A method, system and medium for verifying the reversibility of non-linear transformation of vector maps

Abstract

The present invention provides a method for verifying the reversibility of non - linear transformation of vector maps. The vector map is processed by a non - linear transformation algorithm that meets the condition constraints to obtain a corresponding offset map. The obtained offset map and the non - linear transformation algorithm are received as inputs, and the successive approximation iteration method is used for security verification to restore the original map coordinate information. The number of iterations is recorded, and the security of the non - linear transformation algorithm is judged by the number of iterations. In addition, a security verification method based on linear interpolation is also provided. Using the method provided by the present invention, the reversibility of non - linear transformation of vector maps can be effectively verified, providing an important basis for the numerical analysis of the reversibility of non - linear transformation of vector maps.

Description

Technical Field

[0001] The present invention belongs to the technical field of geographic information security, and in particular relates to a method, system and medium for verifying the reversibility of non-linear transformation of vector maps. Background Art

[0002] In recent years, with the rapid development of technologies such as cloud computing, big data, artificial intelligence, and 5G, the application scope of geographic information has become increasingly broad, giving rise to many new business forms, new products, and new applications. Therefore, it is necessary to focus on developing geographic information confidentiality processing technologies that meet the application requirements of the new development stage.

[0003] Geographic data has specific structural characteristics and accuracy requirements, and is usually divided into vector data, raster data, digital elevation models, 3D models, real-scene data, and navigation electronic maps, etc. Geographic information confidentiality processing technology is a key technology to ensure the secure application of confidential geographic information. It mainly uses specific means to perform decryption processing on the spatial location, accuracy, attributes, adjacent relationships, etc. of confidential geographic information through offset, deformation, forgery, hiding, etc., which is of great significance for maintaining geographic information security and promoting the healthy development of the geographic information industry.

[0004] Non-linear transformation of vector maps is an important technical means for geographic information confidentiality processing. By non-linear coordinate transformation or random perturbation to increase errors, the purpose is to reduce the accuracy of the data plane position and achieve decryption processing. For geographic information confidentiality processing technology, the processed result must meet the actual application and be irreversible, achieving a balance between security and usability. Security means that the non-linear transformation of the vector map has no inverse function, is irreversible, and the complexity of numerical analysis for reverse calculation is high enough so that the time cost and space cost exceed the tolerance of the cracker; usability means that the vector data after non-linear transformation can meet the basic requirements of map applications, and the geometric shape, topological relationship, etc. should remain consistent before and after the transformation.

[0005] At present, the main research directions of non-linear transformation of vector maps include non-linear transformation of vector maps based on feature points or region division, geometric accuracy decryption models of vector data based on trigonometric functions or Chebyshev polynomials, decryption methods of vector data based on bilinear interpolation or grid coordinate transformation, etc. There is no research on geographic information confidentiality processing for numerical analysis of the reversibility of non-linear transformation of vector maps at home and abroad. Summary of the Invention

[0006] As an important geographic information confidentiality processing technology, the security of non-linear transformation of vector maps is related to the security of confidential geographic information. The present invention provides a method, system and medium for verifying the reversibility of non-linear transformation of vector maps, aiming to effectively obtain the security performance of geographic information confidentiality processing, which has relatively important reference value for geographic information confidentiality processing.

[0007] The first aspect of the present invention discloses a method for verifying the reversibility of non - linear transformation of a vector map, and the method includes:

[0008] Step S11: Process the vector map through a non - linear transformation algorithm that meets the condition constraints to obtain a corresponding offset map;

[0009] Step S12: Receive the offset map and the non - linear transformation algorithm obtained in step S11 as inputs;

[0010] Step S13: Based on the inputs in step S12, use the successive approximation iteration method for security verification, restore the original map coordinate information, and record the number of iterations;

[0011] Step S14: Determine the reversibility of the non - linear transformation of the vector map based on whether the number of iterations obtained in step S13 exceeds a threshold. If it exceeds the threshold, determine that the reversibility of the non - linear transformation of the vector map is weak and the security of the non - linear transformation algorithm is strong; otherwise, determine that the reversibility of the non - linear transformation of the vector map is strong and the security of the non - linear transformation algorithm is weak.

[0012] According to the method of the first aspect of the present invention, in step S11, the condition constraints include: Constraint 1: The map vectors maintain the same quantity and attributes; there is a one - to - one mapping relationship between the vectors in the map normed vector spaces V and W. The non - linear transformation cannot delete, add, split, or merge the vectors in V and cannot change the attributes of the vectors; Constraint 2: Maintain the same geometric features and topological relationships; after the non - linear transformation, the map normed vector space W should maintain the same geometric and topological relationships as V, and the vector offsets are smooth and continuous without mutations.

[0013] According to the method of the first aspect of the present invention, in step S13, based on the inputs in step S12, using the successive approximation iteration method for security verification, restoring the original map coordinate information, and recording the number of iterations specifically includes:

[0014] Step S131: The successive approximation iteration method is constructed based on the law that the non - linear transformation of adjacent points in a sufficiently small normed vector space is approximately a direction - preserving isometric transformation. Let the original Figure 1 For adjacent points a(x1, y1) and b(z2, w2), the corresponding points in the offset map are a'(T(x1), T(y1)) and b'(T(z2), T(w2)). Let u=(z2 - x1, w2 - y1), then T(u)=(T(z2)-T(x1), T(w2)-T(y1)). From the approximation of the direction - preserving isometric transformation, we can obtain:

[0015] ||T(u)||≈||u|| (1)

[0016]

[0017] Step S132: Since the height is highly similar in the u direction for T(u), we get:

[0018] x1 - z2 ≈ T(x1) - T(z2) (4)

[0019] y1 - w2 ≈ T(y1) - T(w2) (5)

[0020] Take the coordinates of point a' as the coordinate values of the adjacent point b of point a in the original map. The coordinates of point a' are located in a sufficiently small norm vector space centered at point a with a radius of ε max The coordinate values formed by the further non - linear transformation of point a' are used as the coordinate values of the offset map point b', denoted as (T(T(x1)), T(T(y1))); Use the non - linear transformation functions f(x) and f(y) to replace the non - linear transformations T(x) and T(y);

[0021] Then, from equations (4) and (5), the following equation is derived:

[0022] x1 - f(x1) ≈ f(x1) - f(f(x1)) (6)

[0023] y1 - f(y1) ≈ f(y1) - f(f(y1)) (7)

[0024] f(x1) and f(y1) are known and can be calculated through the map non - linear transformation algorithm. Solving equations (6) and (7) gives the approximate coordinate values of the original map point a as:

[0025] x1 ≈ 2f(x1) - f(f(x1)) (8)

[0026] y1 ≈ 2f(y1) - f(f(y1)) (9)

