CN106055694A - 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, which relates to the field of geographic information science and technology. The invention sequentially completes the identification of bending units, superposition and determination of bending nesting relationships at different scales, establishment of bending hierarchy trees, and deletion of invalid Curvature and the work of measuring the tortuosity of geographic curves based on information entropy theory, using the comprehensive complexity that combines dimensional complexity and hierarchical complexity to describe the tortuosity, completely showing the partial and overall tortuosity of the curve, and at the same time It comprehensively considers the nesting relationship between different levels of bending, overcomes the defects of the existing technology, can better describe the tortuosity of the curve, fully 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 bends, and uses the information entropy theory to measure the complexity, which is easy to operate and realize, and is of great significance to the study of geographical features.

Description

A Measuring Method of Geographic Curve Tortuosity Based on Information Entropy

technical field

The invention relates to the technical field of geographic information science, in particular to an information entropy-based measurement method for the tortuosity of geographic curves.

Background technique

The tortuosity of a geographical curve is a description method of a geographical characteristic curve, which has an important relationship with the geographical characteristics contained in the curve itself, and is of great significance to the study of geographical characteristics. However, the commonly used description methods for geographical characteristics are very unclear, such as coastlines. The tortuosity description of the curve is mostly judged by fuzzy concepts such as "extremely tortuous", "relatively tortuous", and "relatively straight", lacking clear judgment indicators, which is very unfavorable for the determination of the baseline type of territorial sea; application value.

At present, the tortuosity of geographic curves is mainly measured by methods such as tortuosity index, angle-based calculation, and fractal dimension. The tortuosity index is a quantitative index that can reflect the overall shape of line elements. The larger the tortuosity index value, 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 mainly Study the self-similarity of irregular things, but the fractal dimension value is only a statistic, which can only reflect the overall situation of the curve, and cannot correspond to the specific bending unit of the curve. The fractal dimension is only a characteristic of the curve, and cannot be obtained through the fractal dimension Other structural features of the curve cannot fully reflect the curve shape and structural features of the curves of various lengths.

Contents of the invention

(1) Solved technical problems

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 to solve the above problems.

(2) Technical solution

To achieve the above object, the present invention is achieved through the following technical solutions: a method for measuring the tortuosity of geographical curves based on information entropy, comprising the following steps:

1) Recognition of curved units: Carry out glue transformation of different widths on the curve, superimpose the result of glue transformation on the original curve to obtain curved polygons of different scales, connect the curved division points to obtain the curved recognition map, and obtain the result by intersecting with the original geographic curve Bending at each scale, and calculating the quantitative index of each bending unit, stored in the relevant attribute domain;

2) Overlaying determines the nesting relationship of bends at different scales, and establishes a bend hierarchy tree: conducts an overlay analysis of bend polygons at different levels, judges the attribution of each bend polygon, and establishes a bend hierarchy tree for each bend;

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

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

Further, the bending division point is the intersection point of the original curve and the bending transformation line.

Further, superimposition determines the curved nesting relationship at different scales by judging the ownership of each curved polygon at different levels. The curved polygons at each scale are superimposed with the curved polygons at the larger scale of the previous level to determine the small-scale The attribution of polygons, so as to obtain the corresponding nesting relationship, determine the level of each curved unit, and finally establish a curved hierarchical tree with the original curve as the root node.

Further, the method of building a curved hierarchical tree is to loop from bottom to top for each curved hierarchical tree to determine whether the leaf node with a parent node in each layer has a sibling node, and if there is a sibling node, the node Reserve, if there is no sibling node, delete the node; continue to search up the layer to determine whether the leaf node of the layer has a parent node, if not, end the loop, if yes, continue to determine whether it has a sibling node Points until all leaf nodes of each layer are traversed.

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

Furthermore, the method of using the information entropy theory to calculate the tortuosity of the geographical curve represented by the curved hierarchical tree is to measure the tortuosity of the geographical curve by using the comprehensive complexity of size complexity and 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 calculation of the SC takes the bending unit as the basic unit, and the calculation formula is:

$\mathrm{<span\; class="notranslate">\; SC</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; N</span>}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

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

Further, the LC is hierarchical complexity, and the calculation of the LC takes a layer of a hierarchical tree as a basic unit, and the calculation formula is:

$\mathrm{<span\; class="notranslate">\; LC</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; N</span>}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

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

(3) Beneficial effects

The invention provides a method for measuring the tortuosity of geographical curves based on information entropy, which sequentially completes the identification of bending units, superposition and determination of bending nesting relationships at different scales, establishment of bending hierarchy trees, deletion of invalid bending, and measurement of geographic curves based on information entropy theory For the work on the tortuosity of the curve, the tortuosity is described using the comprehensive complexity that combines the dimensional complexity and the hierarchical complexity, which fully shows the partial and overall tortuosity of the curve, and at the same time takes into account the differences between different levels of bending. The nesting relationship overcomes the defects of the existing technology, can better describe the tortuosity of the curve, fully reflect the shape and structural characteristics of the curve, is less affected by the length of the curve, and fully uses the bending hierarchy tree to fully reflect the proximity between the curves Relationship and hierarchical characteristics, and the use of information entropy theory to measure complexity, easy to operate and implement, is of great significance to the study of geographical features.

