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

Voronoi and inverse distance weighting combined density map drawing method Download PDF

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CN103336783A
CN103336783A CN2013102152822A CN201310215282A CN103336783A CN 103336783 A CN103336783 A CN 103336783A CN 2013102152822 A CN2013102152822 A CN 2013102152822A CN 201310215282 A CN201310215282 A CN 201310215282A CN 103336783 A CN103336783 A CN 103336783A
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王结臣
芮一康
崔璨
王豹
周生路
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Nanjing University
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Abstract

联合泰森多边形与反距离加权的密度图制图方法,首先读取离散点数据,以离散点构建V图;然后对整个V图区域进行栅格化,并根据像元与V多边形的隶属关系及V多边形内部像元中心点距离散点的距离来计算像元的密度值。为使结果更加合理,本发明中对生成栅格进行了邻域均值平滑处理,并进行重分类,赋予不同的灰度值;最后,渲染所有栅格。基于离散点构建的V图通过基于最短距离约束的空间划分为每个离散点生成影响范围,在此范围内进行局部密度计算保证了各影响范围之间计算结果的可比性及可靠性;另外,本方法中考虑了影响范围内不同像元点密度的差异,在各点所在的V多边形内部采用了基于距离的密度值分配办法,使结果更加合理准确。

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

联合泰森多边形与反距离加权的密度图制图方法Density Mapping Method Combined Thiessen Polygon and Inverse Distance Weighting

本案是中国发明专利申请201210146965.2,申请日2012-05-11,名称“一种基于Voronoi图的点密度专题图制作方法”的分案申请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".

技术领域technical field

本发明提出的是一种点密度专题地图的制作方法,属于计算几何、地图学与地理信息系统(GIS)中Voronoi图(Voronoi多边形也称为“泰森多边形”)应用技术的交叉领域。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).

背景技术Background technique

点密度的计算是指根据区域内点的分布估计其内任一点所在位置单位面积的点数。在地学应用中,点密度的计算是对居民点密度计算、建筑物密度计算等问题进行归纳与总结后的理论抽象,在人口密度分布、图像处理、聚类分析、判别分析、独立成分分析(ICA)、计算机视觉等领域有着重要的应用。体现点密度分布的最直接的方法就是通过绘制点密度图。点密度图是地图学中专题图的一种,它用点(数)来表现与数据值对应的边界或域对象,一个域对象中点的总个数代表了域对应的数据值。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.

目前,计算任意一点所在处点密度主要有两种常用方法。一种是简单点密度计算方法,即根据落入每个栅格像元周围邻域内的点要素计算单位面积的量级。从概念上讲,它以栅格像元为出发点,对每个栅格像元中心的周围都定义了一个邻域,将邻域内点的数量相加,然后除以邻域面积,最终得到点要素的密度。另一种是核密度估计的方法,它以点要素为出发点,计算点要素在其周围邻域中的密度。该方法认为在每个点上方均覆盖着一个平滑曲面。在点所在位置处表面值最高,随着与点的距离的增大表面值逐渐减小,在与点的距离等于搜索半径的位置处表面值为零。核函数密度估计方法的搜索邻域仅允许使用圆形,每个输出像元的密度均为叠加在栅格像元中心的所有核表面的值之和。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.

这两种方法在点密度计算中都有着广泛的应用,但是搜索邻域的大小对结果有一定影响:当搜索半径参数值越大,生成的密度栅格越平滑且概化程度越高;当参数值越小,生成的栅格所显示的信息越详细。此外,简单点密度计算方法以统一大小的邻域统计离散点的个数,但忽略了邻域范围内点密度的差异;而核密度估计方法以统一大小的圆作为搜索邻域,考虑了邻域内点密度的差异,但忽略了不同离散点“影响范围”的不同。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.

