CN106055694B - Geographic curve tortuosity measuring method based on information entropy - Google Patents

Abstract

The invention discloses a method for measuring the tortuosity of geographic curves based on information entropy, and relates to the technical field of geographic information science. The invention sequentially completes the identification of curved units, superposition and determination of curved nesting relationships at different scales, establishment of a curved hierarchical tree, and deletion of invalidity. The work of bending and measuring the tortuosity of geographic curves based on information entropy theory adopts the comprehensive complexity of combining dimensional complexity and hierarchical complexity to describe the tortuosity, which completely shows the partial and overall tortuosity of the curve. It comprehensively considers the nested relationship between different levels of bending, overcomes the defects of the existing technology, can better describe the tortuosity of the curve, comprehensively reflects the shape and structural characteristics of the curve, is less affected by the length of the curve, and makes full use of the bending The hierarchical tree completely reflects the adjacent relationship and hierarchical characteristics between the bends, and uses the information entropy theory to measure the complexity.

Description

A Measuring Method of Geographical Curve Tortuosity Based on Information Entropy

technical field

The invention relates to the technical field of geographic information science, in particular to a method for measuring the tortuosity of geographic curves based on information entropy.

Background technique

The tortuosity of a geographic curve is a description method of a geographic feature curve, which has an important relationship with the geographic features contained in the curve itself, and is of great significance to the study of geographic features. The description of the tortuosity of the territorial sea is mostly judged by fuzzy concepts such as "extremely tortuous", "relatively tortuous", "relatively straight", and lack of clear judgment indicators, which is very unfavorable for the determination of the type of territorial sea baselines; therefore, quantitative expression of curve tortuosity is of great importance. application value.

At present, the tortuosity of geographic curves is measured by methods based on tortuosity index, angle measurement and fractal dimension. The tortuosity index is a quantitative index that can reflect the overall shape of the line element. The larger the value of the tortuosity index, the more complex the curve is. However, the tortuosity index cannot reflect the tortuosity of complex nested short curves, and cannot fully reflect the shape of the curve. ; The tortuosity measurement method based on angle measurement adds the angle values between the straight line segments on the curve, and uses the addition result to represent the complexity of the curve, but this representation method is affected by the length of the curve; the fractal dimension method is mainly Study the self-similarity of irregular things, but the fractal dimension value is only a statistic, it can only reflect the overall situation of the curve, but cannot correspond to the specific bending element of the curve. Other structural features of the curve cannot comprehensively reflect the curve shape and structural features for curves of various lengths.

SUMMARY OF THE INVENTION

(1) Technical problems solved

The technical problem to be solved by the present invention is to provide a method for measuring the tortuosity of geographic curves based on information entropy, so as to solve the above problems.

(2) Technical solutions

In order to achieve the above purpose, the present invention is achieved through the following technical solutions: a method for measuring the tortuosity of geographic curves based on information entropy, comprising the following steps:

1) Identify the bending unit: perform glue transformation on the curve with different widths, superimpose the result of the glue transformation with the original curve to obtain bending polygons of different scales, connect the bending division points to obtain the bending identification map, and obtain the intersection operation with the original geographic curve. Bending at each scale, and calculate the quantitative index of each bending unit, and store it in the relevant attribute field;

2) Superposition determines the curved nesting relationship at different scales, and establishes a curved hierarchical tree: carry out superposition analysis on curved polygons at different levels, determine the attribution of each curved polygon, and establish a curved hierarchical tree for each curved;

3) Delete invalid bending: delete the invalid bending of each layer, and finally get the bending unit of each layer;

4) Measure the tortuosity of geographic curves based on information entropy theory: Calculate the tortuosity of geographic curves represented by curved hierarchical trees using information entropy theory.

Further, the curved dividing point is the intersection point of the original curve and the curved transformation line.

Further, superposition determines the curved nesting relationship at different scales by judging the attribution of each curved polygon at different levels, and superimposes the curved polygons at each scale with the curved polygons at the larger scale at the previous level to determine the small scale. The attribution of polygons can obtain the corresponding nesting relationship, determine the level of each curved element, and finally establish a curved hierarchical tree with the original curve as the root node.

Further, the method of establishing a curved hierarchical tree is to cycle from bottom to top for each curved hierarchical tree to determine whether the leaf nodes with parent nodes at each level have sibling nodes, and if there are sibling nodes, the node Retain, if there is no sibling node, delete the node; continue to search the upper layer to determine whether the leaf node of this layer has a parent node, if not, end the loop, if so, continue to determine whether it has a sibling node point until all leaf nodes of each layer are traversed.

Further, the invalid bending is the bending directly inherited from the upper-layer bending obtained by splitting the upper-layer bending.

Further, the method of using the information entropy theory to calculate the tortuosity of the geographic curve represented by the curved hierarchical tree is to measure the tortuosity of the geographic curve by using the comprehensive complexity of the size complexity and the hierarchical complexity. The calculation formula of the comprehensive complexity is:

ZC=P ₁ SCA+P ₂ SCL+P ₃ SCW+P ₄ LC,

Among them, respectively represent the weights of different types of complexity, and the sum of the weights is 1.

Further, the SC is the size complexity, and the bending unit is used as the basic unit in the calculation of the SC, and the calculation formula is:

where is the total number of valid bending units in the bending hierarchy tree, and is the number of valid bendings in each class.

Further, the LC is the hierarchical complexity, and in the calculation of the LC, one layer of the hierarchical tree is used as the basic unit, and the calculation formula is:

where is the total number of effective curved elements in the curved hierarchical tree, and is the number of effective curved elements in each level.

(3) Beneficial effects

The invention provides a method for measuring the tortuosity of geographic curves based on information entropy, which sequentially completes the identification of bending units, superposition and determination of bending nesting relationships under different scales, establishment of a bending hierarchy tree, deletion of invalid bends, and measurement of geographic curves based on information entropy theory. The tortuosity of the work is described by the comprehensive complexity that combines the dimensional complexity and the level complexity, which completely shows the part and the whole tortuosity of the curve, and comprehensively considers the bending between different levels. It can better describe the curve tortuosity, comprehensively reflect the shape and structural characteristics of the curve, and is less affected by the length of the curve, making full use of the curve hierarchy tree to completely reflect the proximity between the curves. It is easy to operate and implement, and it is of great significance to the study of geographical features.

