A probability distribution mean sensor fusion method based on limited mixing model optimization

Abstract

The present application relates to the field of sensor information fusion, in particular to a probability distribution mean sensor fusion method based on limited mixed model optimization. Any probability density can be approximately expressed in the form of Gaussian mixture, the method aims to express the posterior distribution of the target state in the form of Gaussian mixture, when performing sensor fusion, the patent solves the KL divergence between Gaussian mixture distribution through the variational method, finds a best fusion weight by minimizing the KL divergence of AA fusion to the real target state, so that the result of sensor fusion is closer to the real target state, and the target is more convenient to estimate and track.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of sensor information fusion, in particular to a probability distribution mean sensor fusion method based on limited mixed model optimization. BACKGROUND

[0002] With the vigorous development of networked sensors, the application of information fusion technology in state estimation problems is also more and more, which has aroused great interest in military and commercial fields; one of the most basic fusion methods is linear fusion, which simply mixes all the information in a linear way, which leads to a limited mixed representation of state probability distribution, such as popular Gaussian mixture, limited mixed (such as Gaussian mixture) is a flexible and powerful probability modeling tool, which is widely involved in the multimodal distribution of many estimation and learning problems. At the same time, it provides an arithmetic mean (AA) fusion method for information fusion, which has the advantages of high computational efficiency, strong recovery ability to sensors (such as false detection), and fault tolerance ability to any degree of node correlation; the fusion algorithm has shown attractive performance in single sensor and multi-sensor estimator design.

[0003] The calculation of fusion weight has also become a big problem. The currently disclosed AA fusion weight solving method simplifies the AA fusion result into a Gaussian distribution, which simplifies the calculation of Kullback-Leibler (KL) divergence. However, since the method approximates the Gaussian mixture to the Gaussian distribution calculation, part of the information is lost in the calculation process, so the calculation of fusion weight under AA fusion of limited mixed model needs further research and expansion. SUMMARY

[0004] In order to solve the problem of fusion weight calculation, the present application provides a probability distribution mean sensor fusion method based on limited mixed model optimization. Any probability density can be approximately expressed in the form of Gaussian mixture, this method aims to express the posterior distribution of the target state into the form of Gaussian mixture, when performing sensor fusion, the present application solves the KL divergence between Gaussian mixture distributions by variational method, finds a best fusion weight by minimizing the KL divergence of AA fusion to the real target state, so that the result of sensor fusion is closer to the real target state, and the target is more convenient to estimate and track.

[0005] The technical scheme adopted by the present application to solve its technical problems is: a probability distribution mean sensor fusion method based on limited mixed model optimization, characterized by the following steps:

[0006] Step one, establish the state equation of target motion:

[0007] x(k)=Ax(k-1)+W(k) (1)

[0008] The sensor measurement model is modeled as:

[0009] z(k) = Hx(k) + V(k) (2)

[0010] where x(k) represents the state of the target at time k, x(k-1) represents the state of the target at time k-1, A is a state transition matrix representing the transformation of the state from the previous time to the current time; W(k) represents the system error (noise). In the measurement equation, z(k) is a measurement vector used to update the system state value; H represents a measurement transformation matrix; V(k) is the measurement error (noise), and the system error and the measurement error are both assumed to be mutually independent Gaussian noise.

[0011] Step two, filtering estimation using Gaussian sum filter. In this patent, Gaussian sum filter is used for prediction and update of the target state, and the Gaussian sum filter considers that any probability density distribution can be approximated by the superposition of N Gaussian distributions:

[0012]

[0013] where α i ≥ 0 represents the weight of the i-th Gaussian distribution, N(x; μ i , Σ i ) represents that x follows a Gaussian distribution with mean μ i and covariance Σ i .

[0014] The one-step prediction probability density distribution and the posterior probability density distribution of the system at time k are represented as:

[0015]

[0016]

[0017] and represent the weight of the i-th Gaussian distribution of one-step prediction and posterior, N(x; μ k , Σ ) represents that x follows a Gaussian distribution with mean μ and covariance Σ , N(x; μ k , Σ ) represents that x follows a Gaussian distribution with mean μ

[0018]

[0019] wherein denotes the weight, and denotes the mean and covariance of the i-th Gaussian component at time k-1, the process noise is approximated as a Gaussian mixture of the form:

[0020]

[0021] where w k denotes the non-Gaussian variable, denotes the weight of the process noise and denotes the mean and covariance.

[0022] Thus, the one-step prediction of the system state probability density function can be expressed as:

[0023]

[0024] where, denotes x k obeys a Gaussian distribution with mean and covariance .

[0025] Similarly, the posterior probability density distribution is expressed as:

[0026]

[0027] where,

[0028] Step 3, sensor fusion is performed, and the fusion parameters are estimated. The posterior distribution of the Gaussian sum filter is a Gaussian mixture, denoted as π α denotes the weight of the a-th Gaussian distribution, N(x, μ α ,Σ α ) denotes a Gaussian distribution with mean μ α and covariance Σ α , and the AA fusion is:

[0029]

[0030] where X denotes the state of a single or multiple targets, ω = [ω1, ω2, …] T is the fusion weight, usually f i (X) is the probability distribution.

