EFFICIENT FACE-BASED NON-SPLIT CONNECTIVITY

COMPRESSION FOR QUAD AND TRIANGLE-QUAD 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: Non-triangular mesh compression, Non-split encoding, Connectivity compression.

Abstract: In this paper we present an efficient face-based connectivity coding technique for the special class of

quadrilateral and the hybrid triangular-quadrilateral meshes. This work extends the main ideas of non-split

encoding presented by the first contribution of the authors (Khattab, Abd El-Latif, Abdel Wahab and Tolba,

2007) for triangle meshes and improves over the compression results provided so far for existing face-based

connectivity compression techniques. It achieves an average compression ratio of 2.17 bits per quad and per

vertex for simple quadrilateral meshes and bit rates of 1.84 bits per polygon and 1.85 bits per vertex for the

simple hybrid triangle-quad meshes.

1 INTRODUCTION

While a picture is often said to be worth a thousand

words, a 3D model could be said to be worth a

thousand pictures. Polygonal meshes are used most

often as surface representation because of their wide

spread support in many file formats and graphic

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. Therefore,

efficient 3D mesh compression algorithms have

been in high demand in the past few years to reduce

the storage room needed for large, detailed 3D

models and to consequently decrease transmission

time over a network.

Much of the work done in the area of single-

rate mesh connectivity compression has focused on

triangle meshes only (Deering, 1995, Chow, 1997,

Gumhold and Strasser 1998, Rossignac, 1999,

Touma and Gotsman, 1998 and Alliez and Desbrun

2001). However there are a significant number of

non-triangular meshes in use. These models contain

a surprisingly small percentage of triangles likewise;

few triangles are generated by tessellation routines

in existing modelling software. The dominant

element in these meshes is the quadrilateral, but

pentagons, hexagons and higher degree faces are

also common (Alliez and Gotsman, 2005). The

extension of the previously mentioned algorithms to

deal with non-triangular meshes is not obvious.

Hence, in practice, non-triangular meshes are coded

by triangulating them, coding the results using one

of the methods of coding triangular meshes and

storing additional information describing the extra

edges introduced during the triangulation stage.

These edges are discarded after decoding to restore

the original polygonal model. This means that the

code of a non-triangular mesh might be larger than

that of the triangular version, instead of being

shorter, as less connectivity information is present.

In addition, it is beneficial for storage purposes

to keep a mesh in its native polygonal representation

than to triangulate it. This is because most meshes

have associated properties such as normal, colour, or

texture information that account for a large portion

of the storage cost. Triangulating a polygon mesh

not only adds an extra processing step, but also

increases the number of faces and corners and

replicates their associated properties.

1.1 Related Work

To address the problem of compressing the

connectivity of non-triangular meshes, several

algorithms have been proposed to encode polygonal

meshes directly without pre-triangulation. In this

31

R. Khattab D., M. Abd El-Latif Y., S. Abdel Wahab M. and F. Tolba M. (2008).

EFFICIENT FACE-BASED NON-SPLIT CONNECTIVITY COMPRESSION FOR QUAD AND TRIANGLE-QUAD MESHES.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 31-38

DOI: 10.5220/0001093900310038

Copyright

c

SciTePress

section we focus on recent coding methods for

manifold meshes that grow a region over the mesh

and incrementally encode the mesh elements and

their incidence relations to the growing region. The

methods are categorized as face-based, edge-based

and vertex-based methods according to the type of

mesh element playing the dominant role in the

compression scheme.

King et al. (King, Rossignac and Szymczak,

1999) first proposed a connectivity coding algorithm

for quad or mixed triangle-quad meshes, by

generalizing the EdgeBreaker algorithm (Rossignac,

1999 and Rossignac and Szymczak, 1999) which is

one of the triangle conquest methods. This method

implicitly triangulates each quadrilateral to two

triangles and uses sequences of the five basic

EdgeBreaker labels (CLERS) to code the different

possibilities which then arise. This introduced a

compressed format based on entropy coding with a

worst case of 2.67 b/v. Isenburg and Snoeyink

(Isenburg and Snoeyink, 2000) proposed an edge-

based technique called the Face-Fixer. They

achieved a connectivity coding cost of 5 b/v for

simple triangular meshes and 4 b/v for simple

quadrilateral meshes.