[0027] Step S133: In the map norm vector space, using the successive approximation method for the local part of the offset map, a high - precision approximate solution of the original map vector can be obtained. Take the numerical values obtained from equations (8) and (9) as the coordinates (x2, y2) of point a2. This point has an adjacent relationship with point a. Calculate the non - linear transformation values of point a2 to get f(x2) and f(y2), keeping f(x1) and f(y1) unchanged. Based on equations (4) and (5), solve to get another set of approximate coordinate values of point a as

[0028] x1 ≈ x2 + f(x1) - f(x2) (10)

[0029] y1 ≈ y2 + f(y1) - f(y2) (11)

[0030] Continue the iteration. Since a(x1, y1), a i (x i , y i ), where i is an integer not equal to 1, and they are a pair of adjacent points, then in a sufficiently small normed vector space, the following equation holds

[0031] x1 ≈ x i + f(x1) - f(x i ) (12)

[0032] y1 ≈ y i + f(y1) - f(y i ) (13)

[0033] Continuously repeat the iteration process until convergence, and we get

[0034] x1 = x n + f(x1) - f(x n ) (14)

[0035] y1 = y n + f(x1) - f(x n ) (15)

[0036] The right - hand sides of equations (14) and (15) are both known values, and the sum value can be calculated; let

[0037]

[0038] f(y1) - y1 = ω1

[0039] f(x n ) - x n = ω n

[0040] Then the right - hand sides of equations (14) and (15) become respectively

[0041]

[0042] y1+(ω1 - ω n ) (17)

[0043] Make ω1 - ω n = 0; then it is proved that the iteration has converged.

[0044] The second aspect of the present invention discloses a method for verifying the reversibility of non - linear transformation of a vector map, which is characterized in that the method includes:

[0045] Step S21: Process the vector map through a non - linear transformation algorithm that meets the condition constraints to obtain a corresponding offset map;

[0046] Step S22: Receive the obtained offset map in step S21 as input;

[0047] Step S23: Based on the input in step S22, use linear interpolation method for security verification, restore the original map coordinate information, and calculate the error between the original map coordinate values obtained by the linear interpolation method and the real map coordinate values;

[0048] Step S24: Determine the reversibility of the non - linear transformation of the vector map based on whether the error obtained in step S23 exceeds the threshold. If it exceeds the threshold, it is determined that the non - linear transformation of the vector map has strong reversibility and the security of the non - linear transformation algorithm is weak; otherwise, it is determined that the non - linear transformation of the vector map has weak reversibility and the security of the non - linear transformation algorithm is strong.

[0049] According to the method of the second aspect of the present invention, in step S21, the conditional constraints include: Constraint 1: The map vectors maintain the same quantity and attributes; there is a one - to - one mapping relationship between the vectors in the map normed vector spaces V and W. The non - linear transformation cannot delete, add, split, or merge the vectors in V and cannot change the attributes of the vectors; Constraint 2: Keep the geometric features and topological relationships consistent; after the non - linear transformation, the map normed vector space W should maintain the same geometric and topological relationships as V, and the vector offset is smooth and continuous without mutations.

[0050] According to the method of the second aspect of the present invention, in step S23, based on the input in step S22, using the linear interpolation method for security verification and restoring the original map coordinate information specifically includes:

[0051] Step S231: Divide the offset map into equidistant grids, and each grid is a square grid with a diagonal length of ε max ;

[0052] Step S232: Let the offset function be f(x). In each square grid, arbitrarily take two real points a'(f(x1), f(y1)) and b'(f(x2), f(y2)) as control points. Through measurement, the corresponding points on the original map are a(x1, y1) and b(x2, y2). Construct a coordinate system, where the x - axis direction is the x value of the original map and the y - axis direction is the f(x) value of the offset map;

[0053] Suppose point c(x, y) and points a, b are in the same small - scale map normed vector space. According to the Lagrange linear interpolation formula, there is

[0054]

[0055] For adjacent points a, b, according to the measurement results, there is Under the condition of maintaining a certain accuracy, we get

[0056] f(x2)-f(x1) = x2 - x1

[0057] Equation (19) simplifies to

[0058] x = x1 + f(x) - f(x1) (20)

[0059] The calculation in the y-axis direction is the same as that in the x-axis direction, and we get

[0060] y = y1 + f(y) - f(y1) (21)

[0061] The third aspect of the present invention discloses a system for verifying the reversibility of non-linear transformation of a vector map. The system includes:

[0062] A first processing module, configured to process a vector map through a non-linear transformation algorithm that meets condition constraints to obtain a corresponding offset map;

[0063] A second processing module, configured to receive the offset map and the non-linear transformation algorithm obtained in the first processing module as inputs;

[0064] A third processing module, configured to perform security verification based on the inputs of the second processing module by using the successive approximation iteration method, restore the original map coordinate information, and record the number of iterations;

[0065] A fourth processing module, based on whether the number of iterations obtained by the third processing module exceeds a threshold, determines the reversibility of the non-linear transformation of the vector map. If it exceeds the threshold, it determines that the reversibility of the non-linear transformation of the vector map is weak and the security of the non-linear transformation algorithm is strong; otherwise, it determines that the reversibility of the non-linear transformation of the vector map is strong and the security of the non-linear transformation algorithm is weak.

[0066] The fourth aspect of the present invention discloses a system for verifying the reversibility of non-linear transformation of a vector map, characterized in that the system includes:

[0067] A first processing module, configured to process a vector map through a non-linear transformation algorithm that meets condition constraints to obtain a corresponding offset map;

[0068] A second processing module, configured to receive the offset map obtained in the first processing module as an input;

[0069] A third processing module, configured to perform security verification based on the input of the second processing module by using the linear interpolation method, restore the original map coordinate information, and calculate the error between the original map coordinate values obtained by the linear interpolation method and the real map coordinate values;

[0070] The fourth processing module determines the reversibility of the non - linear transformation of the vector map based on whether the error obtained by the third processing module exceeds a threshold. If it exceeds the threshold, it is determined that the non - linear transformation of the vector map has strong reversibility and the security of the non - linear transformation algorithm is weak; otherwise, it is determined that the non - linear transformation of the vector map has weak reversibility and the security of the non - linear transformation algorithm is strong.

[0071] The fifth aspect of the present invention discloses an electronic device, which includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps in the method of the first aspect or the second aspect are implemented.

[0072] The sixth aspect of the present invention discloses a computer - readable storage medium storing computer - readable storage instructions for implementing the steps in the method of the first aspect or the second aspect.

