Target tracking mainly uses the tracking algorithm to select the one with the highest similarity to the target object as the target of the next frame, and then obtains the motion trajectory of the target object in the entire video sequence. Therefore, how accurately represent the similarity between the candidate block and the target block is an important factor that determines the accuracy of target tracking. As the research of target tracking and the difficulty of tracking tasks continue to increase, the distance similarity measurement methods used by tracking algorithms are also continuously improved. In image processing and analysis theory, common similarity measurement methods include Euclidean distance, block distance, checkerboard distance, weighted distance, Bart Charlie coefficient, Harsdorf distance, etc. Among them, Euclidean distance is the most applied and simplest.

The most commonly used mathematical expression of the Euclidean distance is the norm, which is mainly divided into four norms. Among them, the target tracking algorithm often uses the L1 norm and the L2 norm to extract sample sparsity characteristics and measure similarity between samples. With the complex changes in the target tracking scene, the Euclidean distance cannot meet the performance requirements of target tracking. Researchers try to apply similarity measurement algorithms in the signal domain to target trackings, such as KL divergence and JS divergence.

When the probability of an event is p (x), its information quantity is −log⁡(p(x)), and the expectation of information quantity is entropy. Relative entropy is also known as KL divergence. If we have two separate probability distributions P (x) and Q (x) for the same random variable x, we can use KL divergence to measure the difference between the two distributions. In machine learning, P is often used to represent the true distribution of the sample, and Q is used to represent the distribution predicted by the model. Then the KL divergence can calculate the difference between the two distributions, that is the loss value.

(1)DKL(p||q)=∑iP(xi)logp(xi)q(xi)

From the KL divergence formula, we can see that the closer the distribution of Q is to P (the more fitting the Q distribution is to P), the smaller the divergence value is, that is, the smaller the loss value is. Because the logarithmic function is convex, the value of KL divergence is non-negative. KL divergence is sometimes called KL distance, but it does not satisfy the property of distance. Because KL divergence is not symmetric and KL divergence does not satisfy triangle inequality. In machine learning, KL divergence can be used to evaluate the gap between label and predictions. In the process of optimization, we only need to pay attention to the cross entropy.

In probability statistics, JS divergence has the same ability to measure the similarity of two probability distributions as KL divergence. Its calculation method is based on KL divergence and inherits the non-negative nature of KL divergence. However, there is one important difference: JS divergence has symmetry. The formula of JS divergence is as follows: set two probability distributions as P and Q, M=0.5×(P+Q). (2)JS(P1||P2)=12KL(P||M)+12KL(Q||M)

If the two distributions P and Q are far apart and completely do not overlap, then the KL divergence value is meaningless, while the JS divergence value is a constant log⁡2. This has a great impact on the deep learning algorithm based on gradient descent, which means that the gradient is 0, that is, the gradient disappears.

Due to the differences in background, light, sensor resolution, and other aspects, the data collected by UAVs in different scenes may be in different data domains. There are different numbers of positive samples and negative samples in each domain, and the number of negative samples (background) is far greater than the number of positive samples (target) in single-target tracking. The problem of sample imbalance will lead to the over-fitting of a large sample proportion, and the prediction will tend to classify with more samples. Therefore, there is a strong need to align the number of samples between different domains and train the data to find samples suitable for the target domain. Inspired by the trilateral weighted sparse coding scheme (TWSC) in [51] and the dual-domain collaboration in [50], we use the method of two-field weighting to increase the weight of samples, that is, to add a larger weight to the samples with a smaller number of samples, and to add a smaller weight to the samples with a larger number of samples.

In the model, we introduce two diagonal matrices, wa and wb, as weighted matrices. wa is a block diagonal matrix, with each block having the same diagonal element to describe sample attributes in different channels. wb is used to describe the sample variance in the corresponding patch. The formula can be expressed as: (3)Lm=12∥wax−wby∥F2

where x is the positive sample, y is the negative sample. Our method also introduces the idea of dual-domain weighting, and the supporting materials show the specific processing operations of wa and wb. In other words, two weight matrices are introduced into the data fidelity term of the sparse coding frame to adaptively represent the statistics of positive and negative samples in each patch of the multi-channel, which can make better use of the sparse prior of the natural image.

