## TDOA maps in localisation problems

### Citations

2842 |
Algebraic geometry
- Hartshorne
- 1977
(Show Context)
Citation Context ...main mathematical tools that are used in this Subsection come from algebraic geometry because a(τ ), b(τ ) and c(τ ) are polynomials with real coefficients (see Appendix B for a short introduction or =-=[13,22,29]-=-). Let us first describe the vanishing locus of c(τ ), over both R, where it is particularly simple, and C. Proposition 6.4 c(τ ) ≥ 0 for every τ ∈ R2. Moreover, c(τ ) = 0 if and only if τ ∈ {R0, R∗, ... |

473 |
H.Nagaoka, Methods of information geometry
- Amari
- 2000
(Show Context)
Citation Context ...on of the previous paragraph, it is well known that the estimated τ̄ ∈ Im(τ∗2 ) given by MLE is the orthogonal projection of τ∗ on the noiseless measurements set, i.e. the projection pH(τ∗) on H (see =-=[6,42]-=- and Fig. 12) therefore x̄ = τ∗2 −1(pH(τ∗)). A similar reasoning applies to the more complex case of τ∗n. In particular, any estimator has a geometrical interpretation and the relative accuracy depend... |

311 |
Lectures on Discrete Geometry
- Matoušek
- 2002
(Show Context)
Citation Context ...similar for both cases of generic and collinear microphone configurations, which is the reason why we collect them together in this Section. For the definition and properties of convex polytopes, see =-=[39]-=- among the many available references. Geometry of TDOA maps 13 Theorem 5.1 Im(τ2) is locally the τ–plane. Proof. Let us assume that x̄ is a point where τ2 is regular, i.e. where the Jacobian matrix J(... |

191 |
The road to reality. A complete guide to the laws of the universe
- Penrose
- 2005
(Show Context)
Citation Context ... 6.16 can be nicely interpreted in terms of the two-dimensional and the three-dimensional intersection problems. Here we use some standard Minkowski and relativistic conventions used, for example, in =-=[3, 43]-=-. (i) τ ∈ E if, and only if, v(τ ) is isotropic, or light–like. In this case, the line L21(τ ) is parallel to a generatrix of the cone, therefore it meets C0 at an ideal point. On the x–plane this mea... |

159 |
Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
- Ideals
- 2006
(Show Context)
Citation Context ...main mathematical tools that are used in this Subsection come from algebraic geometry because a(τ ), b(τ ) and c(τ ) are polynomials with real coefficients (see Appendix B for a short introduction or =-=[13,22,29]-=-). Let us first describe the vanishing locus of c(τ ), over both R, where it is particularly simple, and C. Proposition 6.4 c(τ ) ≥ 0 for every τ ∈ R2. Moreover, c(τ ) = 0 if and only if τ ∈ {R0, R∗, ... |

149 | Statistical theory of passive location system,”
- Torrieri
- 1984
(Show Context)
Citation Context ... a solution to the localization problem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) =-=[16, 26, 53]-=-; Least Squares (LS) [2]; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is n... |

119 |
Position-location solutions by Taylor-series estimation,” Aerospace and Electronic Systems
- Foy
- 1976
(Show Context)
Citation Context ... a solution to the localization problem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) =-=[16, 26, 53]-=-; Least Squares (LS) [2]; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is n... |

94 |
Maximum-likelihood source localization and unknown sensor location estimation for wideband signals in the near-field,”
- Chen, Hudson, et al.
- 2002
(Show Context)
Citation Context ... a solution to the localization problem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) =-=[16, 26, 53]-=-; Least Squares (LS) [2]; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is n... |

63 |
Singular 3-1-6 — A computer algebra system for polynomial computations
- Decker, Greuel, et al.
- 2012
(Show Context)
Citation Context ...). Quite clearly, on the complex Geometry of TDOA maps 23 plane Ẽ is bound to have singular points, as Ẽ is an algebraic rational quintic curve. In Appendix C we include the source code in Singular =-=[23]-=- language for computing the Cartesian equation (further analysis of the properties of the bifurcation curve is contained in [21]). From Fig. 9 and Fig. 10, we immediately recognize what was assessed t... |

58 | Real-time passive source localization: A practical linear-correction least-squares approach. Speech and Audio Processing, - Huang, Benesty, et al. - 2001 |

