CN104751519A

CN104751519A - Efficient construction method for convex hull of planar point set

Info

Publication number: CN104751519A
Application number: CN201510159967.9A
Authority: CN
Inventors: 张立峰; 田金志; 王中辉
Original assignee: Lanzhou Jiaotong University
Current assignee: Lanzhou Jiaotong University
Priority date: 2015-04-07
Filing date: 2015-04-07
Publication date: 2015-07-01

Abstract

The invention provides an efficient construction method for a convex hull of a planar point set. The efficient construction method for the convex hull of the planar point set comprises the following steps of finding out a minimum point and a maximum point of a coordinate value (x); calculating a point which is the farthest to a segment formed by the minimum point and the maximum point; connecting an initial convex hull formed by the point; sequentially calculating points, which are the farthest to edges of the initial convex hull, from point sets of the outer sides of the edges of the initial convex hull; connecting the points and deleting points positioned in the convex hull; updating the initial convex hull; and repeatedly updating the initial convex hull until other discrete points do not exist on the outside of the initial convex hull so that the convex hull is a finally constructed convex hull. On the basis of a test, an analysis result shows that the convex hull generation efficiency is high by the efficient construction method.

Description

A kind of efficient construction method of convex hull of planar point set

Technical field

The present invention can be applicable to the technical field such as cartography and Geographical Information Sciences, GIS spatial data analysis, pattern-recognition, image procossing, CAD/CAM, computer graphics and artificial intelligence, relate to a kind of for massive planar point sets data, ask for the algorithm of its convex hull.

Background technology

Convex hull as in computational geometry the most substantially, one of the most general data structure.Convex hull algorithm, especially planar convex hull algorithm, enjoy Chinese scholars to pay close attention to, and many scholars have done many research work to improve convex hull generating algorithm for this reason, and object is the convex hull formation efficiency in order to improve plane point set.

At present, about the method solving convex hull of planar point set, all in all, two classes can be divided into: direct method and indirect method.Direct method typically has grahame method, maximum baseline angle intelligence approximatioss [10] etc.; Indirect method typically has method of addition, Rolling bed algorithm and divide and conquer etc.These algorithms mostly need to sort to point set, and the angle calculation that some algorithm is used is too complicated; Therefore, when processing mass data, algorithm operational efficiency can obviously reduce.

Summary of the invention

The present invention is directed to existing algorithm process convex hull of planar point set efficiency not high, especially in process massive planar point sets data, the problem that efficiency of algorithm obviously declines, puts forward a kind of new algorithm.The present invention proposes a kind of by remove to greatest extent irrelevant in point ask for algorithm with the convex hull reducing operand, solve the problem that when mass data asks for convex hull, efficiency obviously declines very well.

The inventive method comprises for plane magnanimity point set, the algorithm of its convex hull of quickly and efficiently constructing.

Plane point set asks for convex hull, refers to for given number of planes strong point, obtains the large convex polygon of group be made up of these point set data.Algorithm steps of the present invention is as follows: first, searches 2 points that x coordinate figure is minimum and maximum, and calculate apart from these 2 form line segment point farthest, connect this point and form initial convex hull; Then, concentrate at remaining point, calculate initial convex hull every bar limit points outside successively and concentrate apart from this limit point farthest, connect this point and delete the point being positioned at inside, upgrading initial convex hull; Repeat this step until initial convex hull outside does not have other discrete points, then this convex hull is final required convex hull.

Herein by expanding initial convex hull to greatest extent, point as much as possible is wrapped in wherein, thus is converted into interior point and removes immediately, the quantity making the data point of participating in computing is obvious downtrending, therefore effectively can improve efficiency of algorithm.And when data volume is larger, operational efficiency, without obvious increase, is asked for convex hull for magnanimity point set data and is had very large meaning.

In addition algorithm is for various abnormal conditions herein, and all can process preferably, algorithm robustness is good.

Accompanying drawing explanation

Figure (1) algorithm flow chart

Figure (2) determines left and right extreme point

Figure (3) calculates apsis P _m

Scheme (4) apsis

Figure (5) connects apsis C

Figure (6) removes internal point D, E, F

Figure (7) connects apsis B

Figure (8) removes internal point A

The final required convex hull CH (S) of figure (9)

Figure (10) x coordinate extreme point exists multiple

Figure (11) apsis exists multiple

Embodiment

In order to describe technology contents of the present invention, algorithm characteristics, the object realized and the effect reached in detail, describe in detail below in conjunction with embodiment.

Herein first algorithm finds the left and right extreme point of given plane point set, then to calculate apart from these 2 form the apsis of line segment, connect this apsis and form initial convex hull.Then for this initial convex hull, constantly calculate the apsis outside its every bar limit, connect and upgrade initial convex hull until there is no discrete point outside initial convex hull, namely try to achieve final convex hull.Algorithm main process as figure (1) shown in.

