CN112131994A - Novel anti-noise satellite image small target detection algorithm - 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 an 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 adding an integral term into the Newton-Laverson iterative algorithm model by utilizing the integral control principle of a control theory based on the Newton-Laverson iterative algorithm model so as to provide an anti-noise Newton-Laverson iterative algorithm model applied to the small target detection problem under the noise interference; and then, the convergence of the algorithm under different noise conditions is verified respectively, the feasibility of the algorithm is proved theoretically, finally, a corresponding simulation experiment is carried out, the effectiveness of the proposed anti-noise Newton-Laverson iterative algorithm model is verified, the model is applied to the image processing problem, and the small target detection problem in the satellite image is effectively solved.
Description
Technical Field
The invention relates to the technical field of image processing, in particular to a novel anti-noise satellite image small target detection algorithm.
Background
With the development of earth science and the advancement of technology, human beings have been unable to search the earth merely by looking at drawings and the like. Satellite image technology is gaining more attention as it collects information at high altitude, greatly enhancing the ability of human beings to view the earth. The satellite image enables a human to observe a large area from a high place in a short time, and a plurality of target-related observation data materials are obtained during observation. The study and observation of the satellite images simultaneously can update and acquire the latest data of the observation target in a relatively short time, so that dynamic monitoring can be performed according to the change of the data. The satellite images digitize the information data, which is easier to digitize than traditional maps. The satellite image technology provides a more economic choice for human to observe and research nature in different fields by virtue of the characteristics of wide range, short period, strong data comparability and the like. Due to the great improvement of the computing speed of computer hardware, the birth of a large data set and the development of deep learning, the performance of the target detection algorithm based on the neural network is more excellent. 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 affairs, 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 objects in satellite images has been a difficult problem in image processing. Small objects, whether in number or size, tend to occupy a small proportion of the picture as compared to larger objects. Just because of the low resolution of small objects, the images are relatively blurry and carry too little information to make them poorly characterizable. In addition, the target detection of the current image is mainly the detection of a large target, so that the situation that a small target is passively ignored occurs, and once the situation occurs, the precision of the target detection by people is greatly reduced, and even the target detection fails. However, sometimes, key information is hidden in small targets, and researchers can better interpret images by means of the small targets, so that in some applications, the small targets have certain research value in image recognition, and Harsanyi proposes a CEM algorithm which can reduce interference in other directions as long as characteristic target vectors are extracted. 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 used for performing matching calculation on pixels to be detected to find the target. The target detection algorithm needs prior information of a target spectrum, so that the target detection algorithm has a good detection effect. Even if some wave bands of the target spectrum are changed greatly or the energy of all the wave bands is increased or reduced, the hyperspectral data has a plurality of different wave bands, and the whole wave band direction of the target spectrum is not influenced, namely the detection result is not greatly influenced. Therefore, the algorithm can restrain the background information and keep the target pixel with larger output, thereby effectively separating the target spectrum from the background spectrum. By observation, we can find that solving the CEM model is essentially an optimization problem, and the solution is converted into a problem of solving a linear equation set by a lagrange multiplier method, and an algorithm for 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 Raphson Iterative (NRI) and deformation of Newton method, and the like. Many of the problems in small target detection can be summarized as solving optimization problems with a matrix equation set. The small target detection problem can be combined with a Constrained Energy Minimization (CEM) method based on a linear hybrid model to convert the small target detection problem into a linear matrix equation system to solve the optimization problem. Noise is inevitably generated in the computer operation process, but the influence of the noise is not considered in many methods, and the performance is greatly reduced or even fails under the condition of interference. Therefore, a new method capable of suppressing the noise influence needs to be researched, and based on the new method, a novel anti-noise satellite image small target detection algorithm is designed to solve the 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 purpose, the invention provides the following technical scheme: a novel anti-noise satellite image small target detection algorithm comprises the following steps:
