog analysis. Applying thepair-wise and scene matching algorithms to synthetic obser-vation data allows us to generate estimates of real-worldperformance.

Our simulations are designed to measure matching perfor-mance over the entire sky. Instead of random orientations, westart with the uniform grid of bore-sight directions, and con-sider multiple roll angles for each case. A typical simulationinvolves 106 inertial orientations, generated using 104 uniquebore-sight vectors. This corresponds to a density of approx-imately 70 different bore-sight angles within the FOV ofthe ST-16.

For each simulated orientation, we determine the stars thatwould be visible to the sensor, and determine the observeddetector-frame star vectors sDi

. This process considers theactual rectangular FOV of the sensor. In this study, we areprimarily interested in evaluating the matching performanceof our algorithms rather than detection behavior. Thusinstead of pixel level image simulation, we abstract thedetection process and generate star measurement vectorsdirectly (with appropriate direction and color index noise).We use the following error model:

sDi¼ sDi

þ nλksDi

þ nλk(19)

Λrg ¼ Λrg þ nrg (20)

Λbg ¼ Λbg þ nbg. (21)

The star vector noise, nλ, is distributed in a Rayleigh dis-tribution around the true star vectors (uniform in direction,

effectively Gaussian in both axes perpendicular to the truestar direction). color index noise (i.e., nrg, nbg) is normallydistributed and zero-mean.4 A summary of the simulationparameters is shown in Table 2.

If the simulated scene produces an acceptable match, thesensor orientation is estimated using the Davenport’s q-method,21 giving the estimate of the quaternion attitude qDI.The net error in the estimate, ϕerr is the angle derived fromthe error quaternion:

qerr ¼ qDI ⊗ q−1DI ; (22)

ϕerr ¼ 2 arccosðqserrÞ; (23)

where qserr is the scalar part of the error quaternion.Figure 9 shows a plot of cumulative distribution in ϕerr

for the entire trial. About 90% of the tests have an errorof 0.01 deg or less; 99.3% of the trials had 1 deg or less.

Table 1 Color matching performance.

Test parameters

Catalog pairs(×103)

Bad views(%)

Color resultsMonochromeambiguouspairs (%)

Resultsunresol.pairsM thresh θFOV σpix ðσrg ; σbgÞ

Ambiguouspairs (%)

Unresol.pairs

4.5 7.5 0.200 0.020 7.8 15.92 0.26 0 99.77 44

4.5 7.5 0.408 0.020 7.8 15.92 0.84 0 99.99 187

4.5 7.5 0.408 0.050 7.8 15.92 3.86 0 99.99 187

4.5 10.0 0.200 0.020 13.6 2.75 0.19 0 99.88 9

4.5 10.0 0.408 0.100 13.6 2.75 8.11 0 99.99 9

5.0 7.5 0.200 0.020 24.7 2.19 0.71 0 100 8

5.0 7.5 0.408 0.020 24.7 2.19 2.41 0 100 154

5.0 10.0 0.200 0.020 42.9 0.05 0.60 0 100 0

5.0 10.0 0.200 0.100 42.9 0.05 5.79 0 100 0

5.0 10.0 0.408 0.100 42.9 0.05 20.63 0 100 0

Table 2 Simulation parameters.

Parameter Value

Total poses 106

Unique bore-sight vectors 104

σλ 55 × 10−6 μrad ð≈ 0.2 pixelsÞ

σrg ; σbg 0.05

M thresh 5.0

Half-angle FOV (minor axis) 7.5 deg




While not particularly noteworthy performance for a startracker, it is consistent with the simulated centroid accuracy.

Understanding where the algorithm performs poorlyoffers valuable insight. Table 3 provides a summary of theproblematic orientations (i.e., no match, or a match with higherror). These cases represent about 0.7% of the total trials.Most of high-error solutions result from incorrect matchingassignments—only one test had both high error, and nomatching mistakes. A few cases exist in which a single cor-rect star was matched, but these merely represent two similarstar-pairs with a common star. No cases appeared with twoor more correct matches, combined with one or more badmatches, suggesting that the geometric consistency check ofEq. (9) is functioning correctly.

