CN109934815B - A Tensor Restoration Infrared Weak Small Target Detection Method Combined with ATV Constraints - Google Patents

Abstract

The invention discloses a tensor recovery infrared small and weak target detection method combined with ATV constraint, and relates to the field of infrared image processing and target detection; the method comprises the following steps of 1: constructing a third-order tensor of the original image; step 2: constructing a prior information weight tensor of an original image; and step 3: using the tensor logDet function and the tensor l ₁ Norm, combining with ATV constraint, constructing an objective function, inputting a third-order tensor and a prior information weight tensor into the objective function, and solving the objective function by using ADMM to obtain a background tensor and an objective tensor; and 4, step 4: reconstructing a background image and a target image according to the background tensor and the target tensor; and 5: segmenting the target image to output a target detection result; the method solves the problems of high false alarm rate in infrared weak and small target detection and local optimality caused by nuclear norm due to the fact that the existing method is easily influenced by background edges and noise, improves the target detection and background suppression capability, and improves the accuracy of target detection.

Description

A tensor recovery infrared dim target detection method combined with ATV constraints

Technical Field

The invention relates to the field of infrared image processing and target detection, and in particular to a tensor recovery infrared dim small target detection method combined with ATV constraints.

Background Art

Infrared imaging technology has the characteristics of non-contact and strong ability to capture details. It is not affected by obstacles such as smoke and fog, and can detect long-distance targets continuously during the day and night. Infrared search and tracking IRST (Infrared search and track) systems are widely used in military and civilian fields. Among them, infrared weak target detection technology, as a basic function of IRST system, is of great significance in infrared search, infrared early warning, and long-distance target detection. However, due to the lack of texture and structural information of the target in the infrared band, and the influence of long distance, complex background, and various clutter, infrared targets often appear as spots or dots, or even submerged in the background, which makes infrared weak target detection extremely difficult.

Infrared small target detection technology is divided into two categories: single-frame based small target detection technology and multi-frame based small target detection technology. However, since multi-frame based detection technology needs to capture the target's motion trajectory by combining multiple frames and eliminate noise interference, it requires a huge amount of calculation and storage, and has high hardware requirements, so it is rarely used in actual projects. At present, the commonly used single-frame based detection methods are divided into the following three categories:

(1) Background suppression: Background suppression methods are based on the assumption of background consistency in infrared images. They use filters to predict the background of infrared images, then subtract the background from the original image, and finally perform threshold segmentation to detect weak targets. Maximum median filtering, maximum mean filtering, top hat transformation, two-dimensional minimum mean square filtering, etc. all belong to the category of background suppression. Although this type of method is simple to implement, since the noise does not meet the consistency assumption, the background suppression method is easily affected by noise clutter, resulting in poor suppression effect for most low signal-to-noise ratio infrared images.

(2) Visual saliency: The human visual system (HVS) involves three mechanisms: contrast, visual attention, and eye movement. The most involved mechanism is the contrast mechanism, which assumes that the target is the most salient object in the infrared image. For example, the Gaussian difference filter uses two different Gaussian filters to calculate the saliency map and detect and identify the target; the local contrast-based method uses the characteristics that the local contrast of the small neighborhood containing the target is high, while the local contrast of the background area that does not contain the target is low. By calculating the local contrast map, the target is highlighted and the background is suppressed to achieve the purpose of detection. When the infrared image meets the visual saliency hypothesis, this type of method can achieve excellent results. However, in actual application scenarios, this assumption is difficult to meet. For example, when there is a significant false alarm source, the false detection problem is difficult to overcome, resulting in low accuracy.

(3) Target-background separation: This type of method uses the non-local autocorrelation of the infrared image background and the sparsity of the target to convert the target detection problem into an optimization problem. It can be further divided into methods based on super-complete dictionaries and low-rank representations and methods based on low-rank background and sparse target restoration. The first method requires the construction of a super-complete dictionary of different target sizes and shapes by the Gaussian intensity model in advance. The process of constructing the target dictionary is cumbersome, and the detection results are greatly affected by the dictionary. In addition, if the target size and shape change greatly, the Gaussian intensity model will no longer be applicable. The second method uses the block image model IPI (Infrared Patch-Image) model to obtain the low-rank original block image, and then uses the sparse characteristics of the target to optimize the objective function to simultaneously restore the background and target images, and finally obtain the detection result. The second method has excellent results, but there are two problems: First, since strong edges, some noise, and false alarm sources are also sparse, it will reduce the accuracy of detection; second, since the objective function optimization process requires iteration, it is difficult to achieve real-time performance.

