CN102679980A - Target tracking method based on multi-scale dimensional decomposition - Google Patents

Target tracking method based on multi-scale dimensional decomposition Download PDF

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CN102679980A
CN102679980A CN2011103610725A CN201110361072A CN102679980A CN 102679980 A CN102679980 A CN 102679980A CN 2011103610725 A CN2011103610725 A CN 2011103610725A CN 201110361072 A CN201110361072 A CN 201110361072A CN 102679980 A CN102679980 A CN 102679980A
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mrow
mover
scale
target
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林云
李靖超
李一兵
葛娟
康健
李一晨
叶方
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Harbin Engineering University
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Harbin Engineering University
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Abstract

The invention aims at providing a target tracking method based on multi-scale dimensional decomposition, and the target tracking method comprises the following steps of selecting a wavelet-base function to decompose a target angle or track measurement data onto a scale, predicting and filtering the measurement data on a low-frequency subspace of each scale by utilizing extended kalman filtering (EKF) algorithm to obtain a rough tracking result of the target on different scales, and further eliminating the influence of noise and wildvalue by utilizing the wavelet threshold algorithm on the high-frequency subspace of different scales; and converging the tracking data on different scales through the wavelet reconstruction algorithm to obtain the precise tracking data of the target. The target tracking method can effectively, accurately, reliably and stably track the target in different complicated environment, the multi-scale EKF algorithm is realized by utilizing field programmable gate array (FPGA) parallel processing structure, the wavelet decomposition and rebuilding, the EKF algorithm of different scales and the wavelet threshold denoising are simultaneously implemented, and the real-time performance of the target tracking is ensured.

Description

Target tracking method based on multi-scale dimensional decomposition
Technical Field
The invention relates to a tracking method in the field of target tracking.
Background
Target tracking based on a single sensor is widely applied due to its simplicity, practicality and good economical efficiency. However, due to the limitations of the current increasingly complex target environment and the structure thereof, the tracking accuracy, reliability and stability of a single sensor are limited to a certain extent. Therefore, new tracking algorithms are urgently needed to be developed.
Disclosure of Invention
The invention aims to provide a target tracking method based on multi-scale dimensional decomposition, which can be effective, accurate, reliable and stable in various complex environments.
The purpose of the invention is realized as follows:
the invention relates to a target tracking method based on multi-scale dimensional decomposition, which is characterized by comprising the following steps:
(1) decomposing the measurement data of the target angle or the flight path to a scale by using a wavelet basis function;
(2) predicting and filtering the measurement data by adopting an EKF algorithm on the low-frequency subspace of each scale to obtain the coarse tracking results of the targets on different scales:
the sensor obtains the visual angle signal
Figure BDA0000108619700000011
Capturing the target and realizing accurate tracking, and simultaneously working Kalman Filter-KF (EKF) filter to obtain the estimated value of the relative motion state quantity of the target and the sensor at the kth moment
Figure BDA0000108619700000012
When in
Figure BDA0000108619700000014
After the time sensor loses the target, the signal is obtained by expanding EKF filtering
Figure BDA0000108619700000015
Estimation of time of day observations
Figure BDA0000108619700000016
Figure BDA0000108619700000017
If the system state equation is linear, that is:
Figure BDA0000108619700000018
wherein x (k) is the n-dimensional state vector at time k, which is also the estimated vector;
Figure BDA0000108619700000019
is k toA one-step transition matrix of time instants (order n λ n); w (k) is the system noise at time k;
Figure BDA00001086197000000111
weighting the system noise at time k;
Figure BDA00001086197000000112
is the weighting of the measurement noise;
Figure BDA00001086197000000113
m-dimensional measurement noise at the time k;
if the observation equation is non-linear, i.e.:
firstly, Taylor expansion is carried out on an observation equation at an optimal state, a low-order expansion term is kept as follows,
Figure BDA0000108619700000022
order toAnd setting the Gaussian white noise with the middle and high order micromerities of the expansion as a zero mean value to obtain a linearized observation equation:
Figure BDA0000108619700000024
wherein,
Figure BDA0000108619700000025
is that
Figure BDA0000108619700000026
Is optimally predicted and satisfied
Figure BDA0000108619700000027
Also has a mean value of zero, and
Figure BDA0000108619700000029
is uncorrelated white Gaussian noise and satisfies
