CN111551928A - Through-wall radar imaging method based on wall low-rank sparse constraint - Google Patents

Through-wall radar imaging method based on wall low-rank sparse constraint Download PDF

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CN111551928A
CN111551928A CN202010491482.0A CN202010491482A CN111551928A CN 111551928 A CN111551928 A CN 111551928A CN 202010491482 A CN202010491482 A CN 202010491482A CN 111551928 A CN111551928 A CN 111551928A
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雒明世
方阳
段沛沛
冯建利
闫效莺
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Xian Shiyou University
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Abstract

The invention discloses a through-wall radar imaging method based on wall low-rank sparse constraint, which utilizes two important characteristics of signals in through-wall radar imaging: the invention converts the problems of wall clutter suppression and target reconstruction into a regularized least square optimization problem, wherein the target function comprises a data fidelity (least square) term, a nuclear norm regularization term and a sum of l1Norm punishment item, realizes the integrated processing of wall clutter suppression and target imaging, separates out wall clutter in iterative cycle, and simultaneously carries out the pairThe target signal is subjected to reconstruction imaging, and the imaging quality is improved.

Description

Through-wall radar imaging method based on wall low-rank sparse constraint
Technical Field
The invention belongs to the technical field of microwave imaging, and particularly relates to a through-wall radar imaging method based on wall low-rank sparse constraint.
Background
Through the development of recent decades, the through-wall radar has been greatly developed in the aspects of theory, technology, system, products and the like, transmits electromagnetic wave signals to penetrate through non-transparent barriers such as walls and the like, processes received data, realizes detection, positioning, imaging, identification and the like of hidden targets behind the barriers, and has important application prospects in civil and military fields such as disaster rescue, law enforcement, anti-terrorism and the like.
In through-wall radar imaging, due to the fact that barriers such as non-transparent walls exist between a target and a radar antenna, echo signals received by the antenna contain a large number of wall reflection clutter, imaging quality is seriously affected, and even the target cannot be displayed. In order to facilitate the performance analysis of the imaging algorithm, in some conventional imaging algorithms, wall reflection clutter is not considered or it is assumed that the wall reflection clutter is filtered out before imaging processing, but in practical cases, data of an empty scene is hardly acquired, so that the background subtraction cannot be used.
At present, the existing wall clutter suppression method and imaging method are relatively independent processing processes, the processing of echoes comprises two stages of wall clutter suppression and imaging algorithm processing, and the processing mode has the following two limitations: (1) when echo signals are undersampled or defective, the step processing mode is relatively complex, and clutter suppression is not particularly ideal; (2) the mode does not realize the inherent unification of clutter removal and imaging, so the imaging quality needs to be improved.
Disclosure of Invention
Aiming at the defects of the existing through-wall radar imaging algorithm, the invention provides a through-wall radar imaging method based on wall low-rank sparse constraint, which utilizes two important characteristics of signals in through-wall radar imaging: the low rank of the wall reflection clutter and the sparsity of the target reflection signal. The method provided by the invention converts the problems of wall clutter suppression and target reconstruction into a regularized least square optimization problem, wherein the target function comprisesData fidelity (least squares) term, kernel norm regularization term and
Figure BDA0002521245290000021
and the norm punishment item realizes the integrated processing of wall clutter suppression and target imaging, separates out wall clutter in iterative loop, and reconstructs and images a target signal, thereby improving the imaging quality.
In order to achieve the purpose, the invention adopts the following technical scheme:
a through-wall radar imaging method based on wall low-rank sparse constraint comprises the following steps:
step 1, obtaining total original echo data, namely horizontally moving a through-wall radar system in a single-station step frequency working mode in parallel to the front surface of a wall, setting N equal-interval measuring positions, sending step frequency signals containing M frequency points by an antenna at each measuring position, and obtaining M × 1-dimensional echo signals s at each antenna measuring positionnIncluding reflecting clutter off the wall
Figure BDA0002521245290000022
Echo of a target signal
Figure BDA0002521245290000023
And a noise signal vnStacking the signals of the N antenna measurement positions into a matrix with the dimension of M × N according to columns, namely obtaining total original echo data S;
step 2, acquiring sparse observation signals: constructing a dimension L multiplied by MN random measurement matrix phi, wherein L is less than MN, the matrix is constructed by adopting elements meeting Gaussian distribution, and the original echo signals are compressed and sampled to obtain L multiplied by 1-dimensional observation signals, namely y is phi upsilon (S), wherein upsilon (r) represents vectorization processing on the matrix;
step 3, imaging scene discretization and backscattering coefficient vectorization: carrying out two-dimensional discretization on a scene to be imaged, namely the discrete grid numbers corresponding to the distance direction and the azimuth direction are respectively Wr and WaThe total discrete grid number of the imaging scene is K ═ Wr×WaThe backscattering coefficient matrix in the scene is stacked into a one-dimensional vector of K × 1 according to rows or columns, and is represented as
Figure BDA0002521245290000024
wherein gkRepresenting the backscatter coefficient of the kth discrete scattering point;
step 4, constructing a target dictionary: constructing a target dictionary psi according to the radar parameters and the imaging scene, wherein the dimension of the target dictionary psi is MN multiplied by K;
and step 5, imaging solving: obtaining a sparse echo signal y and a sparse observation matrix phi according to the steps 1 and 2, obtaining a vectorization representation g of an imaging scene according to the step 3, obtaining a target dictionary psi according to the step 4, and solving the constructed imaging model, namely solving the y-phi upsilon (S)w) Solving the + phi psi g + phi upsilon (V) to obtain a wall echo signal SwAnd the imaging scene reflection coefficient vector g, SwThe dimension is M × N, and V is a noise matrix with the dimension of M × N;
specifically, a step-by-step iterative solving mode is adopted, a singular value soft threshold algorithm is used for solving a first item, namely a wall echo signal, in an imaging mathematical model to obtain an estimated value of a wall reflection signal, then a parallel coordinate descent method is adopted for solving a second item, namely a scene reflection coefficient item, in the model to obtain an estimated value of a target scattering coefficient, and then the optimal solution of the target scattering coefficient is realized in an iterative mode.