发明内容Contents of the invention

本发明要解决技术问题是:克服现有技术的上述不足,借助泰森多边形对于空间划分的技术优势,提供一种联合泰森多边形与反距离加权的密度图制图方法,其以每个离散点所对应的Voronoi多边形(泰森多边形)为邻域,通过反距离权重分配的方法实现邻域内点密度的差异赋值,能够快速、合理、准确的计算点密度并输出点密度图。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.

为了解决以上技术问题,本发明提供的一种联合泰森多边形与反距离加权的密度图制图方法,包括以下步骤: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:

第一步、离散点读取——读取作为原始数据的离散点集,所述离散点具有各自的序号和坐标数据;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;

第二步、构建Voronoi图——基于离散点集中的所有离散点构建Voronoi图;The second step is to construct a Voronoi diagram - construct a Voronoi diagram based on all discrete points in the discrete point set;

第三步、Voronoi图栅格化——根据Voronoi图左上角和右下角的坐标,以及给定的划分栅格像元的行、列数,将整个Voronoi图区域划栅格化,生成若干像元并确定每个像元的中心点坐标;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;

第四步、建立像元与Voronoi多边形的隶属关系——根据像元的中心点坐标与Voronoi多边形的拓扑关系,判定像元与Voronoi多边形的隶属关系,当像元的中心点落在某Voronoi多边形内,则判定该像元属于该Voronoi多边形;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;

第五步、计算像元的密度值——第i个Voronoi多边形Vi的总密度值为1/Sg,将该总密度值1/Sg分摊给Voronoi多边形Vi的各像元,使Voronoi多边形Vi内所有像元的密度值总和等于1/Sg,其中,Sg为单个像元的面积,1≤i≤n,n为Voronoi图中Voronoi多边形的个数;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;

第六步、栅格平滑——采用空域平滑滤波的方法重新计算每个像元密度值;The sixth step, grid smoothing - recalculate the density value of each pixel by using the spatial smoothing filter method;

第七步、像元密度值重分类——统计分析栅格图中所有像元的密度值大小,据此将所有像元密度值重新分类,赋予不同灰度值;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;

第八步、绘制密度图——根据各像元的灰度值对栅格进行渲染获得密度图。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.

本发明的点密度专题图制作方法中,基于离散点构建的Voronoi图,使用Voronoi多边形对区域进行分割,使每个Voronoi多边形内有且只有一个离散点(发生元),每个Voronoi多边形可视为相应离散点生成“影响范围”;Voronoi多边形内各像元密度值的计算不与其他Voronoi多边形发生关联,因此Voronoi多边形内像元密度值的计算受干扰程度低,Voronoi多边形内局部密度计算保证了各Voronoi多边形之间计算结果的可比性及可靠性;并且Voronoi多边形的总密度值分摊方式可以根据实际情况进行选择,可选择均分法,也可选择反距离权重分配法,使用更加灵活;基于离散点所构建的Voronoi图具有唯一性,因此本发明方案实现过程中主观因素更少,易于操作。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.

本发明提供了以下三种Voronoi多边形总密度值分摊方案:The present invention provides following three kinds of Voronoi polygon total density value allocation schemes:

方案一:Option One:

在前述方法的第二步中,建立离散点与其所属的Voronoi多边形之间的对应关系;所述第五步中,采用反距离权重分配法分摊Voronoi多边形内的各像元密度值,Voronoi多边形内像元的密度值与该像元中心点至该Voronoi多边形所对应的离散点的距离成反比;Voronoi多边形Vi内的第j个像元的密度值为

Figure BDA00003274423800031
其中Rij、Rik分别表示Voronoi多边形Vi中第j个和第k个像元的中心点至Voronoi多边形Vi所对应的离散点Pi的距离,j≤mi,k≤mi,mi为Voronoi多边形Vi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
Figure BDA00003274423800031
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.

