AN ENHANCED EDGEBREAKER COMPRESSION ALGORITHM

FOR THE CONNECTIVITY OF TRIANGULAR MESHES

D. R. Khattab, Y. M. Abd El-Latif, M. S. Abdel Wahab and M. F. Tolba

Faculty of Computers and Information Sciences, Ain Shams University

Abbassia, 11566, Cairo, Egypt

Keywords: Triangle mesh compression, EdgeBreaker, Connectivity encoding, Linear decoding.

Abstract: Compression of digital geometry models is the answer to an industrial demand. Over the last years, many

exciting ideas and new theoretical insights have been devoted to finding ways of reducing the amount of

storage such models absorb. EdgeBreaker is one of the effective lossless single-rate connectivity

compression techniques for triangular meshes. This paper presents an enhanced EdgeBreaker encoding

algorithm which solves the problem of non-linearity of EdgeBreaker decoding procedure while

reconstructing the mesh triangles in the same order they were traversed during the encoding phase. The new

enhancement is based on the same data structure: the corner-table used by EdgeBreaker however, it

eliminates some of the computational overhead exhibited by EdgeBreaker compression. This enhanced

technique also yields to significantly smaller rates for connectivity compression than EdgeBreaker. It

achieves an average compression ratio of 1.8 bit per triangle and 3.57 bit per vertex for the used benchmark

3D models.

1 INTRODUCTION

Interactive display of 3D content has been

extensively used in many applications ranging from

electronic commerce to the virtual game industry.

Among several representations, polygonal meshes

are used most often as surface representation (Abd

El-Latif, Ghaleb and Hussein, 2006) because of

their wide spread support in many file formats and

graphics libraries. These large and complex meshes

are becoming commonplace because of the

increasing capabilities of the computing

environments, visualization hardware, modern

interactive modelling tools and semi automatic 3D

data acquisition systems. The complexity of these

models poses basic problems of efficient storage in

file servers, transmission over computer networks,

rendering, analysis, processing etc. for these reasons

it was desirable to compress polygonal meshes to

reduce storage and transmission time requirements.

Geometry compression techniques for such very

large 3D geometric models have thus become a

subject of intense study in recent years. The

compression approach is one of the primarily

approaches for reducing the size of a mesh. It

depends on deriving a new encoding for the

polygonal mesh such that the total number of bits

needed in the new encoding is much lower than the

number needed for the uncompressed representation.

Large body of compression research has

concentrated on clever encoding of the mesh

connectivity. Typically (Shikhare, 2000), the

number of triangles in a mesh is roughly twice the

number of vertices and each vertex is referenced in 5

to 7 triangles, which means that large part of the

representation of the model is in the definition of

connectivity. Also the combinatorial graph structure

of the mesh connectivity allows it to be coded

losslessly, so it can be restored to the original model

after decoding.

Prior Works. Much of the work in the area of

single-rate connectivity compression has been

concerned with triangle meshes only (Allie and

Gotsman, 2005) This is because the triangle is the

basic geometric primitive for standard graphics

rendering hardware and also it can be easily derived

from other surface representations. Many

compression schemes have been emerged recently

using different approaches such as focusing on

hardware decoding (Deering, 1995 and Chow,

1997), applying mesh traversal (Gumhold and

Strasser, 1998 and Rossignac, 1999) and using

valence-based incidence compression (Touma and

Gotsman, 1998 and Alliez and Desbrun 2001).

109

R. Khattab D., M. Abd El-Latif Y., S. Abdel Wahab M. and F. Tolba M. (2007).

AN ENHANCED EDGEBREAKER COMPRESSION ALGORITHM FOR THE CONNECTIVITY OF TRIANGULAR MESHES.

In Proceedings of the Second International Conference on Computer Graphics Theory and Applications - GM/R, pages 109-115

DOI: 10.5220/0002074301090115

 SciTePress

Among these techniques, EdgeBreaker (Rossignac,

1999) is considered one of the best connectivity

compression techniques for compressing simple

manifold triangular meshes that are homeomorphic

to a sphere. It achieved bit rates of 2 bit per triangle

(bpt) and 4 bit per vertex (bpv). As this result was

the best achieved so far, many derivatives of the

EdgeBreaker algorithm had appeared. Using a

slightly more complex code, King and Rossignac

(King and Rossignac, 1999) guaranteed that the

compressed file will not exceed 1.83t bits and 3.67v

bits. The algorithm was further optimized for

meshes with regular connectivity (Szymczak, King

and Rossignac, 2001). In (Rossignac, Safonova and

Szymczak, 2002) the same algorithm was introduced

with a simple data structure, the Corner-Table for

representing the connectivity of triangle meshes.