[0073] In summary, the solution proposed by the present invention has the following technical effects:

[0074] To effectively verify the irreversibility of the non - linear transformation of the vector map, analyze the properties of orientation - preserving isometric transformation of the vector map as a normed vector space, analyze the implementation methods and mathematical bases of the inverse analytical solution and numerical solution, and construct an algorithm for the inverse solution accordingly. Verifying the irreversibility of the non - linear transformation software through the algorithm has the characteristics of high efficiency and accuracy, and has strong practical significance for ensuring the security of geographical information confidentiality processing technology and preventing the release of non - linear transformation software of vector maps with reversible risks, which may lead to potential leakage.

[0075] This application abstracts the vector map as a map normed vector space, adopts vector algebra and functional analysis tools, studies and verifies the numerical reversibility of the non - linear transformation of the vector map from the perspective of usability, and gives the successive approximation iteration method and the linear interpolation method, filling the gap in the security detection of geographical information confidentiality processing algorithms in China, and having relatively important reference value for the design of geographical information confidentiality processing technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following will briefly introduce the drawings required for the description of the specific embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0077] Figure 1 It is a process diagram for implementing the method for verifying the reversibility of the non - linear transformation of the vector map based on the successive approximation iteration method according to the present invention;

[0078] Figure 2 It is a process diagram for implementing the method of verifying the reversibility of the non - linear transformation of a vector map based on the linear interpolation method in the present invention;

[0079] Figure 3 It is the vector map vector in the present invention;

[0080] Figure 4 It is a schematic diagram of the addition of map vectors according to the present invention;

[0081] Figure 5 It is a schematic diagram of a map normed vector space small enough according to the present invention;

[0082] Figure 6 It is a schematic diagram of the relationship between the values of x and f(x) in the linear interpolation method according to the present invention;

[0083] Figure 7 It is a schematic diagram of the offset of the non - linear transformation of the vector map according to an embodiment of the present invention. Detailed implementation manners

[0084] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part rather than all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0085] In this application, the following technical terms are used:

[0086] Geographic information confidentiality processing technology: A technical method for de - classifying the spatial location accuracy, attribute content and their mutual relationships of confidential geographic information, which is of great significance for maintaining the security of geographic information and promoting the healthy development of the geographic information industry.

[0087] Non - linear transformation of vector map: An important technical means for geographic information confidentiality processing. By non - linear offset of coordinates or random perturbation to increase errors, the purpose of reducing the accuracy of the data plane position is achieved, and de - classification processing is realized.

[0088] Map normed vector space: A map vector space that satisfies the norm definition and its rules is called a map normed vector space. Let the vector map vector u=(x2 - x1, y2 - y1), where (x1, y1) and (x2, y2) are the coordinate values of the two endpoints of u respectively. Take the Euclidean distance of the vector as the norm, denoted as Then ||u|| satisfies:

[0089] (1) ||0|| = 0;

[0090] (2) ||ku|| = k||u||,

[0091] (3) ||u + v|| ≤ ||u|| + ||v||, u, v ∈ V.

[0092] Orientation-preserving isometry: Vectors in a normed vector space of a map have the property that their lengths and directions remain the same before and after a non-linear transformation.

[0093] Successive approximation iteration method: An iterative calculation method that constructs a recurrence formula for a sequence of numerical approximate solutions of an equation and proves that the limit of this sequence is the solution of the original equation.

[0094] Linear interpolation method: Refers to an interpolation method where the interpolation function is a first-degree polynomial, and its interpolation error at the interpolation nodes is zero. Linear interpolation can be used to approximately replace the original function.

[0095] As Figure 1 shown, the first embodiment provided by the present invention is a method for verifying the reversibility of a non-linear transformation of a vector map. The method includes:

[0096] Step S11: Process the vector map through a non-linear transformation algorithm that meets the condition constraints to obtain a corresponding offset map;

[0097] Among them, vector map data is composed of points, lines, and surfaces, and is usually stored and expressed as coordinate points in a rectangular coordinate system. A line is a line segment formed by connecting the starting and ending coordinate points, and a surface is a polygon formed by line segments connected end to end. In a rectangular coordinate system, points, line segments, and polygons can all be represented by one or a group of vectors starting from the origin. In this application, the points in the vector map are divided into real points and virtual points. Real points represent the points corresponding to actual ground objects on the vector map and constitute the points, lines, and surfaces of the map; virtual points represent the points corresponding to no actual ground objects on the vector map and are the blank points on the map. The real point vector of the map is the vector connecting the coordinate origin and the real point, and the virtual point vector of the map is the vector connecting the coordinate origin and the virtual point. These two types of vectors constitute the map vector space.

[0098] Next, it is proved that the map vector space is a normed vector space of the map. A vector space that satisfies the norm definition and its rules is called a normed vector space. As Figure 3 shown, let the vector of the vector map u = (x2 - x1, y2 - y1), and (x1, y1), (x2, y2) are the coordinate values of the two endpoints of u respectively. Take the Euclidean distance of the vector as the norm, denoted as Then ||u|| satisfies:

[0099] (1) ||0|| = 0; The norm of the zero vector located at the coordinate origin is zero;

[0100] (2) ||ku|| = k||u||,

[0101] Proof: As Figure 3 shown,

[0102] then

[0103] (3) ||u + v|| ≤ ||u|| + ||v||, u, v ∈ V.

[0104] Proof: As Figure 4 shown, construct u + v using the real - point vectors or imaginary - point vectors u, v of the map. u, v, and u + v form the three sides of a triangle, so (3) holds.

[0105] If the map vector norm ||u|| satisfies the above three conditions, then the map vector space is a map - normed vector space.

[0106] Let the vector map before the non - linear transformation (hereinafter referred to as the original map) be the map vector space V, and the vector map formed after the transformation (hereinafter referred to as the offset map) be the map vector space W. V and W respectively satisfy the above three conditions and are both map - normed vector spaces. The non - linear transformation does not change the basic property of the normed vector space, which is the basis of the analysis of the present invention.

[0107] The satisfied constraint conditions are based on the process that the non - linear transformation of the vector map is a mapping from the map - normed vector space V to W. To ensure usability, the real - point vectors of the vector map cannot be randomly and irregularly transformed, and should be subject to the following constraints:

[0108] Constraint 1: The map vectors maintain the same quantity and attributes. There is a one - to - one mapping relationship between the vectors of the map - normed vector spaces V and W. The non - linear transformation cannot delete, add, split, or merge the vectors of V, and cannot change the attributes of the vectors;

[0109] Constraint 2: The geometric features and topological relationships are maintained. After the non - linear transformation, the map - normed vector space W should be consistent with the geometric and topological relationships of V, and the vector offset is smooth and continuous without mutations.