Description of drawings

Fig. 1 is an exploded view of circle sticking transformation of the present invention;

Fig. 2 is a schematic diagram of graphic form changes before and after the adhesion transformation of the present invention;

The bending figure that Fig. 3 different transformation curves of the present invention obtains;

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

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

Fig. 6 is a flowchart of the present invention;

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

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

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

Fig. 10 is the curved hierarchical tree diagram corresponding to the cohesive transformation diagram when the transformation width of the present invention is 6 nautical miles and 3.8 nautical miles;

Fig. 11 is the curved hierarchical tree diagram corresponding to the bending 18 before deleting each layer of leaf nodes without siblings in the present invention;

Fig. 12 is a curved hierarchical tree diagram corresponding to curved 18 after cyclically deleting each layer of leaf nodes without siblings in the present invention;

Fig. 13 is the sticking transformation figure when the transformation width of the present invention is 1 nautical mile;

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

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

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

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

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

Fig. 19 is a bending recognition diagram of curve C of the present invention;

Figure 20 is a schematic diagram of bending superposition of the present invention;

Fig. 21 is a bending hierarchical tree diagram of bending 2 before invalid bending is deleted in the present invention;

Fig. 22 is a bending hierarchical tree diagram of bending 2 after the invalid bending is deleted in the present invention;

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

In the picture:

1-a, original image; 1-b, shelling transformation; 1-c, shelling transformation map; 1-d, molting transformation; 1-e, color map turns black; 1-f, molting transformation map; 1- g. Overlay map;

2-a. Arcs with no change in graphics before and after glue transformation (circle arcs with a central angle not greater than 180 degrees); 2-b. Arcs with graphics changes before and after glue transformation (circular arcs with a central angle greater than 180 degrees); 2 -c, combined graphics of straight lines and arcs;

3-A, the original curve; 3-B, the curved hierarchical tree diagram of the original curve; 3-C, the scale-glue transformation diagram of the original curve; 3-D, the original curve and the superposition diagram after scale-glue transformation; 3-E , the original curve and the superposition graph after scale-one-glue transformation correspond to the curved hierarchical tree diagram; 3-F, the original curve and scale-two-glue transformation graph; 3-G, the original curve and the superposition graph after scale-two-glue transformation; 3-H , the original curve and the overlay graph corresponding to the curved hierarchical tree diagram after scale two-glue transformation; 3-I, the original curve according to scale three-glue transformation graph; 3-J, the original curve and scale three-glue transformation superposition graph; 3-K , the original curve and the superimposed graph corresponding to the scaled three-glue transformation correspond to the curved hierarchical tree graph;

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

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

9-a, L are the curved hierarchical tree diagrams corresponding to the adhesion transformation diagram at 6 nautical miles; 9-b, 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 brothers in the fourth layer, bend the curved hierarchical tree corresponding to 18; 11-b. After deleting the leaf nodes without brothers in the third layer, bend the curved hierarchical tree corresponding to 18 ;11-c. After deleting the leaf nodes without siblings in the second layer, bend the hierarchical tree corresponding to 18; 11-d. After deleting the leaf nodes without siblings in the first layer, bend the bending corresponding to 18 hierarchical tree;

19-a, the bending identification diagram obtained when the original curve C in Figure 18 has a adhesion width of 200km; 19-b, the bending identification diagram obtained when the original curve C in Figure 18 has an adhesion width of 50km; 19-c, the original curve C in Figure 18 The bending identification diagram obtained when the width is 30km; 19-d, the bending identification diagram obtained when the original curve C in Figure 18 is glued with a width of 15km;

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

detailed description

In order to make the purpose, 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, not all examples. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without creative efforts fall within the protection scope of the present invention.

Combined with what is shown in Figures 1 to 23, the following steps are included:

1) Recognition of curved units: Carry out glue transformation of different widths on the curve, superimpose the result of glue transformation on the original curve to obtain curved polygons of different scales, connect the curved division points to obtain the curved recognition map, and obtain the result by intersecting with the original geographic curve Bending at each scale, and calculating the quantitative index of each bending unit, stored in the relevant attribute domain;

2) Overlaying determines the nesting relationship of bends at different scales, and establishes a bend hierarchy tree: conducts an overlay analysis of bend polygons at different levels, judges the attribution of each bend polygon, and establishes a bend hierarchy tree for each bend;

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

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

Preferably, the bending division point is the intersection of the original curve and the bending transformation line.

Preferably, superimposition determines the curved nesting relationship at different scales by judging the attribution of each curved polygon at different levels, superimposing the curved polygons at each scale with the curved polygons at the upper level of a larger scale to determine the small scale The attribution of polygons, so as to obtain the corresponding nesting relationship, determine the level of each curved unit, and finally establish a curved hierarchical tree with the original curve as the root node.

