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Geo-Information Article

Duality and Dimensionality Reduction Discrete Line Generation Algorithm for a Triangular Grid Lingyu Du 1 , Qiuhe Ma 1 , Jin Ben 1, *, Rui Wang 1 and Jiahao Li 2 1 2

*

Institute of Surveying and Mapping, Information Engineering University, Zhengzhou 450001, China; [email protected] (L.D.); [email protected] (Q.M.); [email protected] (R.W.) The Army Infantry Academy, Nanchang 330103, China; [email protected] Correspondence: [email protected]; Tel.: +86-371-8162-3256

Received: 27 June 2018; Accepted: 22 September 2018; Published: 27 September 2018







Abstract: Vectors are a key type of geospatial data, and their discretization, which involves solving the problem of generating a discrete line, is particularly important. In this study, we propose a method for constructing a discrete line mathematical model for a triangular grid based on a “weak duality” hexagonal grid, to overcome the drawbacks of existing discrete line generation algorithms for a triangular grid. First, a weak duality relationship between triangular and hexagonal grids is explored. Second, an equivalent triangular grid model is established based on the hexagonal grid, using this weak duality relationship. Third, the two-dimensional discrete line model is solved by transforming it into a one-dimensional optimal wandering path model. Finally, we design and implement the dimensionality reduction generation algorithm for a discrete line in a triangular grid. The results of our comparative experiment indicate that the proposed algorithm has a computation speed that is approximately 10 times that of similar existing algorithms; in addition, it has better fitting effectiveness. Our proposed algorithm has broad applications, and it can be used for real-time grid transformation of vector data, discrete global grid system (DGGS), and other similar applications. Keywords: vector; grid transformation; triangle; hexagon; duality; dimensionality reduction

1. Introduction Vector and raster are two fundamental spatial data models, each having advantages for various applications. The process of discretizing a vector to its corresponding grid cells under certain criteria is referred to as “gridding” or “rasterization”, with this facilitating the fusion between vector and raster data. In general, vector data models abstract real-world entities into a combination of geometric elements, such as points, lines, and polygons [1]. The discretization of geometric elements therefore forms the basis of vector data discretization. For discretization of line and polygon elements, the grid path or grid boundary of the line or polygon must first be determined. Thus, the core problem of discretization is the generation of a discrete line [2,3]. In the fields of computer graphics and geographical information systems, various efficient discrete line generation algorithms have been developed. Typical examples include the digital differential analyzer (DDA) algorithm [4], the midpoint line algorithm [5], and the Bresenham algorithm [6]. Furthermore, based on the latter, Wu and Rokne proposed a double-step incremental generation algorithm [7], Jia proposed a six-step algorithm for line drawing [8], and Huang and Zhang proposed a self-adaptive step straight-line algorithm [9]. These algorithms were primarily intended for rectangular (quadrangular) grids; however, triangular and hexagonal grids are also used in many applications. It should be noted that although Vince [10] developed a mathematical model for discrete lines based on a multidimensional grid system and designed a corresponding algorithm, this model is applicable only to grid systems obtained by moving the centrosymmetric cell, such ISPRS Int. J. Geo-Inf. 2018, 7, 391; doi:10.3390/ijgi7100391

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as a square, hexagon, or cube; thus, this algorithm is not suitable for a triangular grid, the cells of which are noncentrosymmetric. The triangular grid has been applied to discrete global grid system (DGGS). Representative achievements include Sphere Quad Tree [11] and Quaternary Triangular Mesh (QTM) [12], which are mainly used in spatial data indexing [13] and map generalization models [14]. Both hexagonal and diamond grids can be formed by triangular grid polymerization. During the visualization process, some typical 3D programming interfaces, such as OpenGL, provide the function to directly handle triangular grids after triangulation and striping of complex objects [15]; triangular grids are therefore useful. However, they are not uniformly adjacent; (in the case of grid cells, adjacency may be at edges, vertices, or both—if there is only edge adjacency, the grid is uniformly adjacent, otherwise it is non-uniformly adjacent). Grid cells also have different orientations (including upper and lower triangles), and the geometric structure of triangular grids is therefore quite complex, posing challenges for discrete line generation. There is also a dearth of related research. Freeman proposed a vertex-to-vertex discrete line generation algorithm that selects grid cells based on the distance between the vertex and vector line [16]; consequently, some cells of the discrete line deviate from the vector line. Nagy generated a discrete line using neighborhood sequences in a triangular grid [17]; the constraining conditions, including adjacent relationship of grid cells and length of the shortest path, are not sufficient to guarantee the uniqueness of a path. Sarkar proposed the shortest path search algorithm along the boundary of the triangular grid cell [18]; however, this algorithm ignores the relationship between the cell center and the vector line. Zhang proposed the full-path algorithm, in which, according to the geometric relationship between the vector line and grid cell, all grid cells crossed by the vector line are selected one by one [2]; the discrete line generated by this algorithm is ideal, but the process is complex, and computation speeds are slow. Dutt discussed the case in which a straight line crosses a vertex of the triangular grid cell, and based on this case, proposed a discrete line generation algorithm for a triangular grid [19]; however, the entire process needs to be conducted in a triaxial coordinate system, with the consequence that computation speeds are slow. In summary, for the generation of a discrete line in a triangular grid, there is currently no algorithm that has both satisfactory outcomes and reasonable computation speeds. We therefore propose a new algorithm in order to advance this research field. Compared with a triangular grid, the geometric structure of a hexagonal grid is ideal, making it easy to establish the discrete line model. Based on the structural characteristics of triangular and hexagonal grids, we therefore define a new positional relationship between the two grid systems henceforth referred to as “weak duality”, which ensures maximum overlap between the centers of hexagonal and triangular grids. Based on this weak duality relationship between grid systems, we establish a mathematical model for triangular grid discrete line generation, using the hexagonal discrete line model. In addition, we solve the model using the dimensionality reduction method to theoretically simplify the problem. 2. Definition of Duality and Weak Duality Generally speaking, a duality in mathematics translates concepts, theorems, or mathematical structures into other concepts, theorems, or structures, in a one-to-one fashion [20]. For example, the duality of the grid refers to the unique positional relationship between two different grid systems. For a regular grid, the corresponding duality grid can be obtained by connecting the centers of adjacent cells and considering them as vertices [21]. A duality relationship exists between a triangular grid and a hexagonal grid. Let the side length of the triangular grid cell be lt ; then, the side length of the hexagonal grid cell having a duality relationship with it is lh = √1 lt , and their sides are perpendicular 3 to each other, as shown in Figure 1.