Again the algorithm was extended with a simple

formulation to deal with triangulated surfaces with

handles (Lopes, Rossignac, Safonova, Szymczak

and Tavares, 2003). Finally the authors in (Lewiner,

Lopes, Rossignac, and Vieira 2004) provided

efficient extensions of the EdgeBreaker compression

which enables to use the EdgeBreaker algorithm to

encode the connectivity of a surface, possibly having

any number of connected components, handles or

boundary curves.

While the encoding scheme of EdgeBreaker

(Rossignac, 1999) exhibits a linear storage cost,

some preliminary preprocessing steps were required

in the decoding phase making it exhibits a non-linear

time complexity of O(n

). More recent work

(Rossignac and Szymczak, 1999) eliminated the

need for this look-ahead procedure and improved the

worst case complexity to O(n). However this

algorithm requires multiple traversals of the mesh

triangles for meshes with handles and an initial

traversal of the encoding for meshes with boundary.

Another simple decoding technique (Isenburg and

Snoeyink, 2001) was developed that recreates the

triangles encoded by EdgeBreaker in linear time

unless this was done in the reverse order these

triangles were encoded.

Contributions. The main work of this paper is

based on the work of EdgeBreaker (Rossignac,

Safonova and Szymczak, 2002) developed by

Rossignac et al. The EdgeBreaker encoding

procedure is enhanced to allow the decoding phase

to be implemented in linear time complexity without

any additional preprocessing and in the same order

the mesh is traversed during the encoding phase.

Further more, this enhancement eliminates the

computational overhead implemented by using extra

data structures and also improves the compression

ratio generated by the original procedure (Rossignac,

1999) to 1.8 bpt and 3.57 bpv.

The remainder of this paper is organized as

follows: in the next section the EdgeBreaker

encoding and decoding schemes are explained. A

detailed description of the algorithm can be found in

(Rossignac, 1999 and Rossignac, Safonova and

Szymczak, 2002). Section 3 illustrates the new

improvement done to the encoding algorithm of

EdgeBreaker. The results and discussions are

presented in section 4 and we conclude in section 5.

2 EDGEBREAKER ENCODING

AND DECODING

The EdgeBreaker encoding process visits the

triangles in a spiraling (depth-first) order and

produces a CLERS string. The CLERS string

includes five different operations called C, L, E, R,

and S. The encoding process starts off with picking

an arbitrary triangle of the mesh as an initial active

boundary. One of the three initial boundary edges is

defined to be the gate of the boundary. An initially

empty stack is used to temporarily store boundaries.

This process terminates after t-1 operations, with t

being the number of mesh triangles. Which

operation is chosen depends on how the respective

triangle is attached to the active boundary at the

moment it is processed (Figure 1). If its third vertex

is not on the active boundary then operation C is

used and the new gate is the right edge of the old

gate. If the third vertex is the next boundary vertex

then operation R is used. If it is the previous

boundary vertex then operation L is used. In both R

and L operations the new gate is the inserted

boundary edge. If the third vertex is some other

boundary vertex, then operation S is used. The left

edge of the old gate is pushed on the stack and the

other becomes the active gate. If the third vertex is

the previous and the next boundary vertex then

operation E is used. This can only happen for an

active boundary of length three.

For triangle meshes with v vertices and t

triangles that are homeomorphic to a sphere, t equals

2v-4 (Rossignac, 2003). Because, except for the first

two vertices, there is a one-to-one mapping between

each C triangle and each vertex, the number of C

triangles is v–2. Consequently, the number of non-C

triangles in a simple mesh is t–(v–2), which is also

v–2. Thus exactly half of the triangles are of type C.

using A straight-forward compression scheme that

uses the following simple binary code for the labels

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110

(C=0, L=110, E=111, R=101, S=100) is guaranteed

to use no more than 2t or 4v bits.