并且已知

可分解为：In today's era of information explosion, the dimension of data is no longer limited to one-dimensional and two-dimensional, and the difficulty of processing is increasing. Tensors are used to represent multidimensional information. In fact, tensors are a general concept of multidimensional arrays. For example, one-dimensional arrays are usually called vectors, and two-dimensional arrays are usually called matrices. Robust principal component analysis (RPCA) overcomes the shortcomings of robust principal component analysis that is easily affected by outliers and is more robust. It has been widely used in image completion, image denoising, and face recognition. However, RPCA can only be used directly to process two-dimensional matrices. If high-dimensional data is to be processed, the high-dimensional data must first be converted into two-dimensional data, and then converted to high-dimensional space after processing. This process is not only complicated, but also completely destroys the internal structure of the data and is inefficient. In order to process high-dimensional data more flexibly, tensor-based technologies have gradually developed. Among them, tensor recovery can utilize more data information (structure, color, time, etc.) and performs better than RPCA in sparse low-rank decomposition. Tensor Robust Principal Component Analysis (TRPCA) is a key technology in tensor recovery technology. It is a high-order extension of RPCA and was proposed by Goldfarb and Qin. Assume that a known tensor

And it is known

Can be decomposed into:

为低秩张量，ε为稀疏张量，根据

求解

和ε的问题就是一个张量恢复问题。in,

is a low-rank tensor, ε is a sparse tensor, according to

Solution

The problem with ε is a tensor recovery problem.

The total variation (TV) model is a well-known partial differential equation denoising model. Since the image details and noise are very similar, it is difficult to protect the details while denoising the image. Osher et al. proposed the idea of total variation in 1992. This model can effectively protect the image edges while denoising. TV has been proven to be able to preserve important edges and corners of the image. It is often used as a regular term when accurate estimation of discontinuous parts of the image is required. In other words, TV represents the smoothness of a given image. It is also widely used in image decomposition. It can decompose the image into two parts: one is an unrelated random pattern, and the other is a sharp edge and piecewise smooth component. By minimizing the TV of the image, the smooth inner surface of the image will be preserved while maintaining clear edges. The TV model includes isotropic total variation (ITV) and anisotropic total variation (ATV). However, since ATV has better edge preservation ability than ITV, ATV is increasingly used in image denoising, image reconstruction and other fields. Without loss of generality, given a graph X, ATV is defined as follows:

和

分别表示水平和垂直的二维有限差分算子。in,

and

denote the horizontal and vertical two-dimensional finite difference operators respectively.

在突出背景边缘的同时也突出了目标的边缘，导致检测结果的目标形状减小，甚至会出现检测不到目标的情况。因此，需要一种红外弱小目标检测方法结合张量恢复和ATV克服以上问题。In order to improve the detection capability of infrared dim small targets, considering that the traditional infrared dim small target detection method only considers the local characteristics of the image, and the optimization method only considers the non-local autocorrelation characteristics of the image, the existing literature proposes the RIPT (Reweighted Infrared Patch-Tensor Model) model, that is, on the basis of the block tensor model, the local and non-local characteristics of the infrared image are combined to construct the objective function, and the alternating direction method of multipliers (ADMM) is used to solve the objective function. In most cases, RIPT has better background suppression and target enhancement capabilities, but the tensor nuclear norm used by RIPT is the nuclear norm and SNN (Sum of Nuclear Norms). The literature "A new convex relaxation for tensor completion" points out that SNN is not the optimal convex approximation of the tensor rank. The nuclear norm gives all singular values the same weight, but in actual scenes, the singular values of the target content and noise are different, so RIPT will cause a local optimal solution and increase the false alarm rate in the target image. At the same time, the local structure weight in RIPT

While highlighting the background edge, the target edge is also highlighted, resulting in a reduction in the target shape of the detection result, or even failure to detect the target. Therefore, an infrared dim target detection method combining tensor recovery and ATV is needed to overcome the above problems.

Summary of the invention

The purpose of the present invention is to provide a tensor recovery infrared small target detection method combined with ATV constraints, which overcomes the problem that the existing method is easily affected by background edges and noise, resulting in a high false alarm rate in infrared small target detection and the local optimality problem caused by the nuclear norm, improves target detection and background suppression capabilities, and improves the accuracy of target detection.

The technical solution adopted by the present invention is as follows:

A tensor recovery infrared small target detection method combined with ATV constraints includes the following steps:

Step 1: Construct the third-order tensor of the original image;

Step 2: Extract the prior information of the original image and construct the prior information weight tensor;

Step 3: Use the tensor logDet function and tensor l ₁ norm, combined with the ATV constraint, to construct the objective function, input the third-order tensor and the prior information weight tensor into the objective function, and use ADMM to solve the objective function to obtain the background tensor and target tensor;

Step 4: Reconstruct the background image and the target image according to the background tensor and the target tensor;

Step 5: Perform adaptive threshold segmentation on the target image to determine the location of the target and output the target detection result.

Preferably, step 1 comprises the following steps:

其中，m和n分别表示图像的长和宽；Step 1.1: Get the original image

Among them, m and n represent the length and width of the image respectively;

Step 1.2: Use a sliding window w of size p×p and traverse the original image D with a step size of s;

Step 1.3: Take the image block in each sliding window w as a front slice, and form a third-order tensor after sliding q times

Preferably, step 2 comprises the following steps:

J_ρ定义如下：Step 2.1: Define the structure tensor of the original image D

_Jρ is defined as follows:

表示克罗内克积，

表示求梯度，

表示D_σ沿x方向的梯度，

表示D_σ沿y方向的梯度，J₁₁替代

J₁₂替代K_ρ*I_xI_y，J₂₁替代K_ρ*I_xI_y，J₂₂替代

Among them, K _ρ represents the Gaussian kernel function with variance ρ, * represents the convolution operation, and D _σ represents the Gaussian smoothing filter with variance σ (>0) on the original image.