Isenburg (Isenburg, 2002) and Khodakovsky et

al. (Khodakovsky, Alliez, Desbrun and Schröder,

2002) independently proposed similar vertex-based

algorithms to encode the connectivity of a manifold

polygonal mesh. Their algorithms are extensions of

the valence-driven approaches (Touma and

Gotsman, 1998 and Alliez and Desbrun 2001).

Isenburg’s algorithm provides slightly better

compression performance than Khodakovsky et

al.’s. They both produce better results compared to

the Face-Fixer. Another vertex-based technique is

the Angle-Analyzer (Lee, Alliez and Desbrun,

2002). The algorithm focused on the hybrid triangle-

quad mesh coding. On average, the algorithm yields

40% and 20% better compression ratios for

connectivity and geometry data than the state-of-the-

art triangular mesh coder given in (Touma and

Gotsman, 1998).

As a face-based technique, Kronrod and

Gotsman (Kronrod and Gotsman, 2000) introduced a

general and direct technique for coding the

connectivity of any non-triangular mesh with an

upper bound on the resulting code length. The

algorithm generalizes EdgeBreaker technique

(Rossignac, 1999) for non-triangular meshes. They

proved that for quadrilateral meshes a worst case of

3.5 bits per quad and per vertex can be achieved and

a worst case of 4 bits per polygon can be achieved

for quad meshes with few triangles.

Two enhancements over the bit rates of

Kronrod and Gotsman algorithm were introduced.

The enhanced technique of (Mukhopadhyay and

Jing, 2003) proved with complex calculations that

the bit rates can be reduced to less than 3 b/v. and by

equivalence to the work done in (King, Rossignac

and Szymczak, 1999) can be reduced to 2.67 b/v. the

other enhancement (Kosicki and Mukhopadhyay,

2004) improved the results to 2.4 b/v using

arithmetic coding.

1.2 Overview

In this paper we introduce an efficient face-based

connectivity coding technique that extends the ideas

of non-split coding presented by the authors’ first

contribution (Khattab, Abd El-Latif, Abdel Wahab

and Tolba, 2007) for triangle meshes. The

compression results achieved is the best compared to

the state-of-the-art face-based techniques for

compressing non-triangular meshes. These results

are compared to those introduced by Kronrod and

Gotsman (Kronrod and Gotsman, 2000) and their

enhancements of (Mukhopadhyay and Jing, 2003

and Kosicki and Mukhopadhyay, 2004).

The remainder of this paper is organized as

follows: section 2 explains the encoding scheme of

Kronrod and Gotsman. Section 3 illustrates the

proposed technique for applying the non-split

encoding to non-triangular meshes. The results and

discussions are presented in section 4 and we

conclude in section 5.

2 KRONROD-GOTSMAN

SCHEME

The Kronrod-Gotsman scheme (Kronrod and

Gotsman, 2000) generalizes the CLRES labelling

scheme of EdgeBreaker (Rossignac, 1999) to non-

triangular meshes. Their main observation is that as

we traverse a mesh in depth-first order, the

interaction of each polygon with the rest of the mesh

can be enumerated in a finite number of ways. For

example, in a quad mesh each quad interacts with

the rest of the mesh in one of thirteen types labelled

from Q

1

to Q

13

(figure 1) and hence this interaction

can be coded in a unique manner. It is easy to

enumerate all these interaction types if we note that

each of the remaining three edges of the current

quad either belongs to the mesh boundary or does

not, and so also for the remaining two vertices.

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

32

Figure 1: Interaction of a quad with the mesh. Arrows

indicate the current gate.