s1: processing an 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 is specifically: let the multispectral sample set be omegai＝[Ω1，Ω2，...，ΩN]TN represents the total number of the total pixels in the image, and i is more than or equal to 1 and less than or equal to N; wherein each pixel element omegaiIs a P-dimensional vector; p is the number of the wave bands,
the target spectrum signal to be detected is used as known information, and a filter coefficient is designed to be ═1,2,...,P]TAfter the spectrum signal passes through the filter, the target spectrum can still keep a certain output, and the background spectrum is suppressed, so the filter should satisfy the following constraint:
assuming input spectral data omegaiObtaining corresponding output data O through a finite impulse response linear filteriCan be expressed as:
corresponding to all inputs omegaiAfter passing through the finite impulse response linear filter, the average output energy is:
wherein
For the autocorrelation matrix, combining (1.1) this constraint condition with the minimum average output energy of formula (1.3) constitutes the main problem of the CEM algorithm, and at the same time, the minimum is solved, and the formula can be expressed as:
this problem is used as an optimization problem, and the lagrangian multiplier method is applied, and the expression can be written as:
L(,μ)＝TR+μ(wT-1) (1.5)
where μ is the Lagrangian multiplier, and both are unknowns, making the derivative of equation (1.5) equal to 0 yields:
combining the constraint condition formula in the formula (1.4) and the formula (1.6), obtaining an optimal solution which satisfies the following conditions:
order to
Calculating the CEM
The problem of the method is converted into a linear equation solving problem, and the equation set equation is listed as:
AX＝B (1.8)
from equation (1.8), the error function is constructed as:
E＝AX-B (1.9)
when formula (1.9) has a theoretical solution of X, then E ═ 0; from this, it is possible to obtain the theoretical solution X when E is 0, and to perform iterative operation, it is necessary to discretize the variables, so that the continuous error function (1.9) needs to be sampled as follows:
Ek＝AkXk-Bk (1.10)
where k is denoted as the kth sample and the sampling interval is k tau.
Further, the step S2 is specifically: for E in the conventional processing modelkConducting a derivation, and EkDerivative with respect to time
Finding the approximate root of equation E-0 using the first two terms of the Taylor series of function E, in combination with equation (2.11), for EkTaylor second order expansion is performed at t τ, yielding:
high order infinitesimal residual term O (tau) when the sampling interval tau is infinitesimal2) Negligible, obtained by transposing the formula (2.12) and substituting the formula (2.11):
Ek＝-Ak(Xk+1-Xk) (2.13)
the iteration formula of the Newton-Laverson iteration method is obtained finally as follows:
Xk+1＝Xk-Ak -1Ek (2.14)
because the 2 norm is the square root value of the maximum characteristic root of the product of the matrix and the transposed conjugate matrix thereof, the linear distance between two vector matrixes in space is represented; that is, the accuracy of algorithm convergence can be judged according to the 2 norm, the smaller the norm is, the closer the algorithm output result is to the theoretical value is, the stronger the convergence of the algorithm is, the smaller the 2 norm of the error is, the smaller the error of the algorithm is, and the 2 norm of the error at this time can be obtained according to the equation (2.12):
therefore, the accuracy of Newton's method is O (tau);
based on equation (2.11), in order to suppress noise, an integral term is added, which can be expressed as:
where α >0 is a scaling factor, combining (2.16) and (2.12), the antinoise newton-raphson iterative algorithm can be expressed as:
further, the step S3 is specifically: theoretical analysis and simulation verification.
Further, the theoretical analysis comprises: the precision and the convergence of the anti-noise Newton-Laverson iterative algorithm are analyzed, the precision of the algorithm can be judged by analyzing the error function convergence value of the anti-noise Newton-Laverson iterative algorithm, and the smaller the error function convergence value is, the higher the algorithm precision is; the convergence of the anti-noise Newton-Laverson iterative algorithm can be analyzed by combining the convergence time of the error function; in addition, the performance of the anti-noise Newton-Laverson iterative algorithm in different noise environments is researched, and in the noise environment, the smaller the error function is, the smaller the influence of noise on the algorithm model is, namely the stronger the anti-noise performance is; before the analysis, firstly, converting the anti-noise Newton-Laverson iterative algorithm for constructing an expression of an error function; disturbed by noise by equation (2.16)kThe noise-immune newton-raphson iterative algorithm of pollution can be equivalently expressed as:
correspondingly, the anti-noise newton-raphson iterative algorithm can be rewritten as:
the term is transposed from equation (3.2) to yield:
the combined formulae (2.11) and (3.3) give:
will be provided with
Using euler forward differential expansion yields:
further, the simulation verification specifically includes: and (3) processing and analyzing the specific example by using an anti-noise Newton-Laverson iteration model (3.2) and a Newton-Laverson iteration model (2.14), namely reading original image data into MATLAB software, finally obtaining filter parameters by solving an optimization problem with equation constraint, and carrying out binarization processing on the image after filtering by using a filter.