58 |
Passive source localization employing intersecting spherical surfaces from time-of-arrival differences,”
- Schau, Robinson
- 1987
(Show Context)
Citation Context ...used on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) [2]; and Constrained Least Squares (CLS) =-=[34, 48]-=-, which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is necessary to minimize the number of sensors in use, due to specific sen... |

54 |
An algebraic solution of the GPS equations,
- Bancroft
- 1985
(Show Context)
Citation Context ...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ... |

51 |
On structural identifiability
- Bellman, Astrom
- 1970
(Show Context)
Citation Context ... expressions for the boundaries of the regions of interest. We observe that this approach to the problem fits into the research field of structural identifiability of complex systems (see for example =-=[11, 40]-=-), where one is interested in studying if the parameters of a model (in our case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source loc... |

46 | Exact and approximate solutions of source localization problems,” Signal Processing, - Beck, Stoica, et al. - 2008 |

42 |
A new approach to geometry of range difference location,”
- Schmidt
- 1972
(Show Context)
Citation Context ...d how many there are) as the measurements or the sensor geometry vary. This problem has been partially addressed in the case of the localization of a radio-beacon receiver in LORAN navigation systems =-=[49]-=- and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see [1,8,9,15,18,19,28,30,37,38]). In particular, these studies provide ... |

34 | Audio Signal Processing for Next Generation Multimedia Communication Systems - Huang, Benesty - 2004 |

31 |
Position calibration of microphones and loudspeakers in distributed computing platforms
- Raykar, Kozintsev, et al.
- 2003
(Show Context)
Citation Context ...lacement. This prospective investigation, in fact, is in line with the recently revamped interest of the research community in self-calibrating and self-synchronizing spatial distributions of sensors =-=[16,45,47]-=-. Our analysis starts from [20], where a different perspective on the localization problem is offered through the adoption of the Space–Range Differences (SRD) reference frame, where the wavefront pro... |

27 |
A First Course in Algebraic Topology
- Kosniowski
- 1980
(Show Context)
Citation Context ...ents, and τ2 : Ũ0 → U0 is a cover. Let us now assume that the two inverse images of τ0 ∈ U0 belong to the same connected component of Ũ0. As U0 is path–connected, from the Path Lifting Theorem (see =-=[36]-=-), it follows that the inverse images of any other point τ ∈ U0 belong to the same connected component of Ũ0 as well. Let x ′ be a point in the other connected component of Ũ0, with τ ′ = τ2(x′). He... |

26 | Passive acoustic source localization for video camera steering - Huang, Benesty, et al. - 2000 |

24 |
Conics and Cubics: A Concrete Introduction to Algebraic Curves,
- Bix
- 2006
(Show Context)
Citation Context ...main mathematical tools that are used in this Subsection come from algebraic geometry because a(τ ), b(τ ) and c(τ ) are polynomials with real coefficients (see Appendix B for a short introduction or =-=[13,22,29]-=-). Let us first describe the vanishing locus of c(τ ), over both R, where it is particularly simple, and C. Proposition 6.4 c(τ ) ≥ 0 for every τ ∈ R2. Moreover, c(τ ) = 0 if and only if τ ∈ {R0, R∗, ... |

22 | The spherical interpolation method for closed-form passive source localization using range difference measurements
- Abel, Smith
- 1987
(Show Context)
Citation Context ...roblem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) =-=[2]-=-; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is necessary to minimize the... |

15 |
On identifiability of nonlinear ode models and applications in viral dynamics.
- Miao, Xia, et al.
- 2011
(Show Context)
Citation Context ... expressions for the boundaries of the regions of interest. We observe that this approach to the problem fits into the research field of structural identifiability of complex systems (see for example =-=[11, 40]-=-), where one is interested in studying if the parameters of a model (in our case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source loc... |

14 |
On the exact solutions of pseudorange equations,
- Chaffee, Abel
- 1994
(Show Context)
Citation Context ...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ... |

14 | The Euclidean distance degree of an algebraic variety. arXiv:1309.0049
- Darisma, Horobet, et al.
- 2013
(Show Context)
Citation Context ...ave been developed and applied to to similar situations, in cases where scientific and engineering models are expressed as sets of real solutions to systems of polynomial equations (see, for example, =-=[4, 25, 35]-=-). The somewhat surprising fact that, although the TDOA map is not polynomial, all the loci involved in the analysis of τ∗2 are algebraic or semi–algebraic, is a promising indicator of the effectivene... |