方案二:Option II:

在前述方法的第二步中,建立离散点与其所属的Voronoi多边形之间的对应关系;所述第五步中,采用反距离权重分配法分摊Voronoi多边形内的各像元密度值,Voronoi多边形内像元的密度值与该像元中心点至该Voronoi多边形所对应的离散点的距离的平方成反比;Voronoi多边形Vi内的第j个像元的密度值为

Figure BDA00003274423800041
其中RijRik分别表示Voronoi多边形Vi中第j个和第k个像元的中心点至Voronoi多边形Vi所对应的离散点Pi的距离,j≤mi,k≤mi,mi为Voronoi多边形Vi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
Figure BDA00003274423800041
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.

方案三:third solution:

在前述方法的第五步中,采用均分法分摊Voronoi多边形内的各像元密度值,Voronoi多边形Vi内第j个像元的密度值Bii=1/(Sg*mi),mi为Voronoi多边形Vi内的像元个数,j≤mi,且j为自然数。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.

前述三种方案中,前两种方法采用的是反距离权重分配法,第三种采用的是均分法。均分法计算更加简单,点密度的空间分布可从Voronoi多边形的大小(Voronoi多边形包含像元的数量)上反映出来,Voronoi多边形面积越大,则点密度值越低;反距离权重分配法则考虑了“影响范围”(Voronoi多边形)内不同像元点密度的差异,各像元所在的Voronoi多边形内部采用了基于距离的密度值分配办法,使结果更加合理准确。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.

其中,反距离权重分配法中,点密度的空间分布可从计算得到的Voronoi多边形面积的大小上反映出来,在Voronoi多边形内部进一步考虑了距离因素,将像元的密度值按距离的倒数(或倒数的平方)所占的比重分配到各个像元中,使得计算结果更加合理。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.

附图说明Description of drawings

下面结合附图对本发明作进一步的说明。The present invention will be further described below in conjunction with the accompanying drawings.

图1为本发明方法流程图。Fig. 1 is a flow chart of the method of the present invention.

图2为离散点及其Voronoi图构建示意图。Figure 2 is a schematic diagram of discrete points and their Voronoi diagram construction.

图3为像元隶属关系及密度值计算示意图。Figure 3 is a schematic diagram of pixel membership and density value calculation.

图4为栅格邻域均值平滑示意图。Fig. 4 is a schematic diagram of raster neighborhood mean smoothing.

图5为像元密度值重分类示意图。Figure 5 is a schematic diagram of reclassification of pixel density values.

具体实施方式Detailed ways

下面根据附图详细说明本发明,本发明的目的和效果将变得更加明显。The purpose and effects of the present invention will become more apparent by referring to the accompanying drawings in detail of the present invention.

如图1所示为本发明联合泰森多边形与反距离加权的密度图制图方法的流程图,包括以下步骤: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:

第一步、离散点读取——读取作为原始数据的离散点集,所述离散点具有各自的序号和坐标数据。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.

读取离散点的坐标,并根据屏幕坐标范围与离散点坐标范围的比例关系显示每个点位置(图2左)。本例中将离散点编号作为离散点的属性数据进行存储。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.

第二步、构建Voronoi图——基于离散点集中的所有离散点构建Voronoi图,建立离散点与其所属的Voronoi多边形之间的对应关系;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;

本例中,以离散点集中所有离散点为发生元,通过扫描线算法构建Voronoi图,将离散点的编号值传递给Voronoi多边形的编号值,这样就建立了离散点与其所属的Voronoi多边形之间的对应关系;同时建立发生元(离散点)与Voronoi边、发生元(离散点)与Voronoi多边形间的拓扑关系,Voronoi边的数据结构中记录构成该边的直线方程系数,构成该边的两个端点以及与该边关联的两侧发生元(离散点),Voronoi多边形的数据结构中记录该多边形的边集合;采用这样的数据有利于下面第五步中像元密度值的计算。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.

第三步、Voronoi图栅格化——根据Voronoi图左上角和右下角的坐标,以及给定的划分栅格像元的行、列数,将整个Voronoi图区域划栅格化,生成若干像元并确定每个像元的中心点坐标。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.

