CN103336783A - Voronoi and inverse distance weighting combined density map drawing method - Google Patents

Abstract

Combining Thiessen polygon and inverse distance weighted density map drawing method, first read the discrete point data, and construct the V map with discrete points; then rasterize the entire V map area, and according to the membership relationship between the pixel and the V polygon and The distance between the center point of the pixel inside the V polygon and the scattered point is used to calculate the density value of the pixel. In order to make the result more reasonable, in the present invention, the neighborhood average smoothing process is performed on the generated grids, and reclassification is performed, and different gray values are assigned; finally, all grids are rendered. The V-diagram constructed based on discrete points generates an influence range for each discrete point through the space division based on the shortest distance constraint, and local density calculation within this range ensures the comparability and reliability of the calculation results among the influence ranges; in addition, In this method, the density difference of different pixel points within the influence range is considered, and the density value distribution method based on distance is adopted in the V polygon where each point is located, so that the result is more reasonable and accurate.

Description

Density Mapping Method Combined Thiessen Polygon and Inverse Distance Weighting

This case is a divisional application of the Chinese invention patent application 201210146965.2, the filing date is 2012-05-11, and the name is "a method for making point density thematic maps based on Voronoi diagrams".

technical field

The invention proposes a method for making a point density thematic map, which belongs to the interdisciplinary field of Voronoi diagram (Voronoi polygon is also called "Tyssen polygon") application technology in computational geometry, cartography and geographic information system (GIS).

Background technique

The calculation of point density refers to estimating the number of points per unit area at the location of any point in the area according to the distribution of points in the area. In geoscience applications, the calculation of point density is a theoretical abstraction after summarizing and summarizing issues such as settlement density calculations and building density calculations. ICA), computer vision and other fields have important applications. The most direct way to reflect the point density distribution is by drawing a point density map. The dot density map is a kind of thematic map in cartography. It uses points (numbers) to represent the boundary or domain objects corresponding to the data values. The total number of points in a domain object represents the data value corresponding to the domain.

At present, there are two common methods for calculating the point density at any point. One is a simple point density calculation method, which calculates the magnitude of a unit area based on the point features falling within the neighborhood around each raster cell. Conceptually, it takes raster cells as the starting point, defines a neighborhood around the center of each raster cell, adds up the number of points in the neighborhood, and then divides it by the area of the neighborhood to finally get the point The density of the feature. The other is the method of kernel density estimation, which takes the point feature as the starting point and calculates the density of the point feature in its surrounding neighborhood. This method assumes that each point is covered by a smooth surface. The surface value is the highest at the position of the point, and the surface value gradually decreases as the distance from the point increases, and the surface value is zero at the position where the distance from the point is equal to the search radius. The search neighborhood of the kernel density estimation method allows only circles, and the density of each output cell is the sum of the values of all kernel surfaces superimposed on the center of the raster cell.

These two methods are widely used in the calculation of point density, but the size of the search neighborhood has a certain impact on the results: when the search radius parameter value is larger, the generated density grid is smoother and the degree of generalization is higher; when The smaller the parameter value, the more detailed the information displayed in the resulting raster. In addition, the simple point density calculation method counts the number of discrete points in a neighborhood of uniform size, but ignores the difference in point density within the neighborhood; while the kernel density estimation method uses a circle of uniform size as the search neighborhood, taking into account the The difference in point density within the domain is ignored, but the difference in the "sphere of influence" of different discrete points is ignored.

Contents of the invention

The technical problem to be solved in the present invention is: to overcome the above-mentioned deficiencies in the prior art, and to provide a density map drawing method combining Thiessen polygons and inverse distance weighting with the help of the technical advantages of Thiessen polygons for space division, which uses each discrete point The corresponding Voronoi polygon (Tyssen polygon) is a neighborhood, and the difference assignment of point density in the neighborhood is realized through the method of inverse distance weight distribution, which can quickly, reasonably and accurately calculate the point density and output the point density map.