Further, in step 1, the through-wall radar imaging system receives the following echo signals received by the mth frequency point antenna at the nth observation position;
Figure BDA0002521245290000031
wherein ,
Figure BDA0002521245290000032
represents the clutter signals reflected by the wall body,
Figure BDA0002521245290000033
representing the reflected echo signal of the target, vn,mRepresents a noise signal, N-0, 1 …, N-1, M-0, 1 …, M-1;
the wall reflection echo signal is expressed as:
Figure BDA0002521245290000034
wherein ,
Figure BDA0002521245290000035
units representing imaginary numbers, σwRepresenting the reflection coefficient of the wall, fmAt the m-th operating frequency, τ, of the signalwThe echo time delay between the nth observation antenna and the wall is shown, and for a uniform wall, the received wall echo signals are the same in the uniform movement process of the antenna, namely
Figure BDA0002521245290000036
The target reflected echo signal is represented as:
Figure BDA0002521245290000041
where P represents the total number of targets in the imaging region, σpDenotes the scattering coefficient, τ, of the p-th targetn,pRepresenting the echo time delay from the nth observation antenna position to the pth target;
an M × 1-dimensional echo signal s is obtained at each antenna measurement positionnIncluding reflecting clutter off the wall
Figure BDA0002521245290000042
Echo of a target signal
Figure BDA0002521245290000043
And a noise signal vnAnd stacking the signals of the N antenna measurement positions into a matrix with the dimension of M × N according to columns, namely obtaining the total original echo data as follows:
S＝Sw+St+V
where S is the raw echo data matrix with dimension M × N, SwIs a wall reflection clutter data matrix with dimension M × N, StIs a target echo data matrix of dimension M × N, and V is a noise matrix of dimension M × N.
Further, when the transceiving antenna is at the nth measurement position, the echo time delay from the transceiving antenna to the pth target behind the wall is as follows:
Figure BDA0002521245290000044
in the formula ,lair1,wall、lwall、lwall,air2The single-path propagation distances of the electromagnetic wave from the antenna to the front surface of the wall body, the front surface of the wall body to the back surface of the wall body and the back surface of the wall body to the target point are respectively represented, c represents the propagation speed of an electromagnetic wave signal in the air, v represents the propagation speed of the electromagnetic wave signal in the wall body, and is related to the relative dielectric constant of the wall body, and the calculation formula is as follows:
Figure BDA0002521245290000045
in the formula ,εwrepresenting the relative dielectric constant of the wall.