对于划分的任一p*q的格网,设其整个区域左下角和右上角的坐标分别为(A1,B1)和(A2,B2),其中A1<A2,B1<B2。像元可以二维数组形式存储,排列方式为从左向右,从上向下排列。对于任一像元aij其像元中心点坐标为 ( A 1 + A 2 - A 1 p &times; ( i + 1 2 ) , B 1 + B 2 - B 1 q &times; ( q - i - 1 2 ) ) 其中,0≤i≤p-1,0≤j≤q-1。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 + A 2 - A 1 p &times; ( i + 1 2 ) , B 1 + B 2 - B 1 q &times; ( q - i - 1 2 ) ) Among them, 0≤i≤p-1, 0≤j≤q-1.

第四步、建立像元与Voronoi多边形的隶属关系——根据像元的中心点坐标与Voronoi多边形的拓扑关系,判定像元与Voronoi多边形的隶属关系,当像元的中心点落在某Voronoi多边形内,则判定该像元属于该Voronoi多边形。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.

本例中,像元与Voronoi多边形的隶属关系规定如下:若像元中心点落在哪个Voronoi多边形内,则此像元就隶属于该Voronoi多边形;若像元中心点落在某条Voronoi边上,则查找与该边关联的两个Voronoi多边形,规定该像元隶属于发生元编号较小的Voronoi多边形;若像元中心恰好与某个Voronoi顶点重合,则查找与该顶点相关联的三条Voronoi边,进而找到与该顶点相关联的三个Voronoi多边形,规定该像元隶属于三个Voronoi多边形中发生元编号最小者。图4中实线边框内的像元为隶属于该多边形的像元。对于像元中心落在Voronoi多边形的Voronoi边及Voronoi多边形的顶点上的情况属于特例,这些像元的隶属关系规则可人为定义,没有严格要求,因此本发明中并没有对该部分内容进行限定和详细说明。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.

第五步、计算各像元的密度值——第i个Voronoi多边形Vi的总密度值为1/Sg,将该总密度值1/Sg分摊给Voronoi多边形Vi的各像元,使Voronoi多边形Vi内所有像元的密度值总和等于1/Sg,其中,Sg为单个像元的面积,1≤i≤n,n为Voronoi图中Voronoi多边形的个数;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;

本例中,采用反距离权重分配法分摊Voronoi多边形内的各像元密度值,Voronoi多边形内像元的密度值与该像元中心点至该Voronoi多边形所对应的离散点的距离成反比。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.

如图3所示为像元隶属关系及密度值计算示意图。像元具体密度值的计算方法如下: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:

Voronoi多边形Vi内的第j个像元的密度值为

Figure BDA00003274423800071
其中Rij、Rik分别表示Voronoi多边形Vi中第j个和第k个像元的中心点至Voronoi多边形Vi所对应的离散点Pi的距离,Sg表示单个像元的面积,j≤mi,k≤mi,mi为Voronoi多边形Vi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
Figure BDA00003274423800071
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.

第六步、栅格平滑——采用空域平滑滤波的方法重新计算每个像元密度值。The sixth step, grid smoothing——recalculate the density value of each pixel by using the method of spatial smoothing and filtering.

空域平滑滤波的方法可以为邻域均值平滑法、邻域中值平滑法、邻域极值平滑法等。本例中采用领域平均法,即将格网中一个像元的密度值与周围邻近像元的密度值相加,然后将求得的平均值作为新格网中该像元的密度值;另外根据生成栅格像元数的多少,采用适宜大小的模板(如5*5,9*9,25*25等)通过逐行遍历每个像元进行所求像元邻域内的矩阵相乘,如图4所示,本例选用的是3*3大小的模板,平滑后的栅格中像元密度值见图5右侧。经过栅格平滑后,基本可消除Voronoi边附近的“陡坡”现象。本实施例中,采用了均值平滑法,不排除存在更合理有效的平滑方法。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.

第七步、像元密度值重分类——统计分析栅格图中所有像元的密度值大小,据此将所有像元密度值重新分类,赋予不同灰度值。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.