Compared with the prior art, the invention has the beneficial effects that: the invention provides an anti-noise Newton-Laverson iterative algorithm model applied to the small target detection problem under noise interference on the basis of the Newton-Laverson iterative algorithm model; converting the small target extraction problem into a linear matrix equation system solving problem by combining a CEM algorithm based on a linear mixed model, solving the matrix equation by using an anti-noise Newton-Laverson iterative algorithm model and completing small target detection, and finally performing a simulation experiment on surface water source distribution observation and analysis by using MATLAB software to conclude that the anti-noise Newton-Laverson iterative algorithm can successfully extract the small target under four different noise environments, and the Newton-Laverson iterative algorithm has a lot of noise in an image under the noisy condition, so that the small target extraction can fail; in the aspect of algorithm precision, the anti-noise Newton-Laverson iterative algorithm has higher precision under different noises, and the algorithm error is smaller, so that the feasibility of applying the anti-noise Newton-Laverson iterative algorithm to a small target extraction problem is proved, and the anti-noise Newton-Laverson iterative algorithm has good performance in the aspect 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 used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flow chart of the present invention;
fig. 2 is a graph of a time-surface water source distribution detection extraction result in a noise-free anti-noise newton-raphson iterative algorithm at τ of 0.001s, (a) a satellite original graph; (b) an image filtered by a filter; (c) filtering the image, and performing binarization processing on the filtered image; (d) finally detecting the extracted contour image by the target;
fig. 3 is a graph of a time-table water source distribution detection extraction result when the constant noise is 10, and the anti-noise newton-raphson iterative algorithm and the newton-raphson iterative algorithm are 0.001s, and (a) a small target detection extraction effect graph of the anti-noise newton-raphson iterative algorithm; (b) a small target detection and extraction effect graph of a Newton-Laverson iterative algorithm; (c) detecting an extracted error map of the small target by using an anti-noise Newton-Raphson iterative algorithm; (d) detecting an extracted error map of the small target by a Newton-Raphson iterative algorithm;
fig. 4 is a graph of the anti-noise newton-raphson iterative algorithm and the newton-raphson iterative algorithm extracted result of the water source distribution detection at τ of 0.001s in the presence of linear noise τ, (a) a graph of the small target detection extraction effect of the anti-noise newton-raphson iterative algorithm; (b) a small target detection and extraction effect graph of a Newton-Laverson iterative algorithm; (c) detecting an extracted error map of the small target by using an anti-noise Newton-Raphson iterative algorithm; (d) detecting an extracted error map of the small target by a Newton-Raphson iterative algorithm;
fig. 5 is a graph showing the water source distribution detection extraction result when the bounded random noise m is 5 and the anti-noise newton-raphson iterative algorithm and the newton-raphson iterative algorithm are t 0.001s, (a) a small target detection extraction effect graph of the anti-noise newton-raphson iterative algorithm; (b) a small target detection and extraction effect graph of a Newton-Laverson iterative algorithm; (c) detecting an extracted error map of the small target by using an anti-noise Newton-Raphson iterative algorithm; (d) and detecting the extracted error graph by using a Newton-Laverson iterative algorithm small target.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In the description of the present invention, it is to be understood that the terms "upper", "lower", "front", "rear", "left", "right", "top", "bottom", "inner", "outer", and the like, indicate orientations or positional relationships based on the orientations or positional relationships shown in the drawings, are merely for convenience in describing the present invention and simplifying the description, and do not indicate or imply that the device or element being referred to must have a particular orientation, be constructed and operated in a particular 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 an 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.