Further, the one-way propagation distance l of the electromagnetic wave from the antenna to the front surface of the wall body, the front surface to the rear surface of the wall body, and the rear surface of the wall body to the target pointair1,wall、lwall、lwall,air2Obtained by the following method:
let X be the coordinate of the measurement position of the nth antenna at a certain timetn＝(xtn0), the coordinate of a certain target point behind the wall is P (x, y), and the vertical distance from the radar antenna to the front surface of the wall is y1The thickness of the wall body is d, and the vertical distance from the rear surface of the wall body to the target is y2Consider two extreme cases: when the relative dielectric constant of the wall is equal to that of air, the electromagnetic wave signal will not be transmitted and refracted when entering the wall from the air, and the propagation path is Xtn→A1→ B, wherein B is the break of electromagnetic wave on the back surface of the wallShoot Point, A1Measuring position X for an antennatnThe intersection point of the connecting line between the B and the B on the front surface of the wall body; when the relative dielectric constant of the wall tends to be infinite, the propagation path of the signal is Xtn→A2→B, wherein A2Representing a point on the front surface of the wall corresponding to the position of the B point; since the radar antenna proceeds in a direction parallel to the wall, Δ BA is known from the triangle similarity theorem1A2～ΔBXtnA3, wherein A3Is represented by A2And the intersection point of the line B and the moving track of the antenna, wherein the coordinate of B is B (x)B,y1+d)，A1Has the coordinate of A1(xA1,y1) Therefore, the following relationship is given:
Figure BDA0002521245290000051
in practical cases, the two extreme cases are not present, but are between the two, so that the refraction point a of the electromagnetic wave is between a1 and A2In the space, the coordinate of A is A (x)A,y1) There are the following equations
Figure BDA0002521245290000052
Similarly, the refraction point of the electromagnetic wave signal on the back surface of the wall body is B and is formed by delta AB1B2～ΔAPB3The following two equations are obtained:
Figure BDA0002521245290000053
Figure BDA0002521245290000054
wherein B1A point B representing the intersection of the line AP between the refraction point A of the electromagnetic wave on the front surface of the wall and the target P and the back surface of the wall2Representing the corresponding point, B, on the rear surface of the wall parallel to point A3Is expressed as A point andB2connection line AB of2The extension of (a) and the point P are perpendicular to the intersection point of two lines on the y axis;
solving the equation set by combining the four formulas to obtain the abscissa x of the point A and the point BA and xB：
Figure BDA0002521245290000061
Wherein, the multiplication operation is represented, and the coordinate A (x) of the two refraction points is obtained through the processA,y1)、B(xB,y1+ d), and then according to the transmitting-receiving antenna coordinate Xtn(xtn0) and coordinates P (x, y) of the target point, the propagation distance of the electromagnetic wave is obtained as:
Figure BDA0002521245290000062
further, the elements of the one-dimensional vector g in step 3 are represented as:
Figure BDA0002521245290000063
further, each element of the dictionary Ψ in step 4 is:
ψn(m,k)＝exp(-j2πfmτn,k)
wherein ,
Figure BDA0002521245290000064
units representing imaginary numbers, fmThe m frequency point of the nth measuring antenna is represented; of the thn,kThe echo time delay from the nth measuring antenna position to the kth grid point is represented, and a target echo signal received at the nth measuring antenna position is represented in a matrix form as follows:
Figure BDA0002521245290000065
further, step 5 specifically comprises:
when the observed signal y is alreadyKnown as clutter signal S to wallwAnd the solution of the target backscattering coefficient vector g is estimated by the following unconstrained optimization problem:
Figure BDA0002521245290000066
where Φ is the sparse observation matrix, expressed as
Figure BDA0002521245290000071
wherein ,
Figure BDA0002521245290000072
l1.. L, MN 1.. MN denotes an element in the L-th row and the MN-th column of the sparse observation matrix Φ, v (·) denotes vectorization processing on the matrix, λ and γ denote regularization parameters of a low-rank portion and a sparse portion, respectively, to balance the low-rank term and the sparse term; i | · | purple wind*Representing the kernel norm, i.e. the sum of the singular values of the matrix; i | · | purple wind1To represent
Figure BDA0002521245290000074
Norm, which is the sum of the absolute values of the elements; i | · | purple wind2To represent
Figure BDA0002521245290000075
Norm, the square of the sum of the squares of all elements;
the complex problem is decomposed into sub-problems by using an iteration technology, namely the sub-problems can be solved effectively, and then the problems are converted into the following problems:
Figure BDA0002521245290000076
wherein ,SiIs an intermediate parameter, and the specific expression is
Figure BDA0002521245290000077
Figure BDA0002521245290000078
Representing the estimated wall echo signal at the ith iteration, α is a tuning parameter to ensure
Figure BDA0002521245290000079
α must satisfy
Figure BDA00025212452900000710
Representing the transformation of the vector into a matrix form;
for the sum of nuclear norms therein
Figure BDA00025212452900000711
Separating and solving the norm to obtain:
Figure BDA00025212452900000712
Figure BDA00025212452900000713
wherein, the problem of minimizing the nuclear norm is solved by adopting a singular value soft threshold;
Figure BDA00025212452900000714
and solving the norm minimization problem by adopting a parallel coordinate descent method.
Further, the concrete solving steps are as follows:
initialization: a random measurement matrix phi; a dictionary matrix Ψ; initial wall clutter signal
Figure BDA0002521245290000081
wherein
Figure BDA0002521245290000082
Represents the pseudo-inverse of the measurement matrix Φ; initial target reflection signal g 00; the iteration variable i is 0;
step 5-1: gradient decomposition:
Figure BDA0002521245290000083
step 5-2: and (3) estimating the wall clutter signals by adopting a singular value soft threshold algorithm:
Figure BDA0002521245290000084
step 5-3: estimating a signal reflection signal by adopting a parallel coordinate descent method: gi＝gi-1+μ(es-gi-1) Where μ is a step parameter of a fixed constant;
step 5-4: judging whether iteration is terminated: error when two adjacent iterations
Figure BDA0002521245290000085
Or stopping iteration when the maximum iteration times is reached; otherwise, returning to the step 5-1 when i is equal to i + 1;
step 5-5: obtaining a reconstructed imaging result
Figure BDA0002521245290000086
Compared with the prior art, the invention has the following beneficial technical effects:
(1) the method of the invention is based on compressed sensing, and overcomes the problems of the traditional through-wall radar imaging that high-resolution imaging needs to be realized, such as: too high sampling rate, too large data volume, hard hardware storage and the like. (2) The method fully utilizes the characteristics of low rank of the wall reflection clutter and sparsity of target signal echoes, and carries out iterative unified solution on the optimization problem of low rank sparse constraint through a singular value soft threshold algorithm and a parallel coordinate descent method, thereby realizing recovery and reconstruction of the hidden target behind the wall in the detection environment of the wall reflection clutter. Through experimental simulation verification, the method can effectively inhibit the wall reflection clutter and obtain a high-quality target image.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a simulation point target model of the present invention.