The compression process traverses the mesh in

depth- first order starting with a quad. It then records

a code of the interaction type of the current quad

with the boundary. After the quad is removed from

the boundary, the boundary is updated and the

procedure repeats. If the interaction type of the

removed quad contains gaps (i.e., the quad has at

least one touching vertex with the boundary) the

boundary will split into two or more boundaries. The

boundary is not stored explicitly; otherwise it is

maintained implicitly as a stack of gate edges. The

procedure terminates when the entire mesh is

reduced to nothing. The output of the encoder is

sufficient to reconstruct the connectivity data of the

mesh. The time complexity of the encoding

procedure is linear in the size of the mesh.

The decoding algorithm is inspired by the linear

time Spirale Reversi decoder (Isenburg and

Snoeyink, 2001). The algorithm is simple however;

it processes the mesh in the reverse order of the

encoding process. It means that mesh polygons will

be reconstructed in an order opposite to which they

were encoded. In the encoding process each

interaction type defines exactly how this polygon is

connected to all generated borders. Hence, if the

decoder knows the connectivity in the interior of the

boundary after the encoding step and the nature of

the interaction between the encoded polygon and the

active boundary, it is easy to reconstruct the

connectivity of the interior of the active boundary as

it was before the encoding step.

The authors provided explicit codes for the

interaction types of pure quad mesh and a quad mesh

with minority of triangles. For the case of a mesh

containing only quads it admits thirteen different

interaction types, which requires (log

2

13) = 4

bits/quad when coded using a fixed-length code. But

as not all the interactions occur with equal

frequencies they produced a more efficient variable-

length prefix code which reduced the total code

length to less than 3.5 bits/quad. For the case of

hybrid quad and triangle mesh, it admits eighteen

different interaction types, which are the summation

of the thirteen interaction types of pure quad meshes

and the five interaction types of triangular meshes

generated by EdgeBreaker (Rossignac, 1999). Again

the naïve coding will require (log

2

18) = 5 bits/quad

but using the same variable-length prefix code it was

reduced to less than 4 bits/poly.

3 NON-SPLIT ENCODING

The new encoding scheme presented in this paper

adapts the main idea of not splitting the boundary

while traversing the mesh. This idea was first

presented by the authors in their first contribution by

introducing an enhanced encoding technique for

triangle meshes (Khattab, Abd El-Latif, Abdel

Wahab and Tolba, 2007). This technique had

updated the CLERS string generated by

EdgeBreaker (Rossignac, 1999) to the CLRGF

string. This was done by the elimination of S

interaction type that causes the active boundary to be

split and so the elimination of its delimiter E label.

These two labels were replaced by another two

labels G and F for moving on the active boundary to

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

choice was decided according to which boundary

length is shorter. The next subsections explain how

this idea can be extended for encoding quadrilateral

and hybrid triangular-quadrilateral meshes.

3.1 Encoding Quadrilateral Meshes

For applying this technique on quad meshes, the

algorithm tends to eliminate the six interaction types

EFFICIENT FACE-BASED NON-SPLIT CONNECTIVITY COMPRESSION FOR QUAD AND TRIANGLE-QUAD

MESHES

33

(Q

3

Q

7

Q

8

Q

9

Q

10

Q

11

) that cause the boundary to be

split from the thirteen interaction types defined by

Kronrod and Gotsman (figure 1). This ensures that

the active boundary will never split. The interaction

type of Q

1

which defines the state of the last reached

quad for every generated boundary after splitting

will never occur except only once at the end of mesh

traversal and so it can be replaced by the label Q

4

.

By this way the enhanced technique preserves only

six interaction types from the whole group of

thirteen interaction types defined by Kronrod and

Gotsman. These six interaction types are reordered

again from Q

1

to Q

6

as can be seen in figure 2.