represents the Kronecker product,

represents the gradient,

represents the gradient of _Dσ along the x direction,

represents the gradient of _Dσ along the y direction, J ₁₁ replaces

J ₁₂ replaces K _ρ *I _x I _y , J ₂₁ replaces K _ρ *I _x I _y , J ₂₂ replaces

和

计算如下：Step 2.2: Calculate the eigenvalue matrix of J _ρ

and

The calculation is as follows:

Step 2.3: Calculate the prior information matrix related to the target

Among them, ⊙ represents the Hadamard product;

Step 2.4: Calculate the prior information matrix related to the background

W _m =max(λ ₁ ,λ ₂ );

Step 2.5: Calculate the prior information matrix based on the obtained W _cs and W _m

W _p =W _cs *W _m ;

Normalize W _p as follows:

Wherein, w _min and w _max represent the minimum and maximum values of W _p respectively;

构建方法为：采用大小为p×p的滑动窗口w遍历W_p，把每次滑动窗口w中的图像小块作为一个正面切片，滑动q次后，构成一个三阶张量即先验信息权重张量

Step 2.6: Construct the prior information weight tensor based on the normalized prior information matrix W _p

The construction method is: use a sliding window w of size p×p to traverse W _p , and take the image block in each sliding window w as a front slice. After sliding q times, a third-order tensor, namely the prior information weight tensor, is constructed.

Preferably, constructing the objective function in step 3 comprises the following steps:

包括低秩张量

和稀疏张量

为分离低秩张量

和稀疏张量

将张量logDet函数作为低秩张量的正则化项，张量l₁范数作为稀疏张量的正则化项，结合ATV约束，构建目标函数，公式如下：Step a1: Third-order tensor

Including low-rank tensors

and sparse tensors

To separate low-rank tensors

and sparse tensors

The tensor logDet function is used as the regularization term of the low-rank tensor, the tensor l ₁ norm is used as the regularization term of the sparse tensor, and the objective function is constructed in combination with the ATV constraint. The formula is as follows:

表示，

表示

的第i(1≤i≤p)个奇异值

是

沿第三维做离散傅里叶变换所得到的张量的第l个正面切片，下文的l表示相同的含义)，||·||₁表示张量l₁范数(即张量中所有元素的奇异值之和)，||·||_HTV表示各向异性全变分约束，它的定义为

其中

和

分别表示水平和垂直的二维有限差分算子；Among them, η＞0 represents a small regularization constant, λ and β represent balance coefficients, logDet(·) represents the logDet function of the tensor, and

express,

express

The i-th (1≤i≤p) singular value of

yes

The lth positive slice of the tensor obtained by discrete Fourier transform along the third dimension, l has the same meaning below), ||·|| ₁ represents the l ₁ norm of the tensor (that is, the sum of the singular values of all elements in the tensor), and ||·|| _HTV represents the anisotropic total variation constraint, which is defined as

in

and

denote the horizontal and vertical two-dimensional finite difference operators respectively;

表示稀疏权重张量，根据稀疏权重和先验信息权重张量定义权重张量

公式如下：Step a2: Let

Represents a sparse weight tensor, which defines a weight tensor based on sparse weights and prior information weight tensors

The formula is as follows:

Among them, c and ξ represent positive numbers greater than 0, and ./ represents the division of corresponding elements between two tensors;

改写原目标函数如下：Step a3: Introducing substitution variables

Rewrite the original objective function as follows:

Then the augmented Lagrangian equation of the rewritten objective function is as follows:

和

表示拉格朗日乘子，μ表示非负的惩罚因子，⊙表示哈达马积，<·>表示内积运算，||·||_F表示Frobenius范数。in,

and

represents the Lagrange multiplier, μ represents a non-negative penalty factor, ⊙ represents the Hadamard product, <·> represents the inner product operation, and ||·|| _F represents the Frobenius norm.

Preferably, in step 3, the third-order tensor and the prior information weight tensor are input into the objective function, and solving the objective function using ADMM includes the following steps:

输入待求解的目标函数；Step b1: The third-order tensor constructed from the original image

Input the objective function to be solved;

Step b2: Initialize the parameters of the augmented Lagrangian equation, set the number of iterations k = 0, and the maximum number of iterations is kmax;

更新

计算公式如下：Step b3: In the k+1th iteration, fix

renew

The calculation formula is as follows:

Where S _τ (·) represents the soft threshold shrinkage operator, S _τ (x) = sgn(x)max(|x|-τ,0);

更新

计算公式如下:Step b4: Fixing

renew

The calculation formula is as follows:

为

的奇异值分解，

为f-对角张量，并且

这里，

和

分别表示

和

沿第三维做离散傅里叶变换得到的结果，

表示在第k次迭代中，

的第l个正面切片的第i个奇异值；in,

for

The singular value decomposition of

is an f-diagonal tensor, and

here,

and

Respectively

and

The result of discrete Fourier transform along the third dimension is:

Indicates that in the kth iteration,

The i-th singular value of the l-th positive slice of ;

更新

如下：Step b5: Fixing

renew

as follows:

To solve the above equation, we can divide it into q sub-problems to solve. The sub-problems are as follows:

This problem can be solved by the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA);

更新

如下：Step b6: Fixing

renew

as follows:

更新

和

如下：Step b7: Fixing

renew

and

as follows:

Step b8: Update μ ^k+1 =ρμ ^k , where ρ represents the growth coefficient, ρ≥1;

Step b9: number of iterations k=k+1;

Step b10: Determine whether k is greater than _kmax . If so, stop the iteration and go to step b11. If not, stop the iteration and go to step b11 when the following conditions are met:

If the iteration stop condition is not met and the number of iterations has not reached the maximum value, go to step b3;

和目标张量

Step b11: Find the optimal solution and output the background tensor

and the target tensor

按顺序取出

的q个正面切片

并依次重构获取背景图

对于输入的目标张量

按顺序取出

的q个正面切片

并依次重构获取目标图

Preferably, the specific steps of step 4 are: for the input background tensor

Take out in order

q frontal slices of

And reconstruct the background image in sequence

For the input target tensor

Take out in order

q frontal slices of

And reconstruct the target graph in turn

In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

1. Taking into account the different importance of singular values of different sizes, the present invention uses a non-convex logDet function to constrain the tensor rank, assigning different weights to different singular values. Combined with the non-convex nature of the function, it is closer to the true rank of the tensor, and solves the problem of low target detection accuracy caused by the local optimal solution in the existing method due to the use of the nuclear norm that assigns the same weight to the singular values, thereby improving the capabilities of target detection and background suppression;

2. Due to the complexity and changeability of actual scenes, the present invention introduces ATV to specifically constrain the background edge. ATV can describe the internal smoothness and clarity of the background, which is closely related to the gradient (high-frequency component) of the image. The edge is a high-frequency component. By introducing the ATV regularization term, the edge in the background can be better described, and the sparse edge components in the target component can be suppressed. This solves the problem that the existing method is difficult to suppress the highlight edge with sparse properties, resulting in edge noise in the separated target, and improves the detection capability in non-smooth and non-uniform scenes;

3. The present invention simultaneously utilizes prior information related to the background and the target, and increases the target constraint capability by utilizing a weight that highlights the target more. At the same time, the prior is added to the objective function as a part of the regular term, so that the objective function has stronger conditions to achieve the optimal solution, narrows the scope of the feasible domain, increases the speed of reaching the optimal solution, accelerates the convergence speed of the algorithm, and improves the robustness of the algorithm;

4. The present invention transforms the traditional infrared dim target detection problem into a tensor recovery problem, and can adaptively separate the target and the background without extracting any features. The present invention has wider applicability.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for use in the embodiments are briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other related drawings can be obtained based on these drawings without creative work.

Fig. 1 is a flow chart of the present invention;

FIG2 is an infrared image of a small target according to the present invention;

FIG3 is a third-order tensor constructed from FIG2 according to the present invention;

FIG4 is a priori information graph and a priori information tensor calculated by the present invention from FIG2;

FIG5 is a target tensor separated from FIG3 according to the present invention;

FIG6 is a background tensor separated from FIG3 according to the present invention;

FIG7 is a target image and a background image reconstructed from FIG5 and FIG6 according to the present invention;

FIG8 is a three-dimensional grayscale distribution diagram of the target image in FIG2 and FIG5 of the present invention;

FIG9 is a detection result obtained by adaptive threshold segmentation of the target image in FIG5 according to the present invention;

FIG10 is a detection result diagram and a three-dimensional grayscale diagram of FIG2 using the LoG method;

FIG11 is a diagram showing the detection result of FIG2 using the RLCM method and a three-dimensional grayscale diagram;

FIG12 is a diagram showing the detection result of FIG2 using the IPI method and a three-dimensional grayscale image;

FIG13 is a diagram showing the detection result of FIG2 using the NIPPS method and a three-dimensional grayscale image;

FIG14 is a detection result diagram and a three-dimensional grayscale diagram of FIG2 using the RIPT method;

FIG. 15 is a schematic diagram of the RIPT method and the prior information of the present invention.

DETAILED DESCRIPTION

In order to make the purpose, technical scheme and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention, that is, the embodiments described are only part of the embodiments of the present invention, rather than all of the embodiments. The components of the embodiments of the present invention described and shown in the drawings herein can be arranged and designed in various different configurations.

Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the invention claimed for protection, but merely represents selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative work are within the scope of protection of the present invention.

It should be noted that relational terms such as "first" and "second" are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, the elements defined by the sentence "comprise a ..." do not exclude the existence of other identical elements in the process, method, article or device including the elements.

The features and performance of the present invention are further described in detail below in conjunction with the embodiments.

Example 1

As shown in FIG1-15, a tensor recovery infrared small target detection method combined with ATV constraints includes the following steps:

Step 1: Construct the third-order tensor of the original image;

Step 2: Extract the prior information of the original image and construct the prior information weight tensor;

Step 3: Use the tensor logDet function and tensor l ₁ norm, combined with the ATV constraint, to construct the objective function, input the third-order tensor and the prior information weight tensor into the objective function, and use ADMM to solve the objective function to obtain the background tensor and target tensor;

Step 4: Reconstruct the background image and the target image according to the background tensor and the target tensor;

Step 5: Perform adaptive threshold segmentation on the target image to determine the location of the target and output the target detection result.