Fig. 3 is a through-wall radar imaging geometric model of the invention.
Fig. 4 is an imaging region division diagram of the present invention.
FIG. 5 is a schematic diagram of a through-wall radar imaging signal propagation path.
Fig. 6 is a diagram of imaging effect of two-stage compressed sensing.
Fig. 7 is an image effect diagram of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the following drawings and specific examples, but the embodiments of the present invention are not limited thereto.
The invention provides a through-wall radar imaging algorithm based on wall low-rank sparse constraint based on Compressed Sensing (CS), which can effectively inhibit wall reflection clutter, realize recovery and reconstruction of hidden targets behind walls in a detection environment in which the wall reflection clutter exists, and obtain high-quality target images.
Fig. 1 is a flowchart of a through-wall radar imaging method according to the present invention, and based on this, the step of imaging the echo data generated by using the target model shown in fig. 2 is as follows:
step 1: total echo data is acquired. As shown in fig. 3, a wall-through radar imaging geometric model is provided, N measurement positions are set in a moving direction of a radar antenna, the radar transmitting/receiving co-located antenna moves at a constant speed along a track parallel to a wall surface, and a step signal with M pulse frequencies is sent at each measurement position, so that an echo signal received by a radar system at an M (M) 0,1, M-1) frequency point antenna at an N (N is 0,1, N-1) measurement position is:
Figure BDA0002521245290000091
wherein ,
Figure BDA0002521245290000092
represents the clutter signals reflected by the wall body,
Figure BDA0002521245290000093
representing reflected echo signals of a targetNumber vn,mRepresenting a noise signal.
The wall reflection echo signal is expressed as:
Figure BDA0002521245290000094
wherein ,σwRepresenting the reflection coefficient of the wall, fmAt the m-th operating frequency, τ, of the signalwShowing the echo time delay between the nth measuring antenna and the wall, and tau is constant in the whole measuring process because the vertical distance between the measuring antenna and the wall is constantwIs a constant independent of the measuring position of the antenna, and in addition, for a uniform wall, the received wall echo signals are the same during the uniform movement of the antenna, namely
Figure BDA0002521245290000101
The target reflected echo signal is represented as:
Figure BDA0002521245290000102
where P represents the total number of targets in the imaging region, σpDenotes the scattering coefficient, τ, of the p-th targetn,pRepresenting the echo time delay between the nth measuring antenna position and the pth target.
An M × 1-dimensional echo signal s is obtained at each antenna measurement positionnIncluding reflecting clutter off the wall
Figure BDA0002521245290000103
Echo of a target signal
Figure BDA0002521245290000104
And a noise signal vnAnd stacking the signals of the N antenna measurement positions into a matrix with the dimension of M × N according to columns, namely obtaining the total original echo data as follows:
S＝Sw+St+V (4)
in through-wall imaging, an electromagnetic wave signal is refracted at a boundary between air and a wall, and as shown in a schematic diagram of a signal propagation path of a through-wall radar imaging shown in fig. 4, when a transmitting-receiving antenna is at an nth measurement position, an echo time delay of a pth target after the transmitting-receiving antenna reaches a wall is as follows:
Figure BDA0002521245290000105
in the formula ,lair1,wall、lwall、lwall,air2Respectively represents the single-way propagation distance of the electromagnetic wave from the antenna to the front surface of the wall body, the front surface of the wall body to the back surface of the wall body and the back surface of the wall body to the target point, c represents the propagation speed of the electromagnetic wave signal in the air, namely c is 3 × 108m/s, v represents the propagation velocity of the electromagnetic wave signal in the wall, and is related to the relative dielectric constant of the wall, and the calculation formula is as follows:
Figure BDA0002521245290000111
in the formula ,εwrepresenting the relative dielectric constant of the wall.
