CN111624571B - Non-uniform Weibull background statistical distribution parameter estimation method based on self-adaptive tight frame - Google Patents
Non-uniform Weibull background statistical distribution parameter estimation method based on self-adaptive tight frame Download PDFInfo
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Abstract
A non-uniform Weibull background statistical distribution parameter estimation method based on a self-adaptive tight frame relates to a unit distribution parameter self-adaptive estimation method under a non-uniform background. The invention aims to solve the problem of inaccurate estimation caused by adopting a non-uniform reference unit to estimate the distribution parameters of a background unit in the non-uniform Weibull background in the prior art. The method comprises the following steps: adopting a self-adaptive tight frame to carry out sparse representation on a curved surface formed by unit distribution parameters; constructing a log-likelihood function of a background based on statistical distribution of background units, and combining with sparse representation and full variation constraint of a unit distribution parameter curved surface in the step one to construct a target function of distribution parameter estimation; and optimizing the constructed objective function, and continuously updating the adaptive tight framework in the optimization process to obtain the distribution parameters of each background unit. Belongs to the field of radar signal processing.
Description
Technical Field
The invention relates to a unit distribution parameter self-adaptive estimation method under a non-uniform background, in particular to a non-uniform Weibull background statistical distribution parameter estimation method based on a self-adaptive tight frame.
Background
The echo signal received by the radar may contain reflected signals from a variety of sources. The background consisting of these different signals with different statistical properties will be non-uniform. In the process of performing statistical analysis and target detection on the background, a reference unit in the neighborhood is required to estimate the statistical distribution parameters or detection threshold of the detection unit. However, since the reference cells are non-uniform, especially where the statistical properties of each reference cell are not the same, statistical analysis or calculation of detection thresholds for background cells becomes challenging. In the existing method, the average power or the statistical distribution parameter of the detection unit is directly estimated by using the non-uniform reference unit in the neighborhood, and the non-uniform reference unit itself is not considered or the non-uniform characteristic of the reference unit is not described by using an effective method. Therefore, the non-uniformity of the reference unit causes a large deviation in the average power or statistical distribution parameters of the estimated detection unit, which affects subsequent processing such as target detection. In order to accurately estimate the statistical properties of the background cells, two problems are faced when using reference cells: whether the reference cells are uniform and the number of reference cells are uniform. In the multi-parameter distribution background with different statistical characteristics, the reference units are all non-uniform, and it is difficult to adaptively acquire enough uniform reference units to estimate the background unit distribution parameters. Therefore, in practical applications, a method for adaptively estimating the distribution parameters of the background unit, which is suitable for the environment of the non-uniform background unit, is urgently needed.
Disclosure of Invention
The invention provides a non-uniform Weibull background statistical distribution parameter estimation method based on a self-adaptive tight frame, aiming at solving the problem of inaccurate estimation caused by the fact that non-uniform reference units are adopted to estimate the distribution parameters of background units in the non-uniform Weibull background in the prior art.
The method comprises the following steps of firstly, carrying out sparse representation on a curved surface formed by unit distribution parameters by adopting a self-adaptive tight frame;
secondly, establishing a log-likelihood function of the background based on the statistical distribution of the background units, and combining the log-likelihood function with the sparse representation of the unit distribution parameter curved surface and the full variation TV constraint in the step one to construct a target function of distribution parameter estimation;
and step three, optimizing the constructed objective function, and continuously updating the self-adaptive tight frame in the optimization process to obtain the distribution parameters of each background unit.
Advantageous effects
The invention provides a self-adaptive distribution parameter estimation method by utilizing the characteristics that the distribution parameters of the non-uniform clutter which is continuously distributed in the background are also continuously distributed and have spatial correlation, and the statistical distribution parameters of all units in the non-uniform Weibull clutter background can be accurately estimated. That is, by using the spatial relationship that the distribution parameters of the neighboring non-uniform background cells have, the shape parameters and the scale parameters of the background cells can be accurately estimated using the non-uniform reference cells. In the embodiment of the invention, the statistical distribution parameters of the background units in the non-uniform area with the abscissa of 50 to 150 are accurately estimated by utilizing the spatial relationship of continuous change of the scale parameters of the logarithm of the non-uniform Weibull background units.
Drawings
FIG. 1 is a non-uniform Weibull clutter background with a shape parameter of 1.5;
FIG. 2 is a diagram of shape parameters obtained by performing parameter estimation on the data of FIG. 1 according to the present invention;
FIG. 3 is a graph of scale parameters obtained from the present invention by performing parameter estimation on the data of FIG. 1;
FIG. 4 is a cross-sectional view of an estimated scale parameter;
FIG. 5 is an RMSE of shape parameters;
FIG. 6 is the RMSE for the logarithmic scale parameter.