Wherein, step S1 specifically includes: let the multispectral sample set be omegai＝[Ω1，Ω2，...，ΩN]TN represents the total number of the total pixels in the image, and i is more than or equal to 1 and less than or equal to N; wherein each pixel element omegaiIs a P-dimensional vector; p is the number of the wave bands,
the target spectrum signal to be detected is used as known information, and a filter coefficient is designed to be ═1,2,...,P]TAfter the spectrum signal passes through the filter, the target spectrum can still keep a certain output, and the background spectrum is suppressed, so the filter should satisfy the following constraint:
assuming input spectral data omegaiObtaining corresponding output data O through a finite impulse response linear filteriCan be expressed as:
corresponding to all inputs omegaiAfter passing through the finite impulse response linear filter, the average output energy is:
wherein
For the autocorrelation matrix, combining the constraint condition (1.1) and the minimum average output energy of the formula (1.3) forms the main problem of the CEM algorithm, and simultaneously, the main problem is converted into the minimum calculationThe problem, formulated as:
this problem is used as an optimization problem, and the lagrangian multiplier method is applied, and the expression can be written as:
L(,μ)＝TR+μ(wT-1) (1.5)
where μ is the Lagrangian multiplier, and both are unknowns, making the derivative of equation (1.5) equal to 0 yields:
combining the constraint condition formula in the formula (1.4) and the formula (1.6), obtaining an optimal solution which satisfies the following conditions:
order to
Calculating the CEM
The problem of the method is converted into a linear equation solving problem, and the equation set equation is listed as:
AX＝B (1.8)
from equation (1.8), the error function is constructed as:
E＝AX-B (1.9)
when formula (1.9) has a theoretical solution of X, then E ═ 0; from this, it is possible to obtain the theoretical solution X when E is 0, and to perform iterative operation, it is necessary to discretize the variables, so that the continuous error function (1.9) needs to be sampled as follows:
Ek＝AkXk-Bk (1.10)
where k is denoted as the kth sample and the sampling interval is k tau.
Step S2 specifically includes: for E in the conventional processing modelkConducting a derivation, and EkDerivative with respect to time
Finding the approximate root of equation E-0 using the first two terms of the Taylor series of function E, in combination with equation (2.11), for EkTaylor second order expansion is performed at t τ, yielding:
high order infinitesimal residual term O (tau) when the sampling interval tau is infinitesimal2) Negligible, obtained by transposing the formula (2.12) and substituting the formula (2.11):
Ek＝-Ak(Xk+1-Xk) (2.13)
the iteration formula of the Newton-Laverson iteration method is obtained finally as follows:
Xk+1＝Xk-Ak -1Ek (2.14)
because the 2 norm is the square root value of the maximum characteristic root of the product of the matrix and the transposed conjugate matrix thereof, the linear distance between two vector matrixes in space is represented; that is, the accuracy of algorithm convergence can be judged according to the 2 norm, the smaller the norm is, the closer the algorithm output result is to the theoretical value is, the stronger the convergence of the algorithm is, the smaller the 2 norm of the error is, the smaller the error of the algorithm is, and the 2 norm of the error at this time can be obtained according to the equation (2.12):
therefore, the accuracy of Newton's method is O (tau);
based on equation (2.11), in order to suppress noise, an integral term is added, which can be expressed as:
where α >0 is a scaling factor, combining (2.16) and (2.12), the antinoise newton-raphson iterative algorithm can be expressed as:
step S3 specifically includes: theoretical analysis and simulation verification.
The theoretical analysis comprises the following steps: the precision and the convergence of the anti-noise Newton-Laverson iterative algorithm are analyzed, the precision of the algorithm can be judged by analyzing the error function convergence value of the anti-noise Newton-Laverson iterative algorithm, and the smaller the error function convergence value is, the higher the algorithm precision is; the convergence of the anti-noise Newton-Laverson iterative algorithm can be analyzed by combining the convergence time of the error function; in addition, the performance of the anti-noise Newton-Laverson iterative algorithm in different noise environments is researched, and in the noise environment, the smaller the error function is, the smaller the influence of noise on the algorithm model is, namely the stronger the anti-noise performance is; before the analysis, firstly, converting the anti-noise Newton-Laverson iterative algorithm for constructing an expression of an error function; disturbed by noise by equation (2.16)kThe noise-immune newton-raphson iterative algorithm of pollution can be equivalently expressed as:
correspondingly, the anti-noise newton-raphson iterative algorithm can be rewritten as:
the term is transposed from equation (3.2) to yield:
the combined formulae (2.11) and (3.3) give:
will be provided with
Using euler forward differential expansion yields:
the simulation verification specifically comprises the following steps: the method comprises the steps of processing and analyzing specific examples by using an anti-noise Newton-Laverson iteration model (3.2) and a Newton-Laverson iteration model (2.14), namely reading original image data into MATLAB software, obtaining filter parameters by solving an optimization problem with equation constraint, filtering the image by using a filter, and then performing binarization processing to the image so as to intercept an RGB satellite map near the North town of the Magnapus region of Zyojiang province of Guangdong province in China from a Gaode map, detecting and extracting a surface water source by using the satellite image, wherein spectral information w of the surface water source can be obtained by the satellite map [ 0.2549; 0.3059, respectively; 0.3686], the target object is detected and extracted under different noise working environments by using Newton method and anti-noise Newton-Raphson iterative algorithm.
