IMPLEMENTATION OF 24-ARY GRID REPRESENTATION FOR

RECTANGULAR SOLID DISSECTIONS

Tomokazu Arita

, Satoshi Kishira

, Tomoe Motohashi

, Kenshi Nomaki

Division of Integrated Sciences, J. F. Oberlin University, Machida, Tokyo, Japan

Department of Computer Science and System Analysis, Nihon University, Setagaya, Tokyo, Japan

Department of Network and Multi-Media Engineering, Kanto-gakuin University, Yokohama, Kanagawa, Japan

Kimio Sugita

, Kensei Tsuchida

, Takeo Yaku

Department of Mathematics, Tokai University, Hiratsuka, Kanagawa, Japan

Department of Information and Computer Sciences, Toyo University, Kawagoe, Saitama, Japan

Keywords: Solid and heterogeneous modeling, Octgrids, Tetraicosa-grids, Data structures.

Abstract: In this paper, we propose 24-ary grid graphs corresponding to rectangular solid dissections for ruled line

preserving operations. We also show a data structure called H9CODE that corresponds to the 24-ary grid

graphs. Furthermore, we describe a voxel unification method in the 24-ary grid graphs and show that the 24-

ary grid graphs are an effective model to represent solid graphics.

1 INTRODUCTION

Rectangular solid dissections are commonly used in

solid graphics. As graph representation models of

rectangular solid dissection processing, octrees

(Jackins and Tanimoto, 1980) are well known. The

octree can be used in geometric modeling and space

planning. The octree structure is an extension of the

quadtree structure for the representation of two-

dimensional images. A multi-level boundary search

algorithm is developed to incorporate surface

information into the octree representation. This

algorithm makes the octree representation useful for

graphic displays and object recognition tasks.

We have examined ruled line oriented

transformation of rectangular solid dissections such

as voxel unification in solid graphics. We note that

voxel unification is frequently used in Level of

Detail (LOD) related operation. We previously

proposed octal grids called octgrids (Motohashi et

al., 2002; Motohashi et al., 2002; Arita et al., 2004;

Akagi et al., 2005) for rectangular dissections, and

hexadecimal grids called hexadeci-grids (Kureha et

al., 2007) for multilayer rectangular dissections with

respect to ruled line oriented operations. Several

transformation algorithms for octgrids are more

efficient than for quadtrees (Arita et al. 2004).

In this paper, we introduce 24-ary grid graphs

called “tetraicosa-grids” (Kureha et al., 2007;

Kishira, Tsuchida et al., 2008; Kishira, Kureha et al

2008) that correspond to rectangular solid

dissections. Tetraicosa-grid structure is constructed

by extending the octgrid structure. We also describe

a voxel unification method that runs in O(1) time,

and we show a data structure called H9CODE

corresponding to the 24-ary grid graphs.

In section 2, we review octgrids for rectangular

dissections as preliminaries. Section 3 contains

several definitions of tetraicosa-grids. In section 4,

we show the H9CODE data format corresponding

to the tetraicosa-grids. In section 5, we explain a

voxel unification method using H9CODE, and in

section 6 we explain the concept of rendering

H9CODE.

2 PRELIMINARIES

2.1 Octgrids for Rectangular

Dissections

In this section, we deal with heterogeneous

rectangular dissections. We review the definitions

103

Arita T., Kishira S., Motohashi T., Nomaki K., Sugita K., Tsuchida K. and Yaku T.

IMPLEMENTATION OF 24-ARY GRID REPRESENTATION FOR RECTANGULAR SOLID DISSECTIONS.

DOI: 10.5220/0001799201030106

In Proceedings of the Fourth International Conference on Computer Graphics Theory and Applications (VISIGRAPP 2009), page

ISBN: 978-989-8111-67-8

concerning octgrids (Motohashi et al., 2002,

Motohashi et al., 2003; Arita et al., 2004; Akagi et

al., 2005) that represent rectangular dissections.

Definition 2.1.1

Let D = (T, P, g) be a rectangular dissection, where

T is an (n, m) − table for some n and m, P is a

partition over T, and g is a grid of T. An octgrid G =

, L, E

, A

, α

) for D is a multi-edge undirected

grid graph, where V

is identified by partition P (We

denote a node corresponding to a cell c in P by v

c), L

= {enw, esw, eew, eww }, E

⊆ V

× L × V

is a set

of undirected labeled edges of V

of the form [vc, l,

d], where vc and vd are in V

, and l is in L. Here, E

is defined by rules 1-4 below, and A

= R

and α

→ R

are defined as the location of perimeter cell

c for v

c in V

by α

= (nw(c), sw(c), ew(c), ww(c)).