[0110] Specifically, Constraint 1 is to ensure the orientation - preserving isometric transformation of a sufficiently small map - normed vector space. Under the restriction of the double - constraint conditions, the vectors in the map - normed vector space should maintain the same length and direction before and after the non - linear transformation, that is, approximately an orientation - preserving isometric transformation. For any two points a(x a , y a ) and b(x b , yb ), let u be the vector connecting points a and b, then

[0111] u = (x b - x a , y b - y a )

[0112] After non - linear transformation, assume that points a and b are respectively offset to points a' and b'. The offsets of these two points in the x - axis and y - axis directions are respectively (Δx a , Δy a ), (Δx b , Δy b ). Let u' be the vector connecting a' and b', then

[0113] u’ = (x b' - x a' , y b' - y a' ) = (x b - x a +(Δx b - Δx a ), y b - y a +(Δy b - Δy a ))

[0114] Considering the complexity of non - linear transformation, especially as the spatial scale increases, the orientation - preserving isometric transformation is more interfered. For example, the non - linear transformation parameters in different regions of the map may be different, resulting in a certain degree of change in the length and direction of the transformed vector. Then the following formula holds

[0115] u’≈u, Δx b ≈Δx a , Δy b ≈Δy a

[0116] The above will lead to a reduction in the accuracy of the inverse numerical solution of non - linear transformation. Therefore, this application adopts the method of performing inverse numerical analysis in a sufficiently small norm vector space.

[0117] If for any arbitrarily small positive real number ε, there is always ||u||≤ε, then b is called a neighboring point of a, and these two points can be infinitely close (but not coincident). This application uses the symbol b→a to represent.

[0118] Regardless of the map non - linear transformation algorithm adopted, from the analysis of the offset results, subject to the double - constraint conditions, the offset of the transformed vector is smooth and continuous, and can better maintain the topological relationship. Then the offsets of infinitely close points a and b in the x - axis and y - axis directions are infinitely close, that is

[0119]

[0120] Then

[0121] For the convenience of reversible numerical analysis, point a is selected as a real point, and point b is selected as a real point or an imaginary point. Two such points are used to construct the map vector u. u may not exist in the original map, and it is only an auxiliary tool for reversible numerical analysis. In reversible numerical analysis, both real points and imaginary points can calculate the corresponding offset points through a non-linear transformation algorithm. The offset calculation of imaginary points plays an auxiliary role in reversible numerical analysis.

[0122] As Figure 5 shown, now construct a sufficiently small map norm vector space V i . Construct a circle with the real point a as the center and an infinitesimal positive real number ε as the radius. The area covered by this circle is the sufficiently small map norm vector space V i , then

[0123]

[0124] The points in the sufficiently small norm vector space are all in a neighboring relationship. The non-linear transformation from the sufficiently small map norm vector space V i to the offset map sufficiently small norm vector space W i is denoted as Let Let u ∈ V i , and Tu represents the vector (such as u') generated by the non-linear transformation of the original map vector u. Tu ∈ W i . From the above analysis, it can be obtained that

[0125] Then

[0126] Then

[0127] Under the action of double constraints, to ensure the usability of the map, the non-linear transformation from the sufficiently small norm vector space V i to W i is highly approximated to a direction-preserving isometric transformation, so that the following equations hold under relatively high-precision conditions

[0128] Tu = u (1)

[0129] ||Tu|| = ||u|| (2)

[0130] Under the constraint of map usability, appropriately expand V iTo improve the efficiency of reversible computing, it is necessary and feasible to find the maximum value of ε. Let ε max represent the maximum value. For any point a on the original map and its offset point a', the Euclidean distance between the two that the user can accept should not exceed ε max (generally dozens of meters, and up to hundreds of meters in special cases). Taking the Earth as the reference system, relative to the coordinate value of point a, ε max is undoubtedly a very small value. Therefore, constructing a sufficiently small map norm vector space with ε max as the radius, its non-linear transformation still has a high approximation of orientation-preserving isometric transformation. Then

[0131]

[0132] For Constraint 2, specifically the injectivity and surjectivity of the non-linear transformation, the concept of the null space is first introduced. The null space of the non-linear transformation T of the vector map (denoted as Ker T) refers to the set of vectors in the original map norm vector space V that are transformed into 0 by T:

[0133] Ker T = {u ∈ V: Tu = 0}

[0134] From "Constraint 1", it can be seen that in the non-linear transformation of the vector map, there is no situation where geographical space elements such as points, line segments, and polygons on the original map are transformed into nothing. Only the zero vector of the map itself can be transformed into the zero vector. Therefore, Ker T = {0}. It should be noted that the zero vector cannot be equated with the virtual point vector defined in this application because the former has no corresponding object in the real world represented by the map, while the latter is the opposite. Therefore, the non-linear transformation T: V → W is injective.

[0135] Next, analyze the range of T. For the non-linear transformation T from the map norm vector space V to W, its range (denoted as ImT) is the set formed by all map vectors such as points, line segments, and polygons in V after transformation:

[0136] Im T = {Tu: u ∈ V, u ≠ 0, Tu ≠ 0}

[0137] From the "Constraint 1" and "Constraint 2" of the vector map transformation, it can be seen that there is a one-to-one mapping relationship between vectors in V and W, and the main characteristics of the geometric and topological relationships are maintained. The range of T is equal to the map norm vector space W, that is, Im T = W. Then T is surjective.

[0138] As can be seen from the above, T is both injective and surjective, which lays a mathematical foundation for its reversible numerical calculation

[18] . Otherwise, if the number of points, lines, and surfaces in W is more or less than that in V, and their topological relationships and geometric characteristics change significantly, it is impossible to obtain V from W through reversible numerical calculation.

[0139] Step S12: Receive the offset map and the non - linear transformation algorithm obtained in step S11 as inputs;

[0140] Step S13: Based on the inputs in step S12, use the successive approximation iteration method to perform security verification, restore and obtain the original map coordinate information, and record the number of iterations;

[0141] The successive approximation iteration method is constructed based on the law that the non - linear transformation of adjacent points in a sufficiently small norm vector space is highly approximately a orientation - preserving isometric transformation. Let the original Figure 1 For adjacent points a(x1, y1) and b(z2, w2), their corresponding points on the offset map are a'(T(x1), T(y1)) and b'(T(z2), T(w2)). Let u=(z2 - x1, w2 - y1), then T(u)=(T(z2)-T(x1), T(w2)-T(y1)). From the approximation of the orientation - preserving isometric transformation, we know that:

[0142] ||T(u)||≈||u||

[0143]

[0144] Since the directions of T(u) and u are highly similar, we get

[0145] x1 - z2≈T(x1)-T(z2) (3)

[0146] y1 - w2≈T(y1)-T(w2) (4)

[0147] Among them, T(x1), T(y1) and T(z2), T(w2) are located on the offset map and are known. We need to solve for x1, y1 and z2, w2. Obviously, the system of equations composed of equations (3) and (4) cannot obtain the coordinate values of points a and b. Without loss of generality, take the coordinates of point a' as the coordinate values of the adjacent point b of point a on the original map at the same time (the coordinates of point a' are located in a sufficiently small norm vector space with point a as the center and ε max as the radius), and take the coordinate values formed by the second non - linear transformation of point a' as the coordinate values of point b' on the offset map, denoted as (T(T(x1)), T(T(y1))). For the convenience of inverse solution, below we directly use the non - linear transformation functions f(x) and f(y) to replace the non - linear transformations T(x) and T(y).