Example one
Noise-free convergence and noise-free lower simulation:
a theory 1 is provided on the basis of proving that the anti-noise Newton-Laverson iterative algorithm (3.2) can be converged under the condition by proving that the 2 norm of the error can be converged.
Theory 1: solving linear matrix equation by using anti-noise Newton-Raffson iterative algorithm, when 0<ατ2<1, 2 norms of error E2Converge to O (tau)2)。
And (3) proving that: the j-th subsystem, which can be derived from equation (3.5) to be noiseless (i.e. when 0) at the (k +1) -th sampling instant, can be represented as:
similarly, the value of the j-th subsystem of equation (3.5) at the k-th sampling instant can be expressed as:
then, subtracting (3.7) from equation (3.6) yields:
Ej k+1＝βEj k+O(τ2) (3.8)
wherein β ═ 1- α τ2. According to (3.8) have
E.g. 0 < beta < 1, i.e. 0 < alpha tau2< 1, having:
thus, the following results:
i.e. proving model convergence in the noise-free case; and through the analysis, the anti-noise Newton-Laverson iterative algorithm (3.2) is used for solving the dynamic matrix equation, and the initial state X0Converge to the vicinity of the theoretical solution and have a residual error of O (τ)2) (ii) a Furthermore, the noise-resistant Newton-Rafferson iterative algorithm (3.2) can improve the accuracy to O (τ) compared to the Newton-Rafferson iterative algorithm (2.14) with the accuracy of O (τ)2)；
Under the noiseless condition, an image of a result obtained by performing surface water source distribution detection and extraction by using an anti-noise Newton-Laverson iterative algorithm is shown in fig. 2(a), a sampling interval tau is set to be 0.001s, and beta is set to be 0.1, original image data is read into MATLAB software, filter parameters are finally obtained by solving an optimization problem of band equation constraint, a satellite image is filtered by a filter to be changed into a fig. 2(b), in order to enable the image to be clearer, binarization processing is performed on the fig. 2(b) to obtain a fig. 2(c), and meanwhile, the outline of a white area is drawn based on the fig. 2(c) to obtain a fig. 2 (d); by comparing fig. 2(a) and 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 image can also be clearly shown in fig. 2 (c).
Example two
Robustness to constant noise and constant noise lower simulation:
in order to prove the anti-interference performance of the anti-noise Newton-Raphson iterative algorithm (3.2) under constant noise, a constant noise interference term is added after an error function, and the 2 norm convergence of the error is proved, so that the anti-noise Newton-Raphson iterative algorithm (3.2) can be proved to be converged under the constant noise condition, and the theory 2 is provided on the basis.
