CN112131994B - Anti-noise satellite image small target detection method - Google Patents
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Abstract
The invention discloses a novel anti-noise satellite image small target detection algorithm in the technical field of image processing, which comprises the following steps: processing the original image to obtain a traditional processing model; improving a traditional processing model to obtain an algorithm model with anti-noise performance; finally, analyzing and verifying the accuracy of the model, and utilizing an integral control principle of a control theory to add an integral term into the Newton-Lawson iterative algorithm model based on the Newton-Lawson iterative algorithm model, thereby providing an anti-noise Newton-Lawson iterative algorithm model applied to the problem of small target detection under noise interference; and then respectively verifying the convergence of the algorithm under different noise conditions, theoretically proving the feasibility of the algorithm, finally carrying out a corresponding simulation experiment, verifying the effectiveness of the proposed anti-noise Newton-Lawson iterative algorithm model, and effectively solving the problem of small target detection in satellite images by applying the model to image processing.
Description
Technical Field
The invention relates to the technical field of image processing, in particular to an anti-noise satellite image small target detection method.
Background
With the development of the earth science and the progress of technology, humans have not been able to merely search the earth by examining drawings and the like. Satellite image technology is getting more attention because it collects information at high altitudes, which can greatly enhance the human ability to observe the earth. Satellite images enable humans to observe large areas from high altitudes in a short period of time, in which many target-related observation data materials are obtained. Meanwhile, the research and observation satellite images can update and acquire the latest data of the observation targets in a relatively short time, so that dynamic monitoring can be performed according to the change of the data. The satellite image digitizes information material, which is easier to digitize than conventional maps. The satellite image technology is characterized by wide range, short period, strong data comparability and the like, and provides a more economical choice for human observation and research of nature in different fields. The target detection algorithm based on the neural network has better performance due to the great improvement of the calculation speed of computer hardware, the birth of a large data set and the development of deep learning. The current popular target detection algorithm is carried out by using a convolutional neural network, and plays an important role in the fields of military, industry, traffic, agriculture and the like in the aspects of ship identification, sewage treatment plant identification, road network data quality evaluation, soil humidity detection and the like.
Detection of small targets in satellite images has been a challenge for image processing. Small objects, whether in number or size, tend to occupy a small proportion in a picture as compared to larger objects. Just because of the low resolution of small objects, the image is relatively more blurred and carries too little information resulting in poor characterizations. In addition, the target detection of the current image is mainly the detection of a large target, the situation that a small target is passively ignored can occur, once the situation occurs, the accuracy of target detection by people can be greatly reduced, and even the target detection fails. However, in some cases, key information is hidden in a small target, and researchers can better interpret images by means of the small target, so that in some applications, the small target has a certain research value in image recognition, and Harsanyi proposes a CEM algorithm which only extracts characteristic target vectors and can weaken interference in other directions. The main idea of the algorithm is to extract signals in a specific direction and attenuate signal interference in other directions, so that an interested target is separated from an image, and a CEM operator is utilized to perform matching calculation on pixels to be detected to find the target. The target detection algorithm has good detection effect because the prior information of the target spectrum is needed. Even if some bands of the target spectrum have larger changes or the energy of all bands is increased or decreased, the direction of the whole band of the target spectrum is not affected, i.e. the detection result is not greatly affected, due to the fact that hyperspectral data have a plurality of different bands. Therefore, the algorithm can restrain the background information and enable the target pixel to keep larger output, so that the advantages of the target spectrum and the background spectrum are effectively separated. By observation, we can find that solving the CEM model is essentially an optimization problem, which is converted into a problem of solving a linear equation set by the lagrangian multiplier method, and the algorithm of solving the linear equation set problem can be roughly divided into two types: iterative algorithms and convolutional neural network algorithms, wherein the iterative algorithms include gradient descent methods, newton methods (Newton Raphson Iterative, NRI), and variations of newton methods, and the like. Many of the problems in small target detection can be reduced to solving an optimization problem with a system of matrix equations. The small target detection problem may be converted into a linear matrix system of equations to solve the optimization problem in conjunction with a linear hybrid model-based constrained energy minimization method (Constrained Energy Minimization, CEM). Noise is inevitably generated during computer operation, but many methods do not take the influence of noise into account, and performance is greatly reduced or even disabled in the presence of interference. Therefore, a new method capable of suppressing noise influence is needed to be studied, and based on this, the present invention designs a new anti-noise satellite image small target detection algorithm to solve the above problems.
Disclosure of Invention
The invention aims to provide a novel anti-noise satellite image small target detection algorithm to solve the problems in the background technology.