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Figure 1. Duality relationship between hexagonal and triangular grids. 1.1.Duality between andtriangular triangular grids. Figure Duality relationship between and grids. The center ofFigure the grid cell isrelationship typically used as hexagonal ahexagonal reference point for calculation and location, insteadThe of the entire grid cell [22]. To simplify the process of grid transformation, the centers of the centerof of the typically usedused as a reference point for calculation and location, The center the grid gridcell cellis is typically as a reference point for calculation andinstead location, triangular and hexagonal grid cells should therefore betransformation, as coincident as Itthe is evident that of the of entire cellgrid [22].cell To [22]. simplify the process grid thepossible. centers of triangular instead the grid entire To simplify theofprocess of grid transformation, the centers of the theand duality relationship shown in Figure 1 cannot meet this requirement. However, by adjusting the hexagonal grid cells should therefore as coincident possible. It evident that duality triangular and hexagonal grid cells shouldbetherefore be asascoincident asispossible. It isthe evident that 1 shown in Figure cannot meet this However, byHowever, adjusting the side length therelationship duality shown1grid in Figure 1 cannot meet thisletting requirement. side length relationship of the hexagonal cell to it be parallel to by theadjusting side of the lh requirement. lt , and 1 0 of the hexagonal grid cell to lh = 3 lt , and letting 3it be parallel to the side of the triangular grid cell, 1 thelength centergrid of hexagonal grid cell centered the vertex the triangular grid coincides with that side ofa the hexagonal grid cell to lh at , and letting it be parallel toofthe of the  ltcell triangular cell, the center of anot hexagonal grid notof centered at the vertex theside triangular 3 of the triangular gridthat cell,of as shown in Figuregrid 2. In this this relationship as weak grid coincides with the triangular cell,study, as shown in Figure is2.referred In thistostudy, this triangular gridit cell, the center of acenters hexagonal grid cell nothexagonal centeredgrid at the vertex of the triangular duality, and can ensure that the of triangular and cells coincide, facilitating relationship is referred to as weak duality, and it can ensure that the centers of triangular and grid coincides with that the triangular grid cell, as shown in Figure 2. In this study, this transformation of the gridof system. hexagonal grid cells coincide, facilitating transformation of the grid system. relationship is referred to as weak duality, and it can ensure that the centers of triangular and hexagonal grid cells coincide, facilitating transformation of the grid system.

Figure betweenhexagonal hexagonaland andtriangular triangular grids. Figure2.2.Weak Weakduality dualityrelationship relationship between grids.

Discrete Line Model Triangular Grid 2. Weak duality relationship between hexagonal and triangular grids. 3. 3. Discrete LineFigure Model ofofaaTriangular Grid Because of the non-uniform adjacency (the grid cells of the triangular grid have both edge and

Because of the non-uniform adjacency 3. Discrete Line Model of a Triangular Grid(the grid cells of the triangular grid have both edge and vertex adjacency) and dissimilar directions of cells in a triangular grid, it is difficult to establish a vertex adjacency) and dissimilar directions of cells in a triangular grid, it is difficult to establish a discrete line of model for the grid. Inadjacency contrast, it(the is easier establish discrete line model a hexagonal Because the non-uniform grid to cells of the atriangular grid haveinboth edge and discrete line model for the grid. In contrast, it is easier to establish a discrete line model in a grid. Based on the weak duality relationship between triangular and hexagonal grids, we can simplify vertex adjacency) and dissimilar directions of cells in a triangular grid, it is difficult to establish a hexagonal grid. Based on the weak duality relationship between triangular and hexagonal grids, we discrete line model for the grid. In contrast, it is easier to establish a discrete line model in a can simplify the discrete line model in a triangular grid by transforming the discrete line model of a hexagonal grid. Based on the weak duality relationship between triangular and hexagonal grids, we hexagonal grid into that of a triangular grid. can simplify the discrete line model in a triangular grid by transforming the discrete line model of a hexagonal grid into that of a triangular grid.

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3.1. Discrete Line Model of in a Hexagonal the discrete line model a triangularGrid grid by transforming the discrete line model of a hexagonal grid into that of a triangular grid.

The corresponding discrete line of vector line

ab is a grid path that fits the vector line

3.1. Discrete Model of grid, a Hexagonal Grid accurately. In aLine hexagonal we consider a set of vectors from the center of a cell to the center of

its adjacent as direction vectors, shown Two direction vectors not on the The cells corresponding discrete line of vector in lineFigure ab is a3. grid path that fits the vectorthat line are accurately. sameInline are linearly discrete cell to the ending cell can a hexagonal grid,independent, we consider a and set ofthe vectors fromline the from centerthe of astarting cell to the center of its adjacent cells as direction shownsequence in Figure 3.ofTwo vectors that are not ondirection the same line are In be represented by vectors, the ordered anydirection two linearly-independent vectors. linearly independent, and the discrete line from the starting cell to the ending cell can be represented practice, the two direction vectors that have the smallest angle with the vector line are selected to by thethe ordered sequence of anytwo twodirection linearly-independent In practice, the two generate discrete line; these vectors are direction referred vectors. to as optimal direction vectors



direction vectors that have the smallest angle with the vector line are selected to generate the discrete

0 , v1}are and line; are these denoted by V   {v . Then, mv and is are a unique abtoas optimal v , where two direction vectors referred vectors denotedpositive by vV  mvdirection  0 0 0 0 V = v0 , v1 . Then, ab = ∑v0 ∈V 0 mv0 v , where mv0 is a unique positive integer. integer.