In fig. 5, let the coordinate at the nth antenna measurement position at a certain time be Xtn＝(xtn0), the coordinate of a certain target point behind the wall is P (x, y), and the vertical distance from the radar antenna to the front surface of the wall is y1The thickness of the wall body is d, and the vertical distance from the rear surface of the wall body to the target is y2. Two extreme cases are considered: when the relative dielectric constant of the wall is equal to that of air, the electromagnetic wave signal will not be transmitted and refracted when entering the wall from the air, and the propagation path is Xtn→A1→ B; when the relative dielectric constant of the wall tends to be infinite, the propagation path of the signal is Xtn→A2→ B. Since the radar antenna proceeds in a direction parallel to the wall, Δ BA is known from the triangle similarity theorem1A2～ΔBXtnA3The coordinate of B in the figure is B (x)B,y1+d)，A1Has the coordinate of A1(xA1,y1) Therefore, the following are givenThe relation is as follows:
Figure BDA0002521245290000112
in practical cases, the two extreme cases are not present, but are between the two, so that the refraction point a of the electromagnetic wave is between a1 and A2In the space, the coordinate of A is A (x)A,y1) There are the following equations
Figure BDA0002521245290000113
Similarly, the refraction point of the electromagnetic wave signal on the back surface of the wall body is B, and is formed by delta AB1B2～ΔAPB3The following two equations are available:
Figure BDA0002521245290000114
Figure BDA0002521245290000115
solving the equation system by combining the equations (7), (8), (9) and (10) can obtain the abscissa x of the point A and the point BA and xB：
Figure BDA0002521245290000121
In the formula, a denotes a multiplication operation. The coordinates A (x) of the two refraction points are obtained through the processA,y1)、B(xB,y1+ d), and then according to the transmitting-receiving antenna coordinate Xtn(xtn0) and the coordinates P (x, y) of the target point, the propagation distance of the electromagnetic wave can be obtained as follows:
Figure BDA0002521245290000122
by substituting equation (12) into equation (5), the two-way time delay from the radar antenna to the target behind the wall at each different measurement position can be obtained.
In this example, 41 antenna measurement positions are set, i.e., M is 41, the imaging range of the antenna observation range is 2M × 2M, the distance direction is from 0M to 2M, the azimuth direction is from-1M to 1M, the transmission signal bandwidth is 2-6GHz, and the number of signal frequency points is 201, i.e., N is 201.
Step 2: and acquiring an observation signal. The method comprises the steps of constructing a dimensionality L multiplied by MN random measurement matrix phi, wherein L is less than MN, constructing the matrix by adopting elements meeting Gaussian distribution, carrying out compression sampling on original echo signals to obtain L multiplied by 1-dimensional observation signals, namely y is phi upsilon (S), wherein upsilon (·) represents vectorization processing on the matrix, and phi is a sparse observation matrix.
In this example, 50% of the signal data is randomly acquired as observation data for each measurement location, and L is MN/2.
And 3, discretizing the imaging scene. As shown in fig. 4, a scene meshing diagram is shown. Dividing a scene to be imaged into K ═ W along a distance direction and an azimuth directionr×WaA grid of uniform size, Wr and WaRepresenting the discrete grid number of the distance direction and the azimuth direction respectively, the reflection coefficient matrix in the scene is strung into a K × 1 one-dimensional vector g according to the rows or the columns, and the elements of the vector g are represented as K × one-dimensional vectors g
Figure BDA0002521245290000131
In this example, the imaged scene is equally spaced into 101 x 101 grids, i.e., Wr＝Wa＝101。
And 4, step 4: and constructing a target dictionary. Constructing a target dictionary Ψ according to the test parameters and the target scene, wherein the dimension of the target dictionary Ψ is MN × K, and each element of the dictionary Ψ is as follows:
ψn(m,k)＝exp(-j2πfmτn,k) (14)
wherein ,fmThe m frequency point of the nth measuring antenna is represented; of the thn,kIndicating the echo time delay between the nth measuring antenna position to the kth grid point. The target echo signal received at the nth measuring antenna position is expressed into a matrixThe form is as follows:
Figure BDA0002521245290000132
and 5: and (3) imaging solution: obtaining a sparse echo signal y and a sparse observation matrix phi according to the steps 1 and 2, obtaining a vectorization representation g of an imaging scene according to the step 3, obtaining a target dictionary psi according to the step 4, and solving the constructed imaging model, namely solving the y-phi upsilon (S)w) Solving the + phi psi g + phi upsilon (V) to obtain a wall echo signal SwAnd the imaging scene reflection coefficient vector g, SwSolving a nuclear norm term in an imaging mathematical model by using a Singular Value Soft-threshold (SVST) algorithm through a distributed solving mode to obtain an estimated Value of a wall reflection signal, and then adopting a Parallel Coordinate Descent (PCD) pair
Figure BDA0002521245290000133
And solving the norm term to obtain an estimated value of the scattering coefficient of the target, and then realizing the optimal solution of the scattering coefficient of the target in an iteration mode.
When the observation signal y is known, the kernel norm sum is respectively used according to the low rank of the wall clutter signal and the sparsity of the target reflection signal
Figure BDA0002521245290000134
The norm is restrained to obtain the clutter signal S of the wallwThe solution to the target sparse vector g can be estimated by the following unconstrained optimization problem:
Figure BDA0002521245290000141
where Φ is the sparse observation matrix, expressed as
Figure BDA0002521245290000142
wherein ,
Figure BDA0002521245290000143
l1.. L, MN 1.. MN denotes an element in the L-th row and the MN-th column of the sparse observation matrix Φ, v (·) denotes vectorization processing on the matrix, λ and γ denote regularization parameters of a low-rank portion and a sparse portion, respectively, to balance the low-rank term and the sparse term; i | · | purple wind*Representing the sum of the nuclear norm and the singular value; i | · | purple wind1To represent
Figure BDA0002521245290000145
Norm, which is the sum of the absolute values of the elements; i | · | purple wind2To represent
Figure BDA0002521245290000146
Norm, the square of the sum of the squares of all elements; decomposing a complex problem into sub-problems can be solved efficiently, so the problem can be transformed into:
Figure BDA0002521245290000147
wherein ,
Figure BDA0002521245290000148
to ensure
Figure BDA0002521245290000149
α must satisfy
Figure BDA00025212452900001410
For the sum of nuclear norms in formula (17)
Figure BDA00025212452900001411
The norm is separated and solved to obtain
Figure BDA00025212452900001412
Figure BDA00025212452900001413
The problem of minimizing the kernel norm in equation (18) can be effectively solved by using a singular value soft threshold.
