Fast 3-d imaging of sar using asymmetric taper atom norm model tomography

Abstract

In order to solve the problem of high computational complexity of traditional tomographic SAR atomic norm three-dimensional imaging using semi-definite programming, a tomographic SAR fast meshless three-dimensional imaging method based on asymmetric cone atomic norm model is disclosed.The atomic norm minimization problem is first re-expressed as an asymmetric cone quadratic optimization problem.Secondly, the Lagrange dual model of the problem is constructed, which shows that the number of dual variables to be estimated can be greatly reduced based on the asymmetric cone modeling.Finally, a fast algorithm for solving the dual problem is given, and the tomographic SAR imaging problem is converted into an atomic norm minimization problem, realizing fast meshless three-dimensional imaging.The method aims to provide a fast solution method for tomographic SAR atomic norm meshless three-dimensional imaging, and the effectiveness of the method has been verified by computer simulation and P-band small unmanned aerial vehicle measured data.

Description

Technical Field

[0001] This invention aims to provide a fast solution method for tomographic SAR atomic norm meshless 3D imaging, solving the problems of high complexity and limited large-scale application of traditional atomic norm-based semidefinite programming solutions. It is expected to be applicable to the field of tomographic SAR 3D imaging in spaceborne, airborne, and vehicle-mounted applications, providing a fast solution method for tomographic SAR atomic norm meshless 3D imaging. Background Technology

[0002] Tomographic Synthetic Aperture Radar (TomoSAR) avoids the overlay problem in traditional two-dimensional imaging by uniquely mapping and projecting the target in three-dimensional space. TomoSAR imaging is typically modeled as a spectrum estimation problem, requiring the estimation of the target's height and scattering frequencies and amplitudes. Currently, compressed sensing (CS) methods are widely used to solve TomoSAR problems. However, CS methods discretize the target's elevation distribution during processing, often resulting in a deviation between the estimation results and the grid, leading to decreased estimation accuracy and even artifacts—a phenomenon known as "grid mismatch."

[0003] To overcome the impact of "grid mismatch," a TomoSAR imaging method based on atomic norm minimization (ANM) has recently been proposed. This method can estimate target frequencies in a continuous frequency domain, fundamentally breaking through the limitations of grid points and achieving gridless 3D imaging. However, the optimization problem of ANM is usually transformed into a standard semidefinite programming problem (SDP) to be solved, which has excessively high computational complexity and is not conducive to large-scale tomographic estimation.

[0004] To address the aforementioned problems, this invention proposes a fast, meshless 3D imaging method for tomographic SAR using an asymmetric cone atomic norm model. The asymmetric cone model significantly reduces the number of dual variables, making it possible to quickly solve the dual optimization problem, thereby achieving rapid solution of the TomoSAR atomic norm. The effectiveness of the proposed method has been verified through computer simulations and TomoSAR experiments using a P-band small UAV.

[0005] This invention aims to provide a rapid solution method for tomographic SAR atomic norm meshless 3D imaging, which is expected to be applied to the fields of spaceborne, airborne, and vehicle-mounted tomographic SAR 3D imaging, providing a rapid solution method for tomographic SAR atomic norm meshless 3D imaging. Summary of the Invention

[0006] The purpose of this invention is to address the problems of high complexity and limited large-scale application of traditional atomic norm-based semidefinite programming solutions. To this end, a fast, meshless 3D imaging method for tomographic SAR using an asymmetric cone atomic norm model is proposed. The specific implementation process is as follows: Figure 1 .

[0007] The method of the present invention is achieved through the following steps:

[0008] Step 1: Modeling based on the minimization of asymmetric cone atom norm;

[0009] Step 2: Construct the Lagrange dual optimization model;

[0010] Step 3: Rapid optimization solution to the dual problem;

[0011] Step 4: Meshless 3D imaging using tomographic SAR atomic norm.

[0012] The advantages of this invention are:

[0013] (1) An asymmetric cone atom norm model was proposed, which can quickly realize meshless three-dimensional imaging of tomographic SAR.