Figure ofaahexagonal hexagonal grid. Figure3.3.Direction Direction vectors vectors of grid. 0

0

0

0

0

0

= ( w1 , w2 , · · · , w be any ordered multiset of elements from V ; i.e., w can be any Let Let W W (w , w 2 ,  , w N )N ) be any ordered multiset of elements from V  ;k i.e., w k can be 0 1 0

element of V . Then, the hexagonal grid cells that form the discrete line can be expressed as uk = a +

0 + w0 +of 0 , while any w element . Then, the griddiscrete cells that form discrete line as · · ·V+ w thehexagonal corresponding line can bethe denoted as a = u0 ,can u0 , ·be · · ,expressed u0 = b. 1

2

k

1

2

N

u k  a We w1cannot  w 2 determine   w k the , while corresponding discrete line can denoted unique the discrete line based only on the starting and be ending cells; as a  the u1 ,following u 2 ,  , u Nconstraints  b . are therefore imposed [10]:

1.We cannot There should be a minimum number of discrete line cells; determine the unique discrete line based only on the starting and ending cells; the 2. Of all discrete lines that satisfy (1), the selected discrete line should minimize max d(u0k , ab), following constraints are therefore imposed [10]: 1≤ k ≤ N

1.

where d(u0kbe , aba) minimum is the vertical distance u0k toline ab. cells; There should number offrom discrete

2.

Of all that satisfy the selected discrete line maxamong d (uk , ab) , Thediscrete discretelines line satisfying the (1), constraints is unique and has theshould highestminimize fitting accuracy 1 k  N

all discrete lines satisfying (1), as shown in Figure 4. Fitting accuracy can be described by the vertical d (uk , ab) where between the vertical distance from uk the to vector ab . line. In general, the shorter the distance the is center of the discrete line cell and vertical distance, the higher the fitting accuracy.

The discrete line satisfying the constraints is unique and has the highest fitting accuracy among all discrete lines satisfying (1), as shown in Figure 4. Fitting accuracy can be described by the vertical distance between the center of the discrete line cell and the vector line. In general, the shorter the vertical distance, the higher the fitting accuracy.

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Figure 4. Discrete line in a hexagonal grid.

3.2. Discrete Line Model of a Weak Duality Discrete lineGrid Figure 4.Triangular in a hexagonal grid. TheDiscrete hexagonal Model grid shown in Figure 4 describes the corresponding weak duality triangular 3.2. 3.2. Discrete Line Line Model of of aa Weak Weak Duality Duality Triangular Triangular Grid Grid

grid; there is a conversion relationship between triangular and hexagonal grids. Furthermore, as The hexagonal grid shown inin Figure 4 describes the the corresponding weak duality triangular grid; hexagonal grid 4 describes corresponding duality triangular shown inThe Figure 5, based onshown the weakFigure duality relationship, the grid cellsweak of the hexagonal discrete there is a conversion relationship between triangular and hexagonal grids. Furthermore, as shown grid; there is a conversion relationship between triangular and hexagonal grids. Furthermore, as line are divided into two types; cells belonging to the first type are those with centers coinciding in Figure based5,on the weak duality the gridthe cells of the discrete discrete line are shown in 5, Figure based on the weak relationship, duality relationship, grid cellshexagonal of the hexagonal with divided triangular centers, shown by blue to dots, while cells belonging to thecoinciding second type are those with into two types; cells belonging the first type are those with centers with triangular line are divided into two types; cells belonging to the first type are those with centers coinciding centers coinciding with green dots. Let uwith the with discrete centers, shown by bluetriangular dots, while cells theby second type those centers coinciding k represent with triangular centers, shown byvertices, blue belonging dots,indicated whiletocells belonging toare the second type are those with triangular vertices, indicated by green dots. Let u represent the discrete line cells of the triangular grid, and let uk represent those of the hexagonal grid. For the first cell type, uk = uk ; however, for the line the triangular grid, and letcell uk type, represent those of the hexagonal grid. For the first cell type,second u kcells u of however, the second further analysis is required.  cell furtherfor analysis is required. k ;type, type, u k  u k ; however, for the second cell type, further analysis is required.

centers with grid, triangular indicatedk by green Let u k represent the the discrete line cells of coinciding the triangular and vertices, let uk represent those of dots. the hexagonal grid. For first cell 0 0

Figure 5. Correspondingrelationship relationship between between two line models. Figure 5. Corresponding twodiscrete discrete line models. 0

As ininFigure 6, the discrete lineline cellcell of the grid u A , which belongsbelongs to the second As shown shown Figure discrete of hexagonal the to the As shown in Figure 6, 6, thethediscrete line cell of the hexagonal hexagonalgrid gridu Au, Awhich , which belongs to the

cell type, has two adjacent cells, the centers of which correspond to the triangular cells, with these