Detailed Description
The first specific implementation way is as follows: echo signals received by a radar are subjected to signal processing to obtain data for target detection or information extraction, and the statistical characteristics of the data can be described by Weibull distribution under certain conditions; the non-uniform Weibull background statistical distribution parameter estimation method based on the self-adaptive tight frame comprises the following steps of:
the method comprises the following steps of firstly, carrying out sparse representation on a curved surface formed by unit distribution parameters by adopting a self-adaptive tight frame;
secondly, constructing a log-likelihood function of a background based on statistical distribution of background units (echo signals received by a radar antenna can obtain a background for target detection or information extraction after signal processing, and data units in the background are background units), and combining sparse representation and Total Variation (TV) constraint of unit distribution parameter curved surfaces in the first step to construct a target function for distribution parameter estimation;
and step three, optimizing the constructed objective function, and continuously updating the self-adaptive tight frame in the optimization process to obtain the distribution parameters of each background unit.
The second embodiment is as follows: the first embodiment is different from the first embodiment in that the first step adopts a self-adaptive tight frame to perform sparse representation on the curved surface formed by the unit distribution parameters; the specific process is as follows:
for the non-uniform Weibull background, the curved surface formed by the distribution parameters of the background units is sparsely expressed by adopting an adaptive tight frame as follows:
wherein, theta i Representing a curved surface formed by all unit distribution parameters in the background; i represents a distribution parameter sequence number contained in the statistical distribution of the background unit, and i belongs to {1,2}; t is transposition; theta 1 In the form of a vector of position parameters,
wherein N represents the number of background elements,
is a real number; theta.theta. 2 In order to be a scale parameter vector, the scale parameter vector,
m i is the mean value; w i Is a tight frame; v. of i Is a transform domain coefficient; in the model (1), there are
And the scale parameter is bounded, i.e.
Wherein, theta 2,i Represents the 2 nd distribution parameter in the statistical distribution of the ith background unit,
is a Hilbert space, and is characterized in that,
is the specified range.
Other steps and parameters are the same as those in the first embodiment.
The third concrete implementation mode: the difference between the present embodiment and the first or second embodiment is that, in the second step, a log-likelihood function of the background is constructed based on the statistical distribution of the background units, and is combined with the sparse representation of the unit distribution parameter curved surface and Total Variation (TV) constraint in the first step to construct a target function of distribution parameter estimation; the specific process is as follows:
for random variables that obey Weibull distribution, the logarithm of the random variable's amplitude will obey Gumbel distribution as follows:
f(x|θ)＝θ 2 exp(θ 2 (x-θ 1 )-exp(θ 2 (x-θ 1 ))) (2)
wherein f (x | θ) represents a Gumbel distribution function, θ is a Gumbel distribution parameter, and θ = [ θ ] 1 ,θ 2 ](ii) a x is the logarithmic magnitude of the detection unit; the log-likelihood function available for x is:
ln f(x|θ)＝lnθ 2 +θ 2 (x-θ 1 )-exp(θ 2 (x-θ 1 )) (3)
where ln f (x | θ) is the log-likelihood function of x;
combining the sparse representation of the unit distribution parameters with the log-likelihood function, and adding TV constraint to the estimation parameters, and constructing a target function of unit distribution parameter estimation as follows:
wherein, F (y; W) 1 ,W 2 ,Θ,m 1 ,m 2 ) An objective function representing an estimate of a distribution parameter; y is n Is the nth unit of Weibull background after logarithmic transformation, and y is the unit of all background units y n A constructed vector; alpha (alpha) ("alpha") i 、β i Is a regularization parameter; theta 1,n Is a vector theta 1 The nth element of (1), θ 2,n Is a vector theta 2 The nth element of (1); i | · | live through p Is p-norm (0-norm when p =0, 1-norm when p = 1); w i Is a tight frame, i =1 or 2; m is i And expressing the mean value of the distribution parameter vector, and constructing a parameter estimation target function by combining the self-adaptive tight frame sparse representation of the distribution parameter with a unit log-likelihood function in a regularization mode.
Other steps and parameters are the same as those in the first or second embodiment.
The fourth concrete implementation mode: the difference between the present embodiment and the first to third embodiments is that, in the third step, the constructed objective function is optimized, and the adaptive tight frame is continuously updated in the optimization process to obtain the distribution parameters of each background unit; the specific process is as follows:
first, by introducing an auxiliary variable a 1 And dual variable b 1 Treating the formula (4)
Then split Bregman iteration is used to pair the variable θ 1 Is solved by
Wherein, γ 1 For the introduction of positive parameters, b 1 H is non-subsampled Haar wavelet transform for introduced dual variables; [ n ]]
Is composed of
The nth element of the value vector; k is the number of iterations;
in the minimization sub-problem (6), regarding a 1 The solution of (2) can be directly carried out by using a soft/hard threshold method, i.e.