[0148] Then, from equations (3) and (4), we can deduce the following formula:

[0149] x1 - f(x1)≈f(x1)-f(f(x1)) (5)

[0150] y1 - f(y1) ≈ f(y1) - f(f(y1)) (6)

[0151] f(x1) and f(y1) are known, and f(f(x1)) and f(f(y1)) can be obtained by calculating through map non - linear transformation software or algorithms. Solving equations (5) and (6) gives the approximate coordinate values of the original map point a as

[0152] x1 ≈ 2f(x1) - f(f(x1)) (7)

[0153] y1 ≈ 2f(y1) - f(f(y1)) (8)

[0154] According to the functional generalized inverse function theorem, in the normed vector space of the map, the successive approximation method is adopted for the local part of the offset map

[20] , and a high - precision approximate solution of the original map vector can be obtained. Therefore, taking the numerical values obtained from equations (7) and (8) as the coordinates (x2, y2) of point a2, this point has a neighboring relationship with point a. Calculate the non - linear transformation values of point a2 to obtain f(x2) and f(y2). Keeping f(x1) and f(y1) unchanged, and solving based on equations (3) and (4), another set of approximate coordinate values of point a is

[0155] x1 ≈ x2 + f(x1) - f(x2) (9)

[0156] y1 ≈ y2 + f(y1) - f(y2) (10)

[0157] Continue the iteration. Since a(x1, y1), a i (x i , y i )(i ∈ N and i ≠ 1) are a pair of neighboring points, then the following equation holds in a sufficiently small normed vector space

[0158] x1 ≈ x i + f(x1) - f(x i ) (11)

[0159] y1 ≈ y i + f(y1) - f(y i ) (12)

[0160] Keep repeating the iteration process until convergence, and we get

[0161] x1 = x n + f(x1) - f(x n ) (13)

[0162] y1 = y n + f(x1) - f(x n ) (14)

[0163] The computational complexity of the successive approximation iteration method is O(2M), where M is the computational complexity of the non-linear transformation algorithm. The convergence of the successive approximation iteration is demonstrated below.

[0164] The right-hand sides of Eqs. (13) and (14) are both known values, and the total value can be calculated. Let

[0165]

[0166] f(y1) - y1 = ω1

[0167] f(x n ) - x n = ω n

[0168] Then the right-hand sides of Eqs. (13) and (14) become

[0169]

[0170] y1 + (ω1 - ω n )(16)

[0171] Theoretically, in the normed vector space V of a sufficiently small map i there are infinitely many neighboring points (including virtual points) of the original map point a(x1, y1). By continuously performing successive approximation iteration, according to the analysis of the offset of neighboring points, it can be seen from Eqs. (1) and (2) that under certain accuracy conditions, 1 point a n (x n , y n ) can be found, and the offset values in the x-axis and y-axis directions before and after its transformation are the same as those of point a, so that ω1 - ω n = 0. Continuing the iteration, when the identity x1 ≡ x1 continuously appears, it proves that the iteration has converged.

[0172] From the analysis of Eqs. (15) and (16), the errors in the x-axis and y-axis directions of the i-th iteration calculation are respectively and ω1 - ω i . The iterative convergence rate in the x-axis direction is discussed below. Let Then

[0173]

[0174] Since it can be obtained that

[0175]

[0176] Therefore, the successive approximation iteration method is quadratic convergent in the x-axis direction. Similarly, it is also quadratic convergent in the y-axis direction. This shows that the algorithm has a relatively fast convergence rate.

[0177] For the non - linear transformation algorithm of vector maps, if the reverse solution by the successive approximation iteration method achieves convergence, the fewer the number of iterations, the lower the complexity of the algorithm. Therefore, the number of iterations of the reverse solution by the successive approximation iteration method can be an important indicator for measuring the quality of the transformation algorithm.

[0178] Step S14: Determine the reversibility of the non - linear transformation of the vector map based on whether the number of iterations obtained in step S13 exceeds a threshold. If it exceeds the threshold, determine that the reversibility of the non - linear transformation of the vector map is weak and the security of the non - linear transformation algorithm is strong; otherwise, determine that the reversibility of the non - linear transformation of the vector map is strong and the security of the non - linear transformation algorithm is weak.

[0179] It can be seen that the successive approximation iteration method takes the offset map and the non - linear transformation software or algorithm as known conditions, and takes the original map coordinates as the result of reverse numerical solution. This method can be used to test the security of the non - linear transformation of vector maps when the attacker has obtained the non - linear transformation software or mastered the transformation algorithm.

[0180] As Figure 2 shown, the second embodiment provided by the present invention is a method for verifying the reversibility of the non - linear transformation of vector maps. The method includes:

[0181] Step S21: Process the vector map through a non - linear transformation algorithm that meets the condition constraints to obtain a corresponding offset map;

[0182] Step S22: Receive the offset map obtained in step S21 as input;

[0183] Step S23: Based on the input in step S22, use linear interpolation for security verification, restore the original map coordinate information, and calculate the error between the original map coordinate values obtained by linear interpolation and the real map coordinate values.

[0184] The linear interpolation method only takes the offset map as a known condition and the original map coordinates as the result of reverse numerical solution. This method can be used to test the linearization degree of the non - linear transformation of vector maps when the attacker only has the offset map, so as to judge the security of the algorithm.

[0185] First, divide the offset map into equidistant grids. Each grid is a square with a diagonal length of ε max in a sufficiently small map normed vector space within this square. For squares that are not large enough to be divided into such sizes, their diagonal lengths are less than ε max , meeting the definition of the sufficiently small map normed vector space of this application.

[0186] Based on the analysis of orientation-preserving isometries in Section 2, the reversibility in the x-axis and y-axis directions is studied respectively. Taking the x-axis direction as an example. Let the offset function be f(x).

[0187] As Figure 6 shown, within each square grid, any two real points a'(f(x1), f(y1)) and b'(f(x2), f(y2)) are taken as control points, and the corresponding points on the original map are obtained as a(x1, y1) and b(x2, y2) through measurement. A coordinate system is constructed, with the x-axis direction being the x value of the original map and the y-axis direction being the f(x) value of the offset map.