Preferably, the method of establishing a curved hierarchical tree is to loop from bottom to top for each curved hierarchical tree to determine whether the leaf node with a parent node in each layer has a sibling node, and if there is a sibling node, the node Reserve, if there is no sibling node, delete the node; continue to search up the layer to determine whether the leaf node of the layer has a parent node, if not, end the loop, if yes, continue to determine whether it has a sibling node Points until all leaf nodes of each layer are traversed.

Preferably, the invalid bending is a bending directly inherited from the upper layer bending which is not split from the upper layer bending.

Preferably, the method of calculating the tortuosity of the geographical curve represented by the curved hierarchical tree using information entropy theory is to measure the tortuosity of the geographical curve by using the comprehensive complexity of size complexity and 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 dimensional complexity, and the calculation of the SC takes the bending unit as the basic unit, and the calculation formula is:

$\mathrm{<span\; class="notranslate">\; S</span>}\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; N</span>}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

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

Preferably, the LC is hierarchical complexity, and the calculation of the LC takes a layer of a hierarchical tree as a basic unit, and the calculation formula is:

$\mathrm{<span\; class="notranslate">\; L</span>}\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; N</span>}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

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

For ease of implementation reference, first introduce the sticky transformation, curved hierarchical tree, curved unit quantization index and information entropy (complexity) involved in the present invention:

(1) sticky transformation

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

The specific algorithm is as follows: first, use the corresponding distance scale to directly implement distance transformation (inner and outer) on the entity to obtain the distance of each point in the whole space; then take all the pixels with distance values from 1 to L (L is the buffer width), and extract buffer (inner, outer). At this time, the outer buffer zone is called the shell, and the inner buffer zone is called the skin. The process of adding the outer buffer to the entity is "packing" transformation, and the process of removing the entity to the inner buffer is "molting" transformation.

The packing transformation of graphics X is defined as formula 1:

XK ₀ (L)＝X∪XB ₀ (l,L)＝X+XB ₀ (L),

In the formula, X is the entity set, K ₀ (L) means to implement the packing transformation of L, B ₀ (1, L) means to take the pixel whose distance value is from 1 to L, that is, the buffer zone whose width is L; XB ₀ (L) is the shell, which refers to the shell with thickness L adjacent to the solid surface.

The molting transformation of graphic X is defined as formula 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 whose surface thickness is L.

Using shelling and molting transformation, we can further obtain glue transformation, which can be defined as Equation 3:

X·L(l ₁ , l ₂ )=XK _O (l ₁ )·K _I (l ₂ ),

In the formula, l ₁ , l ₂ are suitable positive integers or 0, and the sticky transformation is to carry out the packing transformation with the width of l ₁ on the graphics first, and then carry out the molting transformation with the width of l ₂ ; in general, take The encapsulation width is equal to the molt width, that is, l ₁ =l ₂ , and collectively referred to as the adhesion width 1.

The shape-preserving effect of sticky transformation on graphic form 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 flatness" and "preserving convexity".

Take the standard graphic circle of the curve as an example, as shown in Figure 1, for a circle with a radius of r, firstly carry out a shelling transformation with a width of L, the transformed figure is still a circle, and its radius is R=r+L, and then transform The resulting figure is subjected to a molt transformation with a width of L, and then the transformed figure 1-f is superimposed on the original figure 1-a to obtain a superimposed figure 1-g, and 1-a and 1-f are completely overlapped, which means The shape of the circle remains unchanged before and after sticky transformation.

b. For the combination of concave, convex and straight lines, it has the characteristics of "convexity preservation", "flatness preservation" and "concavity filling".

In the glue transformation, the change degree of the graphics of the arc before and after the transformation depends on the central angle of the arc. Convex morphological characteristics, as shown in Figure 2-a, when the central angle is greater than 180°, the adhesive transformation shows a reduced concave morphological characteristic, as shown in Figure 2-b; the combined graphics of arcs and straight lines change in the adhesive transformation The degree is related to the included angle between the two. When the included angle is greater than 180°, the straight line and the arc form a convex part, and the graphics remain unchanged before and after the adhesion transformation. When the included angle is less than 180°, the straight line and the arc form a concave part. As the width increases, the concave part is gradually filled up, as shown in Figure 2-c, which further verifies the shape-preserving characteristics of the adhesion transformation, such as convexity preservation, flatness preservation, and concave filling.

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

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

${\mathrm{<span\; class="notranslate">\; L</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; D.</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$

This also means that the critical transformation width L' is back-calculated with the maximum concave width D of the arc, and then the graphics are glued and transformed with the width L', and the result of the transformation will fill up all the concave parts in the graphics. As shown in Table 1, the arcs with a radius of R=20 pixels and different central angles are connected 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 adhesion transformation, and the tendency of convexity preservation, flatness preservation, and concave filling increases with the increase of L; when the width L of the adhesion transformation increases to L=L', if the adhesion with a width of D (D>L) is continued Transform, the graphics will no longer change.

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

Glue transformation has the characteristics of "convexity preservation", "flatness preservation" and "concavity filling" in maintaining the graphic form, and the degree of "concavity filling" can be controlled by the value of the width L of the glue transformation. Using these characteristics, it can be obtained from The automatic recognition of curve bending units is realized on different sides and different levels.