second cell type, has two adjacent cells, the centers of which correspond to the triangular cells, with second cell the type, hasdirection. two adjacent the centerscell of of which correspond cells, with having same Let u0 cells, be the previous u0 , and let vectorsto v00 the andtriangular v10 be the optimal  these having the same direction.B Let u B be the previousA cell of u A , and let vectors v and v1 0 0 B the of line ab, and Let let uCube triangular of ulet can obtain the v1 thesedirection having vectors the same direction. be corresponding the previous cell of u Acell , and vectors v0 and A . We 0 center uC by direction moving the center u0B along thelet vector ab , and u C vbe be the of optimal vectors ofof line 1 . the corresponding triangular cell of u A . abtriangular , and let cells u C be the corresponding triangular cell of be the optimal direction vectors of line the 0 In contrast, as center shown in Figure the centers of u0Bvand  uC correspond u C by7,moving the center ofto which u B along the 0vector We can obtain the of 1. have different uE , uF , and u A . vector By moving G mightofcorrespond u C ubyD ,moving theucenter u B alongtothe v1 .the centers of We can obtain thedirections. center of Cells u0A and u0B along the vector v00 , we can obtain the centers of uD and uE . In a similar manner, the centers 0 along the vector −v0 . Based on the of uF and uG can be obtained by moving the centers of u0A and uC 0 0 positional relationship between the centers of u A , uD , uE , and line ab, the corresponding triangular cells of u0A can be determined. If the centers of u0A , uD , uE are on different sides of the line, the corresponding triangular cells of u0A are uD and uE ; otherwise, the corresponding cells are uF and uG .

u A .

Figure 6. The triangles to which the blue dots correspond have the same direction.

In contrast, as shown in Figure 7, the triangular cells to which the centers of u B and u C

A . By correspond have u13 ISPRS Int. J. Geo-Inf.different 2018, 7, 391 directions. Cells u D , u E , u F , and u G might correspond to6 of ISPRS Int. J. Geo-Inf. 2018, of 7, x FOR of 13 moving the centers u A PEER and REVIEW u B along the vector v0 , we can obtain the centers of u D 6 and

u E . In a similar manner, the centers of u F and u G can be obtained by moving the centers of u A and u C along the vector  v0 . Based on the positional relationship between the centers of u A , u D , u E , and line ab , the corresponding triangular cells of u A can be determined. If the centers of u A , u D , u E are on different sides of the line, the corresponding triangular cells of u A are u D and u E ; otherwise, the corresponding cells are u F and u G . Using the abovementioned corresponding relationships based on the discrete line model of a hexagonal grid and some simple manipulations, we can obtain the triangular grid discrete line of vector line

ab , thus thetotransformation between thesehave twothetypes of discrete Figure 6.achieving The triangles dots correspond have the same direction.line model. Figure 6. The triangles towhich whichthe the blue blue dots correspond same direction.

In contrast, as shown in Figure 7, the triangular cells to which the centers of u B and u C correspond have different directions. Cells u D , u E , u F , and u G might correspond to u A . By moving the centers of u A and u B along the vector v0 , we can obtain the centers of u D and

u E . In a similar manner, the centers of u F and u G can be obtained by moving the centers of u A and u C along the vector  v0 . Based on the positional relationship between the centers of u A , u D , u E , and line ab , the corresponding triangular cells of u A can be determined. If the centers of u A , u D , u E are on different sides of the line, the corresponding triangular cells of u A Figure 7. The triangles toto which correspondhave have different directions. Figure 7. The triangles whichthe theblue blue dots dots correspond different directions. are u D and u E ; otherwise, the corresponding cells are u F and u G . Using the abovementioned corresponding relationships on the discrete line model of a 4. Dimensionality Reduction Algorithm of the Discrete Linebased Model Using the abovementioned corresponding relationships based on the discrete line model of a hexagonal grid and some simple manipulations, we can obtain the triangular grid discrete line of hexagonal grid and simple manipulations, we can obtain thethe triangular grid discrete line of As discussed in some the previous section, the premise of obtaining triangular grid discrete line vector line ab, thus achieving the transformation between these two types of discrete line model.

is solving hexagonal grid discrete line that hastwo a weak relationship with abcorresponding , thus achieving the transformation between these typesduality of discrete line model. vector linethe 4. Dimensionality Reduction Algorithm of the Discrete Line Model

the triangular grid. Given line ab in a two-dimensional hexagonal grid, we can solve the discrete Asthe discussed in direction the previous section, theconstraining premise of obtaining the triangular grid line to line using optimal vectors and conditions. However, it discrete is necessary is solving the corresponding hexagonal grid discrete linecenter that has a weak relationship with the repeatedly calculate and compare the distance from the of the cellduality to the line; this reduces the triangular grid. Given line ab in a two-dimensional hexagonal grid, we can solve the discrete line computation speed of this method. By projecting the optimal direction vectors of a hexagonal grid using the optimal direction vectors and constraining conditions. However, it is necessary to repeatedly

discrete line to the straight perpendicular to line abcell , we transform the discrete line formed calculate and compare theline distance from the center of the to can the line; this reduces the computation by the arrangement of the optimal direction vectors to a grid closed pathlineon speedordered of this method. By projecting the optimal direction vectors of a hexagonal discrete to the one-dimensional straight line, thereby dimension computational complexity. the straight line perpendicular to linereducing ab, we can transform and the discrete line formed by the ordered arrangement of the optimal direction vectors to a closed path on the one-dimensional straight line, thereby reducing dimension computational complexity. 4.1. One-Dimensional Equivalentand Form of the Hexagonal Grid Discrete Line Model Figure 7. The triangles to which the blue dots correspond haveModel different directions. For One-Dimensional a one-dimensional line, the two opposite vectors that originate from the origin are denoted 4.1. Equivalent Form of the Hexagonal Grid Discrete Line



mv v  0 ,  {v , v1} . If there is atwo positive integer mv | v  Vfrom } that makes by the set ForVa one-dimensional line, the opposite vectorsset that{originate the origin are denoted 4. Dimensionality 0Reduction Algorithm of the Discrete Line Model vV by the set V = {v0 , v1 }. If there is a positive integer set {mv |v ∈ V } that makes ∑ mv v = 0,