[0014] (2) Computer simulation and P-band UAV TomoSAR experimental results show that this method is more efficient while maintaining computational accuracy compared to traditional methods. Attached Figure Description

[0015] Figure 1 Asymmetric cone atom norm model tomographic SAR fast meshless 3D imaging workflow

[0016] Figure 2 Schematic diagram of N-time flyby tomographic SAR observations

[0017] Figure 3 Single-target contrast imaging results of traditional method (left) and proposed method (right), from top to bottom: detection rate, estimation accuracy, and computation time.

[0018] Figure 4 The results of dual-target contrast imaging using the traditional method (left) and the proposed method (right), from top to bottom, are detection rate, estimation accuracy, and computation time.

[0019] Figure 5 Comparison of surface target imaging results between traditional method (top) and proposed method (bottom)

[0020] Figure 6 (a) Optical image and (b) Imaging results of the experimental area of ​​Chongqing General Aviation College, China

[0021] Figure 7 Comparison of angular reflection 3D imaging results of traditional CS, traditional ANM and the proposed method Detailed Implementation

[0022] The implementation of the method of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0023] This invention is a fast, meshless 3D imaging method for asymmetric cone atom norm model tomographic SAR. The processing flow of this algorithm is as follows: Figure 1 As shown, the specific steps include:

[0024] Step 1: Modeling based on the minimization of asymmetric cone atom norm;

[0025] Step 2: Construct the Lagrange dual optimization model;

[0026] Step 3: Rapid optimization solution to the dual problem;

[0027] Step 4: Meshless 3D imaging using tomographic SAR atomic norm.

[0028] Step 1: Model based on asymmetric cone atom norm minimization.

[0029] Assuming observation vector

[0030]

[0031] In the formula, the observation vector consists of K components, and the frequency f of each component is... k ∈[0,1] and complex coefficients Atom a of the guiding vector a(f) m (f k )=exp(j2πmf k In the line spectrum estimation for (1), the key lies in obtaining the model order K and the signal frequency f. k .

[0032] The atomic norm provides a way to describe the concept of sparsity in general. It uses sets of atoms. It is defined by [the relevant authority / method]. Each element in the set is called an atom. An atom is the basic building block of a signal, a dictionary with a sparse representation of the signal. Assuming atoms are single-frequency signals of different frequencies, the set of atoms can be represented as... The corresponding atomic norm can be expressed as:

[0033]

[0034] In the formula, conv Represents a set of atoms The convex hull. It can be seen that the atomic norm provides a generalization of the L1 norm to the continuous parameter space f∈[0,1), the atomic set. It can be considered as an infinite dictionary. Similar to minimizing the L1 norm, the model based on minimizing the atomic norm is as follows:

[0035]

[0036] In the formula, It is a variable in the problem. For the observation results, For regularization parameters, This is a weight matrix, with all parameters known. In the atomic norm minimization, w = 2e0, where e0 is a vector with the first element being 1 and the rest being 0. The function T(u) outputs the complex Hermitian Toeplitz matrix with u as the first column.

[0037]

[0038] The atomic norm minimization problem can be solved using standard SDP solvers, such as SDPT3 or the Matlab CVX toolkit. However, using SDP results in high computational complexity, leading to low efficiency when there are many observations or long original signals, making it unsuitable for large-scale problems. To reduce computational complexity, this invention remodels the atomic norm minimization problem using the asymmetric cone modeling approach. Equation (3) can be restated as:

[0039]

[0040] In the formula, It is a cone, defined as:

[0041]

[0042] Dual form for:

[0043]

[0044] In the formula, λ=(ρ,s T , z T ) T yes The dual variables in, where The inner product of λ and μ is defined as <λ, μ> = ρv + Re(s) H x)+Re(z H u). Dual Cone C * It is a set of finite autocorrelation sequences. It is quite clear from equation (7) that... Not self-dual, that is Therefore, equation (5) is an asymmetric cone optimization problem.

[0045] Step 2: Construct the Lagrange dual optimization model;

[0046] Consider the solution to equation (5), its Lagrangian form is:

[0047]

[0048] Meanwhile, its dual form is expressed as:

[0049]

[0050] As can be seen, by constructing the dual of equation (5) instead of equation (3), the number of dual variables is reduced from O(N). 2 The computational efficiency is reduced to O(N). From a computational perspective, using the asymmetric cone modeling method is beneficial for improving the efficiency of solving for the atomic norm.