In order to solve the above technical problems, a density map drawing method of a joint Thiessen polygon and inverse distance weighting provided by the invention comprises the following steps:

The first step, discrete point reading - read the discrete point set as the original data, the discrete point has its own serial number and coordinate data;

The second step is to construct a Voronoi diagram - construct a Voronoi diagram based on all discrete points in the discrete point set;

The third step is to rasterize the Voronoi diagram——according to the coordinates of the upper left corner and the lower right corner of the Voronoi diagram, and the given number of rows and columns for dividing the grid cells, the entire Voronoi diagram area is rasterized to generate several images element and determine the coordinates of the center point of each pixel;

The fourth step is to establish the affiliation relationship between the pixel and the Voronoi polygon——according to the topological relationship between the coordinates of the center point of the pixel and the Voronoi polygon, determine the affiliation relationship between the pixel and the Voronoi polygon. When the center point of the pixel falls on a certain Voronoi polygon , then it is determined that the pixel belongs to the Voronoi polygon;

The fifth step, calculate the density value of the pixel—the total density value of the i-th Voronoi polygon V _i is 1/S _g , and this total density value 1/S _g is allocated to each pixel of the Voronoi polygon V _i , so that The sum of the density values of all pixels in the Voronoi polygon V _i is equal to 1/ _Sg , wherein, _Sg is the area of a single pixel, 1≤i≤n, and n is the number of Voronoi polygons in the Voronoi figure;

The sixth step, grid smoothing - recalculate the density value of each pixel by using the spatial smoothing filter method;

The seventh step, reclassification of pixel density values——statistically analyze the density values of all pixels in the raster image, and accordingly reclassify all pixel density values and assign different gray values;

The eighth step is to draw a density map - to render the grid according to the gray value of each pixel to obtain a density map.

In the point density thematic map making method of the present invention, based on the Voronoi diagram constructed by discrete points, the region is segmented using Voronoi polygons, so that there is and only one discrete point (generating element) in each Voronoi polygon, and each Voronoi polygon is visible. Generate "range of influence" for the corresponding discrete point; the calculation of the density value of each pixel in the Voronoi polygon is not related to other Voronoi polygons, so the calculation of the density value of the pixel in the Voronoi polygon is less disturbed, and the local density calculation in the Voronoi polygon guarantees The comparability and reliability of the calculation results between the Voronoi polygons are guaranteed; and the distribution method of the total density value of the Voronoi polygons can be selected according to the actual situation, and the equal division method or the inverse distance weight distribution method can be selected, which is more flexible to use; The Voronoi diagram constructed based on the discrete points is unique, so there are fewer subjective factors in the implementation process of the solution of the present invention, and it is easy to operate.

The present invention provides following three kinds of Voronoi polygon total density value allocation schemes:

Option One:

其中R_ij、R_ik分别表示Voronoi多边形V_i中第j个和第k个像元的中心点至Voronoi多边形V_i所对应的离散点P_i的距离，j≤m_i，k≤m_i，m_i为Voronoi多边形V_i内的像元个数，i、j、k均为自然数。In the second step of the aforementioned method, the corresponding relationship between the discrete point and the Voronoi polygon to which it belongs is set up; The density value of a pixel is inversely proportional to the distance from the center point of the pixel to the discrete point corresponding to the Voronoi polygon; the density value of the jth pixel in the Voronoi polygon V _i is

Among them, R _ij and R _ik represent the distance from the center point of the jth and kth pixels in the Voronoi polygon V _i to the discrete point P _i corresponding to the Voronoi polygon V _i , j≤m _i , k≤m _i , m _i is the number of pixels in the Voronoi polygon V _i , and i, j, k are all natural numbers.

Option II:

其中R_ijR_ik分别表示Voronoi多边形V_i中第j个和第k个像元的中心点至Voronoi多边形V_i所对应的离散点P_i的距离，j≤m_i，k≤m_i，m_i为Voronoi多边形V_i内的像元个数，i、j、k均为自然数。In the second step of the aforementioned method, the corresponding relationship between the discrete point and the Voronoi polygon to which it belongs is set up; The density value of a pixel is inversely proportional to the square of the distance from the center point of the pixel to the discrete point corresponding to the Voronoi polygon; the density value of the jth pixel in the Voronoi polygon V _i is

Among them, R _ij R _ik represent the distance from the center point of the jth and kth pixel in the Voronoi polygon V _i to the discrete point _{P i} corresponding to the Voronoi polygon V i, j≤m _i , k≤m _i , m _i is the number of pixels in the Voronoi polygon V _i , and i, j, k are all natural numbers.

third solution:

In the fifth step of the aforementioned method, the density value of each pixel in the Voronoi polygon is apportioned by the equal division method, and the density value B _ii of the jth pixel in the Voronoi polygon V _i =1/(S _g * m _i ), m _i is the number of pixels in the Voronoi polygon V _i , j≤m _i , and j is a natural number.