Figure 1: EdgeBreaker encoding operations.

The EdgeBreaker decoding process starts with a

CLERS string and produces a triangulated mesh.

Two traversals of the CLERS string are needed: The

preprocessing phase computes an offset value for

every S operation. The offset value is the distance

between the active gate and the tip vertex along the

active boundary. These offset values are calculated

by adding up the resulting change in boundary

length for all operations following an S operation

until and including its corresponding E operation.

Since pairs of S and E operations are always nested,

the offset values for all S operations can be

computed in a single traversal (Figure 2).

The other decoding phase is the generation phase

which creates the triangles in the same order they

were encoded by the EdgeBreaker encoding process.

It starts with creating the initial triangle. The active

boundary and the gate are identified and the CLERS

string is processed. For the C operation a new vertex

is created. For all other operations a vertex from the

active boundary is used. For the R operation the

third vertex is the next vertex on the active boundary

and for the L operation the third vertex is the

previous vertex on the active boundary. For the S

operation the precomputed offset value is used.

When the E operation occurs, the active boundary

consists of only three boundary edges leaving no

choice for the third vertex. If the active boundary is

maintained in a linear data structure, each S

operation will require a linear search for the vertex

specified by the offset implying an asymptotic worst

case time complexity of O(n

) for the EdgeBreaker

decoding.

Figure 2: EdgeBreaker decoding operations.

3 ENHANCED EDGEBREAKER

ENCODING

The breakthrough of EdgeBreaker lies in the

discovery that the locations of the tips of the S

triangles need to be stored neither as integer

references nor as offsets which separate the gate

from the tip location in the current loop. They

discovered that these offsets can be recomputed by

the decoding algorithm from the CLERS string

itself. This solution minimizes the encoding size and

improves the compression ratio while it leads for the

nonlinearity of the decoding algorithm. The new

enhancement of EdgeBreaker tries to keep

advantage of not saving the offsets as part of the

encoding while in the same time eliminate the need

of computing them as a preprocessing stage during

decompression.

The S triangle in EdgeBreaker compression

scheme splits the active boundary into two, one on

each side of the S triangle. The algorithm then tends

to fill the hole generated in the mesh surface on the

right side of the S triangle before returning to the left

one. This required keeping a stack for storing the left

edge of every S triangle to be the next active gate

(Figure 3a). The new enhancement developed here

performs the same traversal of the EdgeBreaker

compression but only differs in the way it deals with

the S triangles. Whenever an S triangle is reached,

the algorithm tends to ignore encoding it at the

moment and the active gate is changed to another

AN ENHANCED EDGEBREAKER COMPRESSION ALGORITHM FOR THE CONNECTIVITY OF TRIANGULAR

MESHES

111

location and then the traverse keeps going on. This

is done by moving on the active boundary one step

to the right of the current active gate (Figure 3b).

(a)

(b)

Figure 3: Encoding example of the final eighteen

operations of a mesh. (a) EdgeBreaker encoding, The red

edges show the ones being stored in the stack; (b)

Enhanced EdgeBreaker encoding, The arrows show how

the active edge is changed during traversal.

This approach ensures that all the previously

ignored S triangles are going to be revisited again as

long as they are not encoded so far but from another

gate – usually the pervious right edge – which will

change the triangle state to an L triangle. This

enhancement eliminates the case of S and E triangles

from the CLERS string while it presents another

symbol M that does not interpret a triangle case but

only tells the decoding algorithm that the current

active gate will be changed to another location some

where on the active boundary just right to the current

one. Keeping the same notations of the C, L and R

triangles, the encoding string has been changed to

the CLRM string. The elimination of the S case from

the encoding string makes the active boundary never

split. The algorithm in this case needs only to

maintain one circular linear list for active boundary

during compression and decompression. In addition,

the enhanced technique has eliminated both the

recursive overhead exhibited by the original

EdgeBreaker compression algorithm and the

computational overhead needed by the stack to keep

list of new active gates generated after splitting.

During the decompression process it eliminates the

need for the preprocessing step, the algorithm only

traverses the encoded string once to regenerate the

mesh triangles in the same order they were encoded

and thus maintaining a linear time complexity of the

decoding process O(n).