In order to achieve the above purpose, the present invention provides the following technical solutions: a novel anti-noise satellite image small target detection algorithm comprises the following steps:
s1: processing the original image to obtain a traditional processing model;
s2: improving a traditional processing model to obtain an algorithm model with anti-noise performance;
s3: and analyzing and verifying the accuracy of the model.
Further, the step S1 specifically includes: let the multispectral sample set be omega i ＝[Ω 1 ，Ω 2 ，...，Ω N ] T N represents the total number of pixels in the image, andi is more than or equal to 1 and less than or equal to N; wherein each pixel Ω i Is a P-dimensional vector; p is the number of bands in which,the target spectrum signal to be detected is used as the known information, and a filter coefficient is designed as delta= [ delta ] 1 ,δ 2 ,...,δ P ] T After the spectrum signal passes through the filter, the target spectrum can still keep a certain output, and the background spectrum is restrained, so the filter can meet the constraint of the following conditions:
assume that spectral data Ω is input i Obtaining corresponding output data as O through a finite impulse response linear filter i Can be expressed as:
corresponding to all inputs Ω i After passing through the finite impulse response linear filter, the average output energy is:
wherein the method comprises the steps ofFor the autocorrelation matrix, the constraint condition of (1.1) and the average output energy minimization of the formula (1.3) are combined to form the main problem of the CEM algorithm, and meanwhile, the problem of minimum value is converted, and the formula can be expressed as:
this problem is used as an optimization problem, and by using the Lagrangian multiplier method, the expression can be written as:
L(δ,μ)＝δ T Rδ+μ(w T δ-1)(1.5)
where μ is the Lagrangian multiplier, both of which and δ are unknowns, let the derivative of formula (1.5) equal to 0:
combining the constraint equation in equation (1.4) with equation (1.6), the optimal solution is obtained to satisfy the following condition:
order theThe problem of the CEM algorithm is converted into a linear equation solving problem, and the equation set equation is listed as:
ax=b (1.8) from equation (1.8), an error function is constructed as:
e=ax-B (1.9) when the theoretical solution X exists for formula (1.9), there is e=0; it can be concluded from this that when e=0, the theoretical solution X can be obtained, and in order to be able to perform iterative operations, the variables need to be discretized, so the continuous error function (1.9) needs to be sampled as:
E k ＝A k X k -B k (1.10)
where k is denoted as kth sample and the sampling interval is kτ.
Further, the step S2 specifically includes: for E in the traditional processing model k Conduct derivative and E k Derivative with respect to time
The first two terms of the taylor series of function E are used to find the approximate root of equation e=0, combining equation (2.11), for E k Taylor second order expansion is performed at t=kτ, resulting in:
the higher order infinitely small residual term O (τ) when the sampling interval τ is infinitely small 2 ) Negligible, obtained by shifting the term of formula (2.12) and substituting formula (2.11):
E k ＝-A k (X k+1 -X k ) (2.13)
the iteration formula of the Newton-Lawson iteration method is finally obtained as follows:
X k+1 ＝X k -A k -1 E k (2.14)
since 2 norms are the square root of the maximum eigenvalue of the product of the matrix and its transposed conjugate matrix, represent the linear distance of two vector matrices in space; that is, the accuracy of algorithm convergence can be determined according to the 2 norms, when the norms are smaller, the algorithm output result is closer to the theoretical value, the algorithm convergence is stronger, the error 2 norms of the error are smaller, the error 2 norms of the algorithm are smaller, and the error 2 norms are obtained according to the formula (2.12):
the accuracy of Newton's method is therefore O (τ);
based on equation (2.11), to suppress noise, an integral term is added, which can be expressed as:
where alpha >0 is a scale factor, combined with (2.16) and (2.12), anti-noise Newton-Lawson iterations
The algorithm can be expressed as:
further, the step S3 specifically includes: theoretical analysis and simulation verification.
Further, the theoretical analysis includes: the accuracy and the convergence of the anti-noise Newton-Lawson iterative algorithm are analyzed, the accuracy of the algorithm can be judged by analyzing the error function convergence value of the anti-noise Newton-Lawson iterative algorithm, and the smaller the error function convergence value is, the higher the algorithm accuracy is; the convergence of the anti-noise Newton-Lawson iterative algorithm can be analyzed by combining the error function convergence time; in addition, the performance of the anti-noise Newton-Lawson iterative algorithm under different noise environments is also researched, and in the noise environments, the smaller the error function is, the smaller the influence of noise on the algorithm model is represented, namely the stronger the anti-noise performance is; prior to these analyses, the expression for the construction error function is transformed first against the noise newton-radson iterative algorithm; disturbance epsilon by noise from (2.16) k The contaminated anti-noise newton-raffinon iterative algorithm can be equivalently expressed as:
correspondingly, the anti-noise Newton-Lawson iterative algorithm can be rewritten as:
the term transfer by formula (3.2) can be obtained:
combining formulas (2.11) and (3.3) to obtain:
will beThe Euler forward differential expansion is used to obtain:
further, the simulation verification specifically includes: and processing and analyzing the specific example by using an anti-noise Newton-Lawson iteration model (3.2) and a Newton-Lawson iteration model (2.14), namely reading original image data into MATLAB software, finally obtaining filter parameters by solving an optimization problem with equality constraint, and performing binarization processing after the image is filtered by the filter.