(2) Curved Hierarchical Tree

The bending hierarchical tree refers to the identification of bending based on the adhesion 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 the bending unit. In the curved hierarchical tree, the identified bends under a certain sticky transformation scale represent a layer of the hierarchical tree, and each bend represents a node of the layer.

The following will take the curved standard shape—arc as an example to illustrate the basic concept of the curved hierarchical tree. As shown in 3-A in Figure 3, the curve L is nested by four layers of bends of different sizes, and the largest bend 1 (representing curve L) is nested with three bends 1a, 1b, and 1c from left to right, and this The three bends nest 1a.1, 1a.2, 1a.3, 1b.1, 1b.2, 1c.1, and 1c.2 in turn, and the bend 1c.2 nests 1c.2a , 1c.2b two bends. Carry out the cohesive transformation of the curve L at three different scales, and the obtained transformation lines are 3-B, 3-C and 3-D in Figure 3, and the overlay of the transformation line and the original curve is 3 in Figure 3. -E, 3-F and 3-G, each sticky transformation scale corresponds to a layer in the curved hierarchical tree, so as to establish the corresponding curved hierarchical tree at each scale. As shown in 4-B, 4-C, and 4-D of Figure 4, the root node 1 represents the curve L; node 1 has three child nodes, which are nodes 1a, 1b, and 1c in turn, indicating that the curve L is nested with 1a in turn , 1b, and 1c are three bends; nodes 1a, 1b, and 1c have 3 child nodes, 2 child nodes, and 2 child nodes respectively, indicating that bends 1a, 1b, and 1c are nested with 3 bends and 2 bends in sequence and 2 bends; and so on.... Finally, the leaf nodes of the tree represent the smallest bending unit that can be recognized at a certain sticky transform scale.

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

Table 2 The ability of curved hierarchical trees to express the tortuosity of curves

(3) Quantitative index of bending unit

The curved unit is the smallest unit that makes up a curve. It should be a small arc segment. In order to facilitate the measurement of its various indicators, the curved hierarchical tree is replaced by a line, and it is represented by a curved polygon identified by glue transformation. individual bending units. The existing quantitative indicators of the bending unit mainly include the area, length, and width of the bending unit. The quantitative index of the bending unit is introduced in detail below.

a. Bending area S

The bending area S generally refers to the area of the polygon enclosed by the straight line connecting the starting point and the end point of the bending and the curved section, such as the area of the polygon enclosed by the straight line AB and the arc between points A and B in Figure 5.

b. Bending length L

The bending length L is defined as the total length of the bending unit, 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

Generally, the bending width W is defined as the straight-line distance between the starting point and the ending point of the bending, such as the straight-line distance between AB or BC in FIG. 5 . The definition of the bending width based on glue transformation is still used, but the determination of the bending endpoint is based on the glue transformation composite line. The specific operation is: in the step of identifying the bending, the curved polygon is obtained, and the bending width can be obtained by calculating the difference between the circumference of the curved polygon and the bending length.

(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. These two seemingly different concepts are closely related. In this paper, the 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 it is a 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, and 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. Quantification of information can be effectively realized through information entropy, and its specific calculation formula is as follows:

$\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>$

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

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

From the perspective of composition theory, he replaced entropy with complexity, and proposed the concept of generalized sets and the law of complexity. After transformation, the information entropy theory is more popular and easy 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 law of composition. Set language can be used for qualitative analysis, while generalized set language can be used for quantitative analysis. A set is a collection of things with specific properties, and the main focus of the set is what are the two different elements. The generalized set should not only clarify this difference, but also pay attention to the commonality: classify the elements through commonality, and clarify 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 accurately describe the generalized set, two new concepts need to be introduced, that is, sign and individual. Signs refer to the elements that are different in pairs in a set, while individuals refer to the types of elements, and individuals are the basic elements that make up a set. Symbols are used to describe differences, while individuals emphasize the same status among various elements.

A generalized set is a mathematical model, so there will be 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, and the main explanation is 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 clarify 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 Explanation of generalized set examples

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 mountain elevation How many mountains are there at different altitudes? lake each lake area of the lake How many lakes of different sizes are there? land land per square kilometer land type How many different types of land are there? river each branch branch length 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 shown in Equation 6:

$\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

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, that is, there is no difference, the complexity is zero. Expressed in the language of generalized sets, the greater the difference of flag values, the greater the complexity value; the same flag values, 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 finally obtain the uncertainty of the results; the degree of complexity is to analyze things from the perspective of internal differences. This difference is manifested in the presence of different sign values in a generalized set, and the final result is The richness of things, that is, what kind of composition the generalized set has. Analyze information entropy calculation formula 1 and complexity calculation formula 2, where P _i =n _i /N, bring this relationship into complexity calculation formula 2, and combine with formula 1, you can get information entropy and complexity Corresponding relation: C=NH. From this relational expression, it can be known that complexity and information entropy show a positive correlation trend. 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, a lot of knowledge about information entropy is naturally included in the concept of complexity ^[63] . According to this relationship, it can also be concluded that the reason for the degree of uncertainty is that objectively there is a generalized set with complexity, and it is the complexity of the composition that leads to the uncertainty of the result.