[0031] The fusion rule of AA is to minimize the weighted sum of the directional KL distance between the fusion probability distribution and the fusion result:

[0032]

[0033] p(X) represents the true target distribution, D KL Denotes KL divergence, The theoretical derivation is as follows:

[0034]

[0035] The KL divergence is always non-negative, and only if p(X) = f AA When (X), D KL (f AA (X)||p(X)=0, J(p(X)) is minimized, and equation (11) is proved.

[0036] Often the true probability distribution of a target is unknown, decomposition The original problem Transform into:

[0037]

[0038] The key to the problem lies in how to solve for the KL divergence between two Gaussian mixture distributions, and how to calculate D. KL (f i ||f AA This can be represented as: Introduce a variational parameter φ, where φ>0. Using Jensen's inequality, we introduce a variational lower bound, where:

[0039]

[0040] This is The lower bound, by making The optimal bound is obtained by maximizing φ. The maximum value will be achieved at:

[0041]

[0042] Similarly: The optimal solution will be obtained at:

[0043]

[0044] The variational approximation of the KL divergence based on two Gaussian mixtures is:

[0045]

[0046] Substituting the solution of the KL divergence into (12), the original problem ω subopt It then transforms into:

[0047]

[0048] The beneficial effect of the present application is that the present application mainly proposes a probability distribution mean sensor fusion method based on limited mixed model optimization, the main task of which is to find the fusion weight after AA fusion under the limited mixed model, find the optimal fusion scheme, and obtain the state estimation closest to the target. This method is suitable for single sensor and multi-sensor target tracking and estimation.

[0049] The present application will be described in detail below in combination with the drawings and specific embodiments. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1 is the structural flowchart of the method of the present application.

[0051] Figure 2 is the target simulation scene of the method of the present application.

[0052] Figure 3 is the fusion weight of the sensor of the method of the present application.

[0053] Figure 4 is the simulation precision result of the method of the present application. DETAILED DESCRIPTION

[0054] Reference Figures 1-4 The present application is a probability distribution mean sensor fusion method based on limited mixed model optimization, and the specific steps are as follows:

[0055] Step 1, setting a simulation scene.

[0056] In the urban suburban environment, two laser radar sensors are set to track the target, the target detection probability is 98%, the clutter rate r=2 (i.e. there are two clutter points on average per frame), the target moves at a constant speed in a straight line, and the motion model is as follows:

[0057] In the formula, T represents the sensor sampling time interval, which is 1s, respectively represent the position in the x-axis direction, the position in the y-axis direction, the velocity in the x-axis direction, the velocity in the y-axis direction, and w k represents the process noise, x k+1 represents the state at k+1 time

[0058] The measurement model expression of the target is:

[0059]

[0060] In the formula, v k represents the measurement noise of the model, z k is the measurement vector.

[0061] In the simulation scenario, two sensors are designed to track the target, r = 5m for the first sensor and r = 10m for the second sensor, the performance of sensor 1 is better than that of sensor 2. Set the straight line motion model process noise and observation noise respectively:

[0062] q = 5m / s 2 (3)

[0063] r 1 = 5m, r 2 = 10m (4)

[0064] Then after discretization:

[0065]

[0066]

[0067] Step 2, determine the state estimation of Gaussian and filter.

[0068] In the single target tracking in clutter environment, for linear Gaussian model, the following single target tracking algorithm is obtained:

[0069] Given: Subject to Gaussian distribution with mean Covariance P k-1

[0070] Prediction:

[0071]

[0072] Update:

[0073]

[0074] In equations (7), (8), is the state one-step prediction, P k|k-1 represents the prediction mean square error matrix, and is the one-step prediction of measurement, the prediction mean square error matrix is S k , K k is the filter gain, and the state estimation is The state estimation mean square error is P k|k , represents the θth column of z k is the target observation, and for θ, there is Substitute the complete observation equation into the update step to obtain: represents the target observation, and the above equation is transformed into: Thus:

[0075] ​

[0076]

[0077] P D For the probability of observing the target, λ c is the intensity function.

[0078]

[0079] After one time step, P(x k |Z 1:k ) has m k +1 times as many terms as the previous time step; the number of terms in P(x k |Z 1:k ) is reduced by:

[0080] 1. Gating mechanism

[0081] Consider When i.e. when the clutter intensity is approximately constant, Taking the logarithm of both sides, we get We see that when: is large, the corresponding p θ (x) contributes less to P(x k |Z 1:k ), and can be ignored, thus reducing the number of terms in the sum.

[0082] is the Mahalanobis distance, which measures the distance between the actual measurement and . The larger the distance, the less likely it is that the measurement is of the target.