In order to improve the accuracy of small target detection, it is necessary to overcome the local optimality problem caused by the nuclear norm and the influence of background edges and noise in the image on detection. The tensor logDet function is used as the regularization term of the low-rank tensor, and the tensor l ₁ norm is used as the regularization term of the sparse tensor to construct the tensor recovery objective function. Different singular values are assigned different weights. Combined with the non-convex nature of the function, it is closer to the true rank of the tensor, and the problem of local optimal solution caused by the use of the nuclear norm that assigns the same weight to singular values in the existing method is solved; combined with ATV, the edges in the background are better described, and the sparse edge components in the target component are suppressed; the target detection and background suppression capabilities are improved, and the accuracy of infrared small target detection is improved.

图4是由步骤2提取到的先验信息以及对应的先验信息权重张量

图5是经过步骤3分离得到的背景张量

和目标张量

图6是由步骤4重构的背景图像B与目标图像T；图7是原始图像D与目标图像T对应的灰度三维分布，可以看出，分离出的目标图像很好地压制了背景，除去小目标处，其余位置的背景的灰度均为0；图8是最终的检测结果；图9-图14是几种其他的方法(依次是LoG、RLCM、IPI、NIPPS和RIPT)对图2中小目标的检测结果(未阈值分割)，以及对应的灰度三维分布图，可以看到，LoG和IPI(图10和图12)这两种方法对背景边缘和噪声极其敏感，RLCM和NIPPS(图11和图13)中，除去目标以外，还残留较多噪声，虚警率偏高，目标有不同程度的缩小，而RIPT(图14)则未检测到目标。综上，本申请背景抑制能力强，噪声极其小，无失真，目标检测的效果极佳，目标检测准确度大大提高。The effect analysis is carried out according to the attached figures: Figure 2 shows an infrared image with a complex background. In addition to the weak target, there is also a bright white false alarm source; Figure 3 is the third-order tensor constructed from the original image after step 1.

Figure 4 shows the prior information extracted from step 2 and the corresponding prior information weight tensor

Figure 5 is the background tensor obtained after separation in step 3

and the target tensor

Figure 6 is the background image B and target image T reconstructed by step 4; Figure 7 is the grayscale three-dimensional distribution corresponding to the original image D and the target image T. It can be seen that the separated target image suppresses the background well. Except for the small target, the grayscale of the background at other positions is 0; Figure 8 is the final detection result; Figures 9-14 are the detection results (not threshold segmented) of the small target in Figure 2 by several other methods (LoG, RLCM, IPI, NIPPS and RIPT in turn), and the corresponding grayscale three-dimensional distribution diagram. It can be seen that the two methods LoG and IPI (Figures 10 and 12) are extremely sensitive to background edges and noise. In RLCM and NIPPS (Figures 11 and 13), in addition to the target, there is still a lot of noise left, the false alarm rate is high, and the target is reduced to varying degrees, while RIPT (Figure 14) does not detect the target. In summary, the present application has a strong background suppression ability, extremely small noise, no distortion, excellent target detection effect, and greatly improved target detection accuracy.

Example 2

Based on Example 1, the steps of this application are refined and the technical means used to solve the technical problems are recorded in detail: the objective function is constructed by using the tensor logDet function and the tensor l ₁ norm, combined with the ATV constraint, the third-order tensor and the prior information weight tensor are input into the objective function, and the objective function is solved by ADMM to obtain the background tensor and the target tensor.

Step 1 includes the following steps:

Step 1.1: Get the infrared image D∈R ^m×n to be processed, with a size of 245×326;

Step 1.2: Use a sliding window w of size 40×40 and traverse the original image D with a step size of 40, and take the 40×40 matrix in each sliding window w as a front slice;

Step 1.3: Repeat step 1.2 according to the number of window sliding times (63 in this example) until the traversal is completed, and form all the front slices into a new third-order tensor

As shown in Figure 2, it shows an infrared image with a complex background. In addition to the weak target, there is also a white false alarm source with high brightness. As shown in Figure 3, it shows the third-order tensor constructed from the original image in step 1.

Step 2: Extract the prior information of the original image and construct the prior information weight tensor;

Step 3 includes the following steps:

Step 3.1: Use the tensor logDet function, tensor l ₁ norm, and ATV constraints to construct the objective function;

和先验信息权重张量

输入目标函数，利用ADMM求解目标函数，解出背景张量

和目标张量

Step 3.2: Convert the third-order tensor

and the prior information weight tensor

Input the objective function, use ADMM to solve the objective function, and solve the background tensor

and the target tensor

Step 3.1 includes the following steps:

包括低秩张量

和稀疏张量

为分离低秩张量

和稀疏张量

将张量logDet函数作为低秩张量的正则化项，张量l₁范数作为稀疏张量的正则化项，结合ATV约束，构建目标函数，公式如下：Step 3.1.1: Third-order tensor

Including low-rank tensors

and sparse tensors

To separate low-rank tensors

and sparse tensors

The tensor logDet function is used as the regularization term of the low-rank tensor, the tensor l ₁ norm is used as the regularization term of the sparse tensor, and the objective function is constructed in combination with the ATV constraint. The formula is as follows:

表示，

表示

的第i(1≤i≤40)个奇异值

是

沿第三维做离散傅里叶变换所得到的张量的第l个正面切片，下文的l表示相同的含义)，||·||₁表示张量l₁范数(即张量中所有元素的奇异值之和)，||·||_HTV表示各向异性全变分约束，它的定义为

