Novel Techniques for Creating Nearly Uniform Star Catalog

Transcription

1 Paper AAS 3-69 Novel Techniques for Creating Nearly Uniform Star Catalog Malak A. Samaan 1, Christian Bruccoleri 2, Daniele Mortari 3, and John L. Junkins 4 Department of Aerospace Engineering, Texas A&M University, College Station, TX Abstract In order to develop optimize star sensing and star identification with respect to continues operation and reliability, the concept of star catalogs with near uniform angular spacing between stars arises. Such catalogs are not characterized by the usual constant magnitude cutoffs. With a uniform star catalog we want to build a reference star catalog where the expectation of the number of stars that fall in a given field of view is about constant (i.e. 5 or 6) with a minimized standard deviation, independently which region of the sky the sensor optical axis is pointing. This paper compares three different techniques to create uniform star catalogs. These methods mainly depend on creating a uniform distribution of points on the surface of a unit sphere. In particular, the first method divides the unit sphere into a large number of Spherical Patches all having the same area. The second method, which is derived from a mechanical analogy (and supported by consistent results with the known solution cases), considers uniform the equilibrium distribution of charged particle on the unit sphere. Finally, the third method creates a fixed slope Spiral over the unit sphere and then divide it into segments of the same length, that is, of identical areas. Once the uniform point distribution is created, then an approximated uniform star catalog is created for each set of N reference vectors by picks the cataloged star nearest each reference vector, that is brighter than some maximum cutoff magnitude. The tradeoffs between these approaches are studied by evaluation the inter-star angle statistics. 1 Post-Doc Research Associate, Spacecraft Technology Center, Room 127H, Texas A&M University, College Station, TX , Tel. (979) , samaan@tamu.edu 2 Ph.D. Student, Department of Aerospace Engineering, Texas A&M University, College Station, TX , Tel. (979) , bruccoleri@tamu.edu 3 Associate Professor, Department of Aerospace Engineering, Texas A&M University, 3141 TAMU, College Station, Texas Tel. (979) , FAX (979) George J. Eppright Chair Professor, Department of Aerospace Engineering, Texas A&M University, College Station, TX , Tel: (979) , Fax: (979) , junkins@tamu.edu 1

2 Introduction Any star catalog created by a constant cutoff magnitude is far from being uniform. There are regions with few stars (North Galactic Pole), and regions with too many stars (i.e. the Milky Way). A star identification algorithm needs to have, at minimum, three stars in its Field Of View (FOV), in order to have any possibility of a reliable star identification, but it is recommended that at least 4 stars are contained within the FOV in order to bring the frequency of false catalog pattern match below any reasonable doubt (for modern star cameras with 1 s of micro radians or smaller centroiding errors) [1]. For this to happen it is also necessary that at least four of the stars seen within the FOV of the star sensor are contained in the on board catalog. Minimizing the number of cataloged stars, usually, it is necessary to restrict the magnitude of the cataloged stars to 5.5 or 6., depending on the FOV and limiting sensitivity of the camera. The fundamental limitation is the need to reduce the memory needed to store the k-vector look-up tables [2], but also that heavily redundant catalogs give rise to a greater number of false pattern matches. Let us give an example to understand the need for a uniform star catalog. Suppose you have six stars imaged in your FOV, but only two, for instance, are included in the catalog, the others having been excluded because they were too faint: that is their magnitude is less than the chosen catalog magnitude threshold. In this case the star identification would not be possible even though more than a sufficient number of stars are observed. One possible solution could be to add more of the fainter stars to the catalog by increasing the star magnitude threshold. By increasing the magnitude threshold the overall number of stars in the catalog increases exponentially. This implies: 1. the memory required to store the k-vector tables easily becomes greater than that available, 2. the star identification process slows down since the identification process would deal with a higher number of potential or admissible star pairs, and (more important), 3. the probability of false matching also increases. The criteria adopted to build a uniform star catalog is based on the followings. From an ideal point of view, for every possible direction, you can have an adjustable local magnitude threshold as a function of n s, the desired number of stars in the field of view. Whenever the number of visible stars brighter than the current magnitude threshold m h is less than k (for us we define k = 5), you can raise m h to include fainter stars until you have that E{n s } = k. An opposite criterion can be applied when too many stars appear in the field of view: you may choose to exclude stars too faint or too bright 5 in order to maintain the E{n s } = k. Only the stars whose 5 You may want to limit the brightest stars because of the saturation characteristics of your CCD. 2