Q

1

Q

2

Q

3

Q

4

Q

5

Q

6

Figure 2: Remaining six interaction types of quad

interaction with a mesh. Thick black edges show the

current active gates. Thick red edges show the updated

active gates for each interaction type.

The eliminated labels are replaced with G and F

labels which indicate moving on the active boundary

wherever a split case occurs. The encoding string is

now changed to Q

1

Q

2

Q

3

Q

4

Q

5

Q

6

GF. The elimination

of the split casess from encoding string saves for the

encoding technique many advantages. The algorithm

needs only to maintain one circular linked list for

active boundary during compression and

decompression. It also has eliminated both the

recursive overhead and computational overhead

needed by the stack to keep list of new active gates

generated after splitting. The decompression

processes is done in only one forward pass of

traversing the encoding string and reconstructs the

mesh quads in the same order they were encoded.

The pseudo-code of the proposed encoding

procedure is provided in the frame below. After

processing of each quad element, the active gate is

updated to the first edge incident to an unprocessed

element in the counter clock wise direction from the

current active gate. This can be observed by the red

edges shown in figure 2. This approach of updating

the active gate allows the traversal to be done in a

spiralling depth first clock wise order around the

mesh faces.

Input:

Geometry of the mesh as an array of floats to

specify positions.

Connectivity of the mesh as an array of integers

containing indices into the position array to

specify the faces.

Output:

Q

1

Q

2

Q

3

Q

4

Q

5

Q

6

GF string that contains one

label per quad except for the first one.

Procedure compress_quad()

Assign flag to true

While (flag)

If both tip vertices are not visited Then

Append code of Q

1

to the encoding string,

Add both tip vertices to the list of vertices,

Mark quad and tip vertices as visited,

Update active gate.

Else If both tip vertices lies on the boundary

to the left of current active gate Then

Append code of Q

2

to the encoding string,

Mark quad as visited,

Update active gate.

Else If tip vertices lies on the boundary to

the right of current active gate Then

Append code of Q

3

to the encoding string,

Mark quad as visited,

Update active gate.

Else If first tip vertex lies on the boundary

to the left and second vertex lies to the right

of the active gate Then

Append code of Q

4

to the encoding string,

Mark quad as visited,

Update active gate.

Else If first tip vertex lies on the boundary

to the left of active gate and second tip

vertex is not visited Then

Append code of Q

5

to the encoding string,

Mark quad and second tip vertex as

visited,

Add second tip vertex to the list of

vertices,

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

34

Update active gate.

Else If first tip vertex is not visited and

second tip vertex lies on the boundary to the

right of active gate Then

Append code of Q

6

to the encoding string,

Mark quad and first tip vertex as visited,

Add first tip vertex to the list of vertices,

Update active gate.

Else

Calculate right and left boundary lengths

If the right boundary is shorter in length

Then

Append encoding of G to the

encoding string,

Update active gate to the next right

edge on the boundary.

Else If the left boundary is shorter in

length Then

Append encoding of F to the

encoding string,

Update active gate to the next left

edge on the boundary

If all mesh quads are processed then

Assign flag to false

3.2 Encoding Hybrid

Triangular-Quadrilateral Meshes

The only requirement needed in order to extend the

technique for triangle-quad meshes is to generalize

the data structure used by the encoding procedure to

be able to store elements with arbitrary number of

edges. The procedure checks for the current element

type either it is a triangle or a quad. This checking

distinct between two different subroutines

responsible for checking the different cases of each

type of elements and assigns the appropriate code

for each element.

The generated encoding string will be a mixture

of the six interaction types (Q

1

Q

2

Q

3

Q

4

Q

5

Q

6

) for

encoding quad elements and the three interaction

types of (CRL) for encoding triangle elements. By

addition of the two labels of F and G, the generated

encoding string becomes Q

1

Q

2

Q

3

Q

4

Q

5

Q

6

CLRGF.

The general layout of the procedure is explained in

the frame below.