First, define the shrink operator as:
Figure BDA00025212452900001414
where τ is a constant, assuming that the singular value of S is decomposed as:
S＝UΛVH(21)
wherein ,
Figure BDA0002521245290000151
σiand the matrix is a non-zero singular value of S, k is the number of the non-zero singular values, and U and V are a left singular vector matrix and a right singular vector matrix respectively. Singular value shrinkage operator Dτ(S) can be calculated by equation (22):
Dτ(S)＝UΤτ(Λ)VH(22)
in the formula ,Tτ(Λ)＝diag(Tτ(σi)). Similarly, the solution of equation (18) is as follows:
Figure BDA0002521245290000152
in formula (19)
Figure BDA0002521245290000153
And solving the norm minimization problem by adopting a parallel coordinate descent method (PCD).
The concrete solving steps are as follows:
initialization: a random measurement matrix phi; a dictionary matrix Ψ; initial wall clutter signal
Figure BDA0002521245290000154
wherein
Figure BDA0002521245290000155
Representing the pseudo-inverse of the measurement matrix phi, the initial target reflection signal g 00; the iteration variable i is 0;
step 5-1: gradient decomposition:
Figure BDA0002521245290000156
step 5-2: and (3) estimating the wall clutter signals by adopting a singular value soft threshold algorithm:
Figure BDA0002521245290000157
step 5-3: estimating the signal reflection signal by using a PCD algorithm: gi＝gi-1+μ(es-gi-1) Where μ is a step parameter of a fixed constant;
step 5-4: judging whether iteration is terminated: error when two adjacent iterations
Figure BDA0002521245290000158
Or stopping iteration when the maximum iteration times are reached. Otherwise, returning to the step 5-1 when i is equal to i + 1;
step 5-5: obtaining a reconstructed imaging result
Figure BDA0002521245290000159
FIG. 6 is a two-stage compressive sensing imaging result, in which a subspace projection technique is first used to suppress wall clutter, and then a CS algorithm is used to perform imaging; fig. 7 is an imaging algorithm of the present invention, and it can be clearly seen that the imaging effect of the present invention is better. Comparing fig. 6 with fig. 7, it can be found that when a wall reflection clutter exists, the imaging quality of the invention is obviously better than that of two-stage CS algorithm imaging, the wall clutter is reduced a lot, and a target image can be easily distinguished to effectively suppress the wall clutter, so that a reconstructed target is clearer.

Claims (8)

1. A through-wall radar imaging method based on wall low-rank sparse constraint is characterized by comprising the following steps:
step 1, obtaining total original echo data, namely horizontally moving a through-wall radar system in a single-station step frequency working mode in parallel to the front surface of a wall, setting N equal-interval measuring positions, sending step frequency signals containing M frequency points by an antenna at each measuring position, and obtaining M × 1-dimensional echo signals s at each antenna measuring positionnIncluding reflecting clutter off the wall
Figure FDA0002521245280000011
Echo of a target signal
Figure FDA0002521245280000012
And a noise signal vnStacking the signals of the N antenna measurement positions into a matrix with the dimension of M × N according to columns, namely obtaining total original echo data S;
step 2, acquiring sparse observation signals: constructing a dimension L multiplied by MN random measurement matrix phi, wherein L is less than MN, the matrix is constructed by adopting elements meeting Gaussian distribution, and the original echo signals are compressed and sampled to obtain L multiplied by 1-dimensional observation signals, namely y is phi upsilon (S), wherein upsilon (r) represents vectorization processing on the matrix;
step 3, imaging scene discretization and backscattering coefficient vectorization: carrying out two-dimensional discretization on a scene to be imaged, namely the discrete grid numbers corresponding to the distance direction and the azimuth direction are respectively Wr and WaThe total discrete grid number of the imaging scene is K ═ Wr×WaThe backscattering coefficient matrix in the scene is stacked into a one-dimensional vector of K × 1 according to rows or columns, and is represented as
Figure FDA0002521245280000013
wherein gkRepresenting the backscatter coefficient of the kth discrete scattering point;
step 4, constructing a target dictionary: constructing a target dictionary psi according to the radar parameters and the imaging scene, wherein the dimension of the target dictionary psi is MN multiplied by K;
and step 5, imaging solving: obtaining a sparse echo signal y and a sparse observation matrix phi according to the step 1 and the step 2, and obtaining an imaging field according to the step 3And g is represented by vectorization of the scene, a target dictionary psi is obtained according to the step 4, and the constructed imaging model is solved, namely y is equal to phi upsilon (S)w) Solving the + phi psi g + phi upsilon (V) to obtain a wall echo signal SwAnd the imaging scene reflection coefficient vector g, SwThe dimension is M × N, and V is a noise matrix with the dimension of M × N;
specifically, a step-by-step iterative solving mode is adopted, a singular value soft threshold algorithm is used for solving a first item, namely a wall echo signal, in an imaging mathematical model to obtain an estimated value of a wall reflection signal, then a parallel coordinate descent method is adopted for solving a second item, namely a scene reflection coefficient item, in the model to obtain an estimated value of a target scattering coefficient, and then the optimal solution of the target scattering coefficient is realized in an iterative mode.