Description of drawings

Fig. 1 is the exploded view of circular adhesion transformation of the present invention;

Fig. 2 is the schematic diagram of figure shape change before and after adhesion transformation of the present invention;

Fig. 3 bending diagrams obtained by different transformation curves of the present invention;

Fig. 4 is the corresponding hierarchical tree diagram of the bending diagram obtained by different transformation curves of the present invention;

5 is a schematic diagram of the bending area, bending length and bending width of the present invention;

Fig. 6 is the flow chart of the present invention;

Fig. 7 is the original curve C of the present invention and the bending division point connection diagram of different scales;

Fig. 8 is the adhesion transformation diagram when the transformation width of the present invention is 6 nautical miles;

Fig. 9 is the adhesion transformation diagram when the transformation width of the present invention is 3.8 nautical miles;

Fig. 10 is the bending hierarchy tree diagram corresponding to the adhesion transformation diagram when the transformation width of the present invention is 6 nautical miles and 3.8 nautical miles;

Fig. 11 is the bending hierarchical tree diagram corresponding to bending 18 before deleting each layer of unbrothered leaf nodes according to the present invention;

Fig. 12 is the bending hierarchical tree diagram corresponding to bending 18 after the present invention cyclically deletes the leaf nodes without siblings in each layer;

Fig. 13 is the adhesion transformation diagram when the transformation width of the present invention is 1 nautical mile;

Fig. 14 is the adhesion transformation diagram when the transformation width of the present invention is 0.4 nautical miles;

Fig. 15 is the hierarchical tree diagram corresponding to the adhesion transformation diagram when the transformation width of the present invention is 1 nautical mile;

Fig. 16 is the hierarchical tree diagram corresponding to the adhesion transformation diagram when the transformation width of the present invention is 0.4 nautical miles;

Fig. 17 is the hierarchical tree diagram of node 8 in the hierarchical tree diagram corresponding to the adhesion transformation diagram when the transformation width of the present invention is 0.4 nautical miles.

18 is a schematic diagram of the original curve C of the present invention;

Fig. 19 is the curve identification diagram of curve C of the present invention;

FIG. 20 is a schematic diagram of bending and stacking of the present invention;

Figure 21 is the bending hierarchy tree diagram of bending 2 before the invalid bending is deleted in the present invention;

Figure 22 is the bending hierarchy tree diagram of bending 2 after the invalid bending is deleted in the present invention;

FIG. 23 is a bending hierarchy tree diagram of the original curve C after the invalid bending is deleted according to the present invention.

In the picture:

1-a, original image; 1-b, packer transform; 1-c, packer transform; 1-d, molt transform; 1-e, color image blackened; 1-f, molt transform; 1- g. Overlay graph;

2-a. An arc with no change in the figure before and after the glue transformation (a circular arc with a central angle not greater than 180 degrees); 2-b, an arc with a change in the figure before and after the glue transformation (a circular arc with a central angle greater than 180 degrees); 2 -c. Combination graphics of straight line and arc;

3-A, the original curve; 3-B, the original curve bending hierarchical tree diagram; 3-C, the original curve according to the scale-adhesion transformation diagram; 3-D, the original curve and the scale-based glue transformation after the overlay; 3-E , the original curve and the superimposed map after scale-one glue transformation correspond to the curved hierarchical tree map; 3-F, the original curve according to the scale-two adhesion transformation map; 3-G, the original curve and the scale-two glue-transformed superimposed map;

4-A, the curved hierarchical tree corresponding to the original curve 3-A without sticking transformation in Fig. 3; 4-C, the curved hierarchical tree obtained after the original curve 3-A in Fig. 3 is transformed by the glue transformation line 3-C; 4-D, the original curve 3-A in Fig. 3 obtained after the glue transformation line 3-D transformation Bend the hierarchical tree.

7-a, original curve C; 7-b, L is the connection diagram of bending division points when 6 nautical miles; 7-c, L is the connection diagram of bending division points when 3.8 nautical miles; 7-d, L is 2 nautical miles. The connection diagram of bending division points; the connection diagram of bending division points when 7-e and L are 1 nautical mile; the connection diagram of bending division points when 7-f and L are 0.4 nautical miles;

9-a and L are the curved hierarchical tree diagrams corresponding to the adhesion transformation diagram at 6 nautical miles; 9-b and L are the curved hierarchical tree diagrams corresponding to the adhesion transformation diagram at 3.8 nautical miles;

11-a. After deleting the leaf nodes without siblings in the fourth layer, bend the curved hierarchical tree corresponding to 18; 11-b. After deleting the leaf nodes without siblings in the third layer, bend the curved hierarchical tree corresponding to 18 ; 11-c, after deleting the leaf node without siblings in the second layer, bend the bending hierarchy tree corresponding to 18; 11-d, after deleting the leaf node without siblings in the first layer, bend the bending corresponding to 18 Hierarchical tree;

19-a, Fig. 18 The bending identification diagram obtained when the adhesion width of the original curve C is 200km; 19-b, the bending identification diagram obtained when the adhesion width of the original curve C in Fig. 18 is 50 km; 19-c, Fig. 18 The original curve C adhesion The bending identification map obtained when the width is 30km; the bending identification map obtained when the adhesion width of the original curve C in Fig. 19-d and Fig. 18 is 15km;

20-a, Fig. 18 is a schematic diagram of the bending superposition corresponding to the bending identification map obtained when the adhesion width of the original curve C is 200km; 20-b, Fig. 18 is a schematic diagram of the bending superposition corresponding to the bending identification map obtained when the adhesion width of the original curve C is 50km.

Detailed ways

In order to make the purposes, 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 in conjunction with the embodiments of the present invention. Obviously, the described embodiments are part of the present invention. examples, but not all examples. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

With reference to Figures 1 to 23, the following steps are included:

1) Identify the bending unit: perform glue transformation on the curve with different widths, superimpose the result of the glue transformation with the original curve to obtain bending polygons of different scales, connect the bending division points to obtain the bending identification map, and obtain the intersection operation with the original geographic curve. Bending at each scale, and calculate the quantitative index of each bending unit, and store it in the relevant attribute field;

2) Superposition determines the curved nesting relationship at different scales, and establishes a curved hierarchical tree: carry out superposition analysis on curved polygons at different levels, determine the attribution of each curved polygon, and establish a curved hierarchical tree for each curved;

3) Delete invalid bending: delete the invalid bending of each layer, and finally get the bending unit of each layer;

4) Measure the tortuosity of geographic curves based on information entropy theory: Calculate the tortuosity of geographic curves represented by curved hierarchical trees using information entropy theory.

Preferably, the curved dividing point is the intersection of the original curve and the curved transformation line.

Preferably, the superposition to determine the curved nesting relationship at different scales is achieved by judging the attribution of each curved polygon in different levels, and superimposing the curved polygons in each scale with the curved polygons in the larger scale of the previous level to determine the small scale The attribution of polygons can obtain the corresponding nesting relationship, determine the level of each curved element, and finally establish a curved hierarchical tree with the original curve as the root node.

Preferably, the method for establishing a curved hierarchical tree is to cycle from bottom to top for each curved hierarchical tree to determine whether the leaf nodes with parent nodes at each level have sibling nodes, and if there are sibling nodes, the node Retain, if there is no sibling node, delete the node; continue to search the upper layer to determine whether the leaf node of this layer has a parent node, if not, end the loop, if so, continue to determine whether it has a sibling node point until all leaf nodes of each layer are traversed.

Preferably, the invalid bending is a bending directly inherited from the upper-layer bending obtained by splitting the upper-layer bending.

Preferably, the method for calculating the tortuosity of the geographic curve represented by the curved hierarchical tree using the information entropy theory is to measure the tortuosity of the geographic curve by using the comprehensive complexity of the size complexity and the hierarchical complexity, and the calculation formula of the comprehensive complexity is:

ZC=P ₁ SCA+P ₂ SCL+P ₃ SCW+P ₄ LC,

Among them, respectively represent the weights of different types of complexity, and the sum of the weights is 1.