Rule 1

If nw(c) = nw(d), that is, c and d have a common

north wall, and there is no cell between c and d that

has an equal north wall, then [vc, enw, vd] is in E

In this case, [vc, enw, vd] is called a north wall edge.

Rules 2-4

Labeled edges in other directions are similarly

defined. The following figure illustrates a

rectangular dissection and its corresponding octgrids

(Figure 1).

We note the degree of edges is at most eight in

octgrids.

2.2 H3CODE (ARITA AND YAKU,

2006)

H3CODE is a data format that represents octgrids.

The cell of H3CODE corresponds to a rectangle in a

rectangular dissection (See Figure 2(a)). The whole

structure of the H3CODE files is shown in Figure

2(b).

Figure 1: Rectangular dissection (upper) and its

corresponding octgrid (lower).

(a) (b)

Figure 2: (a): H3CODE cell, (b): whole structure of

H3CODE.

3 24-ARY GRID GRAPHS

Next, we introduce a 24-ary grid graph

representation for rectangular solid graphics. Let D

= {S

, S

, ..., S

} be a rectangular solid dissection,

where each S

is a rectangular solid in D. A

tetraicosa-grid for D is an undirected labeled multi-

edge grid graph G

= (V

, L, E

, A), defined as

follows:

(1) V

= {v

| s is in D; v

corresponds to s} is a set

of nodes,

(2) L = {EquivalentUpwardNorthEastCornerPole,

EquivalentDownwardNorthEastCornerPole, ...,

EquivalentBackwardFloorWestBeam} (|L| = 24) is

the set of edge labels,

(3) E is a set of undirected labeled edges defined as

follows; if s and t are the nearest solids in D such

that s and t have an upper north beam in common,

then [s, EquivalentForwardCeilingNorthBeam

(Figure 3), t] is in E

. Edges for other beams and

corner poles are similarly defined.

Figure 4 illustrates links around a node in a

tetraicosa-grid. Furthermore, Figure 5 shows a

rectangular solid dissection (left) and the

corresponding tetraicosa-grid (right).

Suppose that D is of a k-width, l-depth, and m-

height rectangular solid; let G

be the tetraicosa-grid

for D, and i be the number of inner voxels. We have

2 |E

| = 12×8 + 16×4(k-2) + 16×4(l-2) + 16×4(m-2)

+20×2(k-2)(l-2) + 20×2(l-2)(m-2) + 20×2(m-2)(l-2)

+ 24 i (Kishira et al., 2008).

Figure 3: Example of labels for edges.

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104

Figure 4: Links around a node in a tetraicosa-grid (Kishira

S., Kureha A. et al., 2008).

Figure 5: Rectangular solid dissection (left) and its

corresponding tetraicosa-grid (right).

4 DATA FORMAT: H9CODE

In this section, we propose a data format H9CODE

corresponding to tetraicosa-grids. H9CODE is based

on the data format H3CODE (Arita and Yaku, 2006)

corresponding to octgrids.

We first show the field numbers with their

contents, starting with the 33rd field. We note that

fields 1 to 32 in H9CODE are the same as the fields

in H3CODE.

33. EquivalentUpwardNorthEastCornerPole

(See Figure 6 (left))

34. EquivalentDownwardNorthEastCornerPole

33. EquivalentUpwardNorthWestCornerPole

34. EquivalentDownwardNorthWestCornerole

35. EquivalentUpwardSouthEastCornerPole

36. EquivalentDownwardSouthEastCornerPole

37. EquivalentUpwardSouthWestCornerPole

38. EquivalentDownwardSouthWestCornerPole

41. EquivalentForwardCeilingNorthBeam

(See Figure 6 (right))

42. EquivalentBackwardCeilingNorthBeam

43. EquivalentForwardCeilingSouthBeam

44. EquivalentBackwardCeilingSouthBeam

45. EquivalentForwardFloorNorthBeam

46. EquivalentBackwardFloorNorthBeam

47. EquivalentForwardFloorSouthBeam

48. EquivalentBackwardFloorSouthBeam

49. EquivalentForwardCeilingEastBeam

50. EquivalentForwardCeilingWestBeam

51. EquivalentBackwardCeilingEastBeam

52. EquivalentBackwardCeilingWestBeam

53. EquivalentForwardFloorEastBeam

54. EquivalentForwardFloorWestBeam

55. EquivalentBackwardFloorEastBeam

56. EquivalentBackwardFloorWestBeam

Here, we show examples of H9CODE in Figure 6.