Example 1:

The method for describing the tortuosity of the curve using the curved hierarchical tree in the technical solution of the present invention will be described in detail below in conjunction with accompanying drawing 6 and the embodiments. For the curve C shown in 7-a in FIG. 7, the detailed establishment process of the curved hierarchical tree is as follows:

1) Carry out curve synthesis based on cohesive 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 edge transformation line, only the outer transformation line is selected here for illustration, and the inner transformation line is similar to this), and construct a curved polygon with the original curve C .

Specifically, the original curve C is subjected to cohesive transformations 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, and the intersection of the original curve and the curved transformation line is called the curved division points, and connect the bending division points to obtain the bending recognition diagram shown in Figure 7.

Connect the curved division points to the curve C to construct a curved polygon. 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;

Overlay analysis is performed on curved polygons at different levels, the attribution of each curved polygon is judged, and a curved hierarchical tree of each curved is established. Take the curved polygon with the glued transformation width L=3.8 nautical miles, and compare each bend with the curvature of the previous adjacent large-scale glued transformation (L=6 nautical miles), as shown in Figure 9, it can be seen by observation that the curved 18 is split into three There are three bends, namely bends 18.1, 18.2, and 18.3, while bend 8 is split into two bends, respectively bends 8.1 and 8.2, and the other bends are not split.

The above-mentioned process of determining the curved relationship between different layers on the curve is recognized through human observation. In the specific implementation process, it can be realized by judging the ownership of each curved polygon in different layers.

The specific process is as follows, take the curved polygon with the width L1 of the cohesive transformation, and perform overlapping analysis with the curved polygon with the width L2 (L2<L1), and loop to determine the ownership of each curved polygon when the width is L2. Assuming that polygon P2 is a curved polygon when the glue transformation width is L2, and polygon P1 is a curved polygon when the glue transformation width is L1, if polygon P2 belongs to polygon P1, it means that in the curved hierarchy tree, curved polygon P2 The corresponding node C2 is a child node of the corresponding node C1 of the curved polygon P1. The above-mentioned process of judging the ownership of the curved polygons is executed cyclically until all the curved polygons are traversed.

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

In order to clearly mark the relationship between a certain 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 it as 1.1, 1.2, 1.3, ..., 1.n in turn; if 1.n also has multiple child nodes (set as m), then mark it as 1.n.1, 1.n.2, 1. n.3, ... 1.n.m.

According to the method of judging the polygon belonging in 2), the same process is implemented for the bends with transformation width L=6 nautical miles, 3.8 nautical miles, 2 nautical miles, 1 nautical mile and 0.4 nautical miles, and the corresponding hierarchical tree of each bend can be obtained, as shown in the table 3:

Table 3 Curved hierarchical trees with different transform widths

In the curved hierarchical tree, the determination of the parent node of each node depends on the ownership of the corresponding curved node. If node 18.1 corresponds to bend 18.1, superimpose bend 18.1 and the bend corresponding to the node on the previous layer, obviously bend 18.1 belongs to bend 18, then the parent node of node 18.1 is node 18, as shown in 9 in Figure 10 -a and 9-b. In addition, the attribute of each node in the bending hierarchy tree corresponds to the quantization index of the bending unit.

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

For each curved hierarchical tree, loop from bottom to top to determine whether the leaf nodes with parent nodes in each layer have sibling nodes. If there are sibling nodes, the node will be kept; if there is no sibling node, then Delete the node. Continue to search the upper layer to judge whether the leaf node of this layer has a parent node, if not, end the loop; if yes, continue to judge whether it has a sibling node...until all the leaf nodes of 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 is the curved hierarchical tree corresponding to the bend 18 before deleting the leaf nodes without brothers in each layer. Starting from the fourth layer of the curved hierarchical tree, traverse the leaf nodes of each layer from bottom to top to determine whether it has brothers Node. Observe the leaf nodes in the fourth layer of the curved hierarchical tree in Figure 11. The numbers of the nodes all end with the Arabic numeral "1", indicating that all the leaf nodes in this layer have no sibling nodes and need to be deleted. The obtained result As shown in 11-a in Figure 12; continue to circularly search the leaf nodes in the third layer, we can see that there are two node numbers that do not end with the Arabic numeral "1", these two nodes are 18.2.1.2 and 18.2.3.2, delete the last digit of the two node numbers, get the second layer of nodes (ie 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 third layer) have sibling nodes and do not need to be deleted; other leaf nodes on the third layer have no sibling nodes and are deleted Node operation, the result obtained is as shown in 11-b in Figure 12; continue to search upwards to traverse all leaf nodes on the second layer, and use the same judgment method to delete leaf nodes 18.1.1 and 18.3.1 without brothers, 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 brother nodes, no need to delete; there is no leaf node in layer 0, no need to delete Judgment is required. So far, the leaf nodes of each layer in the curved hierarchical tree have been traversed, indicating that the operation of deleting leaf nodes without siblings in each layer has been completed. The hierarchical tree corresponding to curved 18 after the traversal is shown in Figure 12 11-d. Table 4 shows the results after cyclically traversing each curved hierarchical tree and deleting leaf nodes without brothers in each layer.