Figure BDA00001086197000000210
Also:
Figure BDA00001086197000000211
Figure BDA00001086197000000212
the recursion formula for the EKF algorithm can be written as follows:
Figure BDA00001086197000000213
wherein:
Figure BDA00001086197000000215
Figure BDA00001086197000000216
Figure BDA00001086197000000217
initial value:
Figure BDA00001086197000000218
from wavelet theory: the low-frequency subspace (smoothed) signal at the scale i-1 can be obtained from the scale i by a low-pass filter with an impulse response h (l)The high frequency subspace (detail) signal on the scale i-1 can be obtained by a high pass filter with an impulse response g (l)
Figure BDA00001086197000000220
Figure BDA00001086197000000221
Figure BDA00001086197000000222
Decomposing the state equation and the measurement equation from the dimension i to the dimension i-1 according to the equation to obtain the state equation and the measurement equation under the dimension i-1, wherein G (i, k) is taken as a unit matrix:
Figure BDA0000108619700000031
wherein:
Figure BDA0000108619700000032
Figure BDA0000108619700000034
Figure BDA0000108619700000035
Figure BDA0000108619700000036
Figure BDA0000108619700000037
after a state equation and a measurement equation on the i-1 scale are obtained, an EKF algorithm is adopted to carry out time updating and measurement updating on the i-1 scale, and therefore a final filtering state estimation value on the i-1 scale is obtained
Figure BDA0000108619700000038
Sum covariance estimate
Figure BDA0000108619700000039
The final filter state estimated value on the scale i-1 is obtained
Figure BDA00001086197000000310
Sum covariance estimate
Figure BDA00001086197000000311
As a predicted value of the state at EKF filtering on the scale i-2And error covariance prediction
Figure BDA00001086197000000313
Time updating and measurement updating are carried out to obtain the estimated value of the filtering state on the scaleSum error covariance estimation
Figure BDA00001086197000000315
Thereby respectively obtaining a filtering state estimation value and an error covariance estimation value on different scales;
(3) a wavelet threshold algorithm is adopted on high-frequency subspaces with different scales, so that the influence of noise and outliers is further removed;
(4) and fusing the tracking data on different scales through a wavelet reconstruction algorithm to obtain the accurate tracking data of the target.
The invention has the advantages that: the invention can realize the multi-scale EKF algorithm by utilizing the parallel processing structure of the FPGA, and simultaneously carries out wavelet decomposition and reconstruction, EKF algorithms on different scales and wavelet threshold denoising, thereby ensuring the real-time performance of target tracking.
Drawings
FIG. 1 is a block diagram of an implementation of a target tracking apparatus of the present invention;
FIG. 2 is a flow chart of the present invention;
FIG. 3 is a flow chart of the multi-scale EKF tracking algorithm of the present invention.
Detailed Description
The invention will now be described in more detail by way of example with reference to the accompanying drawings in which:
with reference to fig. 1 to 3, the object of the present invention is achieved by: firstly, a single sensor provides measurement data of a target angle or a track, the measurement data is decomposed to a plurality of scales by using a wavelet decomposition method, and then an EKF algorithm is adopted to carry out rough tracking and filtering on the target on different scales. And finally, fusing processing results on different scales, and adopting a wavelet reconstruction algorithm to realize the precise tracking of the target on a unified scale. During reconstruction, the maximum value points of the detail parts under each scale are removed, and noise and outliers are further filtered. Therefore, more accurate target angle or track data can be obtained, and accurate tracking of targets in various complex environments is achieved.
The method comprehensively utilizes the wavelet decomposition and reconstruction algorithm, the wavelet de-noising algorithm and the EKF filtering and tracking algorithm, and can greatly improve the accuracy and reliability of single-sensor target tracking in the complex environment. The invention utilizes L-layer wavelet decomposition algorithm to decompose the measurement data into 2L subspaces, wherein the low-frequency subspace is less influenced by noise, so the EKF algorithm is adopted to obtain the coarse tracking result of the target. The high-frequency subspace is greatly influenced by noise, the influence of the noise and a wild value is removed by adopting a wavelet threshold value method, and the signal-to-noise ratio of the measured data is improved. And finally, fusing the tracking results of different scales by using a wavelet reconstruction algorithm to obtain the tracking result on a unified scale, thereby achieving the purpose of accurate target. The invention adopts EKF recursion algorithm, when the target disappears momentarily, the recursion results on different scales are fused, so that the target can be ensured to be aligned stably in a short time, and the target can be recovered to a normal tracking state after being searched again.