[0051] Step 3: Rapid optimization solution to the dual problem;

[0052] Since f is convex, the Karush-Kuhn-Tucker (KKT) conditions are both necessary and sufficient conditions for solving the primal and dual problems. The KKT conditions are:

[0053]

[0054] We use the primal duality method to solve the above problem, and the specific process is as follows:

[0055] Initialization: Variables Termination judgment Thr = 0.001, optimization target lower limit f LB =-∞, step size t1>0, α>0, maximum number of iterations I.

[0056] Solve iteratively: i = 1, 2, ...

[0057] 1. Use Newton's method to obtain the search direction Δu;

[0058] 2. Update the estimate u i =u i-1 +αΔu;

[0059] 3. Calculate the dual variable <λ i μ i >=ρv+Re(s H x)+Re(z H u), where

[0060] ρ (t) =τ, z (t) =τw,s (t) =2(x (t) -y)

[0061] v (t) =(τt) -1 +(x (t) ) H T -1 (u(t) )x (t)

[0062] x (t) =T(u (t) )T -1 (u (t) +2 -1 τe0)y

[0063] 4. Update the target lower limit:

[0064]

[0065] 5. Update the threshold and check if it is less than the preset threshold. If it is less, stop the loop.

[0066] η i =f(μ i )-f LB

[0067] 6. Update step size:

[0068]

[0069] Output: Solution to the primal and dual problems (λ) i μ i )

[0070] Once the solution is obtained, the Toeplitz matrix of the observation results can be constructed using μ, and then the frequency and phase of the target can be obtained using subspace methods such as Root-Music.

[0071] Step 4: Meshless 3D imaging using tomographic SAR atomic norm.

[0072] TomoSAR observation illustration Figure 2 As shown, its signal model can be represented as:

[0073] g n =∫ Δs α(x, y, s)·exp(j2πξ) n s)ds (11)

[0074] In the formula, α is the scattering function of elevation s, and Δs is the elevation observation range. (x, y) represents the range-azimuth resolution unit after image registration. Spatial frequency ξ n =2b n / λr0 is determined by the vertical baseline b n Let λ and r0 be the radar wavelength and the slant range from the main image to the target, respectively. Since noise ε objectively exists in the observation, the discretized tomographic SAR signal model can be expressed as:

[0075] g=Lα+ε (12)

[0076] In the formula, g is an N×1 observation vector, and N is the number of observations over multiple flights. L is an N×K sampling matrix, where L nk =exp(-j2πξ) n s k ), s k This represents the discrete elevation position, which is discrete into K points in total.

[0077] The spatial frequency of the TomoSAR signal model is normalized to conform to the ANM norm application framework. The support domain of the TomoSAR signal, i.e., its maximum unambiguous height, can be expressed as:

[0078]

[0079] In the formula, b ⊥ Representing the vertical baseline interval, substituting it into ξ n Then the tomographic signal observed at point n can be expressed as:

[0080]

[0081] In summary, the tomographic SAR solution based on minimizing the atomic norm can be expressed as:

[0082]

[0083] Using the method described in step three to solve (15), the Toeplitz matrix T(u) containing the elevation target frequency can be obtained. The frequency estimate f can be obtained using subspace methods, such as Root-Music. k Then the height corresponding to the scattering point is:

[0084] s k =f k H (16)

[0085] Finally, the scattering at elevation is estimated. Assuming K elevation estimates are obtained, the target location is determined by minimizing the penalty function (model order selection); using the location information, the target amplitude and phase are obtained through least squares, thus completing the rapid meshless 3D imaging of asymmetric cone atom norm model tomographic SAR.

[0086] Example

[0087] Computer simulation

[0088] The results of the traditional SDP-based ANM solution method and the proposed method were compared and verified by computer simulation. The simulation parameters are shown in the table.