Theory 2: solving a linear matrix equation by using an anti-noise Newton-Laverson iterative algorithm under constant noise, wherein 2 norms of errors are | | | E | | sweet2Converge to O (tau)2)；
And (3) proving that: the anti-noise newton-raphson iterative algorithm (3.2) disturbed by constant noise can be expressed as:
Ej k+1＝βEj k+τ+O(τ2) (3.11)
similar to the proof of theory 1, the difference between the expressions for the (k +1) th sampling instant and the kth sampling instant for the jth subsystem of equation (3.11) can be expressed as:
Ej k+1＝βEj k+O(τ2) (3.12)
its expression is the same as that of expression (3.8), and therefore, it proves similar to the above-mentioned demonstration of convergence; it is emphasized that noise immunity newton-rapher is achievedThe steady-state residual of the forest iteration algorithm (3.2) is independent of the constant noise, i.e. the accuracy of the anti-noise Newton-Rapfson iteration algorithm is always O (τ) no matter how large the constant noise is2)；
Filtering the satellite picture under constant noise disturbance by respectively utilizing an anti-noise Newton-Laverson iteration model (3.2) and a Newton-Laverson iteration model (2.14), and detecting and extracting water source information to compare the anti-noise performance of the two algorithms; and by comparing the convergence capacity of the anti-noise Newton-Laverson iterative model (3.2) and the convergence capacity of the Newton-Laverson iterative model (2.14) under constant noise, the anti-noise Newton-Laverson iterative algorithm and the Newton-Laverson iterative algorithm are used for detecting and extracting the surface water source distribution when the constant noise is 10 interference, as shown in FIG. 3, the sampling interval tau is set to 0.001s, and the beta is set to 0.1. Comparing fig. 3(a), fig. 3(b) with fig. 2(a) and fig. 2(c), it can be seen that the anti-noise newton-raphson iterative algorithm has excellent anti-noise performance for constant noise, the small target detection extracted image under constant noise interference is the same as that in the case of no noise, and the newton-raphson iterative algorithm has no anti-noise performance, and the generated image has a large difference with the original image, and effective information cannot be obtained from the image; the difference of the two algorithms under constant noise can be seen from the precision of the optimization problem, and by comparing fig. 3(c) with fig. 3(d), we can see that the precision of the anti-noise newton-raphson iterative algorithm is 0.4, and the precision of the anti-noise newton-raphson iterative algorithm reaches 10-15The error can be ignored, and the error solved by the Newton-Laverson iterative algorithm is large, so that the small target detection and extraction are failed.
EXAMPLE III
Robustness to linear noise and linear noise-down simulation:
in order to prove the anti-interference performance of the anti-noise Newton-Laverson iterative algorithm (3.2) under the linear noise, a linear noise interference term is added after an error function, and 2 norms of errors are proved to be converged, so that the Newton-Laverson iterative algorithm (3.2) is proved to be converged under the condition, and theory 3 is provided on the basis.
Theory 3: using noise-resistant cattleTun-Lafferson iteration (3.2) algorithm in presence of linear noisekWhen k τ ζ + is solved for the linear matrix equation, the 2 norm of the error converges to
And (3) proving that: the anti-noise newton-raphson iterative algorithm (3.2) perturbed by the linear noise k τ ζ may be expressed as:
Ek+1＝βEk+kτ2ζ+τ+O(τ2) (3.13)
similar to the proof of theory 1, there is the following from equation (3.13):
Ej k+1＝βEj k+τ2ζ+O(τ2) (3.14)
then, reverse iteration is carried out, and the following results can be obtained:
therefore, when k → ∞, there are:
the 2 norm of the error therefore eventually converges to:
for linear noise, the high accuracy of the anti-noise newton-raphson iterative algorithm (3.2) can be ensured by increasing the scale factor α, and in addition, in the case where ζ is 0, the linear noise can be regarded as constant noise whose accuracy is approximately O (τ) or more2) The correctness of theory 2 is proved;
the satellite picture is filtered under the disturbance of linear noise by respectively utilizing an anti-noise Newton-Laverson iterative model (3.2) and an anti-noise Newton-Laverson iterative model (2.14), and water source information is detected and extracted to compare two algorithmsNoise immunity performance; and judging the convergence capacity of the algorithm under the linear noise by comparing the residual errors of the anti-noise Newton-Laverson iterative model (3.2) and the anti-noise Newton-Laverson iterative model (2.14); the anti-noise Newton-Laverson iterative algorithm and the Newton-Laverson iterative algorithm are used for detecting and extracting the distribution of the surface water source when the linear noise is interfered by k tau, the result is shown in figure 4, the sampling interval tau is set to be 0.001s, and the beta is set to be 0.1; comparing fig. 4(a), fig. 4(b) with fig. 2(a) and fig. 2(c), it can be seen that the anti-noise newton-raphson iterative algorithm has excellent anti-noise performance for linear noise, and an extracted image of a small target under linear noise interference is the same as that in the case of no noise, while the newton-raphson iterative algorithm has no anti-noise performance, and an error diverges, so that effective information cannot be obtained from fig. 4 (b); 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) and fig. 4(d), we can see that the accuracy of the anti-noise newton-raphson iterative algorithm is about 10-5Errors are negligible; the error of the Newton-Raffson 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 are failed.