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

In fact, sticky transforms at any scale can build curved hierarchical trees of height 1. As shown in Figure 8, the glue transformation is performed on the original curve C, and its width L=6 nautical miles, then 18 bends are generated, and these bends are in the same layer of the bend hierarchy tree, and the bend hierarchy with a height of 1 is obtained as shown in Figure 9-a Tree. If one or more cohesive transformations with a width less than 6 nautical miles are continued on the curve C, a curved hierarchical tree with a height not less than 1 can be established, as shown in Figure 9-b and Figure 15, Figure 16 and Figure 17. The selection of the number of times of glue transformation and the width of transformation determines the height of the curved hierarchical tree and the degree of each node. The height of the bending hierarchy tree determines the number of bending nests; the degree of each node indicates the degree of fragmentation of the bending corresponding to the 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 The hierarchical tree after deleting the leaf nodes without brothers in each layer of each curved hierarchical tree

The curved hierarchical tree is based on the expression of the curved hierarchical structure. It can identify 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, there is a neighboring relationship between adjacent nodes in the same layer; a node in the Nth layer has a hierarchical relationship with a parent node in the N-1th layer, which describes a curved nested structure. As shown in Figure 9-a, nodes 1 and 2 are adjacent, and their corresponding bends have an adjacent relationship; as shown in Figure 9-b, between nodes 8.1, 8.2 and node 8, nodes 18.1, 18.2, 18.3 and node 18 Has a hierarchical relationship, and its corresponding bends embody 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 the curve:

First, it can be seen from Figures 16 to 17 that the heights of the curved hierarchical trees corresponding to nodes 8, 12, and 18 are equal to 4, which are the three trees with the largest height among nodes 1-18. It shows that the number of bend nesting in the curve segment corresponding to these three nodes is the largest, which is a relatively complex bend. 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 split respectively Available in 2 bends, 1 bend and 3 bends. Of course, this is a value relative to the comprehensive scale.

Again, the difference in 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 number of subtrees (that is, the degree of nodes), the depth of subtrees (the number of bending nests), the balance of tree node splitting, and the difference in quantitative indicators of nodes, etc. We can preliminarily judge that node 8 is more complex than node 18 than node 12. On the one hand, the number of summary points (7) at node 12 is much smaller than node 8 (16) and node 18 (13). On the other hand, it can be judged from the number of its subtrees, and the difference between its subtrees can also be compared.

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

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

(2) different levels of bending;

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

Secondly, the nesting relationship of different layers before bending can be determined by judging whether they are parent-child nodes. For example, node 18 has three child nodes, namely 18.1, 18.2, and 18.3, indicating that bend 18 nests three bends, respectively 18.1, 18.2, and 18.3.

In short, a curved hierarchical tree corresponds to the tortuosity of a curve, which can describe both the size and shape characteristics of the curved unit itself, and the topological characteristics between the curved units.

Example 2:

The technical scheme of the present invention is described in detail below in conjunction with accompanying drawing 6 and embodiment, for the curve C shown in Figure 18, the detailed establishment process of curved hierarchical tree is as follows:

1) Identify the bending unit.

The original curve C in Fig. 18 is subjected to sticky transformations with widths of 200km, 50km, 30km, and 15km 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, and the curved division points are connected to obtain Bending recognition diagrams shown in 19-a to 19-d in FIG. 19 .

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

Overlay analysis is performed on curved polygons at different levels, the attribution of each curved polygon is judged, and a curved hierarchical tree of each curved is established. For example, for the above curve C, take a curved polygon with a cohesive transformation width L=50km, and compare each curvature with a cohesive transformation L=200km, as shown in Figure 20. It can be seen by observation: the curved 3 splits the curved 3.1 and 3.2, and the curved 8 Splits into bends 8.1 and 8.2, bend 9 splits into bends 9.1 and 9.2, other bends don't split. The above-mentioned process of determining the curved relationship between different layers on the curve is recognized through human observation. In the specific implementation process, it can be realized by judging the ownership of each curved polygon in different layers. The specific process is as follows:

Take the curved polygon with the width L1 of the cohesive transformation, and perform overlapping analysis with the curved polygon with the width L2 (L2<L1), and loop to determine the ownership of each curved polygon when the width is L2. Assuming that polygon P2 is a curved polygon when the glue transformation width is L2, and polygon P1 is a curved polygon when the glue transformation width is L1, if polygon P2 belongs to polygon P1, it means that in the curved hierarchy tree, curved polygon P2 The corresponding node C2 is a child node of the corresponding node C1 of the curved polygon P1.