Preferably, the SC is the size complexity, and in the calculation of the SC, the bending unit is used as the basic unit, and the calculation formula is:

where is the total number of valid bending units in the bending hierarchy tree, and is the number of valid bendings in each class.

Preferably, the LC is the hierarchical complexity, and in the calculation of the LC, one layer of the hierarchical tree is used as the basic unit, and the calculation formula is:

where is the total number of effective curved elements in the curved hierarchical tree, and is the number of effective curved elements in each level.

For the convenience of implementation and reference, the glue transformation, curved hierarchical tree, curved unit quantization index and information entropy (complexity) involved in the present invention are first introduced:

(1) Adhesion transformation

The buffer transformation based on map algebra can easily and quickly obtain the buffers of points, lines, surfaces and complex entities with a buffer width of L, and according to the difference of distance transformation, it can be divided into inner buffer transformation and outer buffer transformation. .

The specific algorithm is: first, use the corresponding distance scale to directly perform distance transformation (inside and outside) on the entity to obtain the distance of each point in the whole space; Buffers (inside, outside). At this time, the outer buffer is called the shell, and the inner buffer is called the skin. The process of adding an entity to the outer buffer is the "packing" transformation, and the process of removing the entity to the inner buffer is the "molding" transformation.

The packing transformation of the graph X is defined as Equation 1:

XK ₀ (L)=X∪XB ₀ (l,L)=X+XB ₀ (L),

In the formula, X is the entity set, K ₀ (L) represents the implementation of the packing transformation of L, and B ₀ (1, L) represents the pixel whose distance value is from 1 to L, that is, the buffer with a width of L; XB ₀ (L) is the shell, which means that the outer surface of the entity is adjacent to the outer shell of thickness L.

The molting transformation of the graph X is defined as Equation 2:

XK _I (L)=X\XB _I (l,L)=X-XB _I (L),

In the formula, K _I (L) represents the peeling transformation of L, and XB _I (L) is the skin, which refers to the layer where the thickness of the solid surface is L.

Using the shelling and molting transformation, the sticking transformation can be further obtained, and the sticking transformation can be defined as Equation 3:

X·L(l ₁ , l ₂ )=XKO ( _{l 1} )·K _I (l ₂ ),

In the formula, l ₁ , l ₂ are suitable positive integers or 0. The sticking transformation is to first perform a shelling transformation with a width of l ₁ on the graph, and then perform a molting transformation with a width of l ₂ on it; in general, take The shelling width is equal to the molting width, that is, l ₁ =l ₂ , and is collectively referred to as the adhesion width 1.

The shape-preserving effect of glue transformation on the shape of graphics has the following characteristics:

a. For basic standard graphics such as circles and straight lines, it has the characteristics of keeping the basic shape unchanged, that is, the characteristics of "preserving the level" and "preserving the convexity".

Take the standard graph circle of a curve as an example, as shown in Figure 1, for a circle with a radius of r, first perform a shell transformation with a width of L, the transformed graph is still a circle, and its radius is R=r+L, and then the transformation is performed. After the molting transformation with a width of L, the molting transformation is carried out, and then the molting-transformed graphics 1-f and the original image 1-a are superimposed to obtain the superimposed image 1-g, and 1-a and 1-f are completely coincident, which means that The shape of the circle remains unchanged before and after the glue transformation.

b. For the concave, convex and straight combined graphics, it has the characteristics of "preserving convex", "preserving level" and "filling concave".

In the sticking transformation, the degree of change of the shape of the arc before and after the transformation depends on the central angle of the arc. When the central angle of the arc is not greater than 180° (represented as a convex shape), the graphics before and after the sticking transformation remain unchanged, reflecting the preservation of Convex morphological characteristics, as shown in Figure 2-a, when the central angle of the circle is greater than 180°, the adhesion transformation shows the morphological characteristics of reducing concave, as shown in Figure 2-b; the combination of arc and straight line graphics changes in the adhesion transformation The degree is related to the angle between the two. When the angle is greater than 180°, the straight line and the arc form a convex portion, and the figure remains unchanged before and after the adhesive transformation. When the angle is less than 180°, the straight line and the arc form a concave portion. As the width increases, the concave portion is gradually filled in, as shown in Fig. 2-c, which further verifies the shape-preserving properties of the sticking transformation to preserve the convexity, preserve the level, and fill the concave.

c. The degree of "filling concave" is controllable.

According to the morphological change characteristics of the adhesive transformation: if the width of the adhesive transformation is L, the concave bending with a width (here, the bending width is defined as the maximum concave width of the bending, denoted as D) less than 2L will be gradually smoothed, and the smaller the bending width , the larger the transformation width is, the more obvious the smoothing effect is. At the same time, the following width of the curvature will be filled, and the relationship between the critical transformation width L' and the maximum concave width D satisfies Equation 4:

This also means that the critical transformation width L' is inversely calculated with the maximum concave width D of the arc, and then the graph is subjected to a sticking transformation of the width L', and the result of the transformation will fill all the concave parts in the graph. As shown in Table 1, the arcs with a radius of R=20 pixels and different central angles are subjected to the adhesion transformation with a width of L, and the transformation results in the three cases of L<L', L=L' and L>L' are respectively taken, it can be seen that , the smoothness of the graph increases after the sticking transformation, and its tendency to maintain convexity, levelling, and filling concave increases with the increase of L; when the width of the sticking transformation L increases to L=L', if the sticking with a width of D (D>L) continues Transform, the graph no longer changes.

Table 1 Radius R = 20 pixels, arc adhesion transformation with different central angles

The sticking transformation has the characteristics of "convex preservation", "level preservation", and "concave filling" in maintaining the shape of the graph, and the degree of "concave filling" can be controlled by the value of the width L of the sticking transformation. Automatic identification of curved and curved elements on different sides and at different levels.

(2) Curved hierarchical tree

Bending hierarchical tree refers to the identification of bending based on the glue transformation of different scales. On the basis of obtaining the bending at different scales, a hierarchical tree is used to describe the nesting relationship of bending units. In a curved hierarchical tree, a curved identified at a certain glue transformation scale represents a level of the hierarchical tree, and each curved represents a node of the level.

The following will take the standard form of bending - arc as an example, to illustrate the basic concept of bending hierarchical tree. As shown in Figure 3-A in Figure 3, the curve L is formed by nesting four layers of bends of different sizes. The largest bend 1 (representing the curve L) is nested with three bends 1a, 1b, and 1c from left to right. Three bends are nested in turn, 1a.1, 1a.2, 1a.3, 1b.1, 1b.2, 1c.1, 1c.2 Seven bends, bend 1c.2 is nested 1c.2a , 1c.2b two bends. Perform the adhesion transformation on the curve L at three different scales, and the obtained transformation lines are 3-B, 3-C and 3-D in Figure 3 in turn, and the superposition diagram of the transformation line and the original curve is in turn 3 in Figure 3. -E, 3-F and 3-G, each glue transformation scale corresponds to a layer in the curved hierarchical tree, thereby establishing the corresponding curved hierarchical tree at each scale. As shown in Figure 4-B, 4-C and 4-D, the root node 1 represents the curve L; the node 1 has three sub-nodes, which are nodes 1a, 1b, and 1c in sequence, indicating that the curve L is nested 1a in turn , 1b, 1c three bends; nodes 1a, 1b, 1c have 3 sub-nodes, 2 sub-nodes and 2 sub-nodes respectively, indicating that bends 1a, 1b, 1c are nested 3 bends, 2 bends in turn and 2 bends; and so on…. Finally, the leaf nodes of the tree represent the smallest bending unit that can be identified at a certain glue transformation scale.