∈V As discussed in the previous section, the premise of obtaining the triangular vgrid discrete line

then V is considered the basic set in the one-dimensional space. Let W = (w1 , w2 , · · · , w N ) be any is solving the corresponding hexagonal grid has a weak duality relationship with uˆ k = w1line + wthat ordered multiset of elements from set V, anddiscrete 2 + w3 + · · · + wk . We can obtain a path on a one-dimensional connecting 0 = uˆ 1 , uˆ 2 , uˆ 3 , . .hexagonal . , uˆ N . In addition, uˆ N solve = 0, the in a two-dimensional grid, weif can thepath discrete the triangular grid. Givenline linebyab ˆ ˆ ˆ ˆ ˆ ˆ ˆ P : ( 0 = u , , , . . . , 0 ) is a closed path. Let w ( P ) = max | u | ( | u | is the distance from u u u = u 2 3 N k k However, it is necessary k to line using the1 optimal direction vectors and constraining conditions. to 1≤ k ≤ N

repeatedly calculate and distance comparethat thepath distance from the of theThen, cell to line; this origin) be the farthest P wanders fromcenter the origin. w(the V) = minw ( Preduces ) is the the optimal wandering for set By V, and the corresponding closed path is vectors referred of to as the optimal grid computation speed of distance this method. projecting the optimal direction a hexagonal wandering path [10].

discrete line to the straight line perpendicular to line ab , we can transform the discrete line formed by the ordered arrangement of the optimal direction vectors to a closed path on the one-dimensional straight line, thereby reducing dimension and computational complexity. 4.1. One-Dimensional Equivalent Form of the Hexagonal Grid Discrete Line Model For a one-dimensional line, the two opposite vectors that originate from the origin are denoted

w(V )  min w( P) is the optimal wandering distance for set V , and the corresponding closed path is referred to as the optimal wandering path [10]. As described in Section 3.1, the ideal discrete line should satisfy the previously mentioned constraints. After determining the optimal direction vectors, the number of discrete line cells will be ISPRS Int. J. Geo-Inf. 2018, 7, 391 minimal. Furthermore, in order to obtain the vertical distance

7 of 13 d (uk , ab) , we consider a straight

H that goes through the starting point a and is perpendicular to line ab ; projH is the As described in Section 3.1, the ideal discrete line should satisfy the previously mentioned H . optimal a Because direction line ab vectors, is perpendicular to H the projections vertical projection operator on the constraints. After determining the number of ,discrete line cellsofwill 0 , ab),set   {v0 , av1straight and the vertical projections the optimal direction } onto and b coincide, be minimal. Furthermore, in order to obtain of the vertical distance d(uvector we V consider k line H that goes through point a and is perpendicular to line ab;ˆproj proj is the vertical   H form the basic set V the  {vstarting 0  projH (v 0 ), v1  projH (v1 )} . In this case, u k  H H (u k ) and projection operator on H. Because line ab is perpendicular to H, the projections of a and b coincide,  k , ab) d (uthe | uˆ k projections | , as shown in optimal Figure direction 8. Therefore, discrete 0 = v0 ,line 0 of the two-dimensional and vertical of the vectorthe set V 0 v1 onto H form the basic set  0 0 pathd (on hexagonal corresponds one-dimensional  , uH (Nv V = v0 = grid proj Ha(v00 )u, v1 ,1u=2 ,proj this case, uˆ to projclosed u0k , the ab) = |uˆ k |, as shown k =the H (uk ) and 1 ) b. In 0 0 0 in Figureline 8. Therefore, line ˆdiscrete ˆ ˆ of the two-dimensional hexagonal grid a = u1 , u2 , · · · , uN = b straight P : (0  uˆthe 1 , u 2 , u 3 , , u N  0) . Hence, solving the ideal discrete line with the shortest corresponds to the closed path on the one-dimensional straight line P : (0 = uˆ 1 , uˆ 2 , uˆ 3 , . . . , uˆ N = 0). k , ab) the max d (u in ideal two-dimensional is equivalent to solving one-dimensional space optimal Hence, solving discrete line space with the shortest max d(u0k , ab)the in two-dimensional is 1 k  N line

1≤ k ≤ N

equivalent to solving the one-dimensional w( P) . optimal wandering path that minimizes w( P). wandering path that minimizes

Figure 8. Schematic of the projection relationship.

4.2. Algorithm for Solving the Discrete Line of a Triangular Grid 4.2. Algorithm for Solving the Discrete Line of a Triangular Grid Based on the above analysis, it is clear that the key to determining the optimal wandering path Based on the above analysis, it is clear that the key to determining the optimal wandering path is to minimize w( P) = max |uˆ k |. To achieve this objective, we adopt the greedy algorithm, i.e., is to minimize w( P )  max | uˆ k | . To achieve this objective, we adopt the greedy algorithm, i.e., 1≤ k ≤ N 1 k  N the selected vector in each step ensures that the current path is closest to the origin. The greedy the selected vector inas each stepofensures thatsolutions the current path by is closest to the origin. The algorithm is heuristic, a result which the obtained this algorithm may only begreedy locally algorithm is heuristic, as a result of which the solutions obtained by this algorithm may only be optimal; however, in this study, the solution obtained by the greedy algorithm is globally optimal, as locally optimal; however, in this study, the solution obtained by the greedy algorithm is globally per the following deduction: optimal, as perthat the{following uˆ 1 , uˆ 2 , uˆ 3 , deduction: . . . , uˆ N } is the optimal wandering path obtained by the greedy algorithm Consider ˆ 1 ,any the optimal by the greedy uˆ 2 , non-optimal uˆ 3 , , uˆ N } is ˆt1 , ˆt2 , ˆt3 , . .that and {Consider . , ˆt N }{u is wandering path.wandering Accordingpath to theobtained greedy algorithm, each ˆ ˆ node in the optimal is non-optimal closest to thewandering origin, so path. |uˆ 1 | |uˆ i | must be true. There are two cases for |ˆti |: algorithm, each node in the 1≤ optimal wandering path is closest to the origin, so k≤ N the first is |ˆti | = max |ˆtk |, in which max |uˆ k | < max |ˆtk |; and the second is |ˆti | 6= max |ˆtk |, in which 1≤ k ≤ N

