Method and apparatus for fusing signals with partially known independent error components

• US 6,829,568 B2
• Filed: 04/26/2002
• Issued: 12/07/2004
• Est. Priority Date: 04/26/2002
• Status: Expired due to Fees

• Current Assignee: Jeffrey Kent Uhlmann, Simon Justin Julier
• Original Assignee: Jeffrey Kent Uhlmann, Simon Justin Julier
• Inventors: Uhlmann, Jeffrey Kent, Julier, Simon Justin
• Primary Examiner(s): Barlow, John
• Assistant Examiner(s): Washburn, Douglas N
• Application Number:
  US10/132,309
• Publication Number:
  US 20030204382A1
• Time in Patent Office: 956 Days
• Field of Search: 702/189, 702/190, 702/196
• US Class Current: 702/189
• CPC Class Codes: G06F 17/18 for evaluating statistical data , e.g. average values, frequency distributions, probability functions, regression analysis forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04

Abstract

A method and apparatus is described for fusing a plurality of signals corresponding to estimates of the state of an object, system, or process. The method and apparatus is specialized or programmed for (1) receiving estimates, each of which can be expressed in the form of a state vector and an error covariance matrix, at least one estimate of which can be decomposed into a sum of an independent error covariance matrix and a potentially correlated error covariance matrix, and (2) transmitting a resulting signal corresponding to an estimate, which can be expressed in the form of a state vector and an error matrix, in order to evoke a physical response from a system receiving the signal. The method and apparatus provides advantages over the prior art by improving the accuracy of fused estimates while guaranteeing consistency.

Claims

1. A method for producing a fused signal from a set of n>

1 signals, each element i, 1≦

i≦

n, of which contains information defining a vector a_i representing the estimated mean state of a physical system and a covariance matrix A_i=$A_i=\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}$

that represents a consistent estimate of the squared error associated with the said vector for any value of ω

_i between 0 and 1, inclusive, where the covariance matrix A_i2 underestimates the expected squared error in the said vector that is known to be statistically independent of the errors in the remaining n−

1 signals, comprising;

forming a fused signal that can be represented as a mean vector c and covariance C defined mathematically by;

$\begin{array}{c}{C}^{-1}=\underset{i=1}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{H_i^T<br><br>\left(\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}\right)}^{-1}<br><br>H\\ c=C<br><br>\underset{i=1}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{H_i^T<br><br>\left(\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}\right)}^{-1}<br><br>a_i\end{array}$

where any summand of the above summations containing ω

_i=0 is defined to be zero and each matrix H_i defines the transformation from the coordinate frame of signal i to the coordinate frame of the fused signal, and the set of real-valued numbers, ω

₁, . . . , ω

_n, sum to one and are computed using an optimization algorithm to minimize some measure of the size of the fused covariance matrix C;

transmitting a signal derived from the fused signal.

2. The method of claim 1 wherein said n is equal to 2 and the fused signal is augmented to include split covariance information in which the independent error covariance component C₂ is defined mathematically as:

$C_2=C^{-1}<br><br>\left(A_1^{-1}<br><br>A_{12}<br><br>A_1^{-1}+A_2^{-1}<br><br>A_{22}<br><br>A_2^{-1}\right)<br><br>C^{-1}$

where $A_1=\frac{1}{{\mathrm{\omega <br><br>}}_1}<br><br>A_{11}+A_{12}$

corresponds to the covariance of the first signal and $A_2=\frac{1}{{\mathrm{\omega <br><br>}}_2}<br><br>A_{21}+A_{22}$

corresponds to the covariance of the second signal, and C₂ is implicitly determined to be C−

C₁.

3. The method of claim 2 in which ω

₁ and ω

₂ are selected so that said covariance matrix C or a sub-block of C is of minimum size according elements of the set of measures that includes the determinant, trace, weighted Lp norms, and maximum eigenvalue.

4. The method of claim 2 wherein the fused signal relates to the state of a moving platform, and one or more of said signals relate to measurements of previously mapped beacons.

5. The method of claim 1 in which said numbers, ω

₁ . . . , ω

_n, are selected so that said covariance matrix C or a sub-block of C is of minimum size according elements of the set of measures that includes the determinant, trace, weighted Lp norms, and maximum eigenvalue.

6. The method of claim 1 wherein the fused signal relates to the state of a moving platform, and one or more of said signals relate to measurements of previously mapped beacons.

7. A computer-readable memory storing instructions for configuring a signal processing system, coupled with means for transmitting a signal and with means for obtaining a set of n>

1 signals containing information obtained in part from a physical measuring device, each element i, 1≦

i≦

n, of which contains information defining a vector ā

_i representing the estimated mean state of a physical system and a covariance matrix $A_i=\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}$

that represents a consistent estimate of the squared error associated with the said vector for any value of ω

_i between 0 and 1, inclusive, where the covariance matrix A_i,2 underestimates the expected squared error in the said vector that is known to be statistically independent of the errors in the remaining n−

1 signals, to perform a method for producing a fused signal comprising;

forming a fused signal that can be represented as a mean vector c and covariance C defined mathematically by;

$\begin{array}{c}{C}^{-1}=\underset{i=1}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{H_i^T<br><br>\left(\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}\right)}^{-1}<br><br>H\\ c=C<br><br>\underset{i=1}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{H_i^T<br><br>\left(\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}\right)}^{-1}<br><br>a_i\end{array}$

where any summand of the above summations containing ω

_i=0 is defined to be zero and each matrix Hi defines the transformation from the coordinate frame of signal i to the coordinate frame of the fused signal, and the set of real-valued numbers, ω

₁, . . . , ω

_n;

sum to one and are computed using an optimization algorithm to minimize some measure of the size of the fused covariance matrix C;

transmitting a signal derived from the fused signal.