Among the above three schemes, the first two methods adopt the inverse distance weight distribution method, and the third one adopts the equal division method. The calculation of the equipartition method is simpler, and the spatial distribution of the point density can be reflected from the size of the Voronoi polygon (the number of pixels contained in the Voronoi polygon). The larger the area of the Voronoi polygon, the lower the point density value; the inverse distance weight distribution rule considers In order to eliminate the difference in the point density of different pixels in the "influence range" (Voronoi polygon), the Voronoi polygon where each pixel is located uses a distance-based density value distribution method to make the result more reasonable and accurate.

Among them, in the inverse distance weight assignment method, the spatial distribution of point density can be reflected from the size of the calculated Voronoi polygon area, and the distance factor is further considered inside the Voronoi polygon, and the density value of the pixel is calculated according to the reciprocal of the distance (or The proportion of the square of the reciprocal) is distributed to each pixel, making the calculation results more reasonable.

Description of drawings

The present invention will be further described below in conjunction with the accompanying drawings.

Fig. 1 is a flow chart of the method of the present invention.

Figure 2 is a schematic diagram of discrete points and their Voronoi diagram construction.

Figure 3 is a schematic diagram of pixel membership and density value calculation.

Fig. 4 is a schematic diagram of raster neighborhood mean smoothing.

Figure 5 is a schematic diagram of reclassification of pixel density values.

Detailed ways

The purpose and effects of the present invention will become more apparent by referring to the accompanying drawings in detail of the present invention.

As shown in Figure 1, it is a flow chart of the density map drawing method of the present invention's joint Thiessen polygon and inverse distance weighting, comprising the following steps:

The first step, reading discrete points—reading a set of discrete points as raw data, the discrete points have their own serial numbers and coordinate data.

Read the coordinates of the discrete points, and display the position of each point according to the proportional relationship between the screen coordinate range and the discrete point coordinate range (Fig. 2 left). In this example, the discrete point number is stored as the attribute data of the discrete point.

The second step is to construct a Voronoi diagram——construct a Voronoi diagram based on all discrete points in the discrete point set, and establish the correspondence between the discrete points and the Voronoi polygons to which they belong;

In this example, all the discrete points in the discrete point set are used as the generator, and the Voronoi diagram is constructed by the scan line algorithm, and the number value of the discrete point is passed to the number value of the Voronoi polygon, thus establishing the relationship between the discrete point and the Voronoi polygon to which it belongs. Correspondence relationship; simultaneously establish the topological relationship between the generator (discrete point) and the Voronoi edge, the generator (discrete point) and the Voronoi polygon, record the linear equation coefficients that form the edge in the data structure of the Voronoi edge, and form the two sides of the edge Endpoints and the occurrence element (discrete point) on both sides associated with this edge, record the edge set of this polygon in the data structure of Voronoi polygon; Adopt such data to be conducive to the calculation of pixel density value in the fifth step below.

The third step is to rasterize the Voronoi diagram——according to the coordinates of the upper left corner and the lower right corner of the Voronoi diagram, and the given number of rows and columns for dividing the grid cells, the entire Voronoi diagram area is rasterized to generate several images and determine the coordinates of the center point of each pixel.

For any p*q grid divided, let the coordinates of the lower left corner and upper right corner of the entire area be (A ₁ , B ₁ ) and (A ₂ , B ₂ ), where A ₁ <A ₂ , B ₁ <B ₂ . Pixels can be stored in the form of a two-dimensional array, arranged from left to right and from top to bottom. For any pixel a _ij, the coordinates of the center point of the pixel are $(A_1+\frac{A_2-A_1}{p}\&times;(i+\frac{1}{2}),{\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HDA00003274423900021.png"\; state="\{\{state\}\}">B</figure-callout>}}_1+\frac{B_2-B_1}{q}\&times;(q-i-\frac{1}{2}))$ Among them, 0≤i≤p-1, 0≤j≤q-1.