重分类的实质是将像元密度值(属性值)进行重新归类或者把输入像元密度值更改为替代值的方法(如图5所示)。首先,逐行遍历每个像元,统计分析所有像元密度值的中最大值、最小值及其频率分布;其次,做出频率分布直方图与频率变化曲线,并据此选择合理的间隔阈值按密度值从小到大的顺序将所有像元分为若干类;最后,对分类后像元赋予不同的灰度值。密度越大,赋予的灰度值越大。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.

第八步、绘制密度图——根据各像元的灰度值对栅格进行渲染获得密度图。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.

本实施例的第五步中,Voronoi多边形内像元的密度值与该像元中心点至该Voronoi多边形所对应的离散点的距离成反比;除此之外,Voronoi多边形内像元的密度值与该像元中心点至该Voronoi多边形所对应的离散点的距离的平方成反比;Voronoi多边形Vi内的第j个像元的密度值为其中RijRik分别表示Voronoi多边形Vi中第j个和第k个像元的中心点至Voronoi多边形Vi所对应的离散点Pi的距离,j≤mi,k≤mi,mi为Voronoi多边形Vi内的像元个数,i、j、k均为自然数。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.

在计算像元密度值时,也可采用均分法分摊Voronoi多边形内的各像元密度值,Voronoi多边形Vi内第j个像元的密度值Bij=1/(Sg*mi),mi为Voronoi多边形Vi内的像元个数,j≤mi,且j为自然数。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.

本实施例基于离散点构建的Voronoi图通过基于最短距离约束的空间划分为每个离散点生成“影响范围”,在此范围内进行局部密度计算保证了各“影响范围”(Voronoi多边形)之间计算结果的可比性及可靠性;另外,本方法中考虑了“影响范围”内不同像元点密度的差异,在各点所在的Voronoi多边形内部采用了基于距离的密度值分配办法,使结果更加合理准确。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.

除上述实施例外,本发明还可以有其他实施方式。凡采用等同替换或等效变换形成的技术方案,均落在本发明要求的保护范围。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)