Compared with the prior art, the invention has the beneficial effects that: the invention provides an anti-noise Newton-Lawson iterative algorithm model applied to the problem of small target detection under noise interference on the basis of the Newton-Lawson iterative algorithm model; the small target extraction problem is converted into a linear matrix equation set solution problem by combining a CEM algorithm based on a linear mixed model, the matrix equation can be solved and small target detection can be completed by utilizing an anti-noise Newton-Lawson iterative algorithm model, finally simulation experiments for surface water source distribution observation and analysis are carried out by using MATLAB software, the anti-noise Newton-Lawson iterative algorithm can successfully extract the small target in four different noise environments, the Newton-Lawson iterative algorithm has a lot of noise in an image under the noisy condition, and the small target extraction can fail; in terms of algorithm precision, the anti-noise Newton-Lawson iterative algorithm also has higher precision under different noises, and algorithm errors are smaller, so that feasibility of applying the anti-noise Newton-Lawson iterative algorithm to a small target extraction problem is demonstrated, and good performance is achieved in terms of noise suppression.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings that are needed for the description of the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of the present invention;
fig. 2 is a diagram of the detection and extraction result of the distribution of the surface water source when τ=0.001 s by the noise-free newton-radson iterative algorithm, (a) satellite original diagram; (b) an image filtered by a filter; (c) filtering the binarized image; (d) final object detection of the extracted contour image;
fig. 3 is a graph of the detection and extraction results of the anti-noise newton-raffinon iterative algorithm and the surface water source distribution detection and extraction result of the newton-raffinon iterative algorithm when the constant noise epsilon=10 is present, and (a) a graph of the small target detection and extraction result of the anti-noise newton-raffinon iterative algorithm; (b) A small target detection extraction effect diagram of the Newton-Lawson iterative algorithm; (c) Detecting an extracted error map by using an anti-noise Newton-Lawson iterative algorithm small target; (d) Detecting and extracting an error map by using a Newton-Lawson iterative algorithm small target;
fig. 4 is a graph of the detection and extraction results of the anti-noise newton-radson iterative algorithm and the newton-radson iterative algorithm on the surface water source distribution when linear noise epsilon=kτ is present, and (a) a graph of the small target detection and extraction results of the anti-noise newton-radson iterative algorithm; (b) A small target detection extraction effect diagram of the Newton-Lawson iterative algorithm; (c) Detecting an extracted error map by using an anti-noise Newton-Lawson iterative algorithm small target; (d) Detecting and extracting an error map by using a Newton-Lawson iterative algorithm small target;
fig. 5 is a graph of the detection and extraction results of the anti-noise newton-radson iterative algorithm and the newton-radson iterative algorithm on the surface water source distribution when τ=0.001 s in the presence of bounded random noise epsilonm=5, (a) a graph of the small target detection and extraction results of the anti-noise newton-radson iterative algorithm; (b) A small target detection extraction effect diagram of the Newton-Lawson iterative algorithm; (c) Detecting an extracted error map by using an anti-noise Newton-Lawson iterative algorithm small target; (d) And detecting and extracting an error map by using a Newton-Lawson iterative algorithm small target.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
In the description of the present invention, it should be understood that the terms "upper," "lower," "front," "rear," "left," "right," "top," "bottom," "inner," "outer," and the like indicate or are based on the orientation or positional relationship shown in the drawings, merely to facilitate description of the present invention and to simplify the description, and do not indicate or imply that the devices or elements referred to must have a specific orientation, be configured and operated in a specific orientation, and thus should not be construed as limiting the present invention.
Referring to fig. 1, the present invention provides a technical solution: a novel anti-noise satellite image small target detection algorithm comprises the following steps:
s1: processing the original image to obtain a traditional processing model;
s2: improving a traditional processing model to obtain an algorithm model with anti-noise performance;
s3: and analyzing and verifying the accuracy of the model.