3 magnitude m satisfy m l < m < m h are therefore kept in the catalog, where m l represents the magnitude threshold for the brightest stars. Thus we have a variable magnitude threshold catalog, and this leads to a much better approximation of a uniform star density. 6 This method is not easily applicable in practice because you must handle the problem of the overlapping FOVs. From these considerations it become clear that we want to add to the star catalog only those stars that ensure us that in any random direction we shall always have at least 4 or 5 stars in the FOV, no less and not much more (we want to keep our standard deviation as low as possible to avoid storing un-needed stars). It is necessary then to have a nearly uniform star catalog and in pursuit of this goal, we consider some methods to generate a set of uniformly distributed points on a sphere, as is described later. Bauer [3] describes a method to build a uniform star catalog using a Voronoi diagram. In this work, however we decided to compare computationally easier methods in order to validate the benefit of uniform catalog for the star identification problem. The problem of comparing the uniformity of the proposed methods with the existing ones, as for instance the Voronoi diagram approach, is not treated in this paper. To apply the above approach a very robust star identification algorithm, such as Pyramid [1], must be used. The robustness is required because the catalog missing stars, when observed, must be recognized as spikes (spurious objects, electronic noise, reflections, etc.). To our knowledge, Pyramid [1] is the most robust algorithm presently available. Its use allow us to remove some stars from the catalog, obtaining the desired density, without disrupting the star identification process. Once again, if a star that is not in the catalog is observed, then Pyramid will just filter it out as an unidentified spurious star. This technique has some practical limitations. In order to sense the dimmer stars you need a sensitive CCD, or APS device, or you might imagine to use a logarithmic sensor whose local sensitivity is adjusted with a local exposure time that depends on sensed energy flux. You must also take into account that longer integration time causes more image smear, and therefore we will always have a practical limitation on the highest magnitude threshold m h that we can actually include on the star catalog. In this paper once the nearly uniform star catalog has been created, then the associated statistics are evaluated for each method. (?) However, doing some statistics analysis for the uniform distributed points on the unit sphere will not help, in this study, to achieve the meaning of uniform star catalog. The standard deviation and the mean for each method of uniform points distribution, for certain FOV, are found to be almost constant. But when adopting the star catalog and using the star vectors instead of the uniform points vectors, the 6 The implicit assumption is that the camera integration time is adjusted to make the number of visible stars equal to k, or a constant integration time is selected to insure that in the worst case, at least k stars can be imaged. 3

4 statistics for each method are different. This done on the last section of this paper to show the different between each method. Building the Nearly Uniform Star Catalog Let us to describe the technique used to build the uniform star catalog by the following pseudo algorithm. Let C be the uniform star catalog, S = [ v i ] be the full catalog, U = [ p i ] be the uniform distribution of N points on the unit sphere, and σ be the angular tolerance. For each p k U Let V = { v i S cos 1 ( v i p k ) < σ} Let b j V be the brightest star V C = C { b j } end From this algorithm, which is a slightly modified version of the one proposed by Kudva in [5], it becomes clear that, once a method to build a uniform distribution of points on a sphere is available, then the problem of building a uniform star catalog is, consequently, easily solved. Note that instead of using the brightest star that is near to the reference point p, we can use a different criteria: for instance we may select the star that is closer to p, without considering its magnitude (so long as it is brighter than cutoff magnitude). The set S can be built so that only stars within our chosen range of maximum and minimum magnitudes are present, thus avoiding a further check in the loop. Our adopted set S, the full catalog, is a subset of the Smithsonian Astrophysical Observatory (SAO) catalog [4], updated to J2 epoch, that contains all the stars up to magnitude 6.5 For practical reasons, we decided to slightly modify the algorithm such that all the brightest stars are always included in the reduced catalog. Thus a preliminary scan of the full catalog is performed and all the stars who are brighter than (our arbitrary adopted) magnitude 4 are always included in the catalog without considering the uniformity of distribution. This choice is dictated by practical considerations and the high probability that all of these stars will actually be measure. Thus only the stars fainter than magnitude 4 are selected, mainly to populate sparse regions, and to achieve near-uniformity. We tested and validated three techniques to build the catalog, based on three different methods to build a uniform distribution of points on a sphere, that is the core of our algorithm: 1. The Spherical Patches method 2. The Charged Particles method 4