The detailed description of algorithm

step 1:for any given plane point set S, first obtain left extreme point P _lwith right extreme point P _r, known, the region between straight line x=Xmin and x=Xmax is institute's survey region, and all planar points are all included in survey region, as shown in figure (2):

step 2:connect and obtain line segment P _lp _r, calculate point set S middle conductor P _lp _rapsis P _m, connect P successively _lp _m, P _rp _mform initial convex hull BCH (S), as shown in figure (3), triangle P _lp _rp _mbe initial convex hull BCH (S):

step 3:remove the interior point of initial convex hull BCH (S).Detect successively outside the every bar limit of BCH (S) and whether have exterior point:

step.1if do not have a little outside this limit, then directly terminate this limit exterior point detection calculations.

As shown in figure (3), first detect limit P _lp _m, there is no exterior point outside this limit, then directly terminate this limit exterior point detection calculations.

If only have a point outside this limit, then connect this point and terminate this limit exterior point detection calculations.

As shown in figure (4); Detect limit P successively _mp _r, only have an exterior point outside this limit, then connect this apsis and line segment two-end-point, then terminate this limit exterior point detection calculations.

If the number of exterior point is more than one outside this limit, then the apsis outside this limit of double counting, connects this apsis successively and forms new initial convex hull, and removes inner point, returns step 3.

As shown in figure (5), detect limit P successively _rp _l, first the exterior point outside this limit more than two, then calculates limit P _rp _lapsis C, connect this apsis and line segment two-end-point successively.

Point D, E, F in removing, as shown in figure (6).

For two the limit P newly obtained _lc, P _rc, repeated execution of steps 3, first detects limit P _lc, more than one exterior point outside this limit, then calculate its apsis B, connects this apsis and line segment two-end-point, as shown in figure (7).

Point A in removing, as shown in figure (8).

Continue successively to detect the both sides P newly obtained _lb, BC, then all without exterior point outside both sides, from the Step.1 of step 3, directly terminate this limit exterior point detection calculations.Then one side P is in addition detected _rc, only has an exterior point outside this limit, from the Step.2 of step 3, connect this point and terminate exterior point detection calculations, as shown in figure (9).Known, polygon P _lp _mhP _rgCBP _lbe final required convex hull CH (S).

Abnormality processing

For algorithm that article is carried, when seeking extreme point, may exist multiple, specifically having following several special circumstances:

, the situation of more than two of extreme point may be there is, if when the extreme point dropping on right boundary exists multiple in 1 extreme point first calculating x coordinate, then get the point that y coordinate figure is minimum respectively, as shown in figure (10), left extreme point exists multiple, then choose the some P that y coordinate figure is minimum _l1.

2, when calculating apsis, there will be the situation that there is multiple apsis equally, if when existing multiple for apsis required by a line segment, then get 2 points that x coordinate figure is minimum and maximum, as shown in figure (11), then choose P _m1p _m32 points, P like this _m2limit P will be dropped on _m1p _m3above become interior point deletion.

Claims

The characterization step asking for convex hull based on plane point set is as follows:

Connect and obtain line segment P _lp _r, calculate point set S middle conductor P _lp _rapsis P _m, connect P successively _lp _m, P _rp _mform initial convex hull BCH (S), as shown in figure (3), triangle P _lp _rp _mbe initial convex hull BCH (S).
Remove the interior point of initial convex hull BCH (S), detect outside the every bar limit of BCH (S) whether have exterior point successively:

If do not have a little outside this limit, then directly terminate this limit exterior point detection calculations;

If only have a point outside this limit, then connect this point and terminate this limit exterior point detection calculations;

If the number of exterior point is more than one outside this limit, then the apsis outside this limit of double counting, connects this apsis successively and forms new initial convex hull, and removes inner point, returns step 3.
Abnormality processing

For algorithm that article is carried, when seeking extreme point, may exist multiple, specifically having following several special circumstances:

, the situation of more than two of extreme point may be there is, if when the extreme point dropping on right boundary exists multiple in 1 extreme point first calculating x coordinate, then get the point that y coordinate figure is minimum respectively, as shown in figure (10), left extreme point exists multiple, then choose the some P that y coordinate figure is minimum _l1;

2, when calculating apsis, there will be the situation that there is multiple apsis equally, if when existing multiple for apsis required by a line segment, then get 2 points that x coordinate figure is minimum and maximum, as shown in figure (11), then choose P _m1p _m32 points, P like this _m2limit P will be dropped on _m1p _m3above become interior point deletion.