12 | Geometric calibration of distributed microphone arrays
- Redondi, Tagliasacchi, et al.
- 2009
(Show Context)
Citation Context ...lacement. This prospective investigation, in fact, is in line with the recently revamped interest of the research community in self-calibrating and self-synchronizing spatial distributions of sensors =-=[16,45,47]-=-. Our analysis starts from [20], where a different perspective on the localization problem is offered through the adoption of the Space–Range Differences (SRD) reference frame, where the wavefront pro... |

11 | H.F.: A Linear Closed-Form Algorithm for Source Localization From Time-Differences of Arrival. Signal Processing Letters, - Gillette - 2008 |

11 | Closed-form formulae for time-difference-ofarrival estimation. Signal Processing,
- So, Chan, et al.
- 2008
(Show Context)
Citation Context ...τ∗2 . The complete model τ̂∗2 (x) takes into account each one of the three TDOA that can be defined between the sensors. This, in fact, becomes necessary when working in a realistic (noisy) situation =-=[50]-=-. We should keep in mind, however, that there is a linear relationship between the pseudoranges (3), which allows us to simplify the deterministic problem. Definition 2.2 Let (τ10, τ20, τ21) be the co... |

9 |
Existence and Uniqueness of GPS Solutions
- Abel, Chaffee
- 1991
(Show Context)
Citation Context ...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ... |

8 | A Hilbert scheme in computer vision
- Aholt, Thomas, et al.
(Show Context)
Citation Context ...ave been developed and applied to to similar situations, in cases where scientific and engineering models are expressed as sets of real solutions to systems of polynomial equations (see, for example, =-=[4, 25, 35]-=-). The somewhat surprising fact that, although the TDOA map is not polynomial, all the loci involved in the analysis of τ∗2 are algebraic or semi–algebraic, is a promising indicator of the effectivene... |

7 | An exact solution to range computation with time delay information for arbitrary array geometries - Reddi - 1993 |

5 | Tracking multiple acoustic sources using particle filtering - Antonacci, Riva, et al. - 2006 |

5 |
Algebraic Solution of GPS Pseudo-Ranging Equations
- Awange, Shan
(Show Context)
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5 | An improved TDOA-based location estimation algorithm for large aperture microphone arrays - Yu, Silverman - 2004 |

4 |
The GPS Equations and the Problem of Apollonius
- Hoshen
- 1996
(Show Context)
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4 | Improved TDOA disambiguation techniques for sound source localization in reverberant environments. - ZANNINI - 2010 |

3 |
Tensor Analysis and Applications. Second Edition
- Manifolds
- 1988
(Show Context)
Citation Context ... 6.16 can be nicely interpreted in terms of the two-dimensional and the three-dimensional intersection problems. Here we use some standard Minkowski and relativistic conventions used, for example, in =-=[3, 43]-=-. (i) τ ∈ E if, and only if, v(τ ) is isotropic, or light–like. In this case, the line L21(τ ) is parallel to a generatrix of the cone, therefore it meets C0 at an ideal point. On the x–plane this mea... |

3 | A Geometric Approach to Sound Source Localization from Time-Delay Estimates
- Alameda-Pineda, Horaud
- 2014
(Show Context)
Citation Context ... case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source localization problem has been proposed and investigated very recently also in =-=[5, 17]-=-, the latter in the context of the TOA–based target tracking. We believe that characterizing the TDOA map to its fullest extent, even in the simplest case of three calibrated and synchronous sensors, ... |

3 |
Acoustic source localization with distributed asynchronous microphone networks.
- Canclini, Antonacci, et al.
- 2013
(Show Context)
Citation Context ...e localization but measurement errors cause the corresponding hyperbola branches not to meet at a single point, thus ruling out simple geometric intersection as a solution to the localization problem =-=[14]-=-. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) [2]; and Co... |

3 |
An Alternative Closed Form Solution to the GPS Pseudorange Equations
- Leva
- 1995
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3 | An exact noniterative linear method for locating sources based on measuring receiver arrival times - Militello, Buenafuente |

2 |
Information geometry of target tracking sensor networks
- Cheng, Wangb, et al.
(Show Context)
Citation Context ... case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source localization problem has been proposed and investigated very recently also in =-=[5, 17]-=-, the latter in the context of the TOA–based target tracking. We believe that characterizing the TDOA map to its fullest extent, even in the simplest case of three calibrated and synchronous sensors, ... |