In the curved hierarchical tree, each curved element in the curve corresponds to a node, the structure of the hierarchical tree reflects the topological characteristics between the curved elements, and the quantitative index of the curved element can be stored in the corresponding node of the curved hierarchical tree. In this way, a curved hierarchical tree can be used to express the topological properties of curved nesting in a curve, and at the same time, it can also describe the size and shape of each curved element. That is, a curved hierarchy tree can be used to describe the twists and turns of a curve. The ability of this curved hierarchy tree to describe the tortuosity of a curve is shown in Table 2:

Table 2 The ability of curved hierarchical tree to express curve tortuosity

(3) Quantitative index of bending unit

The bending unit is the smallest unit that composes a curve. It should be a small segment of arcs. In order to facilitate the measurement of its various indicators, the curved hierarchical tree is replaced by a surface and represented by a curved polygon identified by the glue transformation. individual bending units. The existing quantitative indicators of bending elements mainly include bending element area, length, width and so on. The quantitative indicators of the bending unit are introduced in detail below.

a. Bending area S

The curved area S generally refers to the area of the polygon surrounded by the straight line connecting the starting point and the end point of the curve and the curved segment.

b. Bending length L

The bending length L is defined as the total length of the bending element, such as the total length of the curve between AB (or BC) in Figure 5. In this paper, the bending length can be directly obtained by calculating the length through the attribute calculator in ArcGIS.

c. Bending width W

In general, the bending width W is defined as the straight-line distance between the starting point and the end point of the bending, such as the straight-line distance between AB or BC in Fig. 5 . Glue transform-based bend widths still use this definition, but the determination of the bend endpoints is based on the glue transform composite lines. The specific operation is as follows: in the step of recognizing the bending, the bending polygon is obtained, and the bending width can be obtained by calculating the difference between the perimeter and the bending length of the bending polygon.

(4) Complexity and information entropy

Information entropy is a measure of the usefulness of information. In geoscience research, information entropy is an effective means to study characteristics and distribution, while complexity is a description of composition. Two seemingly different concepts are closely related. In this paper, complexity is used to measure the tortuosity of the curve, so as to quantitatively describe the shape of the curve.

Entropy can measure the degree of concentration or dispersion of a phenomenon or event in space, and is the scientific name for uncertainty. Entropy is a measure of the degree of organization and order of the system, which can be used to characterize the degree of uncertainty of the system. The average information that removes redundancy is information entropy. Information entropy can be used to measure the amount of information and describe the usefulness of information. Information quantification can be effectively achieved through information entropy, and its specific calculation formula is as follows:

Among them, P _i in the formula is the occurrence probability of event _xi , n means there are n events in total, and the logarithmic function takes different bases, and the calculated entropy value results are different.

Information entropy has a wide range of applications, but the theory of entropy is abstract and difficult to understand in the field of science. Zhang Xuewen transformed the mysterious concept of entropy and the principle of entropy into a very popular way in generalized set theory, and at the same time expanded its application field.

He replaced entropy with complexity from the perspective of composition theory, and proposed the concept of generalized sets and the law of complexity. After transformation, the information entropy theory is easier to understand and easier to apply.

Composition theory generalizes all composition problems and uses unified models and laws to study composition problems in different fields. Among them, the generalized set is a mathematical model used to study the unified composition law. Set languages can be used for qualitative analysis, while generalized set languages can be used for quantitative analysis. A set is a collection of things with specific properties, and the main concern in a set is what are the two different elements. The generalized set should not only clarify the difference, but also focus on the commonality: classify the elements by the commonality, and determine how many elements are in each category. If a population can be divided into multiple individuals with the same status, and each individual has a certain attribute value, then the population is called a generalized set. A generalized set can have not only one attribute, but also multiple attributes, which is called a multidimensional generalized set. This paper mainly studies one-dimensional generalized sets. In order to describe the generalized set accurately, two new concepts need to be introduced, namely, sign and individual. The mark refers to the elements that are different in the set, and the individual refers to the category of the element, and the individual is the basic element that composes the set. Symbols are used to describe differences, and individuals focus on the same status among various elements.

A generalized set is a mathematical model, so there is a distribution function corresponding to it. The function here is different from the function in mathematics in the general sense. It can be an empirical formula, which mainly explains the problem of composition. To clarify a composition is to discover an objective law, and each composition can be described by a generalized set, so the distribution function of each generalized set is a law. To clarify each specific generalized set is to define a composition, and to discover an objective law is actually to obtain a distribution function. Table 3 shows some examples of generalized sets:

Table 3 Generalized set example description

generalized set individual name Logo name The problem to be explained by the distribution function population everyone person's age How many people of different ages Mountain every mountain the altitude of the mountain How many mountains are there at different altitudes? lake each lake area of lake How many lakes are there land land per square kilometer land type How many different types of land are there? river each branch length of branch how many branches of different lengths

The complexity can describe the composition of the generalized set and reflect its internal state. The calculation formula of the complexity is as shown in Equation 6:

In the formula, n represents the number of flag values, that is, how many classes there are; n _i represents the number of individuals in each flag value, that is, the number of elements in each class; N represents the total number of individuals. Each generalized set has its own unique complexity. The minimum value of the complexity is 0. At this time, all elements have the same value and there is only one category. It can be seen from the complexity calculation formula that the value of the complexity is related to the base of the logarithmic function used in the calculation. The base of the logarithm is different, and the obtained complexity value is also different.

The greater the individual differences in a generalized set, the greater its complexity. When the characteristics of each individual are the same, there is no difference, and the complexity is zero. Expressed in the language of generalized sets as the greater the difference in flag values, the greater the complexity value; the flag values are exactly the same, the complexity is zero. Popular understanding, the more complex the composition, the greater the complexity.

Information entropy represents uncertainty, and complexity represents richness, and there is a certain relationship between the two. Information entropy is to analyze things from the perspective of random experiments, and the uncertainty of the final result is obtained; the degree of complexity is to analyze things from the perspective of internal differences. The performance of this difference in a generalized set is that there are different sign values, and the final result is The abundance of things, that is, how the generalized set is composed. Analyze the calculation formula 1 of the information entropy and the calculation formula 2 of the complexity, where P _i =n _i /N, bring this relationship into the calculation formula 2 of the complexity, and combine the formula 1 to obtain the information entropy and complexity. The corresponding relationship: C=NH. It can be seen from this relationship that complexity and information entropy are positively correlated. The greater the complexity, the greater the information entropy, that is, the more complex the composition, the more uncertain the outcome. Because of the proportional relationship between the two, much knowledge about information entropy also naturally falls into the concept of complexity ^[63] . According to this relationship, it can also be concluded that the reason for the degree of uncertainty is that there is an objective generalized set with complexity, and it is precisely because of the complexity of the composition that the uncertainty of the result is caused.

