Skip to main content
SpringerLink
Log in

A k-Vector Approach to Sampling, Interpolation, and Approximation

  • Published:
The Journal of the Astronautical Sciences Aims and scope Submit manuscript

Abstract

The k-vector search technique is a method designed to perform extremely fast range searching of large databases at computational cost independent of the size of the database. k-vector search algorithms have historically found application in satellite star-tracker navigation systems which index very large star catalogues repeatedly in the process of attitude estimation. Recently, the k-vector search algorithm has been applied to numerous other problem areas including non-uniform random variate sampling, interpolation of 1-D or 2-D tables, nonlinear function inversion, and solution of systems of nonlinear equations. This paper presents algorithms in which the k-vector search technique is used to solve each of these problems in a computationally-efficient manner. In instances where these tasks must be performed repeatedly on a static (or nearly-static) data set, the proposed k-vector-based algorithms offer an extremely fast solution technique that outperforms standard methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. Figures 1 and 2 has been taken from Ref. [7].

  2. The expression provided for δ ε comes from the proportionality, \(\frac {y_{\max }-y_{\min }}{\delta \varepsilon } = \frac {(y_{\max } - y_{\min })/(n - 1)}{\varepsilon }\).

  3. One for each check given in Eq. (5) plus the comparison needed to identify the E 0≈1 extraneous element.

  4. The number of elements is 65535=216−1 so that each element index can be stored in 2-bytes.

  5. The small difference between experimental and theoretical values comes from the fact that, when k end = k start+1 occurs, then the number of expected extraneous elements becomes E 0/2.

  6. To have an idea of the many applications of just Ei(x), Ref. [10] mentioned: a) Time-dependent heat transfer, b) Nonequilibrium groundwater flow in the Theis solution (called a well function), c) Radiative transfer in stellar atmospheres, d) Radial Diffusivity Equation for transient or unsteady state flow with line sources and sinks, and e) Solutions to the neutron transport equation in simplified 1-D geometries.

  7. The von Mises-Fisher distribution for p = 3, also called the Fisher distribution, was first used to model the interaction of dipoles in an electric field [17]. Other applications are found in geology, bioinformatics, and text mining.

References

  1. Mortari, D.: A Fast On-Board Autonomous Attitude Determination System based on a new Star-ID Technique for a Wide FOV Star Tracker. Adv. Astronaut. Sci. 93 (II), 893–903 (1996)

    Google Scholar 

  2. Mortari, D.: Search-Less Algorithm for Star Pattern Recognition. J. Astronaut. Sci. 45(2), 179–194 (1997)

    MathSciNet  Google Scholar 

  3. Mortari, D.: SP-Search: A New Algorithm for Star Pattern Recognition. Adv. Astronaut. Sci. 102(II), 1165–1174 (1999)

    Google Scholar 

  4. Mortari, D., Pollock, T.C., Junkins, J.L.: Towards the Most Accurate Attitude Determination System Using Star Trackers. Adv. Astronaut. Sci. 99(II), 839–850 (1998)

    Google Scholar 

  5. Mortari, D., Angelucci, M.: Star Pattern Recognition and Mirror Assembly Misalignment for DIGISTAR II and III Star Sensors. Adv. Astronaut. Sci. 102(II), 1175–1184 (1999)

    Google Scholar 

  6. Ju, G., Kim, Y.H., Pollock, C.T., Junkins, L.J., Juang, N.J., Mortari, D.: Lost-In-Space: A Star Pattern Recognition and Attitude Estimation Approach for the Case of No A Priori Attitude Information, Paper AAS 00-004 of the 2000 AAS Guidance & Control Conference, Breckenridge, CO, Feb (2000)

  7. Mortari, D., Neta, B.: k-vector Range Searching Techniques,” 10th Annual AIAA/AAS Space Flight Mechanics Meeting. Paper AAS 00-128, Clearwater, FL (2000)

  8. Mortari, D., Samaan, M.A., Bruccoleri, C., Junkins, J.L.: The Pyramid Star Pattern Recognition Algorithm. ION Navigation 51(3), 171–183 (2004)

    Article  Google Scholar 

  9. Bentley, L.J., Sedgewick, R.: Fast Algorithms for Sorting and Searching Strings. In: Proceedings of the 8-th Annual ACM-SIAM Symposium on Discrete Algorithms, pp 360–369 (1997)

  10. Bell, G.I., Glasstone, S.: Nuclear Reactor Theory. Van Nostrand Reinhold Company (1970)

  11. Liu, J.S., Chen, R.: Sequential Monte Carlo Methods for Dynamics Systems. J. Am. Stat. Assoc. 93(443), 1032–1044 (1998)

    Article  MATH  Google Scholar 

  12. Carpenter, J., Clifford, P., Fearnhead, P.: Improved Particle Filter for Nonlinear Problems. IEE Proc.-Radar, Sonar, Navigation 146(1), 2–7 (1999)

    Article  Google Scholar 

  13. Kitagawa, G.: Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models. J. Comput. Graph. Stat. 5(1), 1–25 (1996)

    MathSciNet  Google Scholar 

  14. Arulampalam, M.S., Maskell, S., Gordon, N., Clapp, T.: A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking. IEEE Trans. Signal Process. 50(2), 174–188 (2002)

    Article  Google Scholar 

  15. von Mises, R.: Mathematical Theory of Probability and Statistics. Academic Press, New York (1964)

    MATH  Google Scholar 

  16. Fisher, R.A.: Dispersion on a Sphere. Proc. Roy. Soc. London Ser. A. 217, 295–305 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mardia, K.V., Jupp, P.E.: Directional Statistics, 2nd edition. Wiley (2000)

  18. von Neumann, J.: Various Techniques used in Connection with Random Digits. Monte Carlo Methods, National Bureau Standards. Appl. Math. Ser. 12, 36–38 (1951)

    MathSciNet  Google Scholar 

  19. Costello, M., Rogers, J.: BOOM: A Computer-Aided Engineering Tool for Exterior Ballistics of Smart Projectiles. Army Research Laboratory Contractor Report ARL-CR-670, Aberdeen Proving Ground, MD (2011)

    Google Scholar 

  20. McCoy, R.: Modern Exterior Ballistics, p 310. Schiffer Publishing Ltd., Atglen (1999)

    Google Scholar 

Download references

Acknowledgments

The authors would like to dedicate this work to Dr Jer-Nan Juang.

Author information

Authors and Affiliations

  1. Aerospace Engineering, Texas A&M University, 746C H.R. Bright Bldg, College Station, TX, 77843-3141, USA

    Daniele Mortari

  2. Mechanical Engineering, Georgia Institute of Technology, 4503 MRDC Building, Atlanta, GA, 30332-0405, USA

    Jonathan Rogers

Authors
  1. Daniele Mortari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonathan Rogers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniele Mortari.

Additional information

Truth is much too complicated to allow anything but approximations (John von Newman, 1947).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mortari, D., Rogers, J. A k-Vector Approach to Sampling, Interpolation, and Approximation. J of Astronaut Sci 60, 686–706 (2013). https://doi.org/10.1007/s40295-015-0065-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40295-015-0065-x

Keywords

Search

Navigation