其中

和

分别表示水平和垂直的二维有限差分算子；Among them, η＞0 represents a small regularization constant, λ and β represent balance coefficients, logDet(·) represents the logDet function of the tensor, and

express,

express

The i-th (1≤i≤40) singular value of

yes

The lth positive slice of the tensor obtained by discrete Fourier transform along the third dimension, l has the same meaning below), ||·|| ₁ represents the l ₁ norm of the tensor (that is, the sum of the singular values of all elements in the tensor), and ||·|| _HTV represents the anisotropic total variation constraint, which is defined as

in

and

denote the horizontal and vertical two-dimensional finite difference operators respectively;

表示稀疏权重张量，则有Step 3.1.2: Order

represents a sparse weight tensor, then

的定义如下：Among them, c and ξ represent positive numbers greater than 0, so the final weight tensor

is defined as follows:

Among them, ./ represents the division of corresponding elements between two tensors;

改写原目标函数如下：Step 3.1.3: Introducing alternative variables

Rewrite the original objective function as follows:

Then the augmented Lagrangian equation of the rewritten objective function is as follows:

和

表示拉格朗日乘子，μ表示非负的惩罚因子，⊙表示哈达马积，<·>表示内积运算，||·||_F表示Frobenius范数。in,

and

represents the Lagrange multiplier, μ represents a non-negative penalty factor, ⊙ represents the Hadamard product, <·> represents the inner product operation, and ||·|| _F represents the Frobenius norm.

Step 3.2 includes the following steps:

输入待求解的目标函数；Step 3.2.1: The third-order tensor constructed from the original image

Input the objective function to be solved;

c＝1，ξ＝0.01,η＝0.2,β＝0.5；Step 3.2.2: Initialize the parameters of the augmented Lagrangian equation, set the number of iterations k = 0, the maximum number of iterations is _kmax = 500, ρ = 1.05, _μ0 = 0.001,

c=1, ξ=0.01, η=0.2, β=0.5;

更新

计算公式如下：Step 3.2.3: In the k+1th iteration, fix

renew

The calculation formula is as follows:

Wherein, S _τ () represents the soft threshold shrinkage operator, S _τ (x) = sgn(x)max(|x|-τ,0);

更新

计算公式如下:Step 3.2.4: Fixation

renew

The calculation formula is as follows:

为

的奇异值分解，

为f-对角张量，并且

这里，

和

分别表示

和

沿第三维做离散傅里叶变换得到的结果，

表示在第k次迭代中，

的第l个正面切片的第i个奇异值；in,

for

The singular value decomposition of

is an f-diagonal tensor, and

here,

and

Respectively

and

The result of discrete Fourier transform along the third dimension is:

Indicates that in the kth iteration,

The i-th singular value of the l-th positive slice of ;

更新

如下：Step 3.2.5: Fixation

renew

as follows:

To solve the above equation, we can divide it into q sub-problems to solve. The sub-problems are as follows:

This problem can be solved by the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA);

更新

如下：Step 3.2.6: Fixation

renew

as follows:

更新

和

如下：Step 3.2.7: Fixation

renew

and

as follows:

Step 3.2.8: Update μ ^k+1 =ρμ ^k , where ρ represents the growth coefficient, ρ ≥ 1;

Step 3.2.9: Iteration number k = k + 1;

Step 3.2.10: Determine whether k is greater than k _max . If so, stop the iteration and go to step 3.2.11. If not, stop the iteration and go to step 3.2.11 when the following conditions are met:

If the iteration stop condition is not met and the number of iterations has not reached the maximum value, go to step 3.2.3;

和目标张量

输出的带*的符号表示最优解，在迭代收敛后得到的B和T的解即分离出的目标张量和背景张量。Step 3.2.11: Find the optimal solution and output the background tensor

and the target tensor

The output symbol with * represents the optimal solution. The solutions of B and T obtained after iterative convergence are the separated target tensor and background tensor.

按顺序取出

的63个正面切片

并依次重构获取背景图

对于输入的目标张量

按顺序取出

的63个正面切片

并依次重构获取目标图

The specific steps of step 4 are: for the input background tensor

Take out in order

63 frontal slices

And reconstruct the background image in sequence

For the input target tensor

Take out in order

63 frontal slices

And reconstruct the target graph in sequence

The specific steps of step 5 are: perform adaptive threshold segmentation on the target image T, the threshold Th = m + c * σ, where m represents the mean of all grayscales in the target image T, σ represents the standard deviation of all grayscales in the target image T, c = 5, and the segmentation is completed to obtain the target detection result.

As shown in Figure 7, the background image is calculated and processed by the method of the present invention to obtain the final target image, which completely suppresses the background, is noise-free, and has no distortion. A non-convex logDet function with stronger low-rank approximation ability than SNN is used to constrain the background. ATV can describe the internal smoothness and clarity of the background, which is closely related to the gradient (high-frequency component) of the image. The edge is a high-frequency component. By introducing the ATV regularization term, the edge in the background can be better described, and the sparse edge components in the target component can be suppressed. This solves the problem that the existing method is difficult to suppress highlighted edges with sparse properties, resulting in edge noise in the separated target, and improves the detection capability in non-smooth and non-uniform scenes, thereby improving the accuracy of target detection.