[0188] Since there is no non-linear transformation software or algorithm as a tool, for the convenience of reversible numerical analysis, an approximate function of non-linear transformation can be constructed by the interpolation method. Let the points c(x, y) and a, b be located in the same sufficiently small normed vector space of the map. According to the Lagrange linear interpolation formula, there is

[0189]

[0190] As known from Equation (17), to calculate the original map coordinates of point c, the offset coordinates of two control points a, b within the grid and their original map coordinate values are required. The former is known, and the latter can be obtained through measurement. Since a vector map is divided into many grids, the measurement work of control point coordinates is relatively heavy. To simplify the calculation and improve efficiency, for adjacent points a, b, according to the measurement results, there is Under the condition of maintaining a certain accuracy, we get

[0191] f(x2) - f(x1) = x2 - x1

[0192] Equation (17) is simplified to

[0193] x = x1 + f(x) - f(x1) (18)

[0194] Equation (18) is consistent with the discussion results in Section 2 of this application, which also shows that if the distance between points a and b is smaller, the similarity of the offset is higher, and the condition for Equation (18) to hold is more sufficient. This means that within each sufficiently small normed vector space or grid of the map, only 1 control point needs to be taken. Given its offset map coordinates, its original map coordinates are obtained by surveying means, and the original map x-axis direction coordinates of any point within the grid can be obtained through Equation (18). The calculation in the y-axis direction is similar to the above, and we get

[0195] y = y1 + f(y) - f(y1) (19)

[0196] As can be seen from the above, the linear interpolation method requires fewer control points than the classical Lagrange linear interpolation method or Newton linear interpolation method The division operation is deleted, and the calculation efficiency is increased by more than two times.

[0197] For any point in the grid, the accuracy of calculating the reverse numerical solution by the linear interpolation method is directly related to the distance. The closer the point is, the higher the accuracy. Generally speaking, the error between the original map coordinates obtained by the linear interpolation method and the true value is smaller as the distance from this point to the control point is smaller, indicating that the linearization feature of the non - linear transformation is more obvious, the complexity of the reverse calculation is lower, and the security of the transformation algorithm is worse.

[0198] Step S24: Determine the reversibility of the non - linear transformation of the vector map based on whether the error obtained in step S23 exceeds the threshold. If it exceeds the threshold, it is determined that the reversibility of the non - linear transformation of the vector map is strong and the security of the non - linear transformation algorithm is weak; otherwise, it is determined that the reversibility of the non - linear transformation of the vector map is weak and the security of the non - linear transformation algorithm is strong.

[0199] First, discuss the verification of the linear interpolation method. In a sufficiently small normed vector space (such as a 2KM×2KM grid), obtain the offset map coordinates and their original map coordinates of one control point, and calculate through equations (18) and (19) to obtain the original map coordinate values of all real points in the grid. Or use existing geographic information processing software tools to represent the original map coordinates of the control points on the offset map, and use the spatial correction function of the software (such as the Rubbersheet method in the ArcGIS Spatial Adjustment tool), select all real points in the grid to move. When the offset coordinates of the control points coincide with the original map coordinates, the other points in the grid are also restored to the original map coordinates synchronously.

[0200] To verify the effectiveness of the successive approximation iteration method, first construct a non - linear transformation algorithm of the vector map that meets the basic requirements of security and usability as the research object. Let the non - linear transformation equations be

[0201] f(x) = x + 9×10 -6 (5 + 0.003x 2 + 0.11x 0.5 + 3(cos(3x)) 2 + 9(sin(0.1x)) 5 + 1.35sin(29x)) (20)

[0202] f(y) = y + 9×10 -6 sec(x)(5 + 0.004y 2 + 0.03y 0.5 + 3(cos(3y)) 2 + 9(sin(0.1y)) 5 + 1.5sin(27y)) (21)

[0203] Equation (20) calculates the latitude value of the normed vector space W (offset map) in degrees, and equation (21) calculates the longitude value of W in degrees.

[0204] Let g(x) = 5 + 0.003x 2 + 0.11x 0.5 + 3(cos(3x)) 2 + 9(sin(0.1x)) 5 + 1.35sin(29x). This part in equation (20) is responsible for calculating the offset of the latitude of the normed vector space V (original map) in meters. g(x) is a transcendental function, as Figure 7 shown (the horizontal axis is the latitude value and the vertical axis is the offset in meters), presenting a typical "high-frequency and small-amplitude" non-linear change. "High-frequency" ensures that enough points on the map are non-linearly offset, and "small-amplitude" ensures the usability of the map after offset. The change range of g(x) within the latitude range of China is approximately between 4 meters and 19.5 meters. Taking the earth's circumference as 4×10 7 meters, then the offset in meters is converted into the latitude change value as 9×10 -6 g(x); f(x) = x + 9×10 -6 g(x) generates the W latitude value.

[0205] The composition of equation (21) is similar to (20), the difference is that it considers the influence of the original map's parallel circumference on the longitude change (9×10 -6 sec(x)).

[0206] Equations (20) and (21) are both fifth-degree transcendental equations. The mathematical community has proven that such equations (systems) cannot obtain radical solutions for x and y, and only numerical methods can be used for solution.

[0207] The non-linear transformation algorithm of the vector map constructed in this application has characteristics such as non-linearity, irreversibility, no analytical solution, and high-frequency and small-amplitude offset. There is no situation where a non-zero vector is transformed into a zero vector, the vector deformation is basically controllable, it meets the double constraint conditions, has typicality, and can be used as the object of numerical analysis of the reversibility of non-linear transformation.

[0208] In addition, the form of the algorithm verification equation adopted in this application can be summarized as f(x) = x + d(x), where x is the longitude or latitude of the original map (it can also be other measurement units), and d(x) is the offset value. For equations with other forms different from this, set as f(x) = h(x), it can be transformed into the form of the equation in this application by f(x) = x + (h(x) - x). Therefore, the form of the algorithm verification equation in this application has universal significance.

[0209] Randomly select 10 different latitude values within China (retaining 8 - digit precision after the decimal point), and use Equation (20) to calculate the W - latitude value and the offset respectively. See Tables 1 and 2 for details. Based only on the W - latitude value, use the successive approximation iteration method to inversely solve the latitude value, count the number of iterations, and compare the convergence result with the V - latitude value to evaluate the convergence speed and accuracy. See Tables 3 and 4 for details.