Our AKLCF optimizes the response approximation in Euclidean distance to the KL divergence to achieve the curve fitting of the filters. In addition, the AKLCF increases the model confidence to improve the reliability of the model. Below we detail the construction of model functions. For simplicity, let Mk=∑d=1DBxkd∗fkd,Mk−1=∑d=1DBxk−1d∗fk−1d, where x∈RM×N×D, fk∈RM×N×D, fk−1∈RM×N×D, y∈RM×N. Here, we get the simplified form: (4)argminf12∥Mk−y∥F2+∑d=1D∥fkd∥22+r2∥Mk[ψp,q]−Mk−1[ψp,q]∥22

Then, we replace the third term of Eq. (4) with the function R(Mk|Mk−1;xk−1) to obtain the loss function of AKLCF: (5)argminf12∥Mk−y∥F2+λ2∑d=1D∥fkd∥22+R(Mk|Mk−1;xk−1)

As discussed above, in the formula with Eq. (5) add of is replaced by the Kullback-Leibler divergence term. In addition, we add a model confidence term to ensure the reliability of the model. Then we improve the loss functions by minimizing the following unconstraint objective: (6)argminf12∥Mk−y∥F2+λ2∑d=1D∥fkd∥22+β2Rmc+α2Rkl

where β and α are penalty parameters.β2Rmc is the model confidence term and Rmc=∑d=1D(xk−1d)T∗ϕkd∗xk−1d, where ϕkd is a covariance matrix.

To better reflect the effect of the KL divergence term, we first analyze it in detail to obtain the specific expression of the KL divergence term. Since the KL divergence is a measure of the asymmetry of the difference between the two probability distributions. In our model, the KL divergence is used to measure the difference between the convolution matrix of the current frame and the previous frame, that is: (7)Rkl=Rkl(N(Mk,Φk)|N(Mk−1,Φk−1))

where ϕk, ϕk−1 are covariance matrixes. Mk=∑d=1DBxkd∗fkd, Mk−1=∑d=1DBxk−1d∗fk−1d are the convolutions of the image matrix and filter coefficients, which denote the response maps from the current and previous frame respectively. k and k−1 represent the current and previous frame, respectively.

For convenience, we assume that the value vectors of filter and image blocks have multivariate Gaussian distributions, then the convolution matrix also has a Gaussian distribution, i.e., M∼N(M0,Φ), where M0 is the mean value vector of the convolution matrix distribution and Φ represents the covariance matrix of distribution. In the previous section, the function Mi=∑d=iDBxd∗fd indicates the convolution prediction of the class label fi and the sample xi in i−th dimension with the given multivariate Gaussian distribution. For better understanding, we regard each value fi in the i−th dimension as the models knowledge about the corresponding feature xi and regard the diagonal entry of covariance matrix Φi,i as the confidence of feature Mi. The smaller the value of Φi,i is, the more confident the learner is in the mean weight value Mi. Expect diagonal values, other covariance terms Φi,j can be took as the correlations between two weight value Mi and Mj for image feature ∑d=iDBxi∗fi and ∑d=jDBxj∗fj. In this way, we model parameter confidence for each element of along all channels with a diagonal Gaussian distribution. Thus, we construct the kullback-leibler divergence term based on the divergence between empirical distribution and probability distribution. As a result, we get the Rkl as following: (8)Rkl(N(Mk,Φk)∣N(Mk−1,Φk−1))=∑d=1D[logdetΦk−1ddetΦkd+tr((Φk−1−1)dΦkd)+(Mk−Mk−1)T(Φk−1−1)d(Mk−Mk−1)]

We substitute the specific expression of the model confidence term and Eq. (8) into Eq. (6) to get: (9)argminf12∥Mk−y∥F2+β2∑d=1D(xk−1d)T∗Φkd∗xk−1d+λ2∑d=1D∥fkd∥22+α2Rkl(N(Mk,Φk)|N(Mk−1,Φk−1))

In general, the first term of the formula is data fidelity, the second is the space penalty, and the third and fourth terms are our improved contents, namely the model confidence term and the KL divergence term.

