Abstract
We propose in this work the use of Dilation theory for non-stationary signals and their time/Doppler spectra to embed the underlying spectral measure on the Special Unitary group SU(n). The Dilation theory gives access to rotation-like matrices built in with partial correlation coefficients. Due to the non-stationary condition, the time/Doppler spectra is associated with a path on SU(n). We use next the Square root Velocity Transform which has been proven to be equivalent to a first order Sobolev metric on the space of shapes. Because the metric in the space of curves naturally extends to the space of shapes, this enables a comparison between curves’ shapes and allows then the classification of time/Doppler spectra.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ammar, G., Gragg, W., Reichel, L.: Constructing a unitary Hessenberg matrix from spectral data. In: Golub, G.H., Van Dooren, P. (eds.) Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, pp. 385–395. Springer, Heidelberg (1991). https://doi.org/10.1007/978-3-642-75536-1_18
Arnaudon, M., Barbaresco, F., Yang, L.: Riemannian medians and means with applications to radar signal processing. IEEE J. Sel. Top. Signal Proces. 7, 595–604 (2013)
Barbaresco, F.: Interactions between symmetric cone and information geometries: Bruhat-Tits and Siegel spaces models for high resolution autoregressive Doppler imagery. In: Nielsen, F. (ed.) ETVC 2008. LNCS, vol. 5416, pp. 124–163. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00826-9_6
Barbaresco, F.: Radar micro-Doppler signal encoding in Siegle unit poly-disk for machine learning in Fisher metric space. In: Proceedings of the 2018 19th International Radar Symposium (IRS), Bonn, Germany, 20–22 June 2018
Bauer, M., Bruveris, M., Michor, P.W.: Why use Sobolev metrics on the space of curves. In: Turaga, P., Srivastava, A. (eds.) Riemannian Computing in Computer Vision. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-22957-7_11
Bingham, N.H.: Szego’s theorem and its probabilistic descendants (2011). http://arxiv.org/abs/1108.0368
Bouleux, G., Dugast, M., Marcon, E.: Information topological characterization of periodically correlated processes by dilation operators. IEEE Trans. Inf. Theor. (2019, in press). https://doi.org/10.1109/TIT.2019.2923217
Celledoni, E., Eslitzbichler, M., Schmeding, A.: Shape analysis on Lie groups with applications in computer animation. J. Geom. Mech. 8, 273–304 (2016)
Constantinescu, T.: Schur Parameters, Factorization and Dilation Problems. Birkhäuser, Basel (1995)
Desbouvries, F.: Unitary Hessenberg and state-space model based methods for the harmonic retrieval problem. IEE Proc. Radar Sonar Navig. 143, 346–348 (1996)
Dégerine, S., Lambert-Lacroix, S.: Characterization of the partial autocorrelation function of a nonstationary time series. J. Multivariate Anal. 2, 1296–1301 (2003)
Delsarte, P., Genin, Y.V., Kamp, Y.G.: Orthogonal polynomial matrices on the unit circle. IEEE Trans. Circ. Syst. 25, 149–160 (1978)
Dugast, M., Bouleux, G., Marcon, E.: Representation and characterization of nonstationary processes by dilation operators and induced shape space manifolds. Entropy 20(9), 717 (2018)
Hofer, M., Pottmann, H.: Energy-minimizing splines in manifolds. ACM Trans. Graph. 23(3), 284–293 (2004)
Masani, P.: Dilations as propagators of Hilbertian varieties. SIAM J. Math. Anal. 9, 414–456 (1978)
Michor, P., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Mathematica 10, 217–245 (2004)
Michor, P.W., Mumford, D.: An overview of the Riemannian metrics on shape spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23, 74–113 (2007)
Shingel, T.: Interpolation in special orthogonal groups. IMA J. Numer. Anal. 29(3), 731–745 (2009)
Simon, B.: Orthogonal Polynomials on the Unit Circle Part 1 and Part 2, vol. 54. American Mathematical Society, Providence (2009)
Simon, B.: CMV matrices: five years after. J. Comput. Appl. Math. 208, 120–154 (2007)
Sz.-Nagy, B., Foias, C., Bercovici, H., KĂ©rchy, L.: Harmonic Analysis of Operators on Hilbert Space. Springer, New York (2010). https://doi.org/10.1007/978-1-4419-6094-8
Van Kortryk, T.S.: Matrix exponentials, \(SU(N)\) group elements, and real polynomial roots. J. Math. Phys. 57, 021701 (2016)
Yang, L., Arnaudon, M., Barbaresco, F.: Riemannian median, geometry of covariance matrices and radar target detection. In: 2010 European Radar Conference (EuRAD), pp. 415–418, September 2010
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Bouleux, G., Barbaresco, F. (2019). Dilation Operator Approach and Square Root Velocity Transform for Time/Doppler Spectra Characterization on SU(n). In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-26980-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26979-1
Online ISBN: 978-3-030-26980-7
eBook Packages: Computer ScienceComputer Science (R0)