1≤ k ≤ N

1≤ k ≤ N

1≤ k ≤ N

|ˆt j | = max |ˆtk |, and |ˆt j | > |ˆti |. Because |ˆti | > |uˆ i |, max |uˆ k | < max |ˆtk | is true. Therefore, according 1≤ k ≤ N

1≤ k ≤ N

1≤ k ≤ N

to the greedy algorithm, we can obtain the optimal wandering path, where w( P) = max |uˆ k | must be 1≤ k ≤ N

the smallest of all paths. In other words, the optimal wandering path obtained by the greedy algorithm in our study is globally optimal. The optimal direction vector set V 0 corresponds to the basic set V; by using the greedy algorithm to find the element of basic set V and the corresponding element of set V 0 in each step, we can therefore obtain the ideal discrete line of the weak duality hexagonal grid. Finally, based on the conversion relationship between the two discrete line models, described in Section 3.2, we obtain the discrete line of the triangular grid. The flowchart for this algorithm is shown in Figure 9.

the conversion relationship between the two discrete line models, described in Section 3.2, we obtain the discrete line of the triangular grid. The flowchart for this algorithm is shown in Figure 9. The above algorithm successfully obtains the one-dimensional optimal wandering path, the two-dimensional discrete line of the weak duality hexagonal grid, and the two-dimensional discrete line of triangular ISPRS Int.the J. Geo-Inf. 2018, 7,grid. 391 Consequently, the proposed method gradually simplifies the problem 8 of of 13 obtaining a triangular grid discrete line, thereby considerably improving operational efficiency.

Figure 9. 9. Flowchart of the the duality duality and and dimensionality dimensionality reduction reduction algorithm. algorithm. Figure Flowchart of

The aboveand algorithm 5. Experiments Analysissuccessfully obtains the one-dimensional optimal wandering path, the two-dimensional discrete line of the weak duality hexagonal grid, and the two-dimensional discrete To verify the effectiveness of our proposed duality and dimensionality reduction discrete line line of the triangular grid. Consequently, the proposed method gradually simplifies the problem of generation algorithm, we compared it with the Freeman [16] and full-path [2] algorithms. In our obtaining a triangular grid discrete line, thereby considerably improving operational efficiency. study, all three algorithms were implemented in C++. Furthermore, except for the different core steps, all operations were kept the same to ensure fair and objective results. Each program was 5. Experiments and Analysis compiled into the Release version and executed on a SAMSUNG 450R5U laptop (Hardware To verify the effectiveness of our proposed duality dimensionality reduction discrete configuration: Intel Core i5-3230M [email protected] GHz, 8 GBand RAM; Operating System: Windows 7 ×line 64 generation algorithm, we compared it with the Freeman [16] and full-path [2] algorithms. In our Ultimate version; Development Tools: Visual Studio 2012). The data for the experiment were study, all three algorithms were implemented C++. Furthermore, theBureau different core steps, obtained from the 1:250,000 basic geographic in information databaseexcept of the for State of Surveying all operations were kept the same to ensure fair and objective results. Each program was compiled and Mapping; these included the vector data of 34 provinces (municipalities directly under the into the Release version and executed on a SAMSUNG 450R5U laptop (Hardware configuration: Intel central government and autonomous regions) in China, with a total of 917,264 coordinate points. Core i5-3230M [email protected] GHz, 8 GB RAM; Operating System: Windows 7 × 64 Ultimate version; Development Tools: Visual Studio 2012). The data for the experiment were obtained from the 1:250,000 basic geographic information database of the State Bureau of Surveying and Mapping; these included the vector data of 34 provinces (municipalities directly under the central government and autonomous regions) in China, with a total of 917,264 coordinate points. Each provincial boundary was regarded as a closed polygon, and it was processed by all three algorithms. The fitting degree between the generated discrete line and the actual vector data is an important criterion for evaluating the effectiveness of the algorithm. In practice, the closer the discrete line that is generated by the algorithm is to vector data, the higher the fitting accuracy. In this study, the average vertical distance between the center of the discrete line grid cell and the actual vector line, i.e., the ratio of the sum of the vertical distance of all discrete line cells to the total number of discrete line cells, was taken as a quantitative indicator to evaluate fitting accuracy between the discrete line and actual vector data, denoted as average distance (AVD), unit: meter (m). Generally, the smaller the AVD value, the higher the fitting accuracy of the discrete line. When the side length of the grid cell was too large, the generated discrete line only consists of a small number of grid cells. In this case, the description of the vector boundary is very approximate and it is of no practical significance for discussing the fitting relationship between the discrete line and vector data. Therefore, in order to clearly compare the overall effectiveness and fitting accuracy of the proposed algorithm and the Freeman algorithm, we set the side length of the grid cell to 3 km