DJSCF adds dual-domain weighted matrices and JS divergence in the response map which need to be considered in the optimization process, therefore the construction of the loss function in DJSCF tracker is described in detail below: (10)argminf12∥∑d=1Dwa(Bxkd∗fkd)−wb⋅y∥F2+λ2∑d=1D∥fkd∥22+γ2∥∑d=1D(Bxk−1d∗fk−1d)[ψp,q]−∑d=1D(Bxkd∗fkd)[ψp,q]∥22

Let Mk=∑d=1DBxkd∗fkd, Mk−1=∑d=1DBxk−1d∗fk−1d. The objective function is converted into: (11)argminf12∥waMk−wby∥F2+λ2∑d=1D∥fkd∥22+γ2∥(Mk−Mk−1)[ψp,q]∥22

Similar to AKLCF, we use JS divergence and model confidence to improve the third term of the objective function, and get the objective function as follows: (12)argminf12∥waMk−wby∥F2+λ2∑d=1D∥fkd∥22+R(Mk|Mk−1;xk−1)

It equals to: (13)argminf12∥waMk−wby∥F2+λ2∑d=1D∥fkd∥22+β2Rmc+α2RJS

Since the JS divergence is proposed based on the KL divergence, we use two covariance matrices (Φkd and Φk−1d) to derive the DJSCF model according to the previous derivation, then RJS=RJS(N(Mk,ϕk)|N(Mk−1,ϕk−1). Finally, we get the loss function as follows: (14)argminf12∥waMk−wby∥F2+β2∑d=1D(xk−1d)T∗Φkd∗xk−1d+λ2∑d=1D∥fkd∥22+α2RJS(N(Mk,Φk)|N(Mk−1,Φk−1))

The AJSCF model is a version that only replaces the KL divergence of AKLCF with JS divergence. The comparison between AJSCF and AKLCF shows the superiority of JS divergence in similarity fitting. At the same time, the latter DJSCF is obtained by adding dual-domain weighting based on AJSCF. And comparing AJSCF with DJSCF can verify the effect of dual-domain weighting on tracking performance.

AKLCF: The model in Eq. (9) is a convex problem and can be minimized with the ADMM algorithm to obtain the global optimal solution. Since Eq. (9) has a convolution calculation in the time domain, Pasival’s theorem is used to facilitate the calculation by converting the problem into the frequency domain. Then we get the objective function in the frequency domain as follows: (15)ε(fk)=12∥M^k−y^∥F2+β2∑d=1D(X^k−1d)TΦkdX^k−1d+λ2∑d=1D∥fkd∥22+α2Rkl(N(M^k,Φk)|N(M^k−1,Φk−1))

where the superscript ∧ denotes a signal that has been performed discrete Fourier transformation (DFT). In Eq. (15), Mk^=∑d=1DXkd^(ID⊗BT)fkd. Xk is the matrix form of input sample. ID is an identity matrix whose size is D ×D. Operator ⊗ and superscript T indicate respectively Kronecker production and conjugate transpose operation. Similar to the BACF tracker, the alternative direction method of multipliers (ADMM) is used to speed up calculation and achieve a global optimal solution of Eq. (15) for its convexity. Therefore, the augmented Lagrangian form of Eq. (15) can be written as follows: (16)12∥X^kg^k−y^∥F2+ξ^T(g^k−∑d=1DN(ID⊗BT)fkd)+β2∑d=1D(X^k−1d)TΦkdX^k−1d+μ2∥g^k−∑d=1DN(ID⊗BT)fkd∥22+λ2∑d=1D∥fkd∥22+α2Rkl(N(M^k,Φk)|N(M^k−1,Φk−1))

According to Eq. (8) and Pasival’s theorem, the KL divergence term in the frequency domain is expressed as follows: (17)Rkl(N(Mk^,Φk)|N(Mk^,Φk−1)=[logdetΦk−1detΦk+tr((Φk−1−1)Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)]