We can identify several distinct types of problematic ori-entations. Generally, matching failures that are diagnosableas such are preferable to potentially incorrect matches:

• No solutions. The sensor matching routines wereunable to calculate a match that satisfied the consis-tency criteria. The most common cause of this responsewere scenes with fewer than two stars in the FOV.‡ Thisclass of problem is of least concern to us, since thealgorithms are aware of their inability to find a match.Although there were 897 distinct orientations thatyielded no-match results there were only 49 uniquebore-sight vectors represented in these results. Thus,no-match problems are strongly correlated with star-poor areas of the sky.

• Poor matching. This arises from trials that returned acandidate match solution using fewer than 40% of theavailable stars. In almost all cases this was an incorrect,two-star match result. We suspect that noise is the pri-mary cause of these problems, but this type of resultrequires further study. This type of result is more seri-ous than the no-match case, but the low match percent-age offers a convenient confidence metric by which wecan discard the estimates.

• Wrong matches (two and three stars). This class ofresponse results from an incorrect star-match solutionreturned by the matching algorithm. Most of thesesolutions report one matched pair, but a small numberof views report three matched stars. This type ofresponse would be very serious on-orbit, and the fre-quency of occurrence (0.44%) is too high to beneglected. Minimizing these errors is the focus ofongoing research, but we note that raising the poor-match threshold to 50% would halve the number ofresults in this class.

Figure 10 shows the spatial distribution of problematicorientations. The coordinates in the figure correspond tothe direction of the bore-sight axis in celestial (inertial) coor-dinates. Each point represents one of the bore-sights evalu-ated and the data aggregated over the whole set of bore-sightangles. “Good” results have low error characteristics for allroll angles. Any other marker represents the worst result forthat direction. Most matching difficulties occur in regionsof low star density (the band of the Milky Way follows arough sinusoid through the green region, starting from highdeclination at the left side of the figure).

5.3 Field Testing

As a complement to our whole-sky simulations, we also con-ducted a set of night-sky tests, evaluating the performance ofthe color star matching algorithms with real detector images.These tests rely on the complete processing chain fromimage acquisition through attitude estimation.

Proper calibration is vital to the collection and interpre-tation of precise star tracker measurements. Not only mustwe remove the effects of optical aberrations using our con-ventional camera calibration (see Sec. 2.2), but we must alsocompensate for the influence of the Earth’s atmosphere onstellar spectra and positions.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.2

0.4

0.6

0.8

1

1.2

Error (deg)

Cum

ulat

ive

Pro

babi

lity

Fig. 9 Simulation results showing cumulative distribution of errorangle.

Table 3 Simulation results.

Total problem orientations 7078

No solution 897

Poor match (<40% of available stars) 1763

Incorrectly matched stars

2 (incorrect) of 2 (total stars) 1013

2 of 3 1244

2 of 4 1247

2 of 5 905

3 of 4 2

3 of 6 4

3 of 7 3‡Or two stars separated by more than the minor-axis arc-length.




5.3.1 Color calibration

Ground tests of any star tracker will be affected by atmos-pheric attenuation and refraction. These phenomena affectdetection thresholds and distort the inter-star geometry inthe images. Color matching routines must also deal withwavelength-dependent atmospheric attenuation of the appar-ent color indices of the observed stars.

As part of the field trials described by McVittie et al.,4 wemeasured color ratios from 21 bright stars, three from eachprimary spectral class. Figure 11 compares the measuredratios to the expected values based on the theoretical sensi-tivity of our detector and the reference spectra from thePickles catalog. The endpoints of each ‘bar’ on the figurerepresent the observed and catalog values for the stars inour survey. As a general trend, atmospheric attenuation anddispersion reduce the observed response, particularly atshorter wavelengths.