(m). Generally, the smaller the AVD value, the higher the fitting accuracy of the discrete line. When the side length of the grid cell was too large, the generated discrete line only consists of a small number of grid cells. In this case, the description of the vector boundary is very approximate and it is of no practical significance for discussing the fitting relationship between the discrete and2018, vector and ISPRS Int. J.line Geo-Inf. 7, 391data. Therefore, in order to clearly compare the overall effectiveness 9 of 13 fitting accuracy of the proposed algorithm and the Freeman algorithm, we set the side length of the grid cell to 3 km and we selected Beijing for provision of test data, allowing for fewer coordinate and we(3661 selected Beijing fortwo provision of test data,used allowing for fewer coordinate (3661ofpoints). points points). The algorithms were to generate discrete lines, points the results which The two algorithms were used to generate discrete lines, the results of which are shown in Figure 10. are shown in Figure 10. According to the calculations, the AVD values of the proposed algorithm According to the calculations, the AVD values of the proposed algorithm and the Freeman and the Freeman algorithm were 522.829 m and 1102.95 m, respectively, indicating higheralgorithm accuracy were 522.829 m and 1102.95 m, respectively, indicating higher accuracy of the former. The discrete of the former. The discrete lines generated by the proposed algorithm are also more continuous and lines generated by the proposed algorithm are also more continuous and smoother. Compared with smoother. Compared with the proposed algorithm, the Freeman algorithm selects discrete line grid the thedistance Freemanbetween algorithm selects line grid on theand vertical cellsproposed based onalgorithm, the vertical vertex anddiscrete line, leading to cells largebased deviations poor distance between vertex and line, leading to large deviations and poor continuity and smoothness. continuity and smoothness.

(A)

(B)

Figure 10. 10. Comparison Comparison of of the the overall overall effectiveness effectiveness of of the the two two algorithms algorithms (side (side length length of of grid grid cells cells is is Figure approximately33km): km):(A) (A)discrete discrete line Beijing boundary generated by proposed the proposed algorithm approximately line of of Beijing boundary generated by the algorithm and anddiscrete (B) discrete line of Beijing boundary generated the Freeman algorithm. (B) line of Beijing boundary generated by thebyFreeman algorithm.

As 1, with improved grid resolution, the AVD both the Asshown shownininTable Table 1, with improved grid resolution, thevalues AVDofvalues of Freeman both thealgorithm Freeman and the proposed algorithm decrease with a reduction in grid cell side length. When the side lengths algorithm and the proposed algorithm decrease with a reduction in grid cell side length. When the of grid cells are the are AVD value ofAVD the proposed algorithm is lower than is that of the Freeman side lengths of equal, grid cells equal, the value of the proposed algorithm lower than that of algorithm. This means that leads to moreleads accurate description vector data the Freeman algorithm. Thishigher meansgrid thatresolution higher grid resolution to more accurate of description of by the discrete line, with better effectiveness using the proposed algorithm. It should be noted that vector data by the discrete line, with better effectiveness using the proposed algorithm. It should be no matter high thehow gridhigh resolution, even when theeven sidewhen lengththe of the cell is 3 m, noted thathow no matter the grid resolution, sidegrid length of reduced the gridtocell is there are still deviations between the discrete line and the actual vector data, as shown in Figure 11. reduced to 3 m, there are still deviations between the discrete line and the actual vector data, as This is because existing discrete line generation including the Freeman algorithm, and our shown in Figure 11. This is because existingalgorithms, discrete line generation algorithms, including the proposed algorithm, and all use center oralgorithm, vertex of aallgrid the reference generate Freeman algorithm, ourthe proposed usecell theas center or vertexpoint of a to grid cell as the the discrete line. However, in practice, the coordinate point of a vector is rarely at the center or vertex reference point to generate the discrete line. However, in practice, the coordinate point of a vectorof is the gridatcell. results in deviations between generated by the reference point and rarely theThis center or vertex of the grid cell. the Thisdiscrete resultsline in deviations between the discrete line actual vector data. This deviation is inevitable in the process of discretization, and determining ways to reduce it will be the focus of our further research. Table 1. Average distance (AVD) statistical table for Freeman and proposed algorithms. Side Length of Grid Cell (m)

Proposed Algorithm (m)

Freeman Algorithm (m)

3000 1500 750 375 188 94 47 23 12 6 3

522.829 277.862 162.739 94.7479 49.917 25.473 12.828 6.292 3.295 1.650 0.823

1102.95 584.722 287.156 143.333 68.807 33.800 17.096 8.253 4.339 2.152 1.075

94 47 23 12 6 ISPRS Int. J. Geo-Inf. 2018, 7, 391 3

25.473 12.828 6.292 3.295 1.650 0.823

(A)

33.800 17.096 8.253 4.339 2.152 1.075

(B)

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(C)

11. Detailed the proposed algorithm length of grid is approximately 3 m), Figure Figure 11. Detailed resultsresults of theofproposed algorithm (side(side length of grid cellcell is approximately 3 m), indicating the fitting relationship between the discrete line and actual vector data (A) for an inclined indicating the fitting relationship between the discrete line and actual vector data (A) for an inclined at a corner; and for a vertical line; (B)line; at a(B) corner; and (C) for(C) a vertical line. line.

Although the full-path and proposed algorithms similar results, a difference significant Although the full-path and proposed algorithms producedproduced similar results, a significant differencespeed in computation speed listed in Table (side length of grid 3cell in computation was observed. As was listedobserved. in Table 2As (side length of grid2 cell is approximately m),is approximately 3 m), for the same data, the average computation speed of our algorithm was for the same data, the average computation speed of our algorithm was approximately 9.309 times that approximately 9.309 times that of the full-path algorithm. This is predominantly because of the full-path algorithm. This is predominantly because the full-path algorithm involves sorting the a full-pathofalgorithm sorting a large number of time-consuming, data points basedwhereas on distance, which is large number data pointsinvolves based on distance, which is highly the proposed highly time-consuming, whereas the proposed algorithm involves trigonometric operations only algorithm involves trigonometric operations only when determining the optimal direction vectors and when determining the optimal direction vectors and corresponding basic set, and subsequent corresponding basic set, and subsequent calculations only involve addition and subtraction because of calculations only involve addition and subtraction because of the dimensionality reduction the dimensionality reduction operation. The computational complexity of the proposed algorithm is therefore considerably lower than that of the full-path algorithm; consequently, its computation speed is higher. Table 2. Computation speed statistics for the full-path and proposed algorithms. Province (Points)

