Abstract
Localizing a radiant source is a problem of great interest to many scientific and technological research areas. Localization based on range measurements is at the core of technologies such as radar, sonar and wireless sensor networks. In this manuscript, we offer an in-depth study of the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummer’s and Cayley’s surfaces. Our work also gives new insights into the localization based on range differences.
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Notes
For a picture of the surface, see, for example, Wikipedia.
According to our choice of the reference receiver, we change to three every subscript 0 appearing in Compagnoni et al. (2014b). As a consequence, we have to correct signs in formulas whenever necessary.
In order to avoid confusion with the range notation, we call \(B_i(\varvec{\tau })\) the set that in Compagnoni et al. (2014b) is called \(A_i(\varvec{\tau })\).
References
Abraham, R., Marsden, J., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Springer-Verlag, New York (1988)
Alameda-Pineda, X., Horaud, R.: A geometric approach to sound source localization from time-delay estimates. IEEE Trans. Audio Speech Lang. Process. (TASLP) 22, 1082–1095 (2014)
Amari, S.: Differential geometry of curved exponential families-curvatures and information loss. Ann. Stat. 10(2), 357–385 (1982)
Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence, RI (2000)
Aspnes, J., Eren, T., Goldenberg, D., Morse, S., Whiteley, W., Yang, R., Anderson, B., Belhumeur, P.: A theory of network localization. IEEE Trans. Mobile Comput. 5, 1663–1678 (2006)
Beck, A., Stoica, P., Li, J.: Exact and approximate solutions of source localization problems. IEEE Trans. Signal Process. (TSP) 56, 1770–1778 (2008)
Bellman, R., Astrom, K.: On structural identifiability. Math. Biosci. 7, 329–339 (1970)
Beltrametti, M., Carletti, E., Gallarati, D., Monti Bragadin, G.: Lectures on Curves, Surfaces and Projective Varieties. A Classical View of Algebraic Geometry. EMS Textbooks in Mathematics. European Mathematical Society, Zurich (2009)
Benesty, J., Huang, Y. (eds.): Audio Signal Processing for Next-Generation Multimedia Communication Systems. Springer, US (2004)
Berry, T.G.: Points at rational distance from the vertices of a triangle. Acta Arith. LXI I(4), 391–398 (1992)
Bestagini, P., Compagnoni, M., Antonacci, F., Sarti, A., Tubaro, S.: TDOA-based acoustic source localization in the space-range reference frame. Multidimens. Syst. Signal Process. 25, 337–359 (2014)
Blekherman, G., Parrilo, P.A., Thomas, R.R.: Semidefinite Optimization and Convex Algebraic Geometry. Number 13 in MOS-SIAM Series on Optimization. SIAM, Philadelphia (2013)
Blumenthal, L.: Theory and Applications of Distance Geometry. Oxford University Press, Oxford (1953)
Borcea, C.: Point Configurations and Cayley–Menger Varieties. arXiv:math/0207110 (2002)
Borcea, C., Streinu, I.: The number of embeddings of minimally rigid graphs. Discrete Comput. Geom. 31(2), 287–303 (2004)
Caffery, J.J.: A new approach to the geometry of toa location. Proc. IEEE Veh. Technol. Conf. (VTC) 4(52), 1943–1949 (2000)
Cheng, Y., Wangb, X., Morelande, M., Moran, B.: Information geometry of target tracking sensor networks. Inf. Fusion 14, 311–326 (2013)
Cheung, K.W., So, H.C., Ma, W.K., Chan, Y.T.: Least squares algorithms for time-of-arrival-based mobile location. IEEE Trans. Signal Process. 52(4), 1121–1128 (2004)
Cheung, K.W., Ma, W.K., So, H.C.: Accurate approximation algorithm for TOA-based maximum likelihood mobile location using semidefinite programming. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), vol. 2, pp. 145–148 (2004b)
Cheung, K.W., So, H.C.: A multidimensional scaling framework for mobile location using time-of-arrival measurements. IEEE Trans. Signal Process. 53(2), 460–470 (2005)
Coll, B., Ferrando, J., Morales-Lladosa, J.: Positioning systems in minkowski space-time: from emission to inertial coordinates. Class. Quantum Gravity 27, 065013 (2010)
Coll, B., Ferrando, J., Morales-Lladosa, J.: Positioning systems in Minkowski space-time: bifurcation problem and observational data. Phys. Rev. D 86, 084036 (2012)
Compagnoni, M., Bestagini, P., Antonacci, F., Sarti, A., Tubaro, S.: Localization of acoustic sources through the fitting of propagation cones using multiple independent arrays. IEEE Trans. Audio Speech Lang. Process. (TASLP) 20, 1964–1975 (2012)
Compagnoni, M., Canclini, A., Bestagini, P., Antonacci, F., Sarti, A.: Source localization and denoising: a perspective from the TDOA space. J. Multi-Dimens. Syst. Signal Process. doi:10.1007/s11045-016-0400-9 (2016a)
Compagnoni, M., Notari, R., Antonacci, F., Sarti, A.: On the statistical model of source localization based on range difference measurements. arXiv:1606.08303 (2016b)
Compagnoni, M., Notari, R.: Tdoa-based localization in two dimension: the bifurcation curve. Fundam. Inform. 135, 199–210 (2014)
Compagnoni, M., Notari, R., Antonacci, F., Sarti, A.: A comprehensive analysis of the geometry of TDOA maps in localization problems. Inverse Probl. 30(3), 035004 (2014)