[0089] Table 1 Key Simulation Parameters

[0090] parameter value unit band P - distance 1000 m Baseline -42-28 m

[0091] First, 500 Monte Carlo simulations were used to compare the detection rate, recovery accuracy (RMSE), and computation speed of the two methods for single targets. Here, a successful detection is defined as the estimated number of targets matching the set value, and the result deviation being less than the resolution length; otherwise, the detection is considered a failure. Statistical results are as follows: Figure 3 As shown, triangles, squares, and circles represent the cases when the number of samples N is 10, 20, and 30, respectively. Figure 3 (a) and (b) show the detection probabilities of the traditional method and the proposed method for the single target case. The detection performance of the two methods is basically the same. Figure 3 (c) and (d) demonstrate the estimation accuracy for the single-target case. It can be seen that the proposed method performs slightly better than the traditional method at low SNR and low sampling number, but the estimation accuracy of the two methods tends to be consistent as the SNR and sampling number increase. Figure 3 (e) and (f) compare the single-run speed of the two methods. Under the same SNR and number of samples, the computational efficiency of the proposed method is improved by at least 6 times, and by 10 to 15 times at a lower number of samples.

[0092] The bi-objective estimation performance of the two methods was compared similarly, and the results are as follows: Figure 4 As shown, the detection rate and estimation accuracy of the two methods are similar. Under certain conditions of SNR and number of samples, the estimation accuracy of the proposed method is slightly better than that of the traditional method. However, as the SNR and number of samples increase, the two methods tend to converge. Figure 4 As shown in (a), (b), (c) and (d).

[0093] Finally, a simulation was performed on an area target to compare and verify the performance of the two methods. The area target size was 50×50 pixels, and the signal-to-noise ratio was 15dB. The comparison estimation results are as follows: Figure 5 As shown in the figure, both methods can recover the structural features of the target. Table 2 shows the estimation accuracy and computation time of the two methods. It can be seen that the proposed method significantly improves computational efficiency while maintaining estimation accuracy.

[0094] Table 2 Comparison of estimation accuracy and computation time between traditional methods and proposed methods

[0095] RMSE(m) Times(s) Traditional methods 0.13 2309.00 The proposed method 0.13 883.43

[0096] P-band UAV TomoSAR Experiment

[0097] In March 2021, TomoSAR experimental data from an unmanned aerial vehicle (UAV) was successfully acquired from Chongqing General Aviation College in China. The system operated in the P-band, acquiring 30 tracks of data between altitudes of 170-260m. The primary observation area was the college campus, and the corresponding optical maps and SAR imaging results are shown below. Figure 6 As shown in (a) and (b), the system's viewing angle θ = 70°, the scene center slant distance is 656.2m, and the baseline length Δb is approximately 90m. The corresponding elevation resolution ρ is... s It is 2.54m. The area selected by the solid line is the opposite corner position.

[0098] The two-dimensional elevation staining results of the angular inverse region are as follows Figure 7 As shown in the figure, both the traditional SDP method for solving the ANM and the proposed method were used for tomography of the angular inverted region. The figures demonstrate that both methods can effectively estimate the elevation of the angular inverted region. Compared to the traditional method, the proposed method shows a reduction in outliers, such as... Figure 7 As shown in (a) and (b). In terms of computational efficiency, the total running time of the proposed method is significantly reduced, as shown in Table 2.

[0099] Table 3 shows the running time of the reverse area.

[0100] Traditional CS Traditional ANM The proposed method Running time (s) 459.93 869.22 282.25

[0101] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A fast, meshless 3D imaging method for asymmetric cone atom norm model tomographic SAR, characterized in that, Includes the following steps: Step 1: Modeling based on the minimization of asymmetric cone atom norm; Step 2: Construct the Lagrange dual optimization model; Step 3: Rapid optimization solution to the dual problem; Step 4: Meshless 3D imaging using tomographic SAR atomic norm; In step one, the model based on minimizing the atomic norm is as follows: (1)； In the formula, , , , It is a variable in the problem. For the observation results, For regularization parameters, It is a weight matrix, and all parameters are known. It is a cone, defined as: (2)； Dual form for: (3)； In the formula, yes The dual variables in, where , , The inner product of λ and μ is defined as Dual cone It is a set of finite autocorrelation sequences; In step two, equation (1) has the following Lagrangian form: (4)； Its dual form is expressed as: (5)； In step three, after the solution is completed, the Toeplitz matrix of the observation results is constructed using μ; In step four, the target location is determined by minimizing the penalty function, i.e., the model order is selected. Using location information, the target amplitude and phase are obtained through least squares, enabling rapid meshless 3D imaging of asymmetric cone atomic norm model tomographic SAR.

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