Example 1:

The following describes in detail the method for describing the tortuosity of a curve using a curved hierarchical tree in the technical solution of the present invention in conjunction with FIG. 6 and the embodiment. For the curve C shown in 7-a in FIG. 7 , the detailed establishment process of the curved hierarchical tree is as follows:

1) Perform curve synthesis based on glue transformation to obtain curved polygons;

Perform glue transformation on the original curve C to obtain the glue transformation line of the curve (divided into inner transformation line and outer transformation line, here only the outer transformation line is selected for description, the inner transformation line is similar), and construct a curved polygon with the original curve C .

Specifically, the original curve C is subjected to adhesion transformation with widths of 6 nautical miles, 3.8 nautical miles, 2 nautical miles, 1 nautical mile and 0.4 nautical miles, respectively, to obtain transformation lines at different scales. The intersection of the original curve and the curved transformation line is called the curved division Point, connect the bending division points to obtain the bending identification diagram shown in Figure 7.

Connect the curved dividing points with curve C to construct curved polygons. Taking the transformation width of 6 nautical miles as an example, 18 curved polygons can be obtained as shown in Figure 8, corresponding to 18 curved units (numbered 1-18).

2) Overlay analysis to determine the ownership of curved polygons;

The overlapping analysis is performed on the curved polygons of different levels, the attribution of each curved polygon is judged, and the curved hierarchical tree of each curve is established. Take a curved polygon with a sticking transformation width L=3.8 nautical miles, and compare each curvature with the curvature of the previous adjacent large-scale sticking transformation (L=6 nautical miles), as shown in Figure 9. It can be seen from observation that the curvature 18 is split into three Bends 18.1, 18.2, 18.3, while Bend 8 is split into two bends, Bends 8.1 and 8.2, and the other bends are not split.

The above process of determining the bending relationship between different layers on the curve is identified through human eye observation. In the specific implementation process, it can be realized by judging the attribution of each curved polygon in different layers.

The specific process is as follows, take the curved polygon with the width L1 of the glue transformation, and perform the superposition analysis with the curved polygon with the width L2 (L2<L1), and cyclically determine the ownership of each curved polygon when the width is L2. Suppose polygon P2 is a curved polygon with a glue transformation width of L2, and polygon P1 is a curved polygon with a glue transformation width of L1. If polygon P2 belongs to polygon P1, it means that in the curved hierarchical tree, curved polygon P2 The corresponding node C2 is a child node of the node C1 corresponding to the curved polygon P1. The above process of judging the ownership of the curved polygon is performed cyclically until all the curves are traversed.

3) Establish a hierarchical tree corresponding to each bend at the maximum scale;

In order to clearly indicate the relationship between a node and its child nodes, it is assumed that a node is marked as 1. If it has only one child node, it is generally marked as 1.1; if it has multiple child nodes (set to n), Then mark them as 1.1, 1.2, 1.3, ..., 1.n in turn; if 1.n also has multiple child nodes (set as m), then mark them as 1.n.1, 1.n.2, 1. n.3, ... 1.n.m.

According to the method of judging the polygon attribution in 2), the same processing process is carried out for the bending under the transformation width L = 6 nautical miles, 3.8 nautical miles, 2 nautical miles, 1 nautical mile and 0.4 nautical miles, and the hierarchical tree corresponding to each curve can be obtained, as shown in the table 3:

Table 3 Curved Hierarchical Trees with Different Transformation Widths

In the curved hierarchical tree, the determination of the parent node of each node depends on the attribution of the corresponding curvature of the node. If the node 18.1 corresponds to the curve 18.1, and the curve 18.1 is superimposed with the curve corresponding to the upper node, it is obvious that the curve 18.1 belongs to the curve 18, and the parent node of the node 18.1 is the node 18, as shown in Figure 10. 9 -a and 9-b. In addition, the attribute of each node in the bending hierarchy tree corresponds to the quantitative index of the bending element.

4) Delete the leaf nodes without siblings in each layer, and establish a curved hierarchical tree of the entire curve;

For each curved hierarchical tree, cycle from bottom to top to determine whether the leaf node with parent nodes at each level has a sibling node, if there is a sibling node, the node is retained; if there is no sibling node, then Delete this node. Continue to search for the upper layer to determine whether the leaf node of this layer has a parent node, if not, end the loop; if so, continue to determine whether it has a sibling node...until all leaf nodes in each layer are traversed .

Now take Bend 18 as an example to illustrate the process of deleting leaf nodes without siblings at each level in the Bend hierarchy tree. Figure 11 shows the curved hierarchical tree corresponding to curved 18 before deleting the leaf nodes without siblings at each level. Starting from the 4th level of the curved hierarchical tree, traverse the leaf nodes of each level from bottom to top to determine whether they have brothers or not. Node. Observe the leaf nodes of the 4th layer of the curved hierarchical tree in Figure 11. The numbers of the nodes all end with the Arabic numeral "1", indicating that all leaf nodes in this layer have no sibling nodes and need to be deleted. The result obtained 11-a in Figure 12; continue to loop upward to search for the leaf nodes in the third layer, it can be seen that there are two nodes whose numbers do not end with the Arabic numeral "1", and the two nodes are 18.2.1.2 and 18.2.1.2 respectively. 18.2.3.2, delete the last digit of the number of these two nodes, get the nodes of the second layer (that is, the parent nodes) 18.2.1 and 18.2.3, you can determine the child nodes under these two nodes (that is, the nodes 18.2.1.1, 18.2.1.2 and 18.2.3.1, 18.2.3.2 on the 3rd layer have sibling nodes and do not need to be deleted; other leaf nodes on the 3rd layer have no sibling nodes and are deleted. Node operation, the result obtained is as shown in Figure 11-b in Figure 12; continue to search upward and traverse all the leaf nodes of the second layer, and use the same judgment method to delete the leaf nodes 18.1.1 and 18.3.1 without siblings, and get The result is shown in 11-c in Figure 12; continue to search upwards and traverse all leaf nodes in the first layer, leaf nodes 18.1 and 18.3 have sibling nodes, and do not need to be deleted; there are no leaf nodes in the 0th layer, no Judgement is required. So far, the leaf nodes of each layer in the curved hierarchical tree have been traversed, indicating that the operation of deleting the leaf nodes without siblings in each layer is completed, and the hierarchical tree corresponding to the curved 18 after the traversal is completed is shown in Figure 12. 11-d. Table 4 is the result of traversing each curved hierarchical tree in a loop and deleting the leaf nodes without siblings at each level.

Finally, add a parent node C (that is, the root node C) to the 18 bends numbered 1-18 to obtain the bending hierarchical tree of the original curve C, as shown in Figures 16-17. Of course, each node in the curved hierarchical tree has a common attribute—the quantitative index of the curved element to describe the size and shape characteristics of the curved element itself.