Substitute Eq. (17) into Eq. (16), and using Pasival’s theorem, then we get the specific optimization function. (18)ε^(fk,g^k,Φk,ξ^)=12∥X^kg^k−y^∥F2+λ2∥fk∥22+ξ^T(g^k−N(ID⊗BT)fk)+μ2∥g^k−N(ID⊗BT)fk∥22+β2(X^k−1)TΦkX^k−1+α2[logdetΦk−1detΦk+tr(Φk−1−1Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)]

where μ is a penalty factor. In the current frame, since the response map in the previous frame is already generated, Mk−1^ can be treated as a constant signal, which can simplify the further calculation. The ADMM algorithm is used to divide the optimization function into several sub-problems as follows: (19){fk+1∗=argminf{λ2∥fk∥22+ξ^T(g^k−N(ID⊗BT)fk)+μ2∥g^k−N(ID⊗BT)fk∥22}g^k+1∗,Φk=argming,Φ{ξ^T(g^k−N(ID⊗BT)fk)+12∥X^kg^k−y^∥F2+μ2∥g^k−N(ID⊗BT)fk∥22+β2(X^k−1)TΦkX^k−1+α2[logdetΦk−1detΦk+tr(Φk−1−1Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)]}

DJSCF: Similar to AKLCF, according to Pasival’s theorem, we get: (20)ε(fk)=12∥WaX^k(ID⊗BT)fk−Wby^∥F2+λ2∥fk∥22+β2(X^k−1)TΦkX^k−1+α2RJS(N(M^k,Φk)|N(M^k−1,Φk−1))

We get the optimization formulation: (21)ε(fk,g^k)=12∥WaX^kg^k−Wby^∥F2+λ2∥fk∥22+ζ^T(g^k−N(ID⊗BT)fk)+μ2∥g^k−N(ID⊗BT)fk∥22+β2X^k−1TΦkX^k−1+α2RJS(N(M^k,Φk)|N(M^k−1,Φk−1))

According to the principle of JS divergence and Eq. (8), we get: (22)RJS(N(M^k,Φk)|N(M^k−1))=12Rkl(N(M^k,Φk)|N(M^k,Φk)+N(M^k−1,Φk−1)2)+12Rkl(N(M^k−1,Φk−1)|N(M^k,Φk)+N(M^k−1,Φk−1)2)=12(logdetΦk−1detΦk+tr(Φk−1−1Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1))+12(logdetΦkdetΦk−1+tr(Φk−1Φk−1)+(M^k−M^k−1)TΦk−1(M^k−M^k−1))

Then the optimization solution of DJSCF is implemented by ADMM and is divided into the following subproblems: (23){fk+1∗=argminf{λ2∥fk∥22+ξ^T(g^k−N(ID⊗BT)fk)+μ2∥g^k−N(ID⊗BT)fk∥22}g^k+1∗,Φk=argming,Φ{12∥WaX^kg^k−Wby^∥F2+ξ^T(g^k−N(ID⊗BT)fk)+μ2∥g^k−N(ID⊗BT)fk∥22+β2(X^k−1)TΦkX^k−1+α2RJS}

From Eqs. (19) and (23), it is easy to find that except for the subproblem gk+1∗, the solutions of other subproblems in the two models are the same. Thus, we talk about the solution integration of the two models and only discuss them separately on subproblem gk+1∗.

Solution to subproblem fk+1∗:

The solution to subproblem fk+1∗ can be easily obtained as follows: (24)fk+1∗−(λ+μN)−1N(ID⊗BT)ξ^+μN(ID⊗BT)g^k−(λN+μ)−1(ξ+μgk)

where

(25){gk=1N(ID⊗BT)g^kξ=1N(ID⊗BT)ξ^k

Solution to subproblem: gk+1∗^ and Φk:

In AKLCF model, the augmented Lagrangian form is expressed as follows: (26)L(g^k+1∗,Φk)=12∥X^kg^k−y^∥F2+ξ^T(g^k−N(ID⊗BT)fk)+μ2∥g^k−N(ID⊗BT)fk∥22+β2(X^k−1)TΦkX^k−1+α2[logdetΦk−1detΦk+tr(Φk−1−1Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)]

Using the ADMM algorithm, two sub-problems can be obtained, namely: (27){L(g^k+1∗)=12∥X^kg^k−y^∥F2+ξ^T(g^k−N(ID⊗BT)fk)+μ2∥g^k−N(ID⊗BT)fk∥22+α2(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)L(Φk)=β2(X^k−1)TΦkX^k−1+α2[logdetΦk−1detΦk+tr(Φk−1−1Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)]

Unfortunately, since subproblem gk+1∗^ contains Xk^gk^, it would be highly time consuming to solve gk+1∗^ subproblem and it is more difficult than to solve subproblem fk∗^. And the calculation of gk+1∗^ needs to be carried out in every ADMM iteration. Therefore, the sparsity of Xk^ is exploited. Each element of y^, i.e., y^(n),n=1,2,⋯,N, is solely dependent on each xk^(n)=[xt1^(n),xt2^(n),⋯,xtD^(n)]T and gt^(n)=[conj(gt1^(n)),conj(gt2^(n)),⋯,conj(gtD^(n))]T. Operator conj(.) denotes the complex conjugate operation.