Wherein the content of the first and second substances,
and
for element-by-element operation, i.e.
Whereas for the minimization sub-problem (7),
the value of (c) can be obtained by the nonlinear equation (13):
wherein, the first and the second end of the pipe are connected with each other,
c 1,n is a vector c 1 The nth element of (1); for the non-linear equation (13), newton iteration can be used to solve;
similarly, a similar approach can be used for θ 2,n And variable a 2 、b 2 Solving is carried out; the corresponding split Bregman iteration is
Wherein the content of the first and second substances,
solving for a according to equation (14) as a function of probability density 2 It can be done directly using soft/hard threshold method, i.e.
Further, as for the minimization problem (15), with respect to θ 2,n The equation of (a) is:
wherein, the first and the second end of the pipe are connected with each other,
c 2,n is a vector c 2 The nth element; for the non-linear equation (20), a Newton iteration may be employed to solve; wherein the variables need to be satisfied
In the process of optimizing the objective function, the adaptive tight framework is required to be updated; in the invention, the sparse representation of the tight frame adopted for the position parameter and the scale parameter has a similar form, so the same method can be adopted when the tight frame is updated, and subscripts of different variables are ignored when updating calculation is carried out;
reference document [1 ]]([1]Cai J F,Ji H,Shen Z W,et al.Data-driven tight frame construction and image denoising[J]Applied and comparative Harmonic Analysis,2014,37 (1): 89-105), solving W (W is not distinguished here 1 And W 2 ) The iteration of (2) is:
D＝ XY T ,XΣY T ＝ QV T (21)
wherein, the first and the second end of the pipe are connected with each other,
where θ' is the through projection matrix R j The resulting r × r matrix is formed, and vectors are formed as a matrix by concatenation
A column vector of (a); wherein L is the total number of the sub-matrixes; filter corresponding to tight frame W
Form a matrix
A column vector of (a);
representing tight-frame transform coefficients; in addition, there are
In summary, solve for W (i.e., solve for W)
) The algorithm flow of (2) is shown in table 1:
TABLE 1 estimation of the W Algorithm Main Process
For the (partial) subscripts that have been ignored for the different parameters and variables in table 1, when updating the different tight frames, the parameters and variables corresponding to the position parameters or the scale parameters, respectively, are used;
wherein k is max =5; for more details of tight frame updates, reference may be made to corresponding document [1 ]]；
In the course of the parameter estimation process,
can be calculated by the following formula: when p =1, there are:
wherein the content of the first and second substances,
to adopt a median estimator pair
Estimating the noise standard deviation obtained by the thinnest layer Haar wavelet coefficient; when p =0, there is
Wherein the content of the first and second substances,
(θ i ) diff ＝θ i -F g (θ i )(F g is a document [2](guiding filters in Kaiming He, jian Sun, xiaoou Tang, guided Image Filter. IEEE Transactions on Pattern Analysis and Machine Analysis, volume 35, issue 6, pp.1397-1409, june), which can directly call the self-guiding filter of default parameters in Matlab when implemented); in addition, other parameters in the algorithm of the present invention can be set as: the size of the tight frame is 8 multiplied by 8, gamma 1 ＝40,γ 2 ＝20，α 1 ＝2.0，α 2 =1.5, maximum number of iterations T =50.
Other steps and parameters are the same as those in one of the first to third embodiments.
Examples
The effectiveness of the method for estimating the distribution parameters of the non-uniform Weibull clutter background units is shown by simulation in an experiment. The simulation scene is as follows: the average power continuously changes the inhomogeneous Weibull background of the clutter edge; weibull simulation data with a shape parameter of 1.5 are shown in FIG. 1, wherein the size parameter of the area with the lowest amplitude is 1, and the parameter of the area with the highest amplitude is 100;
wherein, the scale parameter of the lowest amplitude region is 1, and the highest amplitude region is 100;
the results obtained by the invention by performing parameter estimation on the data as shown in fig. 1 are shown in fig. 2 and 3;
for the simulation scenario, the Root Mean Square Error (RMSE) of the present invention is shown in fig. 5 and 6;
according to the simulation result, the unit statistical distribution parameters in the non-uniform Weibull clutter background can be effectively estimated, and the purpose of the invention is achieved.
Claims (1)
1. The non-uniform Weibull background statistical distribution parameter estimation method based on the self-adaptive tight frame is characterized by comprising the following steps of:
the method comprises the following steps of firstly, carrying out sparse representation on a curved surface formed by unit distribution parameters by adopting a self-adaptive tight frame, and specifically comprising the following steps:
for a non-uniform Weibull background, the surface formed by the distribution parameters of background units is sparsely represented by adopting a self-adaptive tight frame as follows:
wherein, theta i Representing a curved surface formed by all unit distribution parameters in the background; i represents the sequence number of the distribution parameter contained in the statistical distribution of the background unit, and i belongs to {1,2}; t is transposition; theta.theta. 1 Is a vector of the position parameters and is,
wherein N represents the number of background cells,
is a real number; theta 2 In order to be a scale parameter vector, the scale parameter vector,
m i is an average value; w is a group of i The frame is tight; v. of i Is a transform domain coefficient; in the model (1), there are
And the scale parameter is bounded, i.e.