5 3. The Fixed-Slope Spiral method For each of these methods we test the uniformity of the generated catalog using a Monte-carlo approach. In each test a random unit vector r is generated for the sensor boresight, and we compute the number of stars n s that are within the predefined Field Of View (FOV) of θ degrees about r. As already outlined above, our goal is having an expected value E{n s } = 6 and a variance of n s, σ ns = 1. The Spherical Patches Method In this method we will assume that the unit sphere (sky) can be divided uniformly into a large number of an equal area of spherical patches. These patches could be of any shape like: circle, triangles, squares or hexagonal (like the Soccer ball). An example of a uniform distributed triangle patches on a unit sphere is shown on Fig. 1. So, the centers of these patches are nearly uniform distributed over the surface of the unit sphere. For simplicity, we will assume that these patches have spherical squares shape then Nl 2 = 4π (1) where l is the length of each patch and N is the total number of patches. Also around the equator nl = 2π (2) where n is the total number of patch around the equator. From Eqs. (1) and (2) N = n2 π The number of patches N is chosen such that n and N both of them are integers. The algorithm used for the construction of points on a unit sphere is as follows: Set δ(1) = and k =. i-loop: where i goes from 2 to n δ(i) = δ(i 1) + l; where δ(i) is the declination of each square patch. Calculate the total patches area at each δ(i): s = 2π[cos δ(i 1) cos δ(i)]; The number of patches at each δ(i) is calculated by m(i) = round( s/l 2 ); j-loop: where j goes from 1 to m(i) θ(j) = 2π(j 1)/m(i); k = k + 1; x(k) = sin δ(i) cos θ(j); y(k) = sin δ(i) sin θ(j); z(k) = cos δ(i); P (k) = [ x(k), y(k), z(k) ]; j-loop end i-loop end 5 (3)

6 The unit vectors are created from the origin to the center of each patch. x i sin δ i cos θ i v i = y i = sin δ i sin θ i z i cos δ i (4) where θ i and δ i are the right ascension and declination of each patch center Figure 1: A Unit Sphere Surface Divided Into Equal Patches The Charged Particles Method This method begins by approximately locating N equally spared points on a unit sphere. The approach is based on the observation that a number of (positively or negatively) charged particles contrained to remain on a sphere (see [6] and [7]), will repel each other and if a global minimum energy state can be found, will be approximately equally spaces. The problem of finding N approximately uniformly spaced points on a sphere is also known as the Thomson Problem, as stated in [9]. For N large, the uniform spacing can be approached. Thus, in our simulation, we try to find the minimum potential energy V = 1/r ij, associated to the distribution, by moving the particles as for the result of the forces 6