[0210] Table 1 Calculation of the W - latitude value and offset in the normed vector space (1)

[0211]

[0212] Table 2 Calculation of the W - latitude value and offset in the normed vector space (2)

[0213]

[0214] Table 3 Successive approximation iteration calculation (1)

[0215]

[0216] Table 4 Successive approximation iteration calculation (2)

[0217]

[0218] Tables 1 and 2 show that under the action of the non - linear transformation algorithm, the offsets of the 10 latitude values show irregular changes, and the mean square error is 10.28571401 meters. Tables 3 and 4 show that when using the successive approximation iteration method for inverse solution, 9 out of 10 calculations converge in the second iteration, and 1 converges in the first iteration. By comparing Tables 3 and 4 with Tables 1 and 2, the iteration convergence point is the original latitude value (retaining 8 - digit precision after the decimal point). This is because taking the coordinate value of the offset map point as the initial condition (initial value) is very close to the true coordinate value of the original map, which is the solution of the transcendental equation (determined by the map availability). Therefore, both the convergence speed and accuracy of using the successive approximation iteration to solve the transcendental equation are relatively ideal.

[0219] After obtaining the original map latitude value, use the above - mentioned method to inversely solve Equation (21). After iteration, the original map longitude value is obtained, and the process will not be elaborated here.

[0220] Modify the parameters of Equations (20) and (21) to increase the point coordinate offset value, and the mean square errors reach the order of 100 meters and 1000 meters respectively. Repeat the above - mentioned iterative calculation. Under the condition of retaining 8 - digit precision after the decimal point, the number of iterative convergences is no more than 3. In addition, take the coordinates of 1000 points in a certain city in China for non - linear transformation, and use the successive approximation iteration method for inverse solution. The conclusion is the same as above.

[0221] In summary, the technical solution proposed by the present invention has the following technical effects: The present invention can prevent the release of vector map non-linear transformation software with reversible risks into the market, ensuring the security of map use. Moreover, by adopting the verification method provided by the present invention for whether a vector map is reversible, the verification speed is fast and the verification efficiency is high, filling the gap in the security detection of domestic geographic information confidentiality processing algorithms and playing an important guiding role in practical applications.

[0222] Please note that the technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as falling within the scope described in this specification. The above embodiments only represent several implementation manners of the present application, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention patent. It should be pointed out that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent of the present application shall be subject to the appended claims.

Claims

1. A method for verifying the reversibility of non - linear transformation of vector maps, characterized in that, The method includes: Step S11: Process the vector map through a non-linear transformation algorithm that satisfies the condition constraints to obtain a corresponding offset map; Step S12: Receive the offset map and the non-linear transformation algorithm obtained in step S11 as inputs; Step S13: Based on the input in step S12, use the successive approximation iteration method for security verification, restore and obtain the original map coordinate information, and record the number of iterations; Step S14: Determine the reversibility of the non-linear transformation of the vector map based on whether the number of iterations obtained in step S13 exceeds a threshold. If it exceeds the threshold, determine that the reversibility of the non-linear transformation of the vector map is weak and the security of the non-linear transformation algorithm is strong; otherwise, determine that the reversibility of the non-linear transformation of the vector map is strong and the security of the non-linear transformation algorithm is weak; Among them, step S13 includes: Step S131: The successive approximation iteration method is constructed based on the law that the non-linear transformation of adjacent points in a sufficiently small norm vector space is highly approximately orientation-preserving isometric transformation. Let a pair of adjacent points a(x1,y1) and b(z2,w2) on the original map, and the corresponding points on the offset map are a'(T(x1),T(y1)) and b'(T(z2),T(w2)); Step S132: Take the coordinates of point a' as the coordinate values of the neighboring point b of point a on the original map. The coordinates of point a' are located in a sufficiently small norm vector space centered at point a with a radius of ε max and use the coordinate values formed by the second non-linear transformation of point a' as the coordinate values of the offset map point b', denoted as (T(T(x1)), T(T(y1))); Step S133: In the map norm vector space, use the successive approximation method for the local part of the offset map to obtain a high-precision approximate solution of the original map vector. Calculate the approximate coordinate value of point a on the original map as the coordinate (x2,y2) of point a2. This point a2 has an adjacent relationship with point a. Calculate the non-linear transformation value of point a2, and solve to obtain another set of approximate coordinate values of point a. Continuously repeat the iteration process until convergence.

2. The method for verifying the reversibility of non - linear transformation of vector maps according to claim 1, characterized in that, In step S11, the condition constraints include: Constraint 1: The map vectors maintain the same quantity and attributes; there is a one-to-one mapping relationship between the vectors in the map norm vector spaces V and W. The non-linear transformation cannot delete, add, split, or merge the vectors in V and cannot change the attributes of the vectors; Constraint 2: Keep the geometric features and topological relationships consistent; after the non-linear transformation, the map norm vector space W should be consistent with the geometric and topological relationships of V, and the vector offset is smooth and continuous without mutations.

3. The method for verifying the reversibility of non - linear transformation of vector maps according to claim 1, characterized in that, In step S131: Let u = (z2 - x1, w2 - y1), then T(u) = (T(z2) - T(x1), T(w2) - T(y1)). From the approximation of orientation-preserving isometric transformation, we can get: ||T(u)||≈||u|| (1) In step S132: Since the directions of T(u) and u are highly similar, we get: x1 - z2≈T(x1) - T(z2) (4) y1 - w2≈T(y1) - T(w2) (5) Use the non-linear transformation functions f(x) and f(y) to replace the non-linear transformations T(x) and T(y); Then, from equations (4) and (5), we can deduce the following equation: x1 - f(x1)≈f(x1) - f(f(x1)) (6) y1 - f(y1)≈f(y1) - f(f(x1)) (7) f(x1) and f(y1) are known, obtained by calculating through the map non - linear transformation algorithm. Solving equations (6) and (7) gives the approximate coordinate values of the original map point a as: x1≈2f(x1)-f(f(x1)) (8) y1≈2f(y1)-f(f(y1)) (9) In step S133: Calculate the non - linear transformation values of point a2 to obtain f(x2) and f(y2), keeping f(x1) and f(y1) unchanged. Solving based on equations (4) and (5), another set of approximate coordinate values of point a is x1≈x2+f(x1)-f(x2) (10) y1≈y2+f(y1)-f(y2) (11) Continue the iteration. Since \(a(x_1,y_1),a i (x i ,y i ), where \(i\) is an integer not equal to 1, is a pair of neighboring points, the following equation holds in a sufficiently small normed vector space x1≈x i +f(x1)-f(x i ) (12) y1≈y i +f(y1)-f(y i ) (13) Continuously repeat the iterative process until convergence, obtaining x1 = x n + f(x1) - f(x n ) (14) y1 = y n + f(x1) - f(x n ) (15) The right - hand sides of equations (14) and (15) are both known values, and the total value can be calculated; let f(y1)-y1=ω1 f(x n ) - x n = ω n Then the right - hand sides of equations (14) and (15) respectively become y1+(ω1-ω n ) (17) Make ω1 - ω n = 0; then it is proved that the iteration has converged.