2 |
Positioning systems in minkowski space-time: from emission to inertial coordinates
- Coll, Ferrando, et al.
(Show Context)
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2 |
Positioning systems in minkowski space-time: Bifurcation problem and observational data
- Coll, Ferrando, et al.
- 2012
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2 | Localization of acoustic sources through the fitting of propagation cones using multiple independent arrays
- Compagnoni, Bestagini, et al.
- 1975
(Show Context)
Citation Context ...ion, in fact, is in line with the recently revamped interest of the research community in self-calibrating and self-synchronizing spatial distributions of sensors [16,45,47]. Our analysis starts from =-=[20]-=-, where a different perspective on the localization problem is offered through the adoption of the Space–Range Differences (SRD) reference frame, where the wavefront propagation is described by a (pro... |

2 | Robustness of TOA and TDOA positioning under suboptimal weighting conditions - Do, Rabinowitz, et al. - 2007 |

2 |
Nonlinear statistical models, volume 254 of Mathematics and its Applications
- Pázman
- 1993
(Show Context)
Citation Context ...on of the previous paragraph, it is well known that the estimated τ̄ ∈ Im(τ∗2 ) given by MLE is the orthogonal projection of τ∗ on the noiseless measurements set, i.e. the projection pH(τ∗) on H (see =-=[6,42]-=- and Fig. 12) therefore x̄ = τ∗2 −1(pH(τ∗)). A similar reasoning applies to the more complex case of τ∗n. In particular, any estimator has a geometrical interpretation and the relative accuracy depend... |

2 |
The two-dimensional source location problem for time differences of arrival at minimal element monitoring arrays,”
- Spencer
- 2007
(Show Context)
Citation Context ...rs (i.e. with two TDOAs) and they recognize the possibility of dual solutions in some instances, as two different source positions could correspond to the same pair of TDOA measurements. Recently, in =-=[51]-=- the author focused on the assessment of the ill-posedness of the localization problem in the case of 2D minimal sensor configurations, i.e. on quantify how changes in the measurements propagate onto ... |

2 | Disambiguation in multidimensional tracking of multiple acoustic sources using a gaussian likelihood criterion - Teng, Lombard, et al. - 2010 |

1 |
Tdoa-based acoustic source localization in the space–range reference frame
- Bestagini, Compagnoni, et al.
- 2013
(Show Context)
Citation Context ...ie on the surface of the propagation cone, localizing a source in the SRD space corresponds to finding the vertex of the cone that best fits the measured data. The SRD reference frame is also used in =-=[12]-=- to offer geometric interpretations to the underlying principles behind the most common TDOA-based localization solutions. Although not explicitly claimed, the localization problem is described in [12... |

1 |
TDOA-based localization in two dimension: the bifurcation curve
- Compagnoni, Notari
(Show Context)
Citation Context ...nal quintic curve. In Appendix C we include the source code in Singular [23] language for computing the Cartesian equation (further analysis of the properties of the bifurcation curve is contained in =-=[21]-=-). From Fig. 9 and Fig. 10, we immediately recognize what was assessed through simulations and for a specific sensor configuration in [51]. These results, however, have been here derived in closed for... |

1 |
G.Elko. Source Localization, chapter 9
- Huang, Benesty
- 2004
(Show Context)
Citation Context ...used on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) [2]; and Constrained Least Squares (CLS) =-=[34, 48]-=-, which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is necessary to minimize the number of sensors in use, due to specific sen... |

1 |
Computational algebraic methods in efficient estimation
- Kobayashi, Wynn
(Show Context)
Citation Context ...In particular, any estimator has a geometrical interpretation and the relative accuracy depends on the (non trivial) shape of Im(τ∗n). Possible techniques to be applied come from Information Geometry =-=[6, 35]-=-, an approach that proved successful in similar situations and that is based on the careful description of Im(τ∗n). With this in mind, Geometry of TDOA maps 36 notice that our characterization of the ... |

1 |
A direct solution to GPS-type navigation equations
- Kraus
- 1987
(Show Context)
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1 | Tde-based 2d real time high accuracy sound source location calculation using a special microphones arrangement - Pourmohammad, Ahadi - 2010 |