Slant path length (related to altitude and star elevationrelative to the Earth observation site), atmospheric aerosols,and humidity can cause significant variations in astronomicalobservations. As a full atmospheric correction model goesbeyond the current scope of work, we instead present a sim-ple system that has proven useful in validating the colormatching as a proof-of-concept. Our proposed color correc-tion is a six-parameter bilinear transformation of the form:�Λ 0

rg

Λ 0bg

�¼

�β1 β2β3 β4

��Λrg

Λbg

�þ�β5β6

�; (24)

where (Λrg;Λbg) are the observed color ratios and βi arethe model coefficients. The model is a simple low-ordercurve-fit, and the model parameters are not intended to bephysically meaningful. The fitting parameters are determinedusing a least squares minimization, with any obvious outliersremoved from the dataset (see red data point in Fig. 11).Figure 12 shows the calibrated dataset compared to the cata-log values. Although the fit does not provide a perfect cor-rection, it reduces the residual RMS color-error from 0.43(uncorrected) to 0.10 (corrected). A comprehensive atmos-pheric correction combined with larger sample set could seethis error reduced even further.

5.3.2 Atmospheric refraction

To minimize geometric distortions, the star vectors used inthese tests were corrected for atmospheric refraction. Usinga manual star match and the raw star vectors, we calculatethe inertial attitude CDI, of the sensor frame D. Refractionmay bias these estimates but not by more than a degree. Fromthe latitude, longitude, and time of the observations, we cancalculate the rotation, CIG, between the inertial frame I andlocal topocentric frame G. The precise azimuthal definitionof T is immaterial, as long as the ‘up’ direction is clearlyunderstood. These two matrices give us the sensor-to-topocentric rotation:

Fig. 10 Mollweide projection of spatial distribution of matching problems.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Λrg

Λbg

Λcat

Λobs

Fig. 11 Observed color ratios.




CDG ¼ CDICIG. (25)

Once we have calculated, CDT, the star vectors, sD can berotated to G, corrected for refraction using the formulae inBennett,22 and the resulting vector rotated back into D priorto matching.

5.3.3 Matching tests

Once the color-calibration data were processed, we collecteda number of images in order to test the color -matching

routines. Due to limitations of the prototype sensor optics,we had to make some adjustments to the instrument focusto get consistent detection of star color. These modificationsincreased the defocus of the prototype in order to providebetter sampling of the star point spread function (PSF). Gooddetections were limited, to a relatively small region near thecentre of the detector. Consequently, we concentrate on asmall cluster of three known stars, specifically Orion’s belt.Figure 13 shows the three stars of interest.

All three stars are bright and easily identifiable to thenaked-eye. As spectral-type O and B stars, they all havestrong blue-green response. In fact, both δ-Ori and ζ-Orimap to the same reference spectra, and ϵ-Ori, is only slightlydifferent. These properties make this a challenging triplet ofstars to match. The observation data for the stars are shownin Table 4. The Λrg ratios have low variability, but the Λbgratios remain higher than ideal (this catalog was designed forσrg ¼ σbg ¼ 0.05). The mean of the color-corrected ratios isfairly close to the predicted catalog values.

Table 5 shows the accuracy of the matching routines whenapplied to our Orion observations (a total of 70 images). Allof the correct results, successfully matched all three stars; allof the incorrect matches reported an incorrect pair of stars(with the third star unmatched). The initial success rate of50% is much lower than what would be suitable for a flightinstrument. Even attaining this modest performance relied onrelaxing the geometric constraint threshold [from Eq. (9)],and increasing the values of σrg and σbg used in the cataloglook-ups to 0.1, from the design value of 0.05. To understandthis poor performance, we took a closer look at the observedquantities we were using in the matching routines.