The step S1 specifically includes: let the multispectral sample set be omega i ＝[Ω 1 ，Ω 2 ，...，Ω N ] T N represents the total number of pixels in the image, and i is more than or equal to 1 and less than or equal to N; wherein each pixel Ω i Is a P-dimensional vector; p is the number of bands in which,the target spectrum signal to be detected is used as the known information, and a filter coefficient is designed as delta= [ delta ] 1 ,δ 2 ,...,δ P ] T Is of finite impulse response of (a)The linear filter can keep the target spectrum output with a certain size after the spectrum signal passes through the filter, and the background spectrum is restrained, so that the filter can meet the constraint of the following conditions:
assume that spectral data Ω is input i Obtaining corresponding output data as O through a finite impulse response linear filter i Can be expressed as:
corresponding to all inputs Ω i After passing through the finite impulse response linear filter, the average output energy is:
wherein the method comprises the steps ofFor the autocorrelation matrix, the constraint condition of (1.1) and the average output energy minimization of the formula (1.3) are combined to form the main problem of the CEM algorithm, and meanwhile, the problem of minimum value is converted, and the formula can be expressed as:
this problem is used as an optimization problem, and by using the Lagrangian multiplier method, the expression can be written as:
L(δ,μ)＝δ T Rδ+μ(w T δ-1)(1.5)
where μ is the Lagrangian multiplier, both of which and δ are unknowns, let the derivative of formula (1.5) equal to 0:
combining the constraint equation in equation (1.4) with equation (1.6), the optimal solution is obtained to satisfy the following condition:
order theThe problem of the CEM algorithm is converted into a linear equation solving problem, and the equation set equation is listed as:
ax=b (1.8) from equation (1.8), an error function is constructed as:
e=ax-B (1.9) when the theoretical solution X exists for formula (1.9), there is e=0; it can be concluded from this that when e=0, the theoretical solution X can be obtained, and in order to be able to perform iterative operations, the variables need to be discretized, so the continuous error function (1.9) needs to be sampled as:
E k ＝A k X k -B k (1.10)
where k is denoted as kth sample and the sampling interval is kτ.
The step S2 specifically comprises the following steps: for E in the traditional processing model k Conduct derivative and E k Time-to-time guidance
Number of digits
The first two terms of the taylor series of function E are used to find the approximate root of equation e=0, combining equation (2.11), for E k Taylor second order expansion is performed at t=kτ, resulting in:
the higher order infinitely small residual term O (τ) when the sampling interval τ is infinitely small 2 ) Negligible, obtained by shifting the term of formula (2.12) and substituting formula (2.11):
E k ＝-A k (X k+1 -X k ) (2.13)
the iteration formula of the Newton-Lawson iteration method is finally obtained as follows:
X k+1 ＝X k -A k -1 E k (2.14)
since 2 norms are the square root of the maximum eigenvalue of the product of the matrix and its transposed conjugate matrix, represent the linear distance of two vector matrices in space; that is, the accuracy of algorithm convergence can be determined according to the 2 norms, when the norms are smaller, the algorithm output result is closer to the theoretical value, the algorithm convergence is stronger, the error 2 norms of the error are smaller, the error 2 norms of the algorithm are smaller, and the error 2 norms are obtained according to the formula (2.12):
the accuracy of Newton's method is therefore O (τ);
based on equation (2.11), to suppress noise, an integral term is added, which can be expressed as:
where alpha >0 is a scale factor, combined with (2.16) and (2.12), anti-noise Newton-Lawson iterations
The algorithm can be expressed as:
the step S3 specifically comprises the following steps: theoretical analysis and simulation verification.
Theoretical analysis includes: the accuracy and the convergence of the anti-noise Newton-Lawson iterative algorithm are analyzed, the accuracy of the algorithm can be judged by analyzing the error function convergence value of the anti-noise Newton-Lawson iterative algorithm, and the smaller the error function convergence value is, the higher the algorithm accuracy is; the convergence of the anti-noise Newton-Lawson iterative algorithm can be analyzed by combining the error function convergence time; in addition, the performance of the anti-noise Newton-Lawson iterative algorithm under different noise environments is also researched, and in the noise environments, the smaller the error function is, the smaller the influence of noise on the algorithm model is represented, namely the stronger the anti-noise performance is; prior to these analyses, the expression for the construction error function is transformed first against the noise newton-radson iterative algorithm; disturbance epsilon by noise from (2.16) k The contaminated anti-noise newton-raffinon iterative algorithm can be equivalently expressed as:
correspondingly, the anti-noise Newton-Lawson iterative algorithm can be rewritten as:
the term transfer by formula (3.2) can be obtained:
combining formulas (2.11) and (3.3) to obtain:
will beThe Euler forward differential expansion is used to obtain:
the simulation verification is specifically as follows: processing and analyzing a specific example by using an anti-noise Newton-Lawson iteration model (3.2) and a Newton-Lawson iteration model (2.14), namely reading original image data into MATLAB software, finally obtaining filter parameters by solving an optimization problem with equality constraint, filtering an image by using the filter, and performing binarization processing to intercept an RGB satellite map near the mountain north town of Zhanzhang area in Zhanzhangquan in Guangdong province in China from a God map, detecting and extracting a surface water source by using the satellite image, and obtaining spectral information w= [0.2549 of the surface water source by using the satellite map; 0.3059;0.3686 it uses Newton's method and anti-noise Newton-Lawson iterative algorithm to detect and extract the target object under different noise working environments.