4. A method for verifying the reversibility of a non - linear transformation of a vector map, characterized in that, The method includes: Step S21: Process the vector map through a non - linear transformation algorithm that meets the condition constraints to obtain the corresponding offset map; Step S22: Receive the offset map obtained in step S21 as input; Step S23: Based on the input in step S22, use linear interpolation to perform security verification, restore the original map coordinate information, and calculate the error between the original map coordinate values obtained by linear interpolation and the true map coordinate values; Step S24: Based on whether the error obtained in step S23 exceeds the threshold, determine the reversibility of the non - linear transformation of the vector map. If it exceeds the threshold, determine that the reversibility of the non - linear transformation of the vector map is strong and the security of the non - linear transformation algorithm is weak; otherwise, determine that the reversibility of the non - linear transformation of the vector map is weak and the security of the non - linear transformation algorithm is strong; Among them, step S23 includes: Step S231: Divide the offset map into equidistant grids, each grid being a square grid with a diagonal length of ε max ; Step S232: Let the offset function be f(x). In each square grid, arbitrarily take two real points a′(f(x1), f(y1)) and b′(f(x2), f(y2)) as control points. By measurement, the corresponding points on the original map are a(x1, y1) and b(x2, y2). Construct a coordinate system, with the x - axis direction being the x value of the original map and the y - axis direction being the f(x) value of the offset map; Let point c(x,y) and points a, b be located in the same small-scale map normed vector space. According to the Lagrange linear interpolation formula, for adjacent points a, b, based on the measurement results, there are Under the condition of maintaining a certain accuracy, x=x1+f(x)-f(x1) (20) The calculation in the y - axis direction is the same as that in the x - axis direction, obtaining y=y1+f(y)-f(y1) (21).

5. The method for verifying the reversibility of a non - linear transformation of a vector map according to claim 4, characterized in that, In step S21, the condition constraints include: Constraint 1: The map vectors maintain the same quantity and attributes; there is a one - to - one mapping relationship between the vectors in the map normed vector spaces V and W. The non - linear transformation cannot delete, add, split, or merge the vectors in V and cannot change the attributes of the vectors. Constraint 2: Keep the geometric features and topological relationships consistent; after the non - linear transformation, the map normed vector space W should maintain the same geometric and topological relationships as V, and the vector offsets are smooth and continuous without mutations.

6. The method for verifying the reversibility of a non - linear transformation of a vector map according to claim 4, characterized in that, In step S232: According to the Lagrange linear interpolation formula, there is 7. A system for verifying the reversibility of a non - linear transformation of a vector map, characterized in that, The system includes: The first processing module is used to process the vector map through a non - linear transformation algorithm that meets the condition constraints to obtain the corresponding offset map; The second processing module is used to receive the offset map and the non - linear transformation algorithm obtained in the first processing module as inputs; The third processing module is used to perform security verification by using the successive approximation iteration method based on the input of the second processing module, restore the original map coordinate information, and record the number of iterations; The fourth processing module determines the reversibility of the non - linear transformation of the vector map based on whether the number of iterations obtained by the third processing module exceeds a threshold. If it exceeds the threshold, it is determined that the reversibility of the non - linear transformation of the vector map is weak and the security of the non - linear transformation algorithm is strong; otherwise, it is determined that the reversibility of the non - linear transformation of the vector map is strong and the security of the non - linear transformation algorithm is weak; Among them, the third processing module includes: The first sub - module is used for: The successive approximation iteration method is constructed based on the law that the non - linear transformation of adjacent points in a sufficiently small normed vector space is highly approximately orientation - preserving isometric transformation. Let two adjacent points a(x1, y1) and b(z2, w2) on the original map, and their corresponding points on the offset map are a′(T(x1), T(y1)) and b′(T(z2), T(w2)); The second sub-module is used to: take the coordinates of point a' as the coordinate value of the adjacent point b of point a in the original map, where the coordinates of point a' are located in a sufficiently small norm vector space centered at point a with ε max as the radius, and the coordinate value formed by the further non-linear transformation of point a' is used as the coordinate value of the offset map point b', which is represented by (T(T(x1)), T(T(y1))); The third sub - module is used for: In the map normed vector space, the successive approximation method is used for the local part of the offset map to obtain a high - precision approximate solution of the original map vector. Calculate the approximate coordinate value of point a on the original map as the coordinate (x2, y2) of point a2. This point a2 has an adjacent relationship with point a. Calculate the non - linear transformation value of point a2, and solve to obtain another set of approximate coordinate values of point a. Continuously repeat the iteration process until convergence.

8. A system for verifying the reversibility of a non - linear transformation of a vector map, characterized in that, The system includes: The first processing module is used to process the vector map through a non - linear transformation algorithm that meets the condition constraints to obtain the corresponding offset map; The second processing module is used to receive the offset map obtained in the first processing module as an input; The third processing module is used to perform security verification by using the linear interpolation method based on the input of the second processing module, restore the original map coordinate information, and calculate the error between the original map coordinate value obtained by the linear interpolation method and the real map coordinate value; The fourth processing module determines the reversibility of the non - linear transformation of the vector map based on whether the error obtained by the third processing module exceeds a threshold. If it exceeds the threshold, it is determined that the reversibility of the non - linear transformation of the vector map is strong and the security of the non - linear transformation algorithm is weak; otherwise, it is determined that the reversibility of the non - linear transformation of the vector map is weak and the security of the non - linear transformation algorithm is strong; Among them, the third processing module includes: The first sub-module is used to perform equidistant grid division on the offset map, and each grid is a square grid with a diagonal length of ε max ; The second sub - module is used for: Let the offset function be f(x). In each square grid, arbitrarily take two real points a′(f(x1), f(y1)) and b′(f(x2), f(y2)) as control points. By measurement, the corresponding points on the original map are a(x1, y1) and b(x2, y2). Construct a coordinate system, where the x - axis direction is the x value of the original map and the y - axis direction is the f(x) value of the offset map; Let the point c(x,y) and points a, b be located in the same small-scale map normed vector space. According to the Lagrange linear interpolation formula, for the neighboring points a, b, based on the measurement results, there are Under the condition of maintaining a certain accuracy, x = x1 + f(x) - f(x1) (20) The calculation in the y - axis direction is the same as that in the x - axis direction, and we get y = y1 + f(y) - f(y1) (21).

9. An electronic device, characterized in that, The electronic device includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, the steps in the method for verifying the reversibility of the non-linear transformation of the vector map according to any one of claims 1 to 6 are implemented.

10. A computer-readable storage medium storing computer-readable storage instructions, characterized in that, The instructions are used to implement the steps in the method for verifying the reversibility of the non-linear transformation of the vector map according to any one of claims 1 to 6.

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