3) Delete invalid bend

An invalid bend is a bend obtained without splitting, and is represented as a leaf node without sibling nodes in the bend hierarchy tree. For each curved hierarchical tree, loop from bottom to top to determine whether the leaf nodes with parent nodes in each layer have sibling nodes. If there are sibling nodes, the node will be kept; if there is no sibling node, it will be deleted. the node. Continue to search the upper layer to judge whether the leaf node of this layer has a parent node, if not, then terminate the loop; if yes, 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 hierarchy tree. Fig. 21 is the curved hierarchy tree corresponding to bend 2 before deleting leaf nodes without siblings in each layer, and Fig. 22 is the result map of bend 2 after deleting sibling-free nodes in each layer.

Figure 23 shows the bending hierarchy tree corresponding to the original curve C after deleting the invalid bending.

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

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

The tortuosity of the curve, also known as the complexity of the curve, refers to the mutual nesting of bends of different shapes and sizes on the curve 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 and realize the quantitative description of the morphological characteristics of the curve. The basis of the complexity calculation is to classify all the bending units involved in the calculation. In the calculation of this paper, two classification methods are mainly used, one is based on the size of the bending unit (quantitative index representation); the other is based on the hierarchical classification of the bending hierarchical tree itself. Therefore, the complexity of the curved hierarchical tree used to describe the tortuosity of the curve can be divided into two types, namely the size complexity and the hierarchy complexity. Finally, the comprehensive complexity is used to measure the tortuosity of the geographic curve.

(1) Size complexity

Dimensional complexity is based on the calculation of bending units. Each bending unit 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 unit is to consider the composition of the line by taking the bending unit as a unit, and mainly use the number of bending units under each eigenvalue for calculation. The main problem that this calculation method illustrates is how many different sizes of bends there are. Table 4 shows the generalized set language representation of this calculation method. Since the sizes of the bending units are basically different, it is meaningless to calculate the number strictly according to the specific value. Here, the method of classification is used to count the number: according to a certain attribute value (area, length, width) of all bending units, different attribute value intervals are divided according to a certain 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 The area of each bend How many bends of different areas curve each bend the length of each bend how many bends of different lengths curve each bend 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 measurement index of area as an example, the calculation formula of complexity is:

$\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

Among them, n represents the number of subdivided area categories, n _i represents the number of bending units in each interval, and N is the total number of bending units forming the curve.

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

$\mathrm{<span\; class="notranslate">\; SC</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; N</span>}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

Among them, the total number of effective bending units of N bending hierarchical tree, n _i is the number of effective bending in each category. Similarly, the logarithmic function takes different bases, and the calculated complexity results are different. Take the curve C in Figure 7 as an example to calculate the size complexity, and the size complexity result is calculated with base 10. The specific size information of the bending unit of the curve is shown in Table 5:

Table 5 Properties of bending elements of curve C

ID TreeName area Perimeter Baseline Length Radius Olay R 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, which are the area, length and baseline length of the bending element. These three types of indicators are divided into 10 categories according to the equidistant classification method, 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 dimensional complexity of the curve C calculated with the area as the metric is:

SCA＝27*log(27)-10*log(10)-6*log(6)-4*log(4)-3*log(3)-2*log(2)

=19.536,

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

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

The dimensional 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 layer of the hierarchical tree is regarded as a basic unit, that is, an individual, and the number of bends contained in each layer is a symbol value.

This method considers the composition of curves in units of levels, and mainly uses the number of bends contained in each level for calculation. In essence, this calculation method is to use the hierarchy to classify and pay attention to the composition of each category. The main problem explained in the calculation process is how many levels there are in the curved hierarchy tree, and how many bends are there in each level, so as to reflect whether the curve is complex or not. In theory, the more layers, the more complex the curves, and the more uneven the number of bends in the layers, the more complex. This method is described in generalized set language as 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 Amount of bends in each layer How many bending elements are there at different levels

Its complexity calculation formula is:

$\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

Among them, n indicates how many levels the bending hierarchical tree has, n _i indicates how many bending units there are in each level, that is, the number of bending units in each result layer, and N is the total number of bending units forming the curve . The logarithmic function takes different bases, and the calculation results are different.

According to the nature of the logarithmic function, the calculation formula of the hierarchical complexity LC is formula 8:

$\mathrm{<span\; class="notranslate">\; LC</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; N</span>}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; log</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

Among them, N is the total number of effective bending units in the bending hierarchy tree, and n _i is the number of effective bendings 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. Take the curve C in Figure 18 as an example to calculate the size complexity, and the size complexity result is calculated with base 10. For the original curve C, there are five levels in the bending hierarchy tree. There is one valid bending in the first level, 10 valid bendings in the second level, 6 valid bendings in the third level, and 2 valid bendings in the fourth level. There are 9 effective bends in the fifth layer, for a total of 28 effective bends.