While the first version of the developed

enhancement algorithm adopted the idea of moving

to the right of the active gate whenever an S triangle

is reached, a serious problem has emerged. The

original encoding algorithm of EdgeBreaker ensures

that all the triangles in the hole generated to the right

of the S triangle are visited before directing to the

left one. Instead the modification applied here so far

by the enhanced algorithm does not maintain this

feature, meaning that the algorithm can traverse part

of the mesh right to the S triangle and then bypass it

in its reverse course - without encoding - to the other

part of the mesh surface on its left. While the

implemented linear list of boundary edges is

circular, it means that this track can be repeated for

many levels. At each level both areas of the mesh

surface to the left and right of the original S triangle

is being shrinking, giving a chance for other S

triangles to appear until leading to a very long strip

of consecutive S triangles.

(a)

(b)

Figure 4: Steps of mesh generation for the Horse model.

(a) Using the CLRM string; (b) using the CLRGF string.

Notice the long line of un-encoded triangles

generated along the mesh while traverse (figure 4a).

This line begins to appear in parts of the mesh

wherever the boundary tends to meet at a closed

point. In the horse example the line first begins to

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

112

appear in the front left leg, extends to the horse

neck, the back left leg and then to the back right leg

respectively. This long strip will not be encoded

until one of the mesh surfaces on the right or the left

of the original S triangle is encoded completely. This

long strip which is traversed many times will result

in much increase in the encoding string size and

hence lead to minimize the compression ratio instead

of trying to maximize it.

In order to solve the emerging problem of the

long chain of S triangles, the second version of the

enhanced algorithm updated the procedure to allow

moving along the active boundary either to the right

or to the left of the current active gate whenever an S

triangle is reached. This choice is decided according

to which boundary on the right or the left is shorter

in length. The boundary length is calculated by the

number of edges it contains. The M symbol is now

replaced by other two symbols G and F which are

used to differentiate between either to move one step

to the right or to the left of the current active gate.

The final encoding string introduced by the

enhanced EdgeBreaker compression scheme

becomes the CLRGF string. Figure 4b shows that

the problem of the long chain of un-encoded

triangles previously mentioned does not exist any

more. Each of the four parts of the horse example is

encoded completely before directing to another part

of the mesh. The pseudo-code of the proposed

encoding procedure is provided in the frame below.

Input:

Connectivity and geometry of the mesh in

the CornerTable format.

Output:

CLRGF string that contains one symbol per

triangle except for the first triangle.

Procedure Compress()

While (True)

If tip vertex is not visited Then

Append encoding of C to the CLRGF string,

Add tip vertex to the list of vertices,

Mark triangle and tip vertex as visited,

Update active gate to the right edge of the

old gate input

Else If right triangle was visited Then

Append encoding of R to the CLRGF string,

Mark triangle as visited,

If left triangle was visited Then

End Procedure

Else

Update active gate to the new inserted

boundary edge

Else If left triangle was visited Then

Append encoding of L to the CLRGF

string,

Mark triangle as visited,

Update active gate to the new inserted

boundary edge

Else

Calculate right and left boundary lengths

If the right boundary length is less than

the left boundary length Then

Append encoding of G to the CLRGF

string,

Update active gate to the right edge of

the old one

Else

Append encoding of F to the CLRGF

string,

Update active gate to the left edge of

the old one

4 RESULTS AND DISCUSSION

Table 1 presents the data of the test case meshes

shown in Figure 5. All the meshes are triangular,

manifold, without boundary, holes or handles.

Associating the following binary code based upon

Huffman coding (Huffman, 1952) for the labels

(C=0, R=10, L=110, G=1110, F=1111) used in the

CLRGF string leads to the best compression ratio

that can be achieved. This selection is based on a

study of the average percentage each symbol

consumes from an encoding file generated to each of

the 3D models used.

Table 1: Used benchmark 3D models.