Costa, J.A., Patwari, N., Hero, A.O.: Distributed weighted-multidimensional scaling for node localization in sensor networks. ACM Trans. Sens. Netw. (TOSN) 2(1), 39–64 (2006)
do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs, NJ (1976)
Draisma, J., Horobet, E., Ottaviani, G., Sturmfels, B., Thomas, R.R.: The euclidean distance degree. In: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation, pp. 9–16 (2014)
Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics, volume 40 of Oberwolfach Seminars Series. Birkhauser, Basel (2009)
Drton, M., Sullivant, S.: Algebraic statistical models. Stat. Sin. 17(4), 1273–1297 (2007)
Eren, T., Goldenberg, D., Whiteley, W., Yang, Y., Morse, A., Anderson, B., Belhumeur, P.: Rigidity, computation, and randomization in network localization. IEEE Infocom Proc. 4, 2673–2683 (2004)
Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM 24(6), 381–395 (1981)
Forman, G., Zahorjan, J.: The challenges of mobile computing. IEEE Comput. 27, 38–47 (1994)
Friedland, S., Stawiska, M.: Some Approximation Problems in Semi-algebraic Geometry. arXiv:1412.3178v1 [math.AG] (2014)
Hudson, R.W.H.T.: Kummer’s Quartic Surface, with a Forward by W. Cambridge University Press, Barth (1990)
Ichiki, S., Nishimura, T.: Distance-squared mappings. Topol. Appl. 160, 1005–1016 (2013)
Jian, L., Yang, Z., Liu, Y.: Beyond triangle inequality: sifting noisy and outlier distance measurements for localization. In: INFOCOM, 2010 Proceedings IEEE, pp. 1–9 (2010)
Kobayashi, K., Wynn, H.P.: Asymptotically efficient estimators for algebraic statistical manifolds. Lect. Notes Comput. Sci. 8085, 721–728 (2013)
Lasserre, J.B.: An Introduction to Polynomial and Semi-algebraic Optimization. Number 52 in Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2015)
Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)
Menger, K.: Untersuchungen uber allgemeine metrik. Math. Ann. 100, 75–163 (1928)
Menger, K.: New foundation of euclidean geometry. Am. J. Math. 53, 721–745 (1931)
Moran, B., Howard, S.D., Cochran, D.: An information-geometric approach to sensor management. In: 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5261–5264 (2012)
Navidi, W., Murphy, W.S., Hereman, W.: Statistical methods in surveying by trilateration. Comput. Stat. Data Anal. 27(2), 209–227 (1998)
Pistone, G., Riccomagno, E., Wynn, H.P.: Algebraic Statistics. Computational Commutative Algebra in Statistics. Volume 89 of Monographs on Statistics and Applied Probability. Chapman and Hall/CRC, Boca Raton (2001)
Quazi, A.H.: An overview on the time delay estimation in active and passive systems for target localization. IEEE Trans. Acoust. Speech Signal Process. ASSP 29, 527–533 (1981)
Savvides, A., Han, C.C., Strivastava, M.: Dynamic fine-grained localization in ad-hoc networks of sensors. In: Proceedings of the 7th Annual International Conference on Mobile Computing and Networking, pp. 166–179 (2001)
Schoenberg, I.: Remarks to maurice frèchet’s article ”sur la définition axiomatique d’une classe d’espaces distanciès vectoriellement applicable sur l’espace de hilbert”. Ann. Math. 36, 724–732 (1935)
Shen, G., Zetik, R., Thoma, R.S.: Performance comparison of TOA and TDOA based location estimation algorithms in los environment. In: Proceedings of the 5th Workshop on Positioning, Navigation and Communication, pp. 71–78 (2008)
Shin, D.H., Sung, T.K.: Comparisons of error characteristics between TOA and TDOA positioning. IEEE Trans. Aerosp. Electron. Syst. 38(1), 307–311 (2002)
Sippl, M., Scheraga, H.: Cayley-menger coordinates. Proc. Natl. Acad. Sci. USA 83, 2283–2287 (1986)
So, H., Chan, F.: A generalized subspace approach for mobile positioning with time-of-arrival measurements. IEEE Trans. Signal Process. 55(10), 5103–5107 (2007)
Watanabe, S.: Algebraic Analysis for Singular Statistical Estimation. In: 10th International Conference on Algorithmic Learning Theory, ALT’99, volume 1720 of Lecture Notes in Computer Science, pp. 39–50 (1999)
Wei, H.W., Wan, Q., Chen, Z.X., Ye, S.F.: A novel weighted multidimensional scaling analysis for time-of-arrival-based mobile location. IEEE Trans. Signal Process. 56(7), 3018–3022 (2008)
Weiser, M.: Some computer science issues in ubiquitous computing. Commun. ACM 36, 75–84 (1993)
Zhayida, S., Burgess, S., Kuang, Y., Åström, K.: Toa-based self-calibration of dual-microphone array. IEEE J. Sel. Top. Signal Process. 9(5), 791–801 (2015)
Acknowledgments
The authors would like to thank R. Sacco for useful discussions and suggestions during the preparation of this work and the anonymous referees for the valuable remarks on the preliminary version of the paper.
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Communicated by Melvin Leok.
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Compagnoni, M., Notari, R., Ruggiu, A.A. et al. The Algebro-geometric Study of Range Maps. J Nonlinear Sci 27, 99–157 (2017). https://doi.org/10.1007/s00332-016-9327-4
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DOI: https://doi.org/10.1007/s00332-016-9327-4