Example 1
Noise-free convergence and noise-free simulation:
the 2 norms of the error can be proved to be converged, so that the anti-noise Newton-Lawson iterative algorithm (3.2) can be proved to be converged under the condition, and theory 1 is provided on the basis.
Theory 1: solving a linear matrix equation by using an anti-noise Newton-Lawson iterative algorithm, and when 0<ατ 2
<At 1, 2 norms of error E 2 Converging to O (tau) 2 )。
And (3) proving: the jth subsystem that is noiseless at the (k+1) th sampling instant (i.e., epsilon=0) that can be obtained by equation (3.5) can be expressed as:
similarly, the value of the jth subsystem of equation (3.5) at the kth sampling instant may be expressed as:
then, subtracting (3.7) from equation (3.6) yields:
E j k+1 ＝βE j k +Ο(τ 2 ) (3.8)
wherein β=1- ατ 2 . According to (3.8) there is
For example 0 < beta < 1, i.e. 0 < alpha tau 2 < 1, there are:
thus, there is obtained:
that is, the model convergence under the noise-free condition is proved; and through the analysis, the anti-noise Newton-Lawson iterative algorithm (3.2) is proved to solve the dynamic matrix equation, and the initial state X 0 Converging to the vicinity of the theoretical solution, the residual error is O (τ 2 ) The method comprises the steps of carrying out a first treatment on the surface of the In addition, the anti-noise Newton-Lawson iterative algorithm (3.2) can improve the accuracy to O (τ) compared to the Newton-Lawson iterative algorithm (2.14) with the accuracy of O (τ) 2 )；
Under the condition of no noise, the noise-resistant Newton-Lawson iterative algorithm is used for carrying out surface water source distribution detection extraction, a result image is shown in fig. 2 (a), sampling intervals tau=0.001 s, beta=0.1 are set, original image data are read into MATLAB software, filter parameters are finally obtained by solving an optimization problem with equality constraint, a satellite image becomes fig. 2 (b) after filtering by the filter, in order to make the image become clearer, the fig. 2 (b) is subjected to binarization processing to obtain fig. 2 (c), and meanwhile, the outline of a white area is drawn based on the fig. 2 (c) to obtain fig. 2 (d); by comparing fig. 2 (a) with fig. 2 (c), we can clearly obtain the distribution diagram of the surface water source, and the surface water source with smaller area in the original diagram can also be clearly shown in fig. 2 (c).
Example two
Robustness to constant noise and simulation under constant noise:
in order to prove the anti-noise Newton-Lawson iterative algorithm (3.2) anti-interference performance under constant noise, a constant noise interference term is added after an error function, and the 2 norm of the error can be proved to be converged, so that the anti-noise Newton-Lawson iterative algorithm (3.2) can be proved to be converged under the constant noise condition, and theory 2 is proposed on the basis.