Wherein, theta 2,i Represents the 2 nd distribution parameter in the statistical distribution of the ith background unit,
is a Hilbert space, and the method is characterized in that,
is a specified range;
step two, a log-likelihood function of the background is constructed based on the statistical distribution of background units, and is combined with the sparse representation of the unit distribution parameter curved surface and the total variation TV constraint in the step one to construct a target function of distribution parameter estimation, and the specific process is as follows:
for random variables that obey Weibull distribution, the logarithm of the random variable's amplitude will obey Gumbel distribution as follows:
f(x|θ)＝θ 2 exp(θ 2 (x-θ 1 )-exp(θ 2 (x-θ 1 ))) (2)
wherein f (x | θ) represents a Gumbel distribution function, θ is a Gumbel distribution parameter, and θ = [ θ ] 1 ,θ 2 ](ii) a x is the logarithmic magnitude of the detection unit; the log-likelihood function for x is:
ln f(x|θ)＝lnθ 2 +θ 2 (x-θ 1 )-exp(θ 2 (x-θ 1 )) (3)
where lnf (x | θ) is the log-likelihood function of x;
combining the sparse representation of the unit distribution parameters with the log-likelihood function, adding TV constraint to the estimation parameters, and constructing a target function of unit distribution parameter estimation as follows:
wherein, F (y; W) 1 ,W 2 ,Θ,m 1 ,m 2 ) An objective function representing a distribution parameter estimate; y is n Is the nth unit of Weibull background after logarithmic transformation, and y is the unit of all background units y n A constructed vector; alpha is alpha i 、β i Is a regularization parameter; theta 1,n Is a vector theta 1 The nth element of (1), θ 2,n Is a vector theta 2 The nth element of (1); i | · | purple wind p Is a p-norm; w i Is a tight frame, i =1 or 2; m is i Expressing the mean value of the distribution parameter vector, and combining the self-adaptive close frame sparse representation of the distribution parameter with a unit log-likelihood function to jointly construct a target function of parameter estimation in a regularization mode;
step three, optimizing the constructed objective function, and continuously updating the adaptive tight frame in the optimization process to obtain the distribution parameters of each background unit, wherein the specific process is as follows:
by introducing an auxiliary variable a 1 And dual variable b 1 And (4) processing:
then split Bregman iterations are used to pair the variable theta 1 Is solved by
Wherein, gamma is 1 For the introduced positive parameter, b 1 H is non-downsampling Haar wavelet transform for introduced dual variables; [ n ] of]Is composed of
The nth element of the value vector; k is the number of iterations;
in the minimization of sub-problem (6), regarding a 1 The solution of (2) is directly carried out by adopting a soft/hard threshold method, namely:
wherein the content of the first and second substances,
and
for element-by-element operation, i.e.:
for the minimization of the sub-problem (7),
the value of (c) is obtained by the nonlinear equation (13):
wherein the content of the first and second substances,
c 1,n is a vector c 1 The nth element of (1); for non-linearEquation (13), solving by using Newton iteration;
to theta 2,n And variable a 2 、b 2 Solving is carried out; the corresponding split Bregman iteration is:
wherein the content of the first and second substances,
solving for a as a function of probability density according to equation (14) 2 The method is carried out by adopting a soft/hard threshold method, namely:
further, as for the minimization problem (15), with respect to θ 2,n The equation of (a) is:
wherein the content of the first and second substances,
c 2,n is a vector c 2 The nth element; solving the nonlinear equation (20) by using Newton iteration; wherein the content of the first and second substances,
the iteration of the solution W is:
D＝XY T ,XΣY T ＝QV T (21)
wherein the content of the first and second substances,
where θ' is the through projection matrix R j R × r matrices are obtained, and vectors are formed as matrices by concatenation
The column vector of (a); wherein L is the total number of the sub-matrixes; filter corresponding to tight frame W
Form a matrix
A column vector of (a);
representing tight-frame transform coefficients; comprises the following steps:
In the course of the parameter estimation process,
calculated by the following formula: when p =1, there are:
wherein the content of the first and second substances,
to adopt a median estimator pair
Estimating the noise standard deviation obtained by the thinnest layer Haar wavelet coefficient; when p =0, there is:
wherein the content of the first and second substances,
(θ i ) diff ＝θ i -F g (θ i )，F g is a pilot filter.
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