8. The computer-readable memory of claim 7 including instructions for the case in which said n is equal to 2 for computing a fused signal that is augmented to include split covariance information in which the independent error covariance component C₂ is defined mathematically as:

$C_2=C^{-1}<br><br>\left(A_1^{-1}<br><br>A_{12}<br><br>A_1^{-1}+A_2^{-1}<br><br>A_{22}<br><br>A_2^{-1}\right)<br><br>C^{-1}$

where $A_1=\frac{1}{{\mathrm{\omega <br><br>}}_1}<br><br>A_{11}+A_{12}$

corresponds to the covariance of the first signal and $A_2=\frac{1}{{\mathrm{\omega <br><br>}}_2}<br><br>A_{21}+A_{22}$

corresponds to the covariance of the second signal, and C₂ is implicitly determined to be C−

C₁.

9. The computer readable memory of claim 8 including instructions for determining ω

₁ and ω

₂ so that said covariance matrix C or a sub-block of C is of minimum size according elements of the set of measures that includes the determinant, trace, weighted Lp norms, and maximum eigenvalue.

10. The computer readable memory of claim 8 including instructions for computing a fused signal relating to the state of a moving platform when one or more of said signals relate to measurements of previously mapped beacons.

11. The computer readable memory of claim 7 including instructions for determining said numbers, ω

₁, . . . ω

_n, so that said covariance matrix C or a sub-block of C is of minimum size according elements of the set of measures that includes the determinant, trace, weighted Lp norms, and maximum eigenvalue.

12. The computer readable memory of claim 7 including instructions for computing a fused signal relating to the state of a moving platform when one or more of said signals relate to measurements of previously mapped beacons.

13. A specialized or programmed signal processing system for producing a fused signal from a set of n>

1 signals encoding information obtained in part from a physical measuring device, each element i, 1≦

i≦

n, of which contains information defining a vector a_i representing the estimated mean state of a physical system and a covariance matrix $A_i=\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}$

that represents a consistent estimate of the squared error associated with the said vector for any value of ω

_i between 0 and 1, inclusive, where the covariance matrix A_i,2 underestimates the expected squared error in the said vector that is known to be statistically independent of the errors in the remaining n−

1 signals, comprising;

means for forming a fused signal that can be represented as a mean vector c and covariance C defined mathematically by;

$\begin{array}{c}{C}^{-1}=\underset{i=1}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{H_i^T<br><br>\left(\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}\right)}^{-1}<br><br>H\\ c=C<br><br>\underset{i=1}{\overset{n}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{H_i^T<br><br>\left(\frac{1}{{\mathrm{\omega <br><br>}}_i}<br><br>A_{\mathrm{i1}}+A_{\mathrm{i2}}\right)}^{-1}<br><br>a_i\end{array}$

where any summand of the above summations containing ω

_i, 0 is defined to be zero and each matrix H_i defines the transformation from the coordinate frame of signal 1 to the coordinate frame of the fused signal, and the set of real-valued numbers, ω

₁, . . . , ω

_n, sum to one and are computed using an optimization algorithm to minimize some measure of the size of the fused covariance matrix C;

means for transmitting a signal derived from the fused signal.

14. The signal processing system of claim 13 including means for the case in which said n is equal to 2 for producing a fused signal that is augmented to include split covariance information in which the independent error covariance component C₂ is defined mathematically as:

$C_2=C^{-1}<br><br>\left(A_1^{-1}<br><br>A_{12}<br><br>A_1^{-1}+A_2^{-1}<br><br>A_{22}<br><br>A_2^{-1}\right)<br><br>C^{-1}$

where $A_1=\frac{1}{{\mathrm{\omega <br><br>}}_1}<br><br>A_{11}+A_{12}$

corresponds to the covariance of the first signal and $A_2=\frac{1}{{\mathrm{\omega <br><br>}}_2}<br><br>A_{21}+A_{22}$

corresponds to the covariance of the second signal, and C₂ is implicitly determined to be C−

C₁.

15. The signal processing system of claim 14 including means for determining ω

₁ and ω

₂ so that said covariance matrix C or a sub-block of C is of minimum size according elements of the set of measures that includes the determinant, trace, weighted Lp norms, and maximum eigenvalue.

16. The signal processing system of claim 14 including means for computing a fused signal relating to the state of a moving platform when one or more of said signals relate to measurements of previously mapped beacons.

17. The signal processing system of claim 13 including means for determining said numbers, ω

₁ . . . ω

_n, so that said covariance matrix C or a sub-block of C is of minimum size according elements of the set of measures that includes the determinant, trace, weighted Lp norms, and maximum eigenvalue.

18. The signal processing system of claim 13 including means for computing a fused signal relating to the state of a moving platform when one or more of said signals relate to measurements of previously mapped beacons.