Theory 2: solving a linear matrix equation by using an anti-noise Newton-Lawson iterative algorithm under constant noise, wherein the error is 2 norms E 2 Converging to O (tau) 2 )；
And (3) proving: the anti-noise newton-levenson iterative algorithm (3.2) that is subject to constant noise interference can be expressed as:
E j k+1 ＝βE j k +τε+Ο(τ 2 ) (3.11)
similar to proof of theory 1, the difference between the expression of the (k+1) th sampling time and the kth sampling time of the jth subsystem of the formula (3.11) can be expressed as:
E j k+1 ＝βE j k +Ο(τ 2 ) (3.12)
the expression is the same as that of formula (3.8), and therefore, it proves similar to the above proof of convergence;
it is emphasized that the steady state residual of the anti-noise newton-lawsonia iterative algorithm (3.2) is independent of the constant noise, i.e. no matter how large the constant noise is, the accuracy of the anti-noise newton-lawsonia iterative algorithm is always O (τ 2 )；
The anti-noise Newton-Lawson iterative model (3.2) and the Newton-Lawson iterative model (2.14) are used for carrying out filtering treatment on the satellite picture under constant noise disturbance, and water source information is detected and extracted to compare the anti-noise performance of the two algorithms; and by comparing anti-noise Newton-Lawson iterative modelsThe convergence capacity of the residual error judging algorithm of the Newton-Lawson iterative model (2.14) and the Newton-Lawson iterative algorithm under constant noise, the noise immunity Newton-Lawson iterative algorithm and the Newton-Lawson iterative algorithm surface water source distribution detection extraction results are shown in fig. 3 when constant noise epsilon=10 is interfered, and the sampling interval tau=0.001 s and beta=0.1 is set. Comparing fig. 3 (a), fig. 3 (b) with fig. 2 (a), fig. 2 (c) shows that the anti-noise newton-lawsonian iterative algorithm has excellent anti-noise performance for constant noise, the small target detection extraction image under constant noise interference is the same as that when no noise exists, and the newton-lawsonian iterative algorithm has no anti-noise performance, so that the generated image has huge difference from the original image, and effective information cannot be obtained from the generated image; the difference of the two algorithms under constant noise can be seen from the precision of solving the optimization problem, and by comparing the FIG. 3 (c) with the FIG. 3 (d), we can see that the anti-noise Newton-Lawson iterative algorithm has the precision of 0.4, while the anti-noise Newton-Lawson iterative algorithm has the precision of 10 -15 The error is negligible, and the error solved by the Newton-Lawson iterative algorithm is larger, so that the small target detection and extraction fail.
Example III
Robustness to linear noise and simulation under linear noise:
in order to prove the anti-noise performance of the anti-noise Newton-Lawson iterative algorithm (3.2) under the linear noise, a linear noise interference term is added after an error function, and the Newton-Lawson iterative algorithm (3.2) can be proved to be converged by proving that the 2 norm of the error can be converged under the condition, and theory 3 is proposed on the basis.
Theory 3: with anti-noise Newton-Lawson iteration (3.2) algorithm in the presence of linear noise ε k Solving the linear matrix equation when =kτζ+δ, the 2-norm of the error converges to
And (3) proving: noise-resistant Newton-Lawson iterative algorithm (3.2) perturbed by linear noise kτζ
Expressed as:
E k+1 ＝βE k +kτ 2 ζ+τδ+Ο(τ 2 ) (3.13)
like proof of theory 1, there is a rule given by equation (3.13):
E j k+1 ＝βE j k +τ 2 ζ+Ο(τ 2 ) (3.14)
then, reverse iteration is carried out, and the following steps are obtained:
therefore, when k→infinity, there are:
the 2-norm of the error therefore eventually converges to:
for linear noise, high accuracy of the anti-noise newton-larson iterative algorithm (3.2) can be ensured by increasing the scale factor α, and in addition, in the case of ζ=0, the linear noise can be regarded as constant noise, the accuracy of which is approximately O (τ 2 ) The correctness of theory 2 is proved;
the anti-noise Newton-Lawson iterative model (3.2) and the anti-noise Newton-Lawson iterative model (2.14) are used for carrying out filtering treatment on the satellite picture under the condition of linear noise disturbance, and water source information is detected and extracted to compare the anti-noise performance of the two algorithms; and the convergence capacity of the algorithm under linear noise is judged by comparing the residual errors of the anti-noise Newton-Lawson iteration model (3.2) and the anti-noise Newton-Lawson iteration model (2.14); the anti-noise Newton-Lawson iterative algorithm and Newton-Lawson iterative algorithm detect and extract the surface water source distribution when the linear noise epsilon=kτ is interfered, as shown in fig. 4, the sampling interval tau=0.001 s is set, and beta=0.1; the pair of FIG. 4 (a), FIG. 4 (b) and FIG. 2 (a), FIG. 2 (c)The anti-noise Newton-Lawson iterative algorithm has excellent anti-noise performance on linear noise, and the small target detection extraction image under the interference of the linear noise is the same as that of the noise-free image, but the Newton-Lawson iterative algorithm has no anti-noise performance, so that errors are dispersed, and effective information cannot be obtained from the image (b) of FIG. 4; the difference between the two algorithms under linear noise can also be seen from the accuracy of solving the optimization problem, and by comparing FIG. 4 (c) with FIG. 4 (d), we can see that the accuracy of the anti-noise Newton-Lawson iterative algorithm is about 10 -5 The error is negligible; the error of the Newton-Lawson iterative algorithm gradually increases along with the increase of linear noise, and the error solved under the interference of the linear noise is larger, so that the small target detection and extraction fail.