The fourth step is to establish the affiliation relationship between the pixel and the Voronoi polygon——according to the topological relationship between the coordinates of the center point of the pixel and the Voronoi polygon, determine the affiliation relationship between the pixel and the Voronoi polygon. When the center point of the pixel falls on a certain Voronoi polygon , it is determined that the pixel belongs to the Voronoi polygon.

In this example, the affiliation relationship between the pixel and the Voronoi polygon is stipulated as follows: if the center point of the pixel falls in which Voronoi polygon, the pixel belongs to the Voronoi polygon; if the center point of the pixel falls on a Voronoi edge , then find two Voronoi polygons associated with the edge, and stipulate that the pixel belongs to the Voronoi polygon with a smaller occurrence element number; if the center of the pixel coincides with a Voronoi vertex, then find three Voronoi polygons associated with the vertex Edge, and then find three Voronoi polygons associated with the vertex, and stipulate that the pixel belongs to the one with the smallest element number among the three Voronoi polygons. The pixels within the solid line border in Figure 4 belong to the polygon. Belong to special case for the situation that pixel center falls on the Voronoi edge of Voronoi polygon and the vertex of Voronoi polygon, the affiliation rules of these picture elements can be artificially defined, there is no strict requirement, so this part content is not limited in the present invention and Detailed description.

The fifth step is to calculate the density value of each pixel—the total density value of the i-th Voronoi polygon V _i is 1/S _g , and the total density value 1/S _g is allocated to each pixel of the Voronoi polygon V _i , Make the sum of the density values of all pixels _in the Voronoi polygon V equal 1/ _Sg , wherein, _Sg is the area of a single pixel, 1≤i≤n, and n is the number of Voronoi polygons in the Voronoi figure;

In this example, the inverse distance weight distribution method is used to apportion the density value of each pixel in the Voronoi polygon, and the density value of the pixel in the Voronoi polygon is inversely proportional to the distance from the center point of the pixel to the discrete point corresponding to the Voronoi polygon.

Figure 3 is a schematic diagram of pixel affiliation and density value calculation. The calculation method of the specific density value of the pixel is as follows:

其中R_ij、R_ik分别表示Voronoi多边形V_i中第j个和第k个像元的中心点至Voronoi多边形V_i所对应的离散点P_i的距离，S_g表示单个像元的面积，j≤m_i，k≤m_i，m_i为Voronoi多边形V_i内的像元个数，i、j、k均为自然数。像元密度值计算的实质就是在一个Voronoi多边形中，根据其隶属的每一个像元的几何中心到发生元的距离的倒数占所有像元几何中心到发生元距离倒数之和的比例来分配“一个点”的份额，再除以像元面积得到点密度。如图5所示，左侧栅格中的数据则为计算后得到的像元密度值，接下来进行第六步栅格平滑。The density value of the jth pixel in the Voronoi polygon V _i is

Among them, R _ij and R _ik represent the distance from the center point of the jth and kth pixels in the Voronoi polygon V _i to the discrete point P _i corresponding to the Voronoi polygon V _i , S _g represents the area of a single pixel, and j ≤m _i , k≤m _i , m _i is the number of pixels in the Voronoi polygon V _i , and i, j, k are all natural numbers. The essence of calculating the pixel density value is to allocate in a Voronoi polygon according to the ratio of the reciprocal of the distance from the geometric center of each pixel to the occurrence element to the sum of the reciprocal distances from the geometric center of all pixels to the occurrence element. One point", and then divided by the pixel area to get the point density. As shown in Figure 5, the data in the grid on the left is the calculated cell density value, and then the sixth step of grid smoothing is performed.

The sixth step, grid smoothing——recalculate the density value of each pixel by using the method of spatial smoothing and filtering.

The spatial domain smoothing filtering method may be a neighborhood mean smoothing method, a neighborhood median smoothing method, a neighborhood extremum smoothing method, and the like. In this example, the field average method is adopted, that is, the density value of a cell in the grid is added to the density value of the surrounding adjacent cells, and then the obtained average value is used as the density value of the cell in the new grid; in addition, according to The number of generated raster cells, use a suitable size template (such as 5*5, 9*9, 25*25, etc.) to traverse each cell row by row to perform matrix multiplication in the neighborhood of the requested cell, such as As shown in Figure 4, this example uses a template with a size of 3*3, and the pixel density value in the smoothed grid is shown on the right side of Figure 5. After grid smoothing, the "steep slope" phenomenon near the Voronoi edge can be basically eliminated. In this embodiment, the average smoothing method is adopted, and it is not ruled out that there is a more reasonable and effective smoothing method.