Input:

Geometry of the mesh as an array of floats to

specify positions

Connectivity of the mesh as an array of

integers containing indices into the position

array to specify the faces.

Output:

Q

1

Q

2

Q

3

Q

4

Q

5

Q

6

CLRGF string that contains

one label per polygon except for the first one.

Procedure Compress()

Assign flag to true

While (flag)

If the current processed polygon is a triangle

then

Call procedure compress_triangle()

Else if the current processed polygon is a quad

then

Call procedure compress_quad()

If all mesh polygons are processed then

Assign flag to false

4 RESULTS AND DISCUSSION

In this section the compression ratio achieved using

the adapted non-split encoding technique is

evaluated. Sample test cases of pure quad meshes

(figure 3) and of quad meshes with few triangles

(figure 4) were collected for this purpose and

presented in Table 1. The meshes were selected to

have a variety in size and shape except they all share

the common characteristics of being manifold and

simple without boundary, holes or handles.

Table 1: Used benchmark 3D models.

Model

name

File size

No of

Quads

No of

triangles

No of

vertices

Cow 1.35 MB 25,878 - 25,880

Bumpy

sphere

1.41 MB 34,332 - 34,334

Egea 2.43 MB 49,596 - 49,598

Fandisk 3.55 MB 71,892 - 71,894

Head 4.94 MB 98,232 - 98,234

Horse 6.58 MB 119,09 - 119,096

Dente 6.95 MB 131,67 - 131,672

Eros 8.07 MB 153,91 - 153,914

Bimba 849 KB 15,532 238 15,653

Dragon 7.48 MB 126,77 1,540 127,571

EFFICIENT FACE-BASED NON-SPLIT CONNECTIVITY COMPRESSION FOR QUAD AND TRIANGLE-QUAD

MESHES

35

Cow Bumpy sphere

Egea Fandisk

Head Horse

Dente Eros

Figure 3: Benchmark 3D models of Quad meshes.

Bimba Dragon

Figure 4: Benchmark 3D models of Quad meshes with

few triangles.

In order for the compression ratio to be

calculated a binary code has to be given to each of

the encoding labels used for the encoding process.

Following the fact that not all the codes occur with

equal frequencies (Kronrod and Gotsman, 2000), a

study over the average percentage each label

consumes from the Encoding file was generated

(Khattab, Abd El-Latif, Abdel Wahab and Tolba,

2007) to the entire models used (figure 5).

Figure 5 shows the frequency of occurrence of

each encoding label in all meshes and the average

percentage of each label over the whole group of

meshes. The encoding labels are arranged in an

ascending order according to their average

percentage of occurrence. It is apparent from figure

5a that the largest percentages of quad interaction

types are Q

6

, Q

1

and Q

3

respectively. This result

reflects the behaviour of the encoding technique that

emphasis on traversing the mesh in a spiralling clock

wise order of polygons. The same result is obtained

with triangle-quad meshes (figure 5b) again the label

R has the largest percentage among the set of

triangle labels (C, R and L).

Based on this study a variable length binary

code is given for the labels (Q

6

: 0, Q

1

: 10, Q

3

: 110,

F: 1110, Q

4

: 11110, Q

5

: 111110, Q

2

: 1111110, G:

1111111) of quad meshes and (Q

6

: 0, Q

1

: 10, Q

3

:

110, F: 1110, Q

4

: 11110, R: 111110, Q

2

: 1111110,

Q

5

: 11111110, G: 1111111110, C: 1111111110, L:

1111111111) of tri-quad meshes. This code is based

on Huffman coding (Huffman, 1952) and leads to

the optimal compression ratio that can be achieved.

Table 2: Compression results of quad meshes.