Example IV
Robustness of bounded random noise and simulation under bounded random noise
In order to prove the anti-noise performance of the anti-noise Newton-Lawson iteration (3.2) under the bounded random noise, a theory 4 is provided on the basis of the anti-noise performance of the anti-noise Newton-Lawson iteration algorithm (3.2) by adding a bounded random noise interference term after an error function and by proving that the 2 norm of the error can be converged so as to prove that the anti-noise Newton-Lawson iteration algorithm (3.2) can be converged under the condition.
Theory 4: solving linear matrix equation by anti-noise Newton-Lawson iterative algorithm, and bounded random noise epsilon k ∈(-ε m ,ε m ) Pollution, wherein ε m Is the boundary of the random noise and, error 2-norm E 2 The upper boundary is
And (3) proving: starting from equation (3.5), the anti-noise newton-rafreson iterative algorithm (3.2) subject to bounded random noise interference can be expressed as:
E k+1 ＝βE k +τε k +Ο(τ 2 ) (3.17)
similarly, formula (3.17) can be rewritten:
E j k+1 ＝βE j k +τΔε k +Ο(τ 2 ) (3.18)
wherein Δε k ＝ε k -ε k-1 ≤2ε m . Thus, performing the reverse iteration based on equations (3.18) and (3.19) results in:
therefore, when k→infinity, there are:
the 2-norm of the error therefore eventually converges to:
the model keeps good anti-interference performance, and the performance of the algorithm is prevented from being greatly discounted or even disabled under the condition of interference;
the anti-noise Newton-Lawson iterative model (3.2) and the Newton-Lawson iterative model (2.14) are used for carrying out filtering treatment on satellite pictures under the disturbance of bounded random noise, and water source information is detected and extracted to compare the anti-noise performance of two algorithms; the convergence capacity of a residual error judgment algorithm under the bounded random noise is compared with that of an anti-noise Newton-Lafreon iteration model (3.2) and a Newton-Lafreon iteration model (2.14); bounded random noise epsilon m The detection and extraction results of the anti-noise newton-radson iterative algorithm and the newton-radson iterative algorithm when interference is=5 are shown in fig. 5, and the sampling interval τ=0.001 s and β=0.1 are set; comparing FIG. 5 (a), FIG. 5 (b) with FIG. 2 (a), FIG. 2 (c) shows that the anti-noise Newton-Lawson iterative algorithm has excellent anti-noise performance for bounded random noise, the small target detection extraction image under the interference of the bounded random noise is basically the same as that under the noise-free condition, and the Newton-Lawson iterative algorithmBecause the anti-noise performance is not provided, the small target detection extraction image is more influenced by noise; the difference between the two algorithms under bounded random noise can also be seen from the accuracy of solving the optimization problem, and by comparing FIG. 4 (c) with FIG. 4 (d), we can see that the accuracy of the anti-noise Newton-Lawson iteration is about 10 -2 The anti-noise Newton-Lawson iterative algorithm is slightly superior to Newton-Lawson iterative algorithm in resisting random noise because the error solved by Newton-Lawson iterative algorithm is smaller.
In the description of the present specification, the descriptions of the terms "one embodiment," "example," "specific example," and the like, mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The preferred embodiments of the invention disclosed above are intended only to assist in the explanation of the invention. The preferred embodiments are not exhaustive or to limit the invention to the precise form disclosed. Obviously, many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described in order to best explain the principles of the invention and the practical application, to thereby enable others skilled in the art to best understand and utilize the invention. The invention is limited only by the claims and the full scope and equivalents thereof.