The subproblem gk+1∗^ can be thus further divided into N smaller problems as follows solved over n=[1,2,⋯,N]: (28)g^k+1∗(n)=12∥X^k(n)g^k(n)−y^(n)∥F2+ξ^T(g^k(n)−N(ID⊗BT)f^k(n))+μ2∥g^k(n)−N(ID⊗BT)f^k(n)∥22+α2(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)

Each smaller problem can be efficiently calculated and the solution is presented below: (29)g^k+1∗=11+α(x^kT(n)x^k(n)+μ1+αID)−1(x^k(n)y^(n)−ξ^(n)+μf^k(n)+∂Rkl∂g^k)

where

(30){Rkl=logdetΦk−1detΦk+tr(Φk−1−1Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)M^k=BX^kg^k

Finally, the solution of gk+1∗ is expressed as follows: (31)g^k+1∗(n)=11+α(x^k(n)x^kT(n)+μ1+αID)−1(x^k(n)y^(n)−ξ^(n)+μf^k(n)+α2Φk−1−1(x^k(n)g^kT(n)x^k(n)+x^k−1(n)g^k−1(n)))

According to the ADMM algorithm, the ϕk in Eq. (31) appears as ϕk−1, thus we only need to calculate the result of ϕk−1. We first obtain the derivation of ϕk from Eq. (26). (32)∂L∂Φk=α(−Φk−1+Φk−1−1)+βX^k−1TX^k−1=0

Then the solution of ϕk−1 can be obtained. (33)Φk−1=Φk−1−1−(Φk−1−1)TX^k−1βαX^k−1TΦk−1−1I+βαX^k−1TΦk−1−1X^k−1

In the actual calculation process, ϕk−1 is directly substituted into the calculation, thus we allow the solution for Φk−1 to produce a full matrix implicitly, and then project it to a diagonal matrix, where the non-zero off-diagonal entries are dropped. Unlike the AKLCF, DJSCF introduces the dual-domain weighted matrices, therefore the solution of gk∗^ involves the solutions of the weighted matrices Wa or Wb. It can be obtained from Eq. (23) as follows: (34){L(g^k+1∗)=12∥X^kg^k−y^∥F2+ξ^T(g^k−N(ID⊗BT)fk)+α2[12(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)+12(M^k−M^k−1)TΦk−1(M^k−M^k−1)]+μ2∥g^k−N(ID⊗BT)fk∥22L(Φk)=β2(X^k−1)TΦkX^k−1+α4[logdetΦk−1detΦk+tr(Φk−1−1Φk)+(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)]+α4[logdetΦkdetΦk−1+tr(Φk−1Φk−1)+(M^k−M^k−1)TΦk−1(M^k−M^k−1)]

Similar to Eq. (28), the subproblem gk+1∗ of DJSCF also can be divided into N smaller problems as follows: (35)g^k+1∗(n)=12∥X^k(n)g^k(n)−y^(n)∥F2+ξ^T(g^k(n)−N(ID⊗BT)f^k(n))+μ2∥g^k(n)−N(ID⊗BT)f^k(n)∥22+α4(M^k−M^k−1)TΦk−1−1(M^k−M^k−1)+α4(M^k−M^k−1)TΦk−1(M^k−M^k−1)

After iterative calculation, gk+1∗ and ϕk can be updated as follows: (36)g^k+1∗(n)=11+α(Wax^kT(n)x^k(n)+μ1+αID)−1[Wbx^k(n)y^(n)−ξ^(n)+μf^k(n)+α4(Φk−1+ΦkΦk−1Φk)(x^kT(n)g^k(n)x^k(n)−x^k−1(n)g^k−1(n))] (37)αΦk−1TΦk−1+α(M^k−M^k−1)TΦk−1(M^k−M^k−1)αI+2βX^k−1TΦk−1X^k−1