Our algorithms rely on accurate color and geometric mea-surements. Biases in either quantity lead directly to increasesin Δ, and will impair our ability to match to the star catalog.The critical Mahalanobis distance, Δ⋆, explicitly limits thebounds of the search space. To investigate how these errorscontributed to our poor match success, we devised two ideal-ized corrections. The first was a offset to each measuredcolor ratio, to bring the mean color ratio error to zero. Thesecond was correction to the measured arc-length φ betweenδ-Ori and ζ-Ori (the longest arc), again bringing the meanerror to zero (the magnitude of this correction was approx-imately 18 × 10−3 deg ). A final trial combined both ideal-ized corrections. Although these corrections are artificial,they isolate the separate effects of measurement noise andsystematic bias.

When the color and geometric corrections were com-bined, the matching performance greatly improved. The geo-metric correction is particularly significant because we have

0.6 0.7 0.8 0.9 1 1.1 1.20.6

0.7

0.8

0.9

1

1.1

1.2

Λrg

Λbg

Λcat

Λobs

Fig. 12 Calibrated color ratios.

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

1100

1200

1300

1400

1500

1600

1700

1800

ε-Ori (B0Ia)

ζ-Ori (O9.7Ib)

δ-Ori (O9.5II)

η-Ori

σ-Ori

Image Row

Imag

e C

olum

n

Fig. 13 Stylized view of one of a field test image. Axes indicateapproximate image scale and the dashed rectangle indicates starsused in the field trial.

Table 4 Field testing color ratios.

Star σrg σbg

Λbg Λrg

Raw Corrected Catalog Observed Corrected Catalog

δ-Ori 0.045 0.116 1.160 1.206 1.232 0.582 0.639 0.652

ϵ-Ori 0.038 0.096 1.116 1.156 1.161 0.598 0.663 0.674

ζ-Ori 0.030 0.081 1.123 1.166 1.232 0.580 0.652 0.652




only corrected a single arc length (and not the position ofall three stars). Some simple laboratory tests suggest thatthe geometric problems stem from chromatic aberration.Our geometric camera calibration is conducted using afibre-coupled, 3000 K incandescent source. Replacing thebroadband calibration source with a blue light emittingdiode (ζ-Ori is much more blue than our broadband source),introduced differential centroid motion very similar to theobserved geometric error. Some contributions to the mea-surement bias are attributable to testing in the atmosphere(e.g., color bias), whereas others represent limitations ofthe prototype hardware (e.g., chromatic aberrations to geom-etry). Careful attention must be paid to these factors in whendeveloping a revised color star tracker prototype.

Atmospheric scintillation is a likely cause for some ofthe observed color and centroid noise. Observations madewith short exposures and small apertures (as in our sensor),can see large amplitude variations between images, and sig-nificant band-to-band variability as well.23 As an example,separate experiments with the monochrome ST-16 show thataround 75% of the temporal centroid noise during field testsis attributable to scintillation. This effect makes it very dif-ficult to design a field test that assesses both noise perfor-mance and matching effectiveness. Matching performanceduring terrestrial tests will always need to deal with largercolor error than would be encountered on-orbit.

One final insight was gained from a careful inspection ofalgorithm execution. The three stars observed in the fieldtest are very close in spectrum; two are identical and thethird only slightly different. As a result, it is common forcolor noise to lead to a reversed star assignment, particularlywith the initial matched pair. With only a single pair of starsin view, little more can be done to resolve the ambiguity. Ouralgorithm does not use an explicit search and hence, cannotbacktrack through a star assignment. This approach is veryfast if we can make one correct initial match, but does notdeal well with situations where color-noise has reversed astar assignment. An alternate implementation with a limitedamount of backtracking is currently under study, and isexpected to resolve this shortcoming.