[0083] 2. Gaussian sum filtering

[0084] It retains a number of terms of P(x k |Z 1:k ) that is no more than the upper bound N max . Let p(x k-1 |Z 1:k-1 ) be the sum of H k-1 Gaussian distributions at this time, i.e. given :

[0085] Prediction:

[0086]

[0087] is equivalent to selecting the H i best terms of p k-1 (x 1:k-1 )(x) is predicted while the weights remain constant, and the weighted sum is calculated:

[0088]

[0089] renew:

[0090]

[0091] This is equivalent to each item in the prediction results. Update the algorithm and calculate the weighted sum. When p(x) k-1 |Z 1:k-1 When a sum is composed of several terms, each term can be predicted and updated, and then a weighted sum can be calculated.

[0092] Step 3: Sensor fusion parameter estimation.

[0093] To find the optimal fusion scheme, the fusion parameter ω can be estimated by minimizing the KL divergence. The fusion weight is measured by minimizing the KL divergence between the fused distribution and the true target distribution. Since the true target distribution is unknown, solving for the optimal fusion weight is transformed into:

[0094]

[0095] Step 4: Performance index comparison.

[0096] Using the root mean square error (RMSE) as a comparison metric, the RMSE of the target at time t is defined as:

[0097]

[0098] In the formula, x t It is the true state of the target. It is the estimated state after sensor fusion, and N represents the simulation step size.

[0099] The values ​​of the fusion weights in the finite mixture model based on the variational approximation are as follows: Figure 3 As shown, the weights of sensor 1 fluctuate in the range of 0.5 to 0.55, while those of sensor 2 fluctuate in the range of 0.45 to 0.5. This indicates that the weighted solution is meaningful because it assigns a larger weight to the sensor with better quality. The RMSE based on Gaussian approximation and variational approximation are as follows: Figure 4 As shown, the performance of the two is not much different; this patent provides another approach to solving the fusion weights for AA fusion.

Claims

1. A probability distribution mean sensor fusion method based on limited mixing model optimization, characterized in that, The method comprises the following steps: Step one, establishing state equation of target motion: x(k) = Ax(k-1) + W(k) (1) The sensor measurement model is modeled as: z(k) = Hx(k) + V(k) (2) In the formula, x(k) represents the state of the target at time k, x(k-1) represents the state of the target at time k-1, A is a state transition matrix, representing the transformation of the state from the previous time to the current time; W(k) represents system error noise; In the measurement equation, z(k) is a measurement vector, which is used to update the system state value; H represents a measurement transformation matrix; V(k) is measurement error noise, and the system error and the measurement error are assumed to be mutually independent Gaussian noise; Step two, filtering estimation is performed by using a Gaussian sum filter; The Gaussian sum filter believes that any probability density distribution is approximated by the superposition of N Gaussian distributions: wherein α i ≥ 0 represents the weight of the i-th Gaussian distribution, N(x; μ i ,Σ i ) denotes that x follows a Gaussian distribution with mean μ i and covariance Σ i . The one-step prediction probability density distribution and the posterior probability density distribution of the system at time k are respectively represented as: and Wi, i = 1, 2,..., k - 1, represent the weights of the k - 1 steps of prediction and posterior, xk-1| k-1= xk-1, k xk-1| k-1= xk-1, xk-1| k-1= xk-1, xk-1| k-1= xk-1, xk-1| k-1= xk-1, k xk-1| k-1= xk-1, xk-1| k-1= xk-1, xk-1| k-1= xk-1, wherein denotes the weight, and denotes the mean and covariance of the i-th Gaussian component at time k - 1, the process noise is approximated as a Gaussian mixture of the form: where w k denotes a non-Gaussian variable, represents a weight for the process noise and denotes the mean and covariance; Therefore, the one-step prediction value of the system state probability density function is represented as follows: wherein, represents x k subject to a Gaussian distribution with mean covariance covariance Similarly, the posterior probability density distribution is represented as: In the formulae, Step three, sensor fusion is performed, and fusion parameters are estimated; the posterior distribution of Gaussian and filtering is Gaussian mixture, which is simply denoted as π α represents the weight of the αth Gaussian distribution, N(x, μ α ,Σ α ) represents a Gaussian distribution with mean μ α and covariance Σ α , and AA fusion is: where X represents the state of a single or multiple targets, ω = [ω1, ω2, … ] T is a fusion weight, typically f i (X) is a probability distribution; The fusion rule of AA is to minimize the weighted sum of the directional KL distance between the fusion probability distribution and the fusion result: p(X) represents the true target distribution, D KL represents the KL divergence, decomposition reducing the problem to: D KL (f i ||f AA ) is expressed as: Introduce a variational parameter φ, which φ > 0, Using Jensen’s inequality, introduce a variational lower bound, where: This is L fi (f AA ) the lower bound, by maximizing φ to obtain the optimal bound; the maximum will be taken at: By analogy: The optimal solution will be taken at: The variational approximation KL divergence based on two Gaussian mixtures is: Substituting the solution of the KL divergence into (13), the original problem ω subopt into:

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