Example 3

Based on Example 1, this example refines step 2, extracts the prior information of the original image, constructs a prior information weight tensor, and utilizes the prior information related to the background and the target to ensure that the target is not distorted, thereby accelerating the convergence speed of the algorithm and improving the robustness of the algorithm.

Step 2 includes the following steps:

J_ρ定义如下：Step 2.1: Define the structure tensor of the original image D

_Jρ is defined as follows:

表示克罗内克积，

表示求梯度，

表示D_σ沿x方向的梯度，

表示D_σ沿y方向的梯度，J₁₁替代

J₁₂替代K_ρ*I_xI_y，J₂₁替代K_ρ*I_xI_y，J₂₂替代

Among them, K _ρ represents the Gaussian kernel function with variance 2, * represents the convolution operation, and D _σ represents the Gaussian smoothing filter with variance 9 on the original image.

represents the Kronecker product,

represents the gradient,

represents the gradient of _Dσ along the x direction,

represents the gradient of _Dσ along the y direction, J ₁₁ replaces

J ₁₂ replaces K _ρ *I _x I _y , J ₂₁ replaces K _ρ *I _x I _y , J ₂₂ replaces

和

计算如下：Step 2.2: Calculate the eigenvalue matrix of J _ρ

and

The calculation is as follows:

Step 2.3: Calculate the prior information matrix related to the target

Among them, ⊙ represents the Hadamard product;

Step 2.4: Calculate the prior information matrix related to the background

W _m =max(λ ₁ ,λ ₂ );

Step 2.5: Calculate the prior information matrix based on the obtained W _cs and W _m

W _p =W _cs *W _m ;

Normalize W _p as follows:

Wherein, w _min and w _max represent the minimum and maximum values of W _p respectively;

构建方法为：采用大小为40×40的滑动窗口w遍历W_p，把每次滑动窗口w中的图像小块作为一个正面切片，滑动63次后构成一个三阶张量

Step 2.6: Construct the prior information weight tensor based on the normalized prior information matrix W _p

The construction method is: use a sliding window w of size 40×40 to traverse W _p , take the image block in each sliding window w as a front slice, and slide 63 times to form a third-order tensor

As shown in Figure 15, (a) is the prior information graph obtained in RIPT, and (b) is the prior information graph obtained by this method. By observing the two figures, it can be found that the prior information graph of the present invention only highlights the target, while RIPT highlights the edge while highlighting the edge of the target; therefore, the present invention enhances the target constraint ability by utilizing weights that highlight the target more; utilizing the background-related and target-related prior information, the prior is added to the objective function as part of the regularization term, so that the objective function has stronger conditions for reaching the optimal solution, narrows the scope of the feasible domain, increases the speed of reaching the optimal solution, accelerates the convergence speed of the algorithm, and improves the robustness of the algorithm.

The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims (4)

1. A tensor recovery infrared small target detection method combined with ATV constraint is characterized in that: the method comprises the following steps:

step 1: constructing a third-order tensor of the original image;

step 2: extracting prior information of an original image, and constructing a prior information weight tensor;

and 3, step 3: using the tensor logDet function and the tensor l ₁ Norm, combining with ATV constraint, constructing an objective function, inputting a third-order tensor and a prior information weight tensor into the objective function, and solving the objective function by using ADMM to obtain a background tensor and an objective tensor;

and 4, step 4: reconstructing a background image and a target image according to the background tensor and the target tensor;

and 5: performing self-adaptive threshold segmentation on the target image to determine the position of the target, and outputting a target detection result;

specifically, the constructing the objective function in step 3 includes the following steps:

step a1: third order tensor

Comprising a low rank tensor pick>

And sparse tensor>

To separate the low rank tensor B and the sparse tensor T, the tensor logDet function is used as the regularization term of the low rank tensor, tensor l ₁ The norm is used as a regularization term of the sparse tensor, and an objective function is constructed by combining ATV constraint, wherein the formula is as follows:

min logDet(B，η)+λ||W⊙T|| ₁ +β||B|| _HTV

s.t.D＝B+T

where η > 0 represents a very small regularization constant, λ and β represent equilibrium coefficients, logDet (-) represents the logDet function of the tensor, and has

Means for>

Represents->

I (i is more than or equal to 1 and less than or equal to p) th singular value->

Is the ith front slice of the tensor obtained by the discrete Fourier transform of B along the third dimension, wherein l below represents the same meaning, | | · | | survival ₁ The representation tensor l ₁ Norm, | · | luminance _HTV Represents an anisotropic total variation constraint, which is defined as->

Wherein->

D _h And D _v Two-dimensional finite difference operators which respectively represent the horizontal and the vertical;

step a2: order to

Representing a sparse weight tensor, defining a weight tensor based on the sparse weight and the prior information weight tensor>

The formula is as follows:

W＝W _s ./W _p

where c and ξ represent positive numbers greater than 0,/represents the division of the corresponding element between the two tensors;

step a3: introducing a substitute variable Z = B, and rewriting an original objective function as follows:

min logDet(B，η)+λ||W⊙T|| ₁ +β||Z|| _HTV

s.t.D＝B+T，Z＝B

the augmented lagrangian equation for the rewritten objective function is as follows:

wherein, Y ₁ And Y ₂ Indicating a lagrange multiplier, mu indicating a non-negative penalty factor, an, indicating a hadamard product,<·>expressing inner product operation, | · Lixiao _F Represents a Frobenius norm;

in the step 3, the third-order tensor and the prior information weight tensor are input into the objective function, and the step of solving the objective function by using the ADMM comprises the following steps:

step b1: third order tensor to be constructed by original graph

Inputting an objective function to be solved; />

Step b2: initializing parameters of an augmented Lagrange equation, and enabling the iteration number k =0 and the maximum iteration number to be kmax;

and b3: in the k +1 th iteration, B, Z, Y are fixed ₁ ，Y ₂ W, update T ^k+1 The calculation formula is as follows:

wherein S is _τ (. Represents a soft threshold shrink operator, S _τ (x)＝sgn(x)max(|x|-τ，0)；

Step b4: fixing T, Z, Y ₁ ，Y ₂ W, update B ^k+1 The calculation formula is as follows:

wherein,

A＝U*S*V ^T is a singular value decomposition of A, C is an f-diagonal tensor, and ^ s>

Here, a combination of>

And &>

Respectively representing the results of a discrete Fourier transform performed in a third dimension for C and S, based on the first dimension and the second dimension, based on the first dimension>

Indicating that in the kth iteration>

Epsilon is a constant greater than 0 in order to prevent the denominator from being 0;

and b5: fixing B, T, Y ₁ ，Y ₂ W, update Z ^k+1 The following were used:

to solve the above equation, the above equation can be solved by dividing it into q sub-problems, which are as follows:

this problem can be solved by a Fast Iterative threshold Shrinkage Algorithm (FISTA) Algorithm;

step b6: fixing B, T, Z, Y ₁ ，Y ₂ Update W ^k+1 The following were used:

W ^k+1 ＝W _s ^k+1 ./W _p

step b7: fixing B, T, Z, W, updating Y ₁ ^k+1 And Y ₂ ^k+1 The following were used:

Y ₁ ^k+1 ＝Y ₁ ^k +μ ^k (T ^k+1 +B ^k+1 -D)

Y ₂ ^k+1 ＝Y ₂ ^k +μ ^k (B ^k+1 -Z ^k+1 )

step b8: updating mu ^k+1 ＝ρμ ^k Wherein rho represents a growth coefficient, and rho is more than or equal to 1;

step b9: the number of iterations k = k +1;

step b10: judging whether k is larger than kmax, if so, stopping iteration, and turning to the step b11; if not, stopping iteration when the following conditions are met, and turning to the step b11:

if the iteration stop condition is not met and the iteration times are not the maximum value, turning to the step b3;

step b11: finding out optimal solution and outputting background tensor

And the target tensor pick>

2. The method for detecting the tensor recovery infrared small dim target combined with the ATV constraint as set forth in claim 1, wherein: the step 1 comprises the following steps:

step 1.1: obtaining an original image

Wherein m and n represent the length and width of the image, respectively;

step 1.2: traversing an original image D by adopting a sliding window w with the size of P multiplied by P and according to the step length of s;

step 1.3: taking the small image block in the sliding window w as a front slice every time, and establishing a third-order tensor after sliding for q times

3. The method for detecting the infrared weak and small target by combining the tensor recovery constrained by the ATV as claimed in claim 1 or 2, which is characterized in that: the step 2 comprises the following steps:

step 2.1: defining the structure tensor of the original image D

J _ρ The definition is as follows:

wherein, K _ρ A Gaussian kernel function representing the variance ρ representing a convolution operation, D _σ Which means that the original image is subjected to a gaussian smoothing filter with variance, sigma, where sigma is greater than 0,

represents a kronecker product, -is present>

Indicating a gradient is taken over>

Represents D _σ A gradient in the x-direction, is present>

Is shown by D _σ Gradient in the y-direction, J ₁₁ ，J ₁₂ ，J ₂₁ And J ₂₂ Are all auxiliary variables;

step 2.2: calculation of J _ρ Eigenvalue matrix of

And &>

The calculation is as follows:

step 2.3: calculating a prior information matrix associated with the object

Wherein, an indicates a Hadamard product;

step 2.4: calculating a prior information matrix related to the background

W _m ＝max(λ ₁ ，λ ₂ )；

Step 2.5: according to the obtained W _cs And W _m Calculating a prior information matrix

W _p ＝W _cs *W _m ；

To W _p Normalization was performed as follows:

wherein w _min And w _max Respectively represent W _p Minimum and maximum values of (d);

step 2.6: according to a normalized prior information matrix W _p Constructing a priori information weight tensor

The construction method comprises the following steps: traversing W with a sliding window W of size p _p The image small block in the sliding window w is taken as a front section, and after sliding for q times, a third-order tensor, namely prior information weight tensor->

4. The method for detecting the infrared weak and small target by combining the tensor recovery constrained by the ATV as claimed in claim 1 or 2, which is characterized in that: the specific steps of the step 4 are as follows: background tensor for input

Q frontal slices B of B are taken out in order ⁽¹⁾ ，B ^(q) And in turn reconstruct the captured background map>

For an input target tensor->

Taking q frontal slices T of T in order ⁽¹⁾ ，...，T ^(q) And in turn reconstruct the acquired target map>

Images