Proposed Algorithm (ms)

Full-Path Algorithm (ms)

Time Ratio

Inner Mongolia (52,486) Heilongjiang (27,521) Xinjiang (37,855) Tibet (39,184) Liaoning (35,732) Guangdong (28,887) Jilin (21,588) Henan (21,567) Jiangxi (13,621) Hubei (19,335) Hainan (6065) Tianjin (4206) Beijing (3661) Ningxia (9704)

2086.43 1079.67 1306.59 1065.33 676.80 689.23 656.53 491.06 393.64 519.29 203.55 141.69 144.36 298.29

28,999.20 16,511.36 13,895.09 10,135.34 4804.96 5684.01 5563.50 3982.95 3339.67 4543.93 1640.03 1065.81 1175.63 2423.04

13.90 15.29 10.63 9.51 7.10 8.25 8.47 8.11 8.48 8.75 8.06 7.52 8.14 8.12

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6. Application of Our Algorithm in DGGS DGGS is a promising global multiscale data model. Compared with the traditional geospatial data organization mode within a local area, DGGS is more suitable for large-scale applications, and it structurally supports efficient multi-resolution geospatial data processing [22]. Although we have thus far only discussed the application of the algorithm in the plane, the relationship between a plane and a spherical surface can be established by the icosahedron and Snyder projection. In this way, our algorithm can be extended to the spherical surface and applied to DGGS. In the case of a set of spherical vector data, if this is inside a spherical triangle, it can be projected onto the triangular face using the forward projection (transformation from the spherical surface to the icosahedral triangular faces) of the Snyder projection. On the triangular face, the corresponding triangular grid discrete lines can be generated by our algorithm, and the discrete lines can be projected onto the spherical surface by inverse projection (transformation from the icosahedral triangular faces to the spherical surface) of the Snyder projection, thereby generating discrete lines in DGGS. However, spherical vector data is generally located at the junction of spherical triangles and will therefore be split into different spherical triangles. Following the forward projection of the Snyder projection, the triangular face corresponding to each spherical triangle has an independent coordinate system. The spherical vector data in each spherical triangle can only be processed separately, which requires solving the intersection of the spherical vector data and spherical triangle boundaries. Only in this way can the spherical vector data inside each spherical triangle be determined, and can the corresponding discrete lines be generated in DGGS by the means of the Snyder projection and our algorithm. Although we have solved the problem of the intersection of spherical vector data and spherical triangle boundaries, the specific process is more complicated and it is not the focus of this paper; we will elaborate this further in future publications. Taking actual spherical vector data as an example, its multi-resolution display in DGGS can be achieved using our algorithm. Results are shown in Figure 12. After generating discrete lines, spatial operations such as buffer generation [23] and overlay analysis can be performed, providing a reference for solving practical problems. ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW

(A)

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(B)

(C)

Figure12. 12.Multiresolution Multiresolution discretization discretization of spherical vector Figure vector data. data. The The white whitearcs arcsare arethe theboundaries boundaries of the spherical triangles, and the figures represent the following: (A) the side length of of the spherical triangles, and the figures represent following: (A) the side length ofgrid gridcell cellisis approximately 483 km; (B) the side length of grid cell is approximately 242 km; (C) the side length approximately 483 km; (B) the side length of approximately 242 km; (C) the side lengthofof gridcell cellisisapproximately approximately121 121km. km. grid

7.7.Conclusions Conclusionsand andFuture Future Directions Directions Based triangular and and hexagonal hexagonalgrids, grids,as aswell wellas asthe the Basedon onthe theweak weakduality duality relationship relationship between triangular equivalence discrete line line and and one-dimensional one-dimensionaloptimal optimal equivalencerelationship relationship between between the the two-dimensional two-dimensional discrete wandering wewe successfully performed transformations between the discrete line of aline triangular wanderingpath, path, successfully performed transformations between the discrete of a triangular grid, the discrete line of a weak duality hexagonal grid, and the optimal wandering path, thus reducing the complexity of generating a discrete line in a triangular grid. Compared with existing algorithms, our algorithm can generate the ideal discrete line and has clear advantages in terms of computation speed. In addition, the concepts of duality and dimensionality reduction provide a new way to solve related discretization problems; our proposed method can therefore support further research.

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grid, the discrete line of a weak duality hexagonal grid, and the optimal wandering path, thus reducing the complexity of generating a discrete line in a triangular grid. Compared with existing algorithms, our algorithm can generate the ideal discrete line and has clear advantages in terms of computation speed. In addition, the concepts of duality and dimensionality reduction provide a new way to solve related discretization problems; our proposed method can therefore support further research. Our proposed algorithm can be applied to real-time grid transformation of vector data, and it can also be indirectly applied to DGGS by the means of the Snyder projection and icosahedron. Meanwhile, our algorithm is beneficial for interoperation between hexagonal and triangular grids. The idea of duality and dimensionality reduction can be extended to multi-dimensional spaces; therefore, it is of great significance for the study of high-dimensionality problems, suggesting broad application prospects in the future. Author Contributions: Conceptualization, L.D., Q.M., and J.B.; Methodology, L.D.; Software, L.D. and J.B.; Validation, R.W. and J.L.; Formal Analysis, L.D.; Investigation, L.D., R.W.; Resources, J.L.; Data Curation, J.B.; Writing—Original Draft Preparation, L.D., Q.M., and J.B.; Writing—Review & Editing, L.D. and J.B.; Visualization, L.D. and J.B.; Supervision, Q.M. Funding: This research was funded by the National Key Research and Development Program of China, grant number [2018YFB0505301], the Natural Science Foundation of China, grant number [41671410], and the Excellent Young Scholar Foundation of Information Engineering University, grant number [2016610802]. Conflicts of Interest: The authors declare no conflict of interest.

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