2. The wall-through radar imaging method based on the wall low-rank sparse constraint is characterized in that in the step 1, the wall-through radar imaging system receives the echo signals received by the mth frequency point antenna at the nth observation position as follows;
Figure FDA0002521245280000021
wherein ,
Figure FDA0002521245280000022
represents the clutter signals reflected by the wall body,
Figure FDA0002521245280000023
representing the reflected echo signal of the target, vn,mRepresents a noise signal, N-0, 1 …, N-1, M-0, 1 …, M-1;
the wall reflection echo signal is expressed as:
Figure FDA0002521245280000024
wherein ,
Figure FDA0002521245280000025
units representing imaginary numbers, σwRepresenting the reflection coefficient of the wall, fmAt the m-th operating frequency, τ, of the signalwThe echo time delay between the nth observation antenna and the wall is shown, and for a uniform wall, the received wall echo signals are the same in the uniform movement process of the antenna, namely
Figure FDA0002521245280000026
The target reflected echo signal is represented as:
Figure FDA0002521245280000027
where P represents the total number of targets in the imaging region, σpDenotes the scattering coefficient, τ, of the p-th targetn,pRepresenting the echo time delay from the nth observation antenna position to the pth target;
an M × 1-dimensional echo signal s is obtained at each antenna measurement positionnIncluding reflecting clutter off the wall
Figure FDA0002521245280000028
Echo of a target signal
Figure FDA0002521245280000029
And a noise signal vnAnd stacking the signals of the N antenna measurement positions into a matrix with the dimension of M × N according to columns, namely obtaining the total original echo data as follows:
S＝Sw+St+V
where S is the raw echo data matrix with dimension M × N, SwIs a wall reflection clutter data matrix with dimension M × N, StIs a target echo data matrix of dimension M × N, and V is a noise matrix of dimension M × N.
3. The through-wall radar imaging method based on the wall low-rank sparse constraint is characterized in that when the transmitting-receiving antenna is at the nth measurement position, the echo time delay of the transmitting-receiving antenna to the pth target behind the wall is as follows:
Figure FDA0002521245280000031
in the formula ,lair1,wall、lwall、lwall,air2The single-path propagation distances of the electromagnetic wave from the antenna to the front surface of the wall body, the front surface of the wall body to the back surface of the wall body and the back surface of the wall body to the target point are respectively represented, c represents the propagation speed of an electromagnetic wave signal in the air, v represents the propagation speed of the electromagnetic wave signal in the wall body, and is related to the relative dielectric constant of the wall body, and the calculation formula is as follows:
Figure FDA0002521245280000032
in the formula ,εwrepresenting the relative dielectric constant of the wall.