According to formula 8, the hierarchical 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

Comprehensive complexity is used to measure the tortuosity of geographical curves from two aspects of level and size. The comprehensive complexity ZC can be defined as formula 9:

ZC＝P ₁ SCA+P ₂ SCL+P ₃ SCW+P ₄ LC,

Wherein, P _i (i=1, . . . , 4) represent the weights of different types of complexity respectively, 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, and then a 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 embodiment of the present invention has the following beneficial effects: sequentially complete the work of identifying bending units, overlaying and determining bending nesting relationships at different scales, establishing a bending hierarchical tree, deleting invalid bending, and measuring the tortuosity of geographic curves based on information entropy theory , using the combination of dimensional complexity and hierarchical complexity to describe the tortuosity, fully showing the partial and overall tortuosity of the curve, and comprehensively considering the nesting relationship between different levels of bending, It overcomes the defects of the existing technology, can better describe the tortuosity of the curve, fully reflect the shape and structural characteristics of the curve, is less affected by the length of the curve, and fully uses the bending hierarchy tree to fully reflect the adjacent relationship and hierarchical characteristics between the curves. 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 article, 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 that there is a relationship between these entities or operations. There is no such actual relationship or order between them. Furthermore, the term "comprises", "comprises" or any other variation thereof is intended to cover a non-exclusive inclusion such that a process, method, article or apparatus comprising a set of elements includes not only those elements, but also includes elements not expressly listed. other elements of or also include elements inherent in such a process, method, article, or device. Without further limitations, an element defined by the phrase "comprising a ..." does not exclude the presence of additional identical elements in the process, method, article or apparatus comprising said element.

The above embodiments are only used to illustrate the technical solutions of the present invention, rather than 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: it can still be described in the foregoing embodiments Modifications are made to the recorded technical solutions, or equivalent replacements are made to some of the technical features; 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 (8)

1. a geographical line tortuosity measure based on comentropy, it is characterised in that comprise the following steps:

1) bending unit is identified: curve is carried out the adhesion conversion of different in width, the result that adhesion converts is folded with primitive curve Add, obtain the bending polygon of different scale, connect bending division points and be bent identification figure, by entering with original geographical line Row intersects computing and obtains the bending under each yardstick, and calculates the quantizating index of each bending unit, is stored in association attributes territory In；

2) the bending nest relation under superposition determines different scale, sets up and bends hierarchical tree: the bending polygon to different levels It is laid out analyzing, it is judged that the polygonal ownership of each bending, sets up the bending hierarchical tree of each bending；

3) invalid bending is deleted: delete the invalid bending of each layer, finally give the bending unit of each level；

4) tortuosity based on information entropy theory tolerance geographical line: use information entropy theory to calculate the ground that bending hierarchical tree represents The tortuosity of reason curve.

A kind of geographical line tortuosity measure based on comentropy the most according to claim 1, it is characterised in that: institute State the intersection point that bending division points is primitive curve and bending transformation line.

A kind of geographical line tortuosity measure based on comentropy the most according to claim 1, it is characterised in that: folded Add the bending nest relation determined under different scale by judging that in different levels, each polygonal ownership of bending realizes, and incites somebody to action Bending polygon under each yardstick superposes with the bending polygon under upper level large scale, determines that little yardstick is polygonal and returns Belonging to, thus obtain corresponding nest relation, determine the level of each bending unit, final foundation is saved using primitive curve as root The bending hierarchical tree of point.

A kind of geographical line tortuosity measure based on comentropy the most according to claim 1, it is characterised in that: build The method of vertical bending hierarchical tree is the hierarchical tree to each bending, and each layer of cycle criterion from bottom to up has the leaf of parents' node to tie Whether point has sibling, if there being sibling, then this node retains, if without sibling, then deletes this node；Continue up One layer of search, it is judged that whether this layer of leaf node has parents' node, if nothing, then end loop, if having, then continues to judge whether it has Sibling, until having traveled through all leaf nodes of each layer.

A kind of geographical line tortuosity measure based on comentropy the most according to claim 1, it is characterised in that: institute State invalid be bent into non-upper strata bending division obtain direct by upper strata bending inherit and come bending.

A kind of geographical line tortuosity measure based on comentropy the most according to claim 1, it is characterised in that adopt The method calculating the geographical line tortuosity that bending hierarchical tree represents with information entropy theory is to use size complexity and level multiple The compositive complexity of miscellaneous degree measures the tortuosity of geographical line, and the computing formula of compositive complexity is:

ZC=P₁SCA+P₂SCL+P₃SCW+P₄LC,

Wherein, P_i(i=1,2 ..., 4) represent that the weight shared by different types of complexity, weight sum are 1 respectively.

A kind of geographical line tortuosity measure based on comentropy the most according to claim 5, it is characterised in that institute Stating SC is size complexity, and with bending unit as elementary cell in the calculating of described SC, computing formula is:

$SC=Nlog\left(N\right)-\stackrel{n}{\mathrm{\&Sigma;}}{n}_{i}log\left(n_i\right),$

Wherein, N is the sum of effective bending unit of bending hierarchical tree, n_iQuantity for the effectively bending of each apoplexy due to endogenous wind.

A kind of geographical line tortuosity measure based on comentropy the most according to claim 5, it is characterised in that institute Stating LC is level complexity, and with one layer of hierarchical tree as elementary cell in the calculating of described LC, computing formula is:

$LC=Nlog\left(N\right)-\stackrel{n}{\mathrm{\&Sigma;}}{n}_{i}log\left(n_i\right),$

Wherein, N is the sum of effective bending unit of bending hierarchical tree, n_iQuantity for the effectively bending in every layer.