Example four
Robustness and under-bounded random noise simulation
In order to prove the anti-noise Newton-Laverson iteration (3.2) in the anti-interference performance under the bounded random noise, a bounded random noise interference term is added after an error function, and 2 norms of errors are proved to be converged, so that the anti-noise Newton-Laverson iteration algorithm (3.2) is proved to be converged under the condition, and theory 4 is provided on the basis.
Theory 4: solving linear matrix equation by using anti-noise Newton-Raffson iterative algorithm, subject to bounded random noisek∈(-m,m) Contamination of whereinmIs the boundary of random noise, 2 norms of error E2Upper bound of
And (3) proving that: starting from equation (3.5), the anti-noise newton-raphson iterative algorithm (3.2) disturbed by the bounded random noise can be expressed as:
Ek+1＝βEk+τk+O(τ2) (3.17)
similarly, equation (3.17) can be rewritten as:
Ej k+1＝βEj k+τΔk+O(τ2) (3.18)
wherein Δk＝k-k-1≤2m. Therefore, performing a reverse iteration based on equations (3.18) and (3.19) results in:
therefore, when k → ∞, there are:
the 2 norm of the error therefore eventually converges to:
the model keeps better anti-interference performance, and avoids the performance of the algorithm from being greatly discounted or even losing efficacy under the condition of interference;
an anti-noise Newton-Laverson iteration model (3.2) and a Newton-Laverson iteration model (2.14) are respectively utilized to carry out filtering processing on a satellite picture under bounded random noise disturbance, and water source information is detected and extracted to compare the anti-noise performance of the two algorithms; and judging the convergence capacity of the algorithm under the bounded random noise by comparing the residual errors of the anti-noise Newton-Laverson iteration model (3.2) and the Newton-Laverson iteration model (2.14); in bounded random noisemAnti-noise Newton-Laverson iterative algorithm and Newton-Laverson iterative algorithm when interference is 5, and the water source is expressed when tau is 0.001sAs shown in fig. 5, the cloth detection extraction result is set with a sampling interval τ of 0.001s and β of 0.1; comparing fig. 5(a), fig. 5(b) with fig. 2(a) and fig. 2(c), it can be seen that the anti-noise newton-raphson iterative algorithm has excellent anti-noise performance for bounded random noise, and the small target detection extracted image under the interference of the bounded random noise is basically the same as that without noise, while the newton-raphson iterative algorithm has no anti-noise performance, and the small target detection extracted image is influenced by noise more; the difference between the two algorithms under bounded random noise can also be seen from the accuracy with which the optimization problem is solved, by comparing fig. 4(c) and fig. 4(d), we can see that the accuracy of the anti-noise newton-raphson iteration is about 10-2Compared with the error solved by the Newton-Laverson iterative algorithm, the anti-noise Newton-Laverson iterative algorithm is slightly superior to the Newton-Laverson iterative algorithm in the part resisting random noise.
In the description herein, references to the description of "one embodiment," "an example," "a specific example" or the like are intended to 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 invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. 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 to be illustrative only. The preferred embodiments are not intended to be exhaustive or to limit the invention to the precise embodiments 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 utilize the invention. The invention is limited only by the claims and their full scope and equivalents.
Claims (6)
1. A novel anti-noise satellite image small target detection algorithm is characterized by comprising the following steps:
s1: processing an 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.
2. The novel anti-noise satellite image small target detection algorithm according to claim 1, characterized in that: the step S1 specifically includes: let the multispectral sample set be omegai＝[Ω1，Ω2，...，ΩN]TN represents the total number of the total pixels in the image, and i is more than or equal to 1 and less than or equal to N; wherein each pixel element omegaiIs a P-dimensional vector; p is the number of the wave bands,
the target spectrum signal to be detected is used as known information, and a filter coefficient is designed to be ═1,2,...,P]TAfter the spectrum signal passes through the filter, the target spectrum can still keep a certain output, and the background spectrum is suppressed, so the filter should satisfy the following constraint:
assuming input spectral data omegaiObtaining corresponding output data O through a finite impulse response linear filteriCan be expressed as:
corresponding to all inputs omegaiAfter passing through the finite impulse response linear filter, the average output energy is:
wherein
For the autocorrelation matrix, combine (1.1) this constraint with equation (1.3)
The minimum average output energy constitutes the main problem of the CEM algorithm, and is converted into the problem of solving the minimum value, which can be expressed by the formula:
this problem is used as an optimization problem, and the lagrangian multiplier method is applied, and the expression can be written as:
L(,μ)＝TR+μ(wT-1) (1.5)
where μ is the Lagrangian multiplier, and both are unknowns, making the derivative of equation (1.5) equal to 0 yields:
combining the constraint condition formula in the formula (1.4) and the formula (1.6), obtaining an optimal solution which satisfies the following conditions:
order to
Converting the problem of the CEM algorithm into a linear equation solving problem, and arranging the equation set equation as:
AX＝B (1.8)
from equation (1.8), the error function is constructed as:
E＝AX-B (1.9)
when formula (1.9) has a theoretical solution of X, then E ═ 0; from this, it is possible to obtain the theoretical solution X when E is 0, and to perform iterative operation, it is necessary to discretize the variables, so that the continuous error function (1.9) needs to be sampled as follows:
Ek＝AkXk-Bk (1.10)
where k is denoted as the kth sample and the sampling interval is k tau.