1.一种联合泰森多边形与反距离加权的密度图制图方法,包括以下步骤:1. A density map drawing method of joint Thiessen polygon and inverse distance weighting, comprising the following steps: 第一步、离散点读取——读取作为原始数据的离散点集,所述离散点具有各自的序号和坐标数据;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; 第二步、构建Voronoi图——基于离散点集中的所有离散点构建Voronoi图;The second step is to construct a Voronoi diagram - construct a Voronoi diagram based on all discrete points in the discrete point set; 所述第二步中,建立离散点与其所属的Voronoi多边形之间的对应关系;In the second step, establish the correspondence between discrete points and their Voronoi polygons; 第三步、Voronoi图栅格化——根据Voronoi图左上角和右下角的坐标,以及给定的划分栅格像元的行、列数,将整个Voronoi图区域划栅格化,生成若干像元并确定每个像元的中心点坐标;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; 第四步、建立像元与Voronoi多边形的隶属关系——根据像元的中心点坐标与Voronoi多边形的拓扑关系,判定像元与Voronoi多边形的隶属关系,当像元的中心点落在某Voronoi多边形内,则判定该像元属于该Voronoi多边形;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; 第五步、计算像元的密度值——第i个Voronoi多边形Vi的总密度值为1/Sg,将该总密度值1/Sg分摊给Voronoi多边形Vi的各像元,使Voronoi多边形Vi内所有像元的密度值总和等于1/Sg,其中,Sg为单个像元的面积,1≤i≤n,n为Voronoi图中Voronoi多边形的个数;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; 所述第五步中,采用反距离权重分配法分摊Voronoi多边形内的各像元密度值,Voronoi多边形内像元的密度值与该像元中心点至该Voronoi多边形所对应的离散点的距离成反比;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; 第六步、栅格平滑——采用空域平滑滤波的方法重新计算每个像元密度值;The sixth step, grid smoothing - recalculate the density value of each pixel by using the spatial smoothing filter method; 第七步、像元密度值重分类——统计分析栅格图中所有像元的密度值大小,据此将所有像元密度值重新分类,赋予不同灰度值;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; 第八步、绘制密度图——根据各像元的灰度值对栅格进行渲染获得密度图。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.根据权利要求1所述的联合泰森多边形与反距离加权的密度图制图方法,其特征在于:Voronoi多边形Vi内的第j个像元的密度值为
Figure FDA00003274423700021
其中Rij、Rik分别表示Voronoi多边形Vi中第j个和第k个像元的中心点至Voronoi多边形Vi所对应的离散点Pi的距离,j≤mi,k≤mi,mi为Voronoi多边形Vi内的像元个数,i、j、k均为自然数。
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
Figure FDA00003274423700021
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.根据权利要求1-2任一项所述的联合泰森多边形与反距离加权的密度图制图方法,其特征在于:所述第六步中,空域平滑滤波的方法为邻域均值平滑法、邻域中值平滑法、邻域极值平滑法中的一种。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. 4.一种联合泰森多边形与反距离加权的密度图制图方法,包括以下步骤:4. A density map drawing method of joint Thiessen polygon and inverse distance weighting, comprising the following steps: 第一步、离散点读取——读取作为原始数据的离散点集,所述离散点具有各自的序号和坐标数据;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; 第二步、构建Voronoi图——基于离散点集中的所有离散点构建Voronoi图;The second step is to construct a Voronoi diagram - construct a Voronoi diagram based on all discrete points in the discrete point set; 所述第二步中,建立离散点与其所属的Voronoi多边形之间的对应关系;In the second step, establish the correspondence between discrete points and their Voronoi polygons; 第三步、Voronoi图栅格化——根据Voronoi图左上角和右下角的坐标,以及给定的划分栅格像元的行、列数,将整个Voronoi图区域划栅格化,生成若干像元并确定每个像元的中心点坐标;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; 第四步、建立像元与Voronoi多边形的隶属关系——根据像元的中心点坐标与Voronoi多边形的拓扑关系,判定像元与Voronoi多边形的隶属关系,当像元的中心点落在某Voronoi多边形内,则判定该像元属于该Voronoi多边形;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; 第五步、计算像元的密度值——第i个Voronoi多边形Vi的总密度值为1/Sg,将该总密度值1/Sg分摊给Voronoi多边形Vi的各像元,使Voronoi多边形Vi内所有像元的密度值总和等于1/Sg,其中,Sg为单个像元的面积,1≤i≤n,n为Voronoi图中Voronoi多边形的个数;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; 所述第五步中,采用反距离权重分配法分摊Voronoi多边形内的各像元密度值,Voronoi多边形内像元的密度值与该像元中心点至该Voronoi多边形所对应的离散点的距离的平方成反比;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; 第六步、栅格平滑——采用空域平滑滤波的方法重新计算每个像元密度值;The sixth step, grid smoothing - recalculate the density value of each pixel by using the spatial smoothing filter method; 第七步、像元密度值重分类——统计分析栅格图中所有像元的密度值大小,据此将所有像元密度值重新分类,赋予不同灰度值;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; 第八步、绘制密度图——根据各像元的灰度值对栅格进行渲染获得密度图。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.根据权利要求4所述的联合泰森多边形与反距离加权的密度图制图方法,其特征在于:Voronoi多边形Vi内的第j个像元的密度值为
Figure FDA00003274423700031
其中RijRik分别表示Voronoi多边形Vi中第j个和第k个像元的中心点至Voronoi多边形Vi所对应的离散点Pi的距离,j≤mi,k≤mi,mi为Voronoi多边形Vi内的像元个数,i、j、k均为自然数。
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
Figure FDA00003274423700031
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.根据权利要求4-5任一项所述的联合泰森多边形与反距离加权的密度图制图方法,其特征在于:所述第六步中,空域平滑滤波的方法为邻域均值平滑法、邻域中值平滑法、邻域极值平滑法中的一种。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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