4. The through-wall radar imaging method based on wall low-rank sparse constraint according to claim 3, wherein the single-pass propagation distances l from the antenna to the front surface of the wall, from the front surface of the wall to the rear surface and from the rear surface of the wall to the target pointair1,wall、lwall、lwall,air2Obtained by the following method:
let X be the coordinate of the measurement position of the nth antenna at a certain timetn＝(xtn0), the coordinate of a certain target point behind the wall is P (x, y), and the vertical distance from the radar antenna to the front surface of the wall is y1The thickness of the wall body is d, and the vertical distance from the rear surface of the wall body to the target is y2Consider two extreme cases: when the relative dielectric constant of the wall is equal to that of air, the electromagnetic wave signal will not be transmitted and refracted when entering the wall from the air, and the propagation path is Xtn→A1→ B, where B is the refraction point of the electromagnetic wave on the back surface of the wall, A1Measuring position X for an antennatnThe intersection point of the connecting line between the B and the B on the front surface of the wall body; when the relative dielectric constant of the wall tends to be infinite, the propagation path of the signal is Xtn→A2→B, wherein A2Representing a point on the front surface of the wall corresponding to the position of the B point; since the radar antenna proceeds in a direction parallel to the wall, Δ BA is known from the triangle similarity theorem1A2～ΔBXtnA3, wherein A3Is represented by A2And the intersection point of the line B and the moving track of the antenna, wherein the coordinate of B is B (x)B,y1+d)，A1Has the coordinate of A1(xA1,y1) Therefore, the following relationship is given:
Figure FDA0002521245280000041
in practical cases, the two extreme cases are not present, but are between the two, so that the refraction point a of the electromagnetic wave is between a1 and A2In the space, the coordinate of A is A (x)A,y1) There are the following equations
Figure FDA0002521245280000042
Similarly, the refraction point of the electromagnetic wave signal on the back surface of the wall body is B and is formed by delta AB1B2～ΔAPB3The following two equations are obtained:
Figure FDA0002521245280000043
Figure FDA0002521245280000044
wherein B1A point B representing the intersection of the line AP between the refraction point A of the electromagnetic wave on the front surface of the wall and the target P and the back surface of the wall2Representing the corresponding point, B, on the rear surface of the wall parallel to point A3Is represented by A point and B point2Connection line AB of2The extension of (a) and the point P are perpendicular to the intersection point of two lines on the y axis;
solving the equation set by combining the four formulas to obtain the abscissa x of the point A and the point BA and xB：
Figure FDA0002521245280000045
Wherein, the multiplication operation is represented, and the coordinate A (x) of the two refraction points is obtained through the processA,y1)、B(xB,y1+ d), and then according to the transmitting-receiving antenna coordinate Xtn(xtn0) and coordinates P (x, y) of the target point, the propagation distance of the electromagnetic wave is obtained as:
Figure FDA0002521245280000051
5. the through-the-wall radar imaging method based on the wall low-rank sparse constraint is characterized in that in the step 3, the element of the one-dimensional vector g is represented as:
Figure FDA0002521245280000052
6. the through-wall radar imaging method based on the wall low-rank sparse constraint is characterized in that each element of the dictionary Ψ in the step 4 is as follows:
ψn(m,k)＝exp(-j2πfmτn,k)
wherein ,
Figure FDA0002521245280000053
units representing imaginary numbers, fmThe m frequency point of the nth measuring antenna is represented; of the thn,kRepresenting the echo time delay from the nth measuring antenna position to the kth grid point, and the target echo signal received at the nth measuring antenna positionThe numbers are shown in matrix form as:
Figure FDA0002521245280000054
7. the wall-through radar imaging method based on the wall low-rank sparse constraint according to claim 2, wherein the step 5 specifically comprises:
when the observation signal y is known, the clutter signal S to the wall bodywAnd the solution of the target backscattering coefficient vector g is estimated by the following unconstrained optimization problem:
Figure FDA0002521245280000055
where Φ is the sparse observation matrix, expressed as
Figure FDA0002521245280000061
wherein ,
Figure FDA0002521245280000062
expressing the elements of the ith row and the mn column of the sparse observation matrix phi, expressing that upsilon (·) is used for vectorizing the matrix, and expressing the regularization parameters of a low-rank part and a sparse part respectively by lambda and gamma for balancing the low-rank term and the sparse term; i | · | purple wind*Representing the kernel norm, i.e. the sum of the singular values of the matrix; i | · | purple wind1Is represented by1Norm, which is the sum of the absolute values of the elements; i | · | purple wind2Is represented by2Norm, the square of the sum of the squares of all elements;
the complex problem is decomposed into sub-problems by using an iteration technology, namely the sub-problems can be solved effectively, and then the problems are converted into the following problems:
Figure FDA0002521245280000063
wherein ,SiIs one inAn inter-parameter, a specific expression is
Figure FDA0002521245280000064
Figure FDA0002521245280000065
Representing the estimated wall echo signal at the ith iteration, α is a tuning parameter to ensure
Figure FDA0002521245280000066
α must satisfy
Figure FDA0002521245280000067
υ*(. -) represents transforming the vector into a matrix form;
for the nuclear norm sum l therein1Separating and solving the norm to obtain:
Figure FDA0002521245280000068
Figure FDA0002521245280000069
wherein, the problem of minimizing the nuclear norm is solved by adopting a singular value soft threshold; l1And solving the norm minimization problem by adopting a parallel coordinate descent method.
8. The wall-through radar imaging method based on the wall low-rank sparse constraint according to claim 7, wherein the concrete solving steps are as follows:
initialization: a random measurement matrix phi; a dictionary matrix Ψ; initial wall clutter signal
Figure FDA0002521245280000071
wherein
Figure FDA0002521245280000072
Represents the pseudo-inverse of the measurement matrix Φ; initial target reflection signal g00; the iteration variable i is 0;
step 5-1: gradient decomposition:
Figure FDA0002521245280000073
step 5-2: and (3) estimating the wall clutter signals by adopting a singular value soft threshold algorithm:
Figure FDA0002521245280000074
step 5-3: estimating a signal reflection signal by adopting a parallel coordinate descent method: gi＝gi-1+μ(es-gi-1) Where μ is a step parameter of a fixed constant;
step 5-4: judging whether iteration is terminated: error when two adjacent iterations
Figure FDA0002521245280000075
Or stopping iteration when the maximum iteration times is reached; otherwise, returning to the step 5-1 when i is equal to i + 1;
step 5-5: obtaining a reconstructed imaging result
Figure FDA0002521245280000076
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