The tracking device consists of a sensor 1, a serial port chip 2, a digital signal processor DSP3, a programmable logic device FPGA4 and a control, display and storage device 5.
When the sensor 1 searches a target, the sensor transmits the measurement data 0 of the target to the serial port chip 2, and the serial port chip 2 quantizes the data 0 into data 1 and transmits the data 1 to the digital signal processor DSP 3. The DSP3 estimates the angle or track of the target according to the transmitted measured data, and transmits the processed data 2 to the FPGA4, the FPGA decomposes the data 2 to a plurality of scales by using wavelet decomposition algorithm, then predicts and filters the data of the low frequency subspace by using EKF algorithm, obtains the rough tracking structure of the target on different scales, and simultaneously removes the maximum value points of the detail part under each scale, further filters the noise and outlier. The tracking device simultaneously completes data processing on different scales by utilizing the FPGA parallel processing structure, and the real-time performance of the tracking device is improved. And finally, fusing the tracking data on different scales by adopting a wavelet reconstruction algorithm to obtain an accurate tracking result of the target, namely data 3. And the data 3 is transmitted to equipment for controlling, displaying, storing, information fusion and the like so as to meet the requirements of different occasions. The tracking device adopts a single sensor to complete the tracking of the target, has simple structure and convenient use, and can ensure effective, accurate, stable, reliable and real-time tracking of the target under various complex environments. In addition, in the case of a short-term loss of the target, the tracking device can continue to stably track the target for a period of time until the sensor reacquires the target.
Fig. 2 is a block diagram of signal processing of a tracking device.
The basic idea of the algorithm is as follows:
1. according to the actual situation, proper wavelet basis functions are selected to decompose the measured data of the target angle or the flight path into a plurality of scales;
2. predicting and filtering the measurement data by adopting an EKF algorithm on the low-frequency subspace of each scale to obtain the coarse tracking results of the targets on different scales;
3. a proper wavelet threshold algorithm is adopted on high-frequency subspaces with different scales, so that the influence of noise and outliers is further removed, and the variance of observation noise is adjusted, so that the filtering is more stable;
4. and fusing the tracking data on different scales through a wavelet reconstruction algorithm to obtain the accurate tracking data of the target.
5. And finally, transmitting the obtained tracking data to control, display, storage, information fusion and other equipment so as to meet the requirements of different occasions.
The EKF algorithm and the wavelet threshold denoising algorithm on different scales are processed in parallel by using an FPGA (field programmable gate array) and are completed simultaneously, so that the requirement on the real-time performance of the tracking device is met.
Principle of multiscale EKF tracking algorithm:
the sensor obtains the visual angle signal
Figure BDA0000108619700000051
Capturing targets and achieving accurate tracking. EKF filters (Kalman Filter-KF) work simultaneously to obtain the estimated value of the relative motion state quantity of the target and the sensor at the kth moment
Figure BDA0000108619700000052
Namely, it is
When in
Figure BDA0000108619700000054
After the moment sensor loses the target, although the sensor loses the observation information, the EKF filtering (EKF) can still be extended
Figure BDA0000108619700000055
Estimation of time of day observations
Figure BDA0000108619700000056
Namely, it is
Figure BDA0000108619700000057
Figure BDA0000108619700000058
Thereby continuing to stably track the target.
If the system state equation is linear, that is:
Figure BDA0000108619700000061
wherein x (k) is the n-dimensional state vector at time k, which is also the estimated vector;
Figure BDA0000108619700000062
is k to
Figure BDA0000108619700000063
A one-step transition matrix of time instants (order n λ n); w (k) is the system noise at time k;
Figure BDA0000108619700000064
weighting the system noise at time k;
Figure BDA0000108619700000065
is the weighting of the measurement noise;
Figure BDA0000108619700000066
the noise is measured for the m dimension at time k.