In fact, the glue transformation at any scale can build a curved hierarchical tree of height 1. As shown in Figure 8, the original curve C is subjected to glue transformation, and its width L = 6 nautical miles, then 18 bends are generated, and these bends are in the same layer of the bending hierarchy tree, and a bending hierarchy with a height of 1 is obtained as shown in Figure 9-a. Tree. If you continue to perform one or more glue transformations on curve C with a width less than 6 nautical miles, a curved hierarchical tree with a height of not less than 1 can be established, as shown in Figure 9-b and Figure 15, Figure 16, and Figure 17. The choice of the number of glue transformations and the transformation width determines the height of the curved hierarchical tree and the degree of each node. The height of the bend hierarchy tree determines the number of bend nestings; the degree of each node indicates the degree of fragmentation of the bend corresponding to that node. These two values can be used as one of the manifestations of the ability of the curved hierarchical tree to describe the tortuosity of the curve.

Table 4. Hierarchical tree after deleting leaf nodes without siblings at each level in each curved hierarchical tree

The curved hierarchical tree is based on the expression of the curved hierarchical structure, and recognizes the curved based on the glue transformation method, which can completely reflect the adjacent relationship and hierarchical characteristics between the curved. In the structure tree, adjacent nodes at the same level have an adjacent relationship; a node at the Nth level has a hierarchical relationship with the parent node at the N-1th level, which describes a curved nested structure. As shown in Figure 9-a, nodes 1 and 2 are adjacent, and their corresponding bends have adjacent relationships; as shown in Figure 9-b, between nodes 8.1, 8.2 and node 8, and between nodes 18.1, 18.2, 18.3 and node 18 With a hierarchical relationship, its corresponding bends reflect the nested structure between bends.

(1) Bending at the same level;

Take node 8, node 12, and node 18 as examples to illustrate the ability of the curved hierarchical tree to describe the tortuosity of a curve:

First of all, it can be seen from Figures 16 to 17 that the heights of the curved hierarchical trees corresponding to node 8, node 12 and node 18 are equal, all of which are 4, which are the three trees with the highest height among nodes 1-18. It shows that the curve segments corresponding to these three nodes have the most bending nesting times, which is relatively complex bending. This conclusion is consistent with the results of human eye recognition in Figure 8.

Secondly, the degrees of node 8, node 12 and node 18 are 2, 1 and 3 respectively, indicating that their subtrees are 2, 1 and 3 respectively, that is to say, bend 8, bend 12 and bend 18 are split respectively for 2 bends, 1 bend and 3 bends. Of course, this is a value relative to the composite scale.

Again, the difference of the tortuosity of the curve segment 8, the curve segment 12 and the curve segment 18 can be measured by the difference of the subtrees corresponding to these three nodes. For example, the tree of the subtree (ie, the degree of the node), the depth of the subtree (the number of times of bending and nesting), the balance of tree node splitting, the difference of the quantitative index of the node, and so on. We can preliminarily judge that node 8 is more complicated than node 18. On the one hand, it is because the number of nodes (7) summarized by node 12 is much smaller than that of node 8 (16) and node 18 (13). On the other hand, it can be judged from the number of its subtrees, and the differences of its subtrees can also be compared.

Then, the equilibrium of the tortuosity of the curve can be judged according to the equilibrium of the splitting of the tree nodes. For node 8 and node 18, it is obvious that the split of node 18 is more balanced, which can be measured by the degree (or average value) of the subtree corresponding to each level of node.

In addition, adjacent nodes in the same layer have an adjacent relationship, such as node 8.1 and node 8.2.

(2) Bending at different levels;

First, the different levels of bending nodes indicate that bending elements are nested differently throughout the curve. For example, in Figures 16 to 17, the levels of node 8.1 and node 8.1.2.1 are 1 and 3 respectively, while the height of the tree corresponding to node 8 is 4, indicating that the number of nesting times of bend 8.1 in the entire curve is 3 (the difference between the height of the tree and the level it is in); the number of nestings of Bend 8.1.2.1 is 1. This is consistent with the results in Figures 16-17.

Second, the nesting relationship before bending of different layers can be determined by judging whether it is a parent-child node. For example, node 18 has three child nodes, 18.1, 18.2, and 18.3, indicating that bend 18 nests three bends, 18.1, 18.2, and 18.3.

In a word, a curved hierarchical tree corresponds to the tortuosity of a curve, which can describe both the size and morphological characteristics of the curved elements themselves, and the topological characteristics between the curved elements.

Example 2:

The technical solution of the present invention will be described in detail below in conjunction with FIG. 6 and the embodiments. For the curve C shown in FIG. 18 , the detailed process of establishing the curved hierarchical tree is as follows:

1) Identify the bending element.

The original curve C in Fig. 18 is subjected to adhesion transformation with widths of 200km, 50km, 30km and 15km, respectively, to obtain transformation lines at different scales. The intersection of the original curve and the bending transformation line is called the bending division point. Bending identification diagrams shown in 19-a to 19-d in FIG. 19 .

2) Superposition determines the curved nesting relationship at different scales, and establishes a curved hierarchical tree.

The overlapping analysis is performed on the curved polygons of different levels, the attribution of each curved polygon is judged, and the curved hierarchical tree of each curve is established. For the above curve C, take a curved polygon with a sticking transformation width L=50km, and compare each bend with the sticking transform L=200km curvature, as shown in Figure 20, it can be seen by observation: Bend 3 splits Bend 3.1 and 3.2, Bend 8 Split into bends 8.1 and 8.2, bend 9 split into bends 9.1 and 9.2, other bends do not split. The above process of determining the bending relationship between different layers on the curve is identified through human eye observation. In the specific implementation process, it can be realized by judging the attribution of each curved polygon in different layers. The specific process is as follows:

Take the curved polygon whose width is L1 in the glue transformation, and perform the superposition analysis with the curved polygon whose width is L2 (L2<L1), and determine the ownership of each curved polygon when the width is L2. Suppose polygon P2 is a curved polygon with a glue transformation width of L2, and polygon P1 is a curved polygon with a glue transformation width of L1. If polygon P2 belongs to polygon P1, it means that in the curved hierarchical tree, curved polygon P2 The corresponding node C2 is a child node of the node C1 corresponding to the curved polygon P1.

3) Remove invalid bends

Invalid bends are bends obtained without splitting and appear as leaf nodes without siblings in the bend hierarchy tree. For each curved hierarchical tree, cycle from bottom to top to determine whether the leaf nodes with parent nodes at each level have sibling nodes. If there are sibling nodes, the node is retained; if there is no sibling node, delete it the node. Continue to search for the upper layer to judge whether the leaf node of this layer has a parent node, if not, terminate the loop; if so, continue to judge whether it has a sibling node...until all the leaf nodes of each layer are traversed .

Now take Bend 2 as an example to illustrate the process of deleting leaf nodes without siblings at each level in the Bend tree. Fig. 21 is the curved hierarchical tree corresponding to Bend 2 before deleting the leaf nodes without siblings at each level, and Fig. 22 is the result diagram of Bend 2 after deleting the non-sibling nodes at each level.