6 ConclusionsIn this paper, we have established a framework for the starmatching routines necessary for color -based star tracking.Color star tracking is a novel concept that can be used toleverage advances in commercial imaging systems todevelop new types of satellite attitude sensors. This tech-nology can be employed in dedicated star trackers, or as acontingency mode of operation for a payload camera. Our

approach combines coarse spectral information availablefrom a CMOS CFA detector and conventional inter-stargeometry. In many cases, the color and geometry of a singlepair of stars is sufficient for an unambiguous match.

The color star matching algorithms use a star-pair com-parison based on a Mahalanobis-distance classifier. Whenavailable, we match additional stars by employing aRANSAC-derived consensus algorithm. The RANSAC ap-proach is robust to false stars and other outliers. We havealso examined some of the practical challenges of star cata-log organization, including color representation, choice ofdata sources, and catalog formatting. A series of simulationand field trials have validated the effectiveness of thesecontributions.

Our experimental results are encouraging. Although thereference point design is not well-suited for color applica-tions, catalog analysis shows that evolutionary designchanges can lead to better performance. Comprehensive sim-ulations demonstrate good accuracy and robustness in themajority of sensor orientations; these tests also help to iden-tify cases where the algorithm performance can be improvedto prevent incorrect matches. Finally, our field-tests demon-strate that the matching routines will work with the noiseproperties encountered in a real sensor.

Notwithstanding the positive outcomes of our tests, ourexperiments do suggest the need for some crucial hardware,software, and testing refinements. First, improved opticalperformance is necessary for future sensors. Better achro-matic optics and explicit color corrections in the sensor cal-ibrations represent promising initial paths to achieving thisgoal. Second, although the bilinear color corrections improvethe correspondence between catalog and measured colorvalues, better approaches are necessary to reduce residualerrors. Finally, we recognize that the need for complicatedcorrections for local terrestrial viewing conditions wouldmake validation of any flight-instrument troublesome. Anadaptive, on-line algorithm in which the sensor graduallylearned the specific color ratios for each cataloged star orspectral type is currently under investigation.

We recognize that the algorithms and optical designspresented here are not optimal. Rather they demonstratethe essential feasibility of this novel star tracking concept.Measurements from simple color cameras can be used tomatch stars. Although color sensors are not as sensitive astheir monochrome counterparts, the information gain fromcolor information makes two-star lost-in-space solutionspractical. We feel that performance in these tests is quitepromising and that the next generation of color star trackerswill provide performance competitive with traditional mono-nchrome sensors.

AcknowledgmentsThe authors would like to thank Mr. Doug Sinclair fromSinclair Interplanetary for the donation of prototypehardware.

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Table 5 Field test matching results.

Corrections Correct matches Incorrect matches

Basic color 35 35

Ideal color 40 30

Chromatic aberration 58 12

Ideal colorþ chromatic 65 5




http://dx.doi.org/10.3390/a2010093

http://dx.doi.org/10.1109/TAES.2006.248184


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23. D. Dravins et al., “Atmospheric intensity scintillation of stars II: depend-ence on optical wavelength,” Publ. Astron. Soc. Pac. 109(736), 725–737 (1997).

John Enright holds a BASc (1997) from theUniversity of Toronto (Engineering Science:Aerospace) and a MS (1999) and a PhD(2002) from MIT in Aerospace Systems. Heis currently an associate professor inAerospace Engineering at RyersonUniversity in Toronto. Having joined the fac-ulty at Ryerson University in 2003, he is nowthe Principal Investigator of the SpaceAvionics and Instrumentation Laboratory(SAIL). While at MIT (1999 to 2003), he led

the software development for the SPHERES flight project, and theGFLOPS real-time spacecraft simulation testbed. His research inter-ests include spacecraft avionics and sensor processing, systemsengineering and flight software. He is a member of the AIAA,CASI, and the IEEE.

Geoffrey R. McVittie received his ME fromRyerson University, in 2008. He is currentlya doctoral student studying AerospaceEngineering in the Space Avionics andInstrumentation Laboratory (SAIL) atRyerson University. His research focuseson the design and development of new gen-eration star tracker sensors.




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