7 that acts on each particle, due to all the other particles. The process is repeated iteratively starting from a random distribution and ends when any perturbation of the configuration leads to an increase of the potential energy or a number of predefined steps has been already completed. The near-uniformity is easily tested by statistical evaluation of the distribution of points. An example of such distribution is shown in Fig. 2 A pseudo-algorithm used to build such distribution can be summarized as follows: 1. Start with N randomly distributed points on a unit-radius sphere, 2. Let E = N i,j E ij where E ij = k/r ij for (i j), 3. Calculate the net force acting on every particle F i, (i = 1,..N) 4. Let F r be the radial component of F i 5. Fi = F i F r 6. Move the i th particle in the direction of the net force F i 7. Re-scale the i th particle position to enforce a unit radius, 8. Let E 1 be then energy of the updated configuration (evaluated as in step 2) 9. if E 1 < E then {E = E 1 and goto 3} else roll back changes. 1. end The Fixed Slope Spiral Method In order to generate approximately uniformly spaced points on the unit sphere we can use a fixed slope spiral around the sphere as shown in Fig. 3. If the slope of the spiral around the unit sphere M = tan µ = constant and N is the total number of points on the spiral that is uniformly divided then θ i = Mφ i (5) where θ i and φ i are the right ascension and declination of each point on the spiral. The distance between any two points on the spiral d = 2π/M. The total length of the spiral is S = C π dφ 2 + dθ 2 sin 2 φ dφ = 1 + M 2 sin 2 φ dφ (6) 7

8 Figure 2: Distribution on sphere obtained with Charged Particles Method For a large spiral slope the above integral is equal to 2M S = 2M = Nd = N 2π M (7) then the slope is solved to be M = Nπ. So, the uniform vectors are generated around the unit sphere by using v i = x i y i z i = sin φ i cos θ i sin φ i sin θ i cos φ i (8) Uniform Star Catalog Results Each one of the above methods is used to generate the uniform star catalog using the SAO star catalog that initially contains about 12, stars in it. In these catalog generations we assume the star sensor has an 8 8 field of view. As we describe earlier, all the stars that are brighter than magnitude 4. are always added to the new catalog without considering the uniformity of distribution, this is the initial star catalog we will start with. The condition is essentially because we don t want to lose any bright star resulting from the image processing while considering less bright stars. The number of stars over the whole sky that are brighter than magnitude 4. is about 9 stars. 8

9 1 v i v i Figure 3: Uniform Spaced Points on Unit Sphere Using the Spiral Method In order to chose the star candidates using the above methods, a cone with very small half angle and centered at each uniformly distributed points is created. Each candidate star for the on board catalog is chosen such that this star lies inside this cone. If more than one star are found, we chose the one with higher magnitude. The cone half angle is calculated such that there will be no gaps between the uniformly distributed points when we scan the sky. The Spherical Patches method is used with n = 24, and N = 3, 57. The cone angle is used to be equal to the patch length (θ = d). These uniformly distributed points are used to update the initial star catalog. The redundant stars will be removed from the resulting catalog and the final catalog size will be about 3,23 stars. The same thing will be used for the Charged Particles method and the Fixed slope Spiral method in which we start our algorithms by choosing the number of points equal to 3, in each one of them and the same cone half angle is used also here. The final catalog sizes for both methods are 3,258 stars and 3,396 stars respectively. Figure 4 shows the distribution of the stars in the sky in terms of the right ascension (α) and declination (δ). In Fig. 4(a) the uniform on-board star catalog distribution using the Spherical Patches method is shown, while Fig. 4(b) shows the star distribution using the regular star catalog with 5.5 magnitude threshold. 9

10 δ (deg) α (deg) (a) Using the uniform catalog The distribution of the stars for 5.5 magnitude threshould δ (deg) α (deg) (b) Using 5.5 Magnitude threshold catalog Figure 4: The distribution of the stars in the sky 1