The seventh step, reclassification of pixel density values——statistically analyze the density values of all pixels in the raster image, and reclassify all pixel density values accordingly, assigning different gray values.

The essence of reclassification is to reclassify the pixel density value (attribute value) or change the input pixel density value to a substitute value (as shown in Figure 5). First, traverse each pixel line by line, statistically analyze the maximum value, minimum value and frequency distribution of all pixel density values; secondly, make a frequency distribution histogram and frequency change curve, and select a reasonable interval threshold accordingly All pixels are divided into several categories according to the order of density value from small to large; finally, different gray values are assigned to the classified pixels. The greater the density, the greater the gray value assigned.

The eighth step is to draw a density map - to render the grid according to the gray value of each pixel to obtain a density map.

In the fifth step of the present embodiment, the density value of the pixel in the Voronoi polygon is inversely proportional to the distance from the pixel center point to the corresponding discrete point of the Voronoi polygon; in addition, the density value of the pixel in the Voronoi polygon It is inversely proportional to the square of the distance from the pixel center point to the discrete point corresponding to the Voronoi polygon; the density value of the jth pixel in the Voronoi polygon V _i is Among them, R _ij R _ik represent the distance from the center point of the jth and kth pixel in the Voronoi polygon V _i to the discrete point _{P i} corresponding to the Voronoi polygon V i, j≤m _i , k≤m _i , m _i is the number of pixels in the Voronoi polygon V _i , and i, j, k are all natural numbers.

When calculating the pixel density value, the average division method can also be used to apportion each pixel density value in the Voronoi polygon, and the density value B _ij of the jth pixel in the Voronoi polygon V _i =1/(S _g *m _i ) , m _i is the number of pixels in the Voronoi polygon V _i , j≤m _i , and j is a natural number.

The Voronoi diagram constructed based on discrete points in this embodiment generates an "influence range" for each discrete point through space division based on the shortest distance constraint, and local density calculations are performed within this range to ensure that the distance between each "influence range" (Voronoi polygon) The comparability and reliability of the calculation results; in addition, this method considers the difference in the density of different pixel points in the "influence range", and adopts a distance-based density value distribution method inside the Voronoi polygon where each point is located to make the result more accurate. reasonably accurate.

In addition to the above-mentioned embodiments, the present invention can also have other implementations. All technical solutions formed by equivalent replacement or equivalent transformation fall within the scope of protection required by the present invention.

Claims (6)

1. A density map drawing method of joint Thiessen polygon and inverse distance weighting, comprising the following steps: The first step, discrete point reading - read the discrete point set as the original data, the discrete point has its own serial number and coordinate data; The second step is to construct a Voronoi diagram - construct a Voronoi diagram based on all discrete points in the discrete point set; In the second step, establish the correspondence between discrete points and their Voronoi polygons; The third step is to rasterize the Voronoi diagram——according to the coordinates of the upper left corner and the lower right corner of the Voronoi diagram, and the given number of rows and columns for dividing the grid cells, the entire Voronoi diagram area is rasterized to generate several images element and determine the coordinates of the center point of each pixel; The fourth step is to establish the affiliation relationship between the pixel and the Voronoi polygon——according to the topological relationship between the coordinates of the center point of the pixel and the Voronoi polygon, determine the affiliation relationship between the pixel and the Voronoi polygon. When the center point of the pixel falls on a certain Voronoi polygon , then it is determined that the pixel belongs to the Voronoi polygon; The fifth step, calculate the density value of the pixel—the total density value of the i-th Voronoi polygon V _i is 1/S _g , and this total density value 1/S _g is allocated to each pixel of the Voronoi polygon V _i , so that The sum of the density values of all pixels in the Voronoi polygon V _i is equal to 1/ _Sg , wherein, _Sg is the area of a single pixel, 1≤i≤n, and n is the number of Voronoi polygons in the Voronoi figure; In the 5th step, adopt inverse distance weight distribution method to apportion each pixel density value in the Voronoi polygon, the density value of the pixel in the Voronoi polygon is proportional to the distance from the pixel center point to the corresponding discrete point of the Voronoi polygon Inverse ratio; The sixth step, grid smoothing - recalculate the density value of each pixel by using the spatial smoothing filter method; The seventh step, reclassification of pixel density values——statistically analyze the density values of all pixels in the raster image, and accordingly reclassify all pixel density values and assign different gray values; The eighth step is to draw a density map - to render the grid according to the gray value of each pixel to obtain a density map. 2. the density diagram drawing method of joint Thiessen polygon and inverse distance weighting according to claim 1, is characterized in that: the density value of the j pixel _in Voronoi polygon V is