Connectivity size

Compression

ratio

File

name

Unencoded Encoded bpq bpv

Head 2.49 MB 18.44 KB 1.53 1.53

Fandisk 1.81 KB 13.92 KB 1.58 1.58

Bumpy

sphere

866 KB 7.3 KB 1.74 1.74

Dente 3.51 MB 30 KB 1.86 1.86

Egea 1.23 KB 11.58 KB 1.9 1.9

Eros 4.16 MB 50.94 KB 2.7 2.7

Horse 3.13 MB 42.93 KB 2.9 2.9

Cow 643 KB 9.8 KB 3.1 3.1

Average 2.17 2.17

Table 3: Compression results of triangle-quad meshes.

Connectivity size

Compression

ratio

File

name

Unencoded Encoded bpp bpv

Bimba 386 KB 511 KB 1.76 1.77

Dragon 3.35 MB 3.741 KB 1.92 1.93

Average 1.84 1.85

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

36

Average

Percentage

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

Head Fandisk Bumpy

sphere

Dente Egea Eros Horse Cow

Per cent sge %

Mesh Name

G

Q2

Q5

Q4

F

Q3

Q1

Q6

(a)

0.26%

1.00%

1.21%

4.9%

5.99%

14.6%

20.36%

51.81%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

Bi mba Dr ago n

Per cent age %

Mesh Name

L

C

G

Q5

Q2

R

Q4

F

Q3

Q1

Q6

(b)

0.06%

0.36%

0.38%

0.41%

0.67%

0.86%

4.27%

4.63%

6.93%

12.15%

69.28%

Figure 5: Frequency percentage each file consumes from the encoding data file.

Table 2 and 3 list the results of the bit rates of

compression ratio per polygon (bpp) and per vertex

(bpv) for quad meshes and the hybrid triangle-quad

meshes respectively. The results in both tables are

arranged in ascending order according to the

compression ratio which varies from the smallest bit

rate of 1.53 (Head example) to the largest bit rate of

3.1 (Cow example) for quad meshes.

This variation in results can be estimated by

referring to figure 5a which shows the degradation

in the percentage of Q

6

labels whose bit code length

is the least. This degradation is consumed in other

labels having larger bit code lengths. Another

parameter that affects the compression ratio is the

shape characteristics of the models (Khattab, Abd

El-Latif, Abdel Wahab and Tolba, 2007). The

largest percentage of split cases during mesh

traversal, the largest percentage of F and G labels

added to the encoding file which finally results in

much increase of bit rates per polygon for these

meshes.

According to the binary code associated to each

label, an average compression ratio of 2.17 bit per

quad and per vertex is achieved for quad meshes and

1.84 bit per polygon and 1.85 bit per vertex for

triangle-quad meshes. This result is more efficient

EFFICIENT FACE-BASED NON-SPLIT CONNECTIVITY COMPRESSION FOR QUAD AND TRIANGLE-QUAD

MESHES

37

compared to the late results obtained by Kronrod

and Gotsman (Kronrod and Gotsman, 2000) and its

enhancements of (Mukhopadhyay and Jing, 2003

and Kosicki and Mukhopadhyay, 2004) directly and

without applying any complex or arithmetic coding.

5 CONCLUSIONS

In this paper, we present an efficient face-based

connectivity encoding technique for compressing

non-triangular meshes. The presented technique

extends the previous work done by the authors in

their first contribution (Khattab, Abd El-Latif, Abdel

Wahab and Tolba, 2007) for compressing triangular

meshes to the special class of pure quad and hybrid

triangle-quad meshes. The presented technique

reduced the interaction types introduced by Kronrod

and Gotsman (Kronrod and Gotsman, 2000) from

thirteen to six by elimination of the interaction types

that causes the boundary to be split. This approach

saves for the encoding technique its simplicity and

efficiency. This reduction of interaction types

improved the compression ratio over the state-of-

the-art face-based techniques for compressing non-

triangular meshes. It is believed that applying

entropy or arithmetic coding to the achieved results

will lead to further increase in compression ratio.

The future work is to apply this efficient non-split

encoding technique for meshes with arbitrary

topology such as boundary and holes. The work in

this direction is under progress and the initial results

are promising.

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