Figure 6: Voxels linked with 33rd field (left) and with

41st field (right).

5 VOXEL UNIFICATION

In this section, we show a voxel unification method

with H9CODE. This method unifies properly

neighboring voxels and is executed as depicted in

Figure 7.

UNIFYVOXEL (See Figure 7)

INPUT

: 24-ary grid graph representation for a

rectangular solid dissection D in H9CODE

: a voxel in G

; vc and vd have four

horizontal beams in common

OUTPUT

: 24-ary grid graph representation for a

rectangular solid dissection E in H9CODE,

where v

is unified to v

Method

Change links on x-axis in v

2. Change links on y-axis in v

Change links on z-axis in v

4. Delete v

The UNIFYVOXEL method unifies two

neighbour voxels into one voxel by replacing

neighbour edges around their voxels based on 24-ary

grid graph representations.

IMPLEMENTATION OF 24-ARY GRID REPRESENTATION FOR RECTANGULAR SOLID DISSECTIONS

105

Figure 8: Output image of sphere using above programs

with H9CODE.

Figure 7: Input and output of voxel unification.

We are now implementing rendering systems for

the 24-ary grid graphs and plan to construct the

unification system for the 24-ary grid graphs.

We note that the time complexity of the method

is O(1), since the number of links around c and d are

bounded by 96 = 48 × 2.

REFERENCES

6 RENDERING WITH H9CODE

Arita, T., Tsuchida, K., Motohashi, T., and Yaku, T.

(2004). An Octet Degree Graph Representation for the

Rectangular Dissections. Joint Conference on Applied

Mathematics, pages 131-136.

In this section, we present the concepts of rendering

programs and an output image created using

H9CODE.

First, we show programs to render coordinate

values of voxels with VRML and H9CODE.

Arita, T. and Yaku, T., (2006). H3-Code 2.3 Reference

Manual. http://www.waap.gr.jp/waap-rr/waap-rr-06-

001/index.html

Program

Akagi, G., .Motohashi, T., Nomaki, K., and Yaku, T.

(2005). Octal Graph Representation for Multi-

Resolution 3D Landform Maps and Its Application. In

Proceedings: Applied Mathematics Symposium, pages

27-32.

 A conversion program of coordinate values

of voxels to H9CODE

 Program Name : cv2h9

 Input: coordinate values

Jackins, C. L. and Tanimoto, S. L. (1980). Oct-Trees and

Their Use in Representing Three-Dimensional Object.

In Proceedings: Computer Graphics and Image

Processing, Vol.14, No.3, pages 249-270.

 Output: H9CODE

 Programming language : C

 A conversion program of H9CODE to

VRML

Kureha, A., Kishira, S., Motohashi, T., Tsuchida, K., and

Yaku, T. (2007). Hexadecimal Grid Graph

Representation of Multilayer Rectangular Dissections

and Its Applications. 10th Society for Industrial and

Applied Mathematics Conf. in Geometric Design &

Computing, Abstract , Texas, USA, page 32.

 Program name : h92vrml

 Input: H9CODE

 Output: VRML

 Programming language : C

Kishira S., Kureha A., Motohashi T., Tsuchida K., and

Yaku T. (2008), 24-ary Grid Graph Representation for

the Rectangular Solid Dissections, Proceedings of the

2008 IEICE General Conference, page 171.

An output image of a sphere using the above

programs and rendered with H9CODE is shown in

Figure 8.

Kishira, S., Tsuchida, K., Motohashi, T., and Yaku, T.

(2008). Tetra-icosa Grid Graph Representation for the

Rectangular Solid Dissection. JSIAM, pages 65-66 (in

Japanese).

7 CONCLUSIONS

Motohashi, T., Tsuchida, K,, and Yaku, T. (2002). Table

Processing Based on Attribute Graphs. In

Proceedings: IASTED SEA5, pages 317-322.

We introduced the basic concepts of a graph

representation method called “tetraicosa-grid” for

volume graphics. We introduced a data format

H9CODE for 24-ary grid graphs. We also showed a

voxel unification method using H9CODE. This

method unifies two voxels into one voxel and runs in

constant time.

Motohashi, T., Tsuchida, K., and Yaku, T. (2002).

Attribute Graphs for Tables and Their Algorithms. In

Proceedings: Foundation of Software Engineering,

pages 183-186.

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