Among them, R _ij and R _ik represent the distance from the center point of the jth and kth pixels in the Voronoi polygon V _i to the discrete point P _i corresponding to the Voronoi polygon V _i , j≤m _i , k≤m _i , m _i is the number of pixels in the Voronoi polygon V _i , and i, j, k are all natural numbers. 3. according to the density map drawing method of the joint Thiessen polygon described in any one of claim 1-2 and inverse distance weighting, it is characterized in that: in the described 6th step, the method for spatial domain smoothing filter is neighborhood mean value smoothing method , neighborhood median smoothing method, neighborhood extremum smoothing method. 4. A density map drawing method of joint Thiessen polygon and inverse distance weighting, comprising the following steps: The first step, discrete point reading - read the discrete point set as the original data, the discrete point has its own serial number and coordinate data; The second step is to construct a Voronoi diagram - construct a Voronoi diagram based on all discrete points in the discrete point set; In the second step, establish the correspondence between discrete points and their Voronoi polygons; The third step is to rasterize the Voronoi diagram——according to the coordinates of the upper left corner and the lower right corner of the Voronoi diagram, and the given number of rows and columns for dividing the grid cells, the entire Voronoi diagram area is rasterized to generate several images element and determine the coordinates of the center point of each pixel; The fourth step is to establish the affiliation relationship between the pixel and the Voronoi polygon——according to the topological relationship between the coordinates of the center point of the pixel and the Voronoi polygon, determine the affiliation relationship between the pixel and the Voronoi polygon. When the center point of the pixel falls on a certain Voronoi polygon , then it is determined that the pixel belongs to the Voronoi polygon; The fifth step, calculate the density value of the pixel—the total density value of the i-th Voronoi polygon V _i is 1/S _g , and this total density value 1/S _g is allocated to each pixel of the Voronoi polygon V _i , so that The sum of the density values of all pixels in the Voronoi polygon V _i is equal to 1/ _Sg , wherein, _Sg is the area of a single pixel, 1≤i≤n, and n is the number of Voronoi polygons in the Voronoi figure; In the 5th step, adopt inverse distance weight distribution method to apportion each pixel density value in the Voronoi polygon, the density value of the pixel in the Voronoi polygon and the distance between the pixel center point and the corresponding discrete point of the Voronoi polygon inversely proportional to the square; The sixth step, grid smoothing - recalculate the density value of each pixel by using the spatial smoothing filter method; The seventh step, reclassification of pixel density values——statistically analyze the density values of all pixels in the raster image, and accordingly reclassify all pixel density values and assign different gray values; The eighth step is to draw a density map - to render the grid according to the gray value of each pixel to obtain a density map. 5. the density map drawing method of joint Thiessen polygon and inverse distance weighting according to claim 4, is characterized in that: the density value of the j pixel _in Voronoi polygon V is

Among them, R _ij R _ik represent the distance from the center point of the jth and kth pixel in the Voronoi polygon V _i to the discrete point _{P i} corresponding to the Voronoi polygon V i, j≤m _i , k≤m _i , m _i is the number of pixels in the Voronoi polygon V _i , and i, j, k are all natural numbers. 6. according to the density map drawing method of the joint Thiessen polygon described in any one of claim 4-5 and inverse distance weighting, it is characterized in that: in the described 6th step, the method for spatial domain smoothing filtering is neighborhood mean value smoothing method , neighborhood median smoothing method, neighborhood extremum smoothing method.

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