If the observation equation is non-linear, i.e.:
Figure BDA0000108619700000067
firstly, Taylor expansion is carried out on an observation equation at an optimal state, a low-order expansion term is kept as follows,
order to
Figure BDA0000108619700000069
And supposing the Gaussian white noise with the high-order microminiature of zero mean in the expansion formula to obtain a linearized observation equation, namely
Figure BDA00001086197000000610
Wherein,
Figure BDA00001086197000000611
is that
Figure BDA00001086197000000612
Is optimally predicted and satisfied
Figure BDA00001086197000000614
Also has a mean value of zero, andis uncorrelated white Gaussian noise and satisfies
Also, in the same manner as above,
Figure BDA00001086197000000617
Figure BDA00001086197000000618
the recurrence formula for the EKF algorithm can be written as follows:
Figure BDA00001086197000000619
wherein:
Figure BDA0000108619700000071
the initial value of the algorithm is as follows:
Figure BDA0000108619700000072
from wavelet theory, it is known that a low-frequency subspace (smooth) signal at the scale i-1 can be obtained from the scale i by a low-pass filter with an impulse response h (l)
Figure BDA0000108619700000073
The high frequency subspace (detail) signal on the scale i-1 can be obtained by a high pass filter with an impulse response g (l)
Figure BDA0000108619700000074
Figure BDA0000108619700000075
Figure BDA0000108619700000076
Decomposing the state equation and the measurement equation from the dimension i to the dimension i-1 according to the equation to obtain the state equation and the measurement equation under the dimension i-1, wherein G (i, k) is taken as a unit matrix:
wherein:
Figure BDA0000108619700000078
after a state equation and a measurement equation on the i-1 scale are obtained, an EKF algorithm is adopted to carry out time updating and measurement updating on the i-1 scale, and therefore a final filtering state estimation value on the i-1 scale is obtained
Figure BDA0000108619700000079
Sum covariance estimate
Figure BDA00001086197000000710
The final filter state estimated value on the scale i-1 is obtained
Figure BDA00001086197000000711
Sum covariance estimate
Figure BDA00001086197000000712
As a predicted value of the state at EKF filtering on the scale i-2And error covariance predictionTime updating and measurement updating are carried out to obtain the estimated value of the filtering state on the scale
Figure BDA0000108619700000082
Sum error covariance estimationAnd analogizing in sequence to respectively obtain a filtering state estimation value and an error covariance estimation value on different scales.
And finally, fusing the predicted and filtered data on each scale through a wavelet reconstruction algorithm to obtain a fusion result of the original measured data on different scales. During reconstruction, a wavelet threshold algorithm is adopted to remove the maximum value points of the high-frequency subspace on each scale, and noise and outlier points are further filtered.
FIG. 3 is a flow chart of a multi-scale EKF tracking algorithm.

Claims (1)

1. A target tracking method based on multi-scale dimensional decomposition is characterized in that:
(1) decomposing the measurement data of the target angle or the flight path to a scale by using a wavelet basis function;
(2) predicting and filtering the measurement data by adopting an EKF algorithm on the low-frequency subspace of each scale to obtain the coarse tracking results of the targets on different scales:
the sensor acquires a target and realizes accurate tracking by acquiring a line-of-sight angle signal Z (X), and a Kalman Filter-KF (EKF) works simultaneously to obtain a target at the kth moment andestimation of relative motion state quantity of sensor
Figure FDA0000108619690000011
<math> <mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mo>&RightArrow;</mo> <mi>KF</mi> </mover> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>
When the sensor loses the target at the k +1 th moment, the estimation value of the observed quantity at the k + i moment is obtained through extended EKF filtering
<math> <mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mo>&RightArrow;</mo> <mi>EKF</mi> </mover> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mi>i</mi> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&RightArrow;</mo> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mi>i</mi> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </math> i＝1，2，3...；
If the system state equation is linear, namely:
X(k+1)＝Ф(k+1，k)X(k)+G(k+1，k)U(k)+Γ(k+1)W(k)，
wherein x (k) is the n-dimensional state vector at time k, which is also the estimated vector; phi (k +1, k) is a one-step transfer matrix (n multiplied by n order) from k to k + 1; w (k) is the system noise at time k; Γ (k +1) is the weighting of the system noise at time k; g (k +1, k) is the weighting of the measurement noise; u (k) is m-dimensional measurement noise at the time k;
if the observation equation is non-linear, i.e.:
Z(k+1)＝h(X(k+1))+V′(k+1)，
firstly, Taylor expansion is carried out on an observation equation at an optimal state, a low-order expansion term is kept as follows,
<math> <mrow> <mi>h</mi> <mrow> <mo>(</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>h</mi> <mrow> <mo>(</mo> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mo>&PartialD;</mo> <mi>h</mi> </mrow> <mrow> <mo>&PartialD;</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mo>&dtri;</mo> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>X</mi> <mo>^</mo> </mover> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math>
order to
Figure FDA0000108619690000016
And setting the Gaussian white noise with the middle and high order micromerities of the expansion as a zero mean value to obtain a linearized observation equation:
Z ( k + 1 ) = H ( k + 1 ) X ( k + 1 ) + h ( X ^ ( k + 1 / k ) ) - H ( k + 1 ) X ^ ( k + 1 / k ) + V ( k + 1 ) ,
wherein,
Figure FDA0000108619690000018
is a one-step optimal prediction of X (k +1) and satisfies
<math> <mrow> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&Phi;</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>U</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>
V (k +1) is also zero mean, is uncorrelated with W (k), is white Gaussian noise, and satisfies
<math> <mrow> <mi>V</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>V</mi> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msup> <mo>&dtri;</mo> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>X</mi> <mo>^</mo> </mover> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> </mrow> </math>
Also:
Figure FDA0000108619690000022
Figure FDA0000108619690000023
the recursion formula for the EKF algorithm can be written as follows:
<math> <mrow> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&Phi;</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>U</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>[</mo> <mi>Z</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mi>h</mi> <mrow> <mo>(</mo> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </math>
wherein:
K(k+1)＝P(k+1/k)HT(k+1)G-1(k+1)
G(k+1)＝H(k+1)P(k+1/k)HT(k+1)+R(k+1)
P(k+1/k)＝Ф(k+1，k)P(k)ФT(k+1，k)+Γ(k+1，k)Q(k)ΓT(k+1，k)
P(k+1)＝(I-K(k+1)H(k+1))P(k+1/k)
initial value: X ^ ( 0 ) = E [ X ( 0 ) ] , P ( 0 ) = var [ X ( 0 ) ] ,
from wavelet theory: the low frequency subspace (smoothed) signal x at scale i-1 is obtained from scale i by a low pass filter with an impulse response h (l)L(i-1, k) the high frequency subspace (detail) signal x on the scale i-1 is obtained by means of a high pass filter with an impulse response g (l)H(i-1，k)：
xL(i-1，k)＝∑lh(l)x(i，2k-l)
xH(i-1，k)＝∑lg(l)x(i，2k-l)，
Decomposing the state equation and the measurement equation from the dimension i to the dimension i-1 according to the equation to obtain the state equation and the measurement equation under the dimension i-1, wherein G (i, k) is taken as a unit matrix:
<math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>X</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>&Phi;</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>w</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Z</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow> </math>
wherein:
Ф(i-1，k+1/k)＝Ф(i，k+1/k)Ф(i，k+1/k)
w(i-1，k)＝Ф(i，k+1/k)·∑lh(l)w(i，2k-l)+∑lh(l)w(i，2k-l+1)
Q(i-1，k)＝Ф(i)∑lh2(l)Q(i，2k-l)ΦT(i)+∑lh2(l)Q(i，2k+1-l)
H(i-1，k)＝H(i，k)
v(i-1，k)＝v(i，k)
<math> <mrow> <mi>R</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&Sigma;</mi> <mi>l</mi> </msub> <msup> <mi>h</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mi>R</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mi>l</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>R</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>
after the state equation and the measurement equation on the i-1 scale are obtained, E is adoptedThe KF algorithm carries out time updating and measurement updating on the scale i-1 so as to obtain a final filtering state estimation value on the scale i-1And a covariance estimate P (i-1, k/k);
the final filter state estimated value on the scale i-1 is obtained
Figure FDA0000108619690000032
And covariance estimate P (i-1, k/k) as the state predictor at EKF filtering on scale i-2
Figure FDA0000108619690000033
And carrying out time updating and measurement updating with the error covariance predicted value P (i-2, k/k-1) to obtain a filter state estimated value on the scale
Figure FDA0000108619690000034
And an error covariance estimation value P (i-2, k/k), thereby respectively obtaining a filtering state estimation value and an error covariance estimation value on different scales;
(3) a wavelet threshold algorithm is adopted on high-frequency subspaces with different scales, so that the influence of noise and outliers is further removed;
(4) and fusing the tracking data on different scales through a wavelet reconstruction algorithm to obtain the accurate tracking data of the target.
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