11 The resulting on-board star catalog from each one of the above methods is checked for uniformity using 5, Monte Carlo random tests. In each one of these tests a random unit vector is assumed to be the bore sight vector for an 8 8 field of view sensor. The number of stars lie inside the corners of FOV centered at each one these random unit vectors are analyzed for the check of uniformity. The histogram of the number of stars found at each one of these tests are shown in Fig. 5. We also show in the Fig. 5 the histogram for the regular star catalog with 5.5 magnitude threshold to show the differences with the uniform star catalog using the pre-described three methods. Finally, some statistical analysis is also done to each one of these catalogs. the mean and the standard deviation of the number of stars are calculated along with the ones for the 5.5 and 5. magnitude threshold regular star catalogs. The results of these statistical analysis are summarized in table 1. The mean of the stars in the FOV is bounded by 7 or 8 stars using the uniform catalogs while it is about 11.6 stars for the 5.5 magnitude threshold catalog. On the other hand the standard deviation is about 1.6 while it is about 4.9 stars for the 5.5 magnitude threshold and 3.25 stars for the 5. magnitude threshold catalogs. The probability of having less the 5 stars is also addressed in this table to be one of the most basis to create these uniform catalogs. This probability is found to be.7% or less using the three uniform catalogs while it is 2.28% for the 5.5 magnitude threshold catalog and reach about 26.8% the 5. magnitude threshold catalog. The last column in table 1 is the k-vector length associated with each one of these catalogs. The k-vector [2] is the heart of the Pyramid Lost-In-Space star identification algorithm [1], this vector contains the indices of each star pair lies inside the camera FOV sorted in ascending order with respect to the interstar angle of this pair. The k-vector is store on-board to by used for the star identification algorithm. Now, by using the uniform star catalog we can save about more than half of the memory required for the k-vector as shown in table 1. Conclusion In this paper we build and compare three different techniques for creating a uniform star catalog. These methods mainly depend on creating a uniform distribution of vectors around the surface of a unit sphere. In the first method we divide the unit sphere into large number of Spherical Patches which have the same area. The second uses the idea of randomly distribute the points on the unit sphere and then let the particles acts as electrically charged, thus generating a repulsive force that tends to uniformly distribute the points on the sphere. The last method is to create a fixed slope Spiral around the unit sphere and then divide it into segments of the same length. The informality condition for each one of these method is used to test some criterion for each one of these uniform catalogs. These criterion are the mean and 11

12 14 Histogram for the number of stars using the Patches method 14 Histogram for the number of stars using sprial method Occurance 8 6 Occurance Number of stars Number of stars (a) Using Spherical Patches (c) Using Fixed Slope Spiral 14 Histogram for the number of stars using the Charged particles 5 Histogram for the number of stars using 5.5 magnitude threshould Occurance 8 6 Occurance number of stars number of stars (b) Using Charged Particles (d) Using 5.5 Magnitude threshold catalog Figure 5: Histogram for the number of stars occurrence 12

13 Spherical Patches Charged Particles Fixed slope Spiral 5.5 Magnitude threshold 5. Magnitude threshold Catalog Size Mean Standard Deviation Probability of Less than 5 stars k-vector Size 3, % 49,555 3, % 5,361 3, % 54,915 4, % 12, 2, % 39,746 Table 1: Comparison between the different on-board star catalogs the standard deviation of the stars in the FOV. Also the uniform catalogs are shown to be efficient in saving memory for the on-board catalog and the k-vector used for the star identification algorithms. References [1] Mortari, D., Junkins, J.L., and Samaan, M.A. Lost-In-Space Pyramid Algorithm for Robust Star Pattern Recognition, Paper AAS 1-4 Guidance and Control Conference, Breckenridge, Colorado, 31 Jan. - 4 Feb. 21. [2] Mortari, D. Search-Less Algorithm for Star Pattern Recognition, Journal of the Astronautical Sciences, Vol. 45, No. 2, April-June 1997, pp [3] Bauer R., Distribution of Points on a Sphere with Application to Star Catalogs, Journal of Guidance and Control, Vol. 23, No. 1, Jan-Feb 2. [4] Smithsonian Astrophysical Observatory, [5] Kudva, P., Flight Star Catalog Development for EOS-AMI, NASA Goddard Space Flight Center Contract NAS5-3259, Task 71-3, TM , March [6] Altschuler, E. L., Williams, T. J., Ratner, E. R., Dowla, F., and Wooten, F. Method of Constrained Global Optimization, Phys. Rev. Let. 72, , [7] Wilkin, A. P., and Heckbert, P. S., Using Particles to Sample and Control Implicit Surfaces, Computer Graphics volume 28, ,

14 [8] Whyte, L. L. Unique Arrangement of Points on a Sphere, Amer. Math. Monthly 59, , [9] Ashby, N. and Brittin, W. E. Thomson s Problem, Amer. J. Phys. 54, ,