Claims (4)
1. The anti-noise satellite image small target detection method is characterized by comprising the following steps of:
s1: processing the original image to obtain a traditional processing model;
the method comprises the following steps: let the multispectral sample set be omega i ＝[Ω 1 ，Ω 2 ，...，Ω N ] T N represents the total number of pixels in the image, and i is more than or equal to 1 and less than or equal to N; wherein the method comprises the steps ofEach pixel omega i Is a P-dimensional vector; p is the number of bands in which,the target spectrum signal to be detected is used as the known information, and a filter coefficient is designed as delta= [ delta ] 1 ,δ 2 ,...,δ P ] T After the spectrum signal passes through the filter, the target spectrum can still keep a certain output, and the background spectrum is restrained, so the filter can meet the constraint of the following conditions:
assume that spectral data Ω is input i Obtaining corresponding output data as O through a finite impulse response linear filter i Can be expressed as:
corresponding to all inputs Ω i After passing through the finite impulse response linear filter, the average output energy is:
wherein the method comprises the steps ofFor the autocorrelation matrix, the constraint condition of (1.1) and the average output energy minimization of the formula (1.3) are combined to form the main problem of the CEM algorithm, and meanwhile, the problem of minimum value is converted, and the formula can be expressed as:
this problem is used as an optimization problem, and by using the Lagrangian multiplier method, the expression can be written as:
L(δ,μ)＝δ T Rδ+μ(w T δ-1) (1.5)
where μ is the Lagrangian multiplier, both of which and δ are unknowns, let the derivative of formula (1.5) equal to 0:
combining the constraint equation in equation (1.4) with equation (1.6), the optimal solution is obtained to satisfy the following condition:
order theThe problem of the CEM algorithm is converted into a linear equation solving problem, and the equation set equation is listed as:
AX＝B (1.8)
from equation (1.8), an error function is constructed as:
E＝AX-B (1.9)
when the theoretical solution X exists for formula (1.9), there is e=0; it can be concluded from this that when e=0, the theoretical solution X can be obtained, and in order to be able to perform iterative operations, the variables need to be discretized, so the continuous error function (1.9) needs to be sampled as:
E k ＝A k X k -B k (1.10)
where k is denoted as the kth sample and the sampling interval is kτ;
s2: improving a traditional processing model to obtain an algorithm model with anti-noise performance;
the method comprises the following steps: for E in the traditional processing model k Conduct the derivation andE k derivative with respect to time
The first two terms of the taylor series of function E are used to find the approximate root of equation e=0, combining equation (2.11), for E k Taylor second order expansion is performed at t=kτ, resulting in:
high-order infinitely small residual terms O (τ) when the sampling interval τ is infinitely small 2 ) Negligible, obtained by shifting the term of formula (2.12) and substituting formula (2.11):
E k ＝-A k (X k+1 -X k ) (2.13)
the iteration formula of the Newton-Lawson iteration method is finally obtained as follows:
X k+1 ＝X k -A k -1 E k (2.14)
since 2 norms are the square root of the maximum eigenvalue of the product of the matrix and its transposed conjugate matrix, represent the linear distance of two vector matrices in space; that is, the accuracy of algorithm convergence can be determined according to the 2 norms, when the norms are smaller, the algorithm output result is closer to the theoretical value, the algorithm convergence is stronger, the error 2 norms of the error are smaller, the error 2 norms of the algorithm are smaller, and the error 2 norms are obtained according to the formula (2.12):
the accuracy of Newton's method is therefore O (τ);
based on equation (2.11), to suppress noise, an integral term is added, which can be expressed as:
where α >0 is a scale factor, and in combination (2.16) and (2.12), the anti-noise Newton-Lawson iterative algorithm can be expressed as:
s3: and analyzing and verifying the accuracy of the model.
2. The method for detecting the small target of the anti-noise satellite image according to claim 1, wherein the method comprises the following steps: the step S3 specifically comprises the following steps: theoretical analysis and simulation verification.
3. The method for detecting the small target of the anti-noise satellite image according to claim 2, wherein the method comprises the following steps: the theoretical analysis includes: the accuracy and the convergence of the anti-noise Newton-Lawson iterative algorithm are analyzed, the accuracy of the algorithm can be judged by analyzing the error function convergence value of the anti-noise Newton-Lawson iterative algorithm, and the smaller the error function convergence value is, the higher the algorithm accuracy is; the convergence of the anti-noise Newton-Lawson iterative algorithm can be analyzed by combining the error function convergence time; in addition, the performance of the anti-noise Newton-Lawson iterative algorithm under different noise environments is also researched, and in the noise environments, the smaller the error function is, the smaller the influence of noise on the algorithm model is represented, namely the stronger the anti-noise performance is; prior to these analyses, the expression for the construction error function is transformed first against the noise newton-radson iterative algorithm; disturbance epsilon by noise from (2.16) k The contaminated anti-noise newton-raffinon iterative algorithm can be equivalently expressed as:
correspondingly, the anti-noise Newton-Lawson iterative algorithm can be rewritten as:
the term transfer by formula (3.2) can be obtained:
combining formulas (2.11) and (3.3) to obtain:
will beThe Euler forward differential expansion is used to obtain:
4. the method for detecting the small target of the anti-noise satellite image according to claim 2, wherein the method comprises the following steps: the simulation verification is specifically as follows: and processing and analyzing the specific example by using an anti-noise Newton-Lawson iteration model (3.2) and a Newton-Lawson iteration model (2.14), namely reading original image data into MATLAB software, finally obtaining filter parameters by solving an optimization problem with equality constraint, and performing binarization processing after the image is filtered by the filter.
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