After deleting the invalid bend, the original curve C corresponds to the bend hierarchy tree shown in Figure 23.

The height of the curved hierarchical tree determines the number of curved nestings, and the degree of each node indicates the degree of fragmentation of the corresponding curvature of the node. These two values can be used to measure the curved tortuosity of the curved hierarchical tree.

4) Measure the tortuosity of geographic curves based on complexity (information entropy) theory

The tortuosity of a curve, also known as curve complexity, refers to the inter-nesting of curves of different shapes and sizes at different levels. A curved hierarchical tree can effectively describe the morphological characteristics of a curve, and a curved hierarchical tree is a curve. The curve is regarded as a generalized set, and the complexity of the curved hierarchical tree is calculated to represent the tortuosity of the curve, so as to quantitatively describe the morphological characteristics of the curve. The basis of the complexity calculation is to classify all bending elements involved in the calculation. In the calculation of this paper, two classification methods are mainly used, one is classification according to the size of the bending unit (represented by quantitative indicators); the other is the hierarchical classification based on the bending hierarchical tree itself. Therefore, the curved hierarchical tree complexity used to describe the tortuosity of a curve can be divided into two types, namely size complexity and level complexity. Finally, the comprehensive complexity is used to measure the tortuosity of the geographic curve.

(1) Size complexity

The dimensional complexity is based on the calculation of bending elements. Each bending element is regarded as the basic element of the curve, that is, an individual, and the quantitative index of bending is used as the characteristic of the individual.

The calculation method based on the bending element is to consider the bending element as a unit to consider the composition of the line, and mainly use the number of bending elements under each eigenvalue for calculation. The main problem of this calculation method is how much bending of different sizes. Table 4 shows the generalized set language representation of this calculation method. Since the size of the bending elements is basically different, it is meaningless to calculate the number strictly according to the specific value. Here, the classification method is used to count the number: according to a certain attribute value (area, length, width) of all bending elements, different attribute value intervals are divided according to a specific principle, and each interval is a category.

Table 4 Generalized set representation curve

generalized set individual name Logo name The problem to be explained by the distribution function curve each bend each curved area How many bends are there in different areas? curve each bend the length of each bend how many bends of different lengths curve each bend the height of each bend how many bends at different heights curve each bend the width of each bend How many bends of different widths

Taking the measure of area as an example, the formula for calculating complexity is:

Among them, n represents the number of divided area categories, _ni represents the number of curved elements in each interval, and N is the total number of curved elements composing the curve.

According to the properties of the logarithmic function, the calculation formula of the size complexity SC is Equation 7:

where N is the total number of valid bending units in the tree of bends, and _ni is the number of valid bends in each class. Similarly, the logarithmic function takes different bases, and the calculated complexity results are different. Taking curve C in FIG. 7 as an example, the size complexity is calculated, and the size complexity result is calculated with the base 10. The specific size information of the bending element of the curve is shown in Table 5:

Table 5 Bending element properties for curve C

Id TreeName Area Perimeter Baseline Length Radius OlayR 1 1 6135.29 471.31 75.34 395.98 200 0 2 2 7304.83 477.05 132.67 344.38 200 0 3 3 2323.95 302.15 88.83 213.33 200 0 4 4 30.41 50.43 24.90 25.53 200 0 5 5 1867.01 236.28 101.10 135.17 200 0 6 6 6500.34 579.97 74.82 505.15 200 0 7 7 988.07 147.90 43.77 104.14 200 0 8 8 2787.62 347.55 86.38 261.16 200 0 9 9 5219.08 431.02 126.85 304.17 200 0 10 10 2324.24 242.83 82.58 160.25 200 0 11 3.2 961.70 150.87 40.90 109.97 50 200 12 3.1 601.65 121.22 37.45 83.77 50 200 13 9.2 1745.74 182.80 55.43 127.37 50 200 14 9.1 1838.18 211.00 61.16 149.84 50 200 15 8.1 1548.95 192.77 39.97 152.80 50 200 16 8.2 788.26 148.59 45.56 103.03 50 200 17 2.2.2 1308.36 184.13 59.00 125.13 30 50 18 2.2.1 1092.67 156.96 46.79 110.17 30 50 19 1.1.1.1 459.41 112.48 37.27 75.21 15 30 20 1.1.1.2 103.30 76.01 34.01 42.00 15 30 twenty one 1.1.1.3 672.62 116.45 31.50 84.95 15 30 twenty two 6.1.1.1 157.76 71.41 25.66 45.74 15 30 twenty three 6.1.1.2 209.58 72.33 22.83 49.50 15 30 twenty four 6.1.1.3 820.77 113.11 17.68 95.43 15 30 25 6.1.1.4 429.63 89.26 19.82 69.45 15 30 26 6.1.1.5 693.95 119.70 25.06 94.64 15 30 27 6.1.1.6 178.77 78.24 29.65 48.58 15 30

In the calculation experiment of size complexity, this paper mainly uses three metrics to participate in the calculation, namely the area, length and baseline length of the bending element. The three categories of indicators are divided into 10 categories according to the method of equidistant classification, and the classification interval is one tenth of the difference between the maximum value and the minimum value. The classification results are shown in Table 6:

Table 6 Classification of quantitative indicators

category area range number length range number width range number 1 30.14—757.85 10 25.53—73.49 6 17.68 - 29.18 6 2 757.85—1485.29 6 73.49—121.25 9 29.18 - 40.68 6 3 1485.29—2212.73 4 121.25—169.14 6 40.68 - 52.18 4 4 2212.73—2940.17 3 169.14—217.00 1 52.18 - 63.68 3 5 2940.17—3667.61 0 217.00—264.86 1 63.68 - 75.18 1 6 3667.61—4395.05 0 264.86—312.72 1 75.18 - 86.68 3 7 4395.05—5122.49 0 312.72—360.58 1 86.68—98.18 1 8 5122.49—5849.93 1 360.58—408.44 1 98.18—109.68 1 9 5849.93—6577.37 2 408.44—456.30 0 109.68—121.18 0 10 6577.37—7304.83 1 456.30—505.15 1 121.18—132.67 2

According to formula 7, the size complexity of the curve C calculated by taking the area as the metric index is:

The size complexity of the curve C calculated with the length as the metric index is:

SCL=27*log(27)-9*log(9)-6*log(1)=20.721,

The size complexity of the curve C calculated with the width as the metric is:

SCW=27*log(27)-12*log(6)-9*log(9)-6*log(1)=23.436.

(2) Hierarchical complexity

Each level of the hierarchical tree is regarded as a basic unit, that is, an individual, and the number of bends contained in each level is a marker value.