File name File size

No of

triangles

No of

vertices

Retinal 240 KB 7,282 3,643

Cow2 304 KB 8,626 4,315

Smooth-

feature

416 KB 12,350 6,177

Egea 534 KB 16,532 8,268

Head 1 MB 32,744 16,374

Horse 4.62 MB 96,966 48,485

Armadillo 11.9 MB 345,944 172,974

Vase-Lion 14.6 MB 400,000 200,002

AN ENHANCED EDGEBREAKER COMPRESSION ALGORITHM FOR THE CONNECTIVITY OF TRIANGULAR

MESHES

113

It is apparent from figure 6 that the largest

percentage of the triangles is of type C; this is

because each C triangle is related with the

introduction of a new vertex in the sequence of mesh

vertices. The percentage of R triangles is very close

from the C one; they both consume about 88.72 %

on the average from the total number of symbols

used to encode a mesh. The rest of the symbols is

distributed between L triangles and symbols G and F

which give an indication of finding an S triangle and

so the need of movement along the boundary list.

The two percentages of L; and G plus F together are

very close. This ensures that most of the S triangles

which were ignored for the first time during

traversal are encoded afterwards as L triangles.

Table 2 lists the compressed file size written in

binary format and the compression ratio achieved in

file size, per triangle and per vertex for every test

case of the 3D models used. The compression results

vary from 1.52 bpt and 3.04 bpv (Head example) to

2.16 bpt and 4.31 bpv (Cow2 example). This

variation is due to the different percentage of S

triangles occurrence during mesh traversal which

leads to the addition of symbols G or F to the

encoding string (Figure 6).

Figure 6: Frequency percentage each symbol consumes

from encoding data file.

This percentage of occurrence depends mainly

on the shape characteristics and it represents the

main parameter controlling the compression ratio

that can be achieved to the mesh. It appears from the

used examples that the largest occurrence which

leads to the smallest compression ratio in file size

happens to the Cow2, Armadillo and Horse

examples respectively. This is due to their shape

characteristics; legs of the Horse and Cow2

examples and legs and arms of the Armadillo

example.

Table 2: Comprssion results.

Connectivity

size

Compression ratio

File name

Unco

mpres

sed

Comp

ressed

file

size %

bpt bpv

Retinal

124

1.47 98.8 1.61 3.23

Cow2

172

2.3 98.65 2.16 4.31

Smooth-

feature

222

2.35 98.94 1.52 3.05

Egea

304

4 98.68 1.89 3.77

Head

608

6 99.01 1.52 3.04

Horse

1.87

24.21 98.7 1.99 3.99

Armadillo

6.95

93.04 98.66 2.15 4.30

Vase-Lion

8.9

91.26 98.97 1.83 3.65

Average

98.92 1.8 3.57

According to the binary code associated to each

symbol, an average compression ratio of 1.8 bit per

triangle and 3.57 bit per vertex is achieved. This

result is improved over the results obtained by

EdgeBreaker (Rossignac, 1999) and its derivative

(King and Rossignac, 1999). Connectivity

information is compressed according to the file size

with an average of 98.92%. The encoding algorithm

is also barely sensitive to the seed triangle; therefore

a random face can be selected without affecting the

achieved results much. The compression ratios of

connectivity file size, in bit per triangle and in bit

per vertex are calculated according to equations 1, 2

and 3 respectively.

bits ed uncompress of No.

bits compressed of no. - bits ed uncompress of No.

(1)

trianglesof No.

bitsity connectiv compressed of No.

(2)

verticesof No.

bitsity connectiv compressed of No.

(3)

Retinal Cow2

Smooth-

feature

Egea

Head

Horse Armadillo Vase-Lion

Figure 5: Benchmark 3D models.

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114

5 CONCLUSION

In this paper, we described an enhanced lossless

single-resolution connectivity encoding algorithm

that is based on the algorithm of EdgeBreaker

(Rossignac, 1999) and uses the same data structure

of (Rossignac, Safonova and Szymczak, 2002). The

enhanced algorithm allowed the decoding procedure

to run in linear time complexity and the triangles to

be generated in the same order they were encoded. It

eliminates the computational overhead consumed by

stack operation and the recursive procedure of

traversing. The new enhancement led to the

elimination of both S and E cases and introducing

new symbols G and F which results in changing the

encoding string used to the CLRGF string. The

achieved result was encouraging as it improved the

late achieved results into 1.8t and 3.57v bits for

representing the mesh connectivity. This

enhancement can be further applied to meshes with

boundary, holes and non-manifold meshes. The

future work is to extend the algorithm to non-

triangular meshes.

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