3. The novel anti-noise satellite image small target detection algorithm according to claim 1, characterized in that: the step S2 specifically includes: for E in the conventional processing modelkConducting a derivation, and EkDerivative with respect to time
Finding the approximate root of equation E-0 using the first two terms of the Taylor series of function E, in combination with equation (2.11), for EkTaylor second order expansion is performed at t τ, yielding:
high order infinitesimal residual term O (tau) when the sampling interval tau is infinitesimal2) Negligible, obtained by transposing the formula (2.12) and substituting the formula (2.11):
Ek＝-Ak(Xk+1-Xk) (2.13)
the iteration formula of the Newton-Laverson iteration method is obtained finally as follows:
Xk+1＝Xk-Ak -1Ek (2.14)
because the 2 norm is the square root value of the maximum characteristic root of the product of the matrix and the transposed conjugate matrix thereof, the linear distance between two vector matrixes in space is represented; that is, the accuracy of algorithm convergence can be judged according to the 2 norm, the smaller the norm is, the closer the algorithm output result is to the theoretical value is, the stronger the convergence of the algorithm is, the smaller the 2 norm of the error is, the smaller the error of the algorithm is, and the 2 norm of the error at this time can be obtained according to the equation (2.12):
therefore, the accuracy of Newton's method is O (tau);
based on equation (2.11), in order to suppress noise, an integral term is added, which can be expressed as:
where α >0 is a scaling factor, combining (2.16) and (2.12), the antinoise newton-raphson iterative algorithm can be expressed as:
4. the novel anti-noise satellite image small target detection algorithm according to claim 1, characterized in that: the step S3 specifically includes: theoretical analysis and simulation verification.
5. The novel anti-noise satellite image small target detection algorithm according to claim 4, characterized in that: the theoretical analysis comprises the following steps: the precision and the convergence of the anti-noise Newton-Laverson iterative algorithm are analyzed, the precision of the algorithm can be judged by analyzing the error function convergence value of the anti-noise Newton-Laverson iterative algorithm, and the smaller the error function convergence value is, the higher the algorithm precision is; the anti-noise Newton-Lafferson can be analyzed by combining the convergence time of the error functionConvergence of the iterative algorithm; in addition, the performance of the anti-noise Newton-Laverson iterative algorithm in different noise environments is researched, and in the noise environment, the smaller the error function is, the smaller the influence of noise on the algorithm model is, namely the stronger the anti-noise performance is; before the analysis, firstly, converting the anti-noise Newton-Laverson iterative algorithm for constructing an expression of an error function; disturbed by noise by equation (2.16)kThe noise-immune newton-raphson iterative algorithm of pollution can be equivalently expressed as:
correspondingly, the anti-noise newton-raphson iterative algorithm can be rewritten as:
the term is transposed from equation (3.2) to yield:
the combined formulae (2.11) and (3.3) give:
will be provided with
Using euler forward differential expansion yields:
6. the novel anti-noise satellite image small target detection algorithm according to claim 4, characterized in that: the simulation verification specifically comprises the following steps: and (3) processing and analyzing the specific example by using an anti-noise Newton-Laverson iteration model (3.2) and a Newton-Laverson iteration model (2.14), namely reading original image data into MATLAB software, finally obtaining filter parameters by solving an optimization problem with equation constraint, and carrying out binarization processing on the image after filtering by using a filter.
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