This method considers the composition of curves in units of layers, and mainly uses the number of bends contained in each layer for calculation. In essence, this calculation method is to use the hierarchy to classify, and pay attention to the composition of each class. The main problem explained in the calculation process is how many levels there are in the curved hierarchical tree, and how many curves are in each level, so as to reflect whether the curve is complicated. In theory, the more layers, the more complex the curve, and the more jagged and complex the number of bends in the layers. The generalized set language to describe this approach is shown in Table 7:

Table 7 Generalized set representation curve

generalized set individual name Logo name The problem to be explained by the distribution function curve each level The number of bends in each level How many curved elements are there at different levels

Its complexity calculation formula is:

Among them, n represents how many levels the curved hierarchical tree has, _ni represents the number of curved elements in each level, that is, the number of curved elements in each result layer, and N is the total number of curved elements that make up the curve . The logarithmic function takes different bases, and the calculation results are different.

According to the properties of the logarithmic function, the calculation formula of the level complexity LC is Equation 8:

Among them, N is the total number of effective curved units in the curved hierarchical tree, and ni is the number of effective curved elements in each layer.

It can be seen from the formula that the logarithmic function takes different bases, and the calculated hierarchical complexity results are different. Taking curve C in FIG. 18 as an example, the size complexity is calculated, and the size complexity result is calculated based on 10. For the original curve C, there are five levels in this bending hierarchy tree, with one valid bend in the first level, 10 valid bends in the second level, 6 valid bends in the third level, and 2 valid bends in the fourth level Bends, there are 9 active bends in the fifth layer for a total of 28 active bends.

According to formula 8, the level complexity of curve C is:

LC=28*log(28)-10*log(10)-6*log(6)-2*log(2)-9*log(9)=16.661.

(3) Comprehensive complexity

The comprehensive complexity is used to measure the tortuosity of the geographic curve from two aspects: level and size. The comprehensive complexity ZC can be defined as Equation 9:

ZC=P ₁ SCA+P ₂ SCL+P ₃ SCW+P ₄ LC,

Among them, P _i (i=1, . . . , 4) respectively represent the weights occupied by different types of complexity, and the sum of the weights is 1. In the actual experiment, multiple sets of weight values can be designed to obtain the comprehensive complexity of different curves respectively, and then a more reasonable set of weight values can be determined according to the empirical method, so the complexity expression method for measuring geographic curves can be obtained.

To sum up, the embodiments of the present invention have the following beneficial effects: the work of identifying bending units, superimposing and determining the bending nesting relationship at different scales, establishing a bending hierarchical tree, deleting invalid bending, and measuring the tortuosity of geographic curves based on information entropy theory are completed in sequence. , the tortuosity is described by the comprehensive complexity that combines the size complexity and the level complexity, which completely shows the partial and overall tortuosity of the curve, and at the same time comprehensively considers the nesting relationship between different levels of the curve. Overcoming the defects of the prior art, it can better describe the tortuosity of the curve, comprehensively reflect the shape and structural characteristics of the curve, and is less affected by the length of the curve. And the information entropy theory is used to measure the complexity, which is easy to operate and realize, and is of great significance to the study of geographical features.

It should be noted that, in this document, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any relationship between these entities or operations. any such actual relationship or sequence exists. Moreover, the terms "comprising", "comprising" or any other variation thereof are intended to encompass non-exclusive inclusion such that a process, method, article or device comprising a list of elements includes not only those elements, but also includes not explicitly listed or other elements inherent to such a process, method, article or apparatus. Without further limitation, an element qualified by the phrase "comprising a..." does not preclude the presence of additional identical elements in a process, method, article or apparatus that includes the element.

The above embodiments are only used to illustrate the technical solutions of the present invention, but not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: The recorded technical solutions are modified, or some technical features thereof are equivalently replaced; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims (7)

1. a geographical curve tortuosity measurement method based on information entropy, is characterized in that, comprises the following steps: 1) Identify the bending unit: perform glue transformation of different widths on the curve, superimpose the result of the glue transformation with the original curve to obtain curved polygons of different scales, connect the bending division points to obtain the bending identification map, and obtain the intersection operation with the original geographic curve. Bending at each scale, and calculate the quantitative index of each bending unit, and store it in the relevant attribute field; 2) Superposition determines the curved nesting relationship at different scales, and establishes a curved hierarchical tree: carry out superposition analysis on curved polygons at different levels, determine the attribution of each curved polygon, and establish a curved hierarchical tree for each curved; 3) Delete invalid bending: delete the invalid bending of each layer, and finally get the bending unit of each layer; 4) Measure the tortuosity of the geographic curve based on the information entropy theory: use the information entropy theory to calculate the tortuosity of the geographic curve represented by the curved hierarchical tree; Among them, the method of using the information entropy theory to calculate the tortuosity of the geographical curve represented by the curved hierarchical tree is to use the comprehensive complexity of the size complexity and the hierarchical complexity to measure the tortuosity of the geographical curve. The calculation formula of the comprehensive complexity is: ZC=P ₁ SCA+P ₂ SCL+P ₃ SCW+P ₄ LC, Among them, P _i (i=1, 2, ..., 4) respectively represent the weights occupied by different types of complexity, and the sum of the weights is 1; SCA is the curve calculated with the area as the metric index. Size complexity; SCL is the size complexity of the curve calculated with the length as the metric; SCW is the size complexity of the curve calculated with the width as the metric; LC is the level complexity. 2 . The method for measuring the tortuosity of geographic curves based on information entropy according to claim 1 , wherein the curved dividing point is the intersection of the original curve and the curved transformation line. 3 . 3. a kind of geographical curve tortuosity measurement method based on information entropy according to claim 1, it is characterized in that: superposition determines the curved nested relation under different scales to realize by judging the attribution of each curved polygon in different levels, will The curved polygons at each scale are superimposed with the curved polygons of the larger scale at the previous level to determine the attribution of the small-scale polygons, so as to obtain the corresponding nesting relationship, determine the level of each curved unit, and finally establish the original curve as the root. A curved hierarchical tree of nodes. 4. a kind of geographical curve tortuosity measurement method based on information entropy according to claim 1, is characterized in that: the method for setting up curved hierarchical tree is to each curved hierarchical tree, from bottom to top, it is judged that each layer has Whether the leaf node of the parent node has a sibling node, if there is a sibling node, the node is retained, if there is no sibling node, the node is deleted; continue to search the upper layer to determine whether the leaf node at this layer is If there is a parent node, if there is no parent node, end the loop, if there is, continue to judge whether it has a sibling node, until all leaf nodes in each layer are traversed. 5 . The method for measuring the tortuosity of geographic curves based on information entropy according to claim 1 , wherein the invalid bending is a bending directly inherited from the upper-layer bending obtained by splitting the non-upper-layer bending. 6 . 6. a kind of geographical curve tortuosity measurement method based on information entropy according to claim 5, is characterized in that, described SC is size complexity, in the calculation of described SC, the bending unit is used as the basic unit, and the calculation formula is : where N is the total number of valid curved elements in the curved hierarchical tree, and _ni is the number of valid curved elements in each class. 7. a kind of geographical curve tortuosity measurement method based on information entropy according to claim 5, is characterized in that, described LC is hierarchical complexity, and in the calculation of described LC, one layer of hierarchical tree is a basic unit, The calculation formula is: where N is the total number of effective curved elements of the curved hierarchical tree, and _ni is the number of effective curved elements in each level.