Simple and Fast Compression of 3D Meshes

Giseok Son, Byeongwook Min, Daiyong Kim, Hyungyu Kim, and Euee S. Jang Digital Media Lab., Hanyang University

17 Haengdang-dong, Seongdong-gu, Seoul, 133-791, Korea {gidol@hanyang.ac.kr, esjang@hanyang.ac.kr}

Abstract

For transmission and storage purposes,

compression of 3D models is an indispensable component in graphics applications. Many existing compression schemes sacrificed computational complexity for compression efficiency. On the other hand, low computational complexity is the most sought functionality from the industry as of today. In this paper, we proposed a simple and fast compression method with low complexity. With a slight increase of compressed data size, the proposed method showed about four times faster in encoding time and about twice faster in decoding from the experimental results. 1. Introduction

3D models are being used in many fields, including games, commercials, animation movies, design applications, and medical imaging. Moreover, a very rapid growth in the usage of 3D models can be observed in many mobile applications. With limited computing resources, efficient compression with low complexity is a very important functionality.

For transmission and storage purposes, compression of 3D models is an indispensable component in graphics applications. Despite the fact that there have been many compression schemes researched thus far, most compression schemes sacrificed computational complexity for compression efficiency [1]-[4].

Low computational complexity, so far, is the most sought functionality, which is to produce the best possible rendering with the given time limit. Therefore, it is often the case that high-performance compression technologies are not used due to the limit of system performance or capacity. So, low quality 3D models with a small number of polygons are used frequently instead of high quality 3D models. This awkward situation is resulted mostly from the fact that the state-of-the-art compression technologies are too complex to be embedded in most graphics applications.

Research on 3D mesh compression should consider that the demand from industry is a compression scheme with very low complexity. Complexity concerns have been addressed from early researches [5][6]. In [6], compression for fast rendering was

identified as one of the important requirements. As of today, compression itself is a great burden to the rendering process due to the increased computational complexity. Therefore, the better use of compression can be found in transmission and storage purposes.

Compression of 3D models requires an initial delay in loading process of 3D assets. This is a similar phenomenon to digital TV in that it requires an initial delay to decode compressed video. Therefore, low complex 3D model compression with a minimal initial delay will remain in the wish list for 3D game industry for a while.

Early researches on 3D mesh compression concentrated more on the compression of connectivity information [1][6][8]. As a result, it has been possible to compress the connectivity information less than two bits per triangle. However, high compression efficiency on connectivity resulted in inefficiency on the compression of mesh properties such as geometry, colors, surface normals, and texture coordinates. The inefficiency is mainly caused from topological analysis process. Topological analysis is to analyze a given 3D mesh into an easily traceable representation like simple polygon, which will yield the better compression of (usually) orientable and manifold models.

Topological analysis brings two major drawbacks in 3D mesh compression: 1) change of vertex/polygon order and 2) increased complexity. Traversal (or coding) order of vertex (or polygon), after topological analysis, is usually different from the original order of vertex (or polygon). This change is critical when a compressed mesh object is to be animated or edited. The result of rendering of decompressed mesh using topological surgery may be chaotic, since the update of some field values are based on the original order of vertex (or polygon).

Topological analysis at the encoder side increases the encoder complexity. Furthermore, the decoder complexity will be increased if we wish to maintain original order in the decompressed mesh. Our early work addressed this problem and proposed the better compression algorithm for coding the order information [7]. Nevertheless, the increased decoder complexity due to coding the order information was not solved in our previous work.

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The more focus has to be given to the compression of properties such as geometry, colors, surface normals, and texture coordinates. For example, we need 32×3=96 bits for three floating point numbers to represent a vertex position in a Cartesian coordinate system. The connectivity information is very marginal in size, regardless of whether it is compressed or not. Additional process adopted for topological analysis is meaningful mostly for connectivity coding, not necessarily for coding of other properties.

In this paper, we removed topological analysis from the encoding process to investigate how we can decrease the computation complexity of existing 3D model coding technologies. As a test bed, we used MPEG-4 3D mesh coding (3DMC) tool [3][4]. We analyzed 3DMC to locate where most complex computing elements lie in decoding. This analysis led us to define a new coding scheme, which simplifies the encoding and decoding process of 3DMC.

The remainder of this paper is as follows. Section 2 describes the background of 3D mesh representation and mesh compression. Some results and analysis on MPEG-4 3D mesh coding (3DMC) are also addressed. The proposed method is explained in Section 3. In Section 4, experimental results are provided. Finally, we summarize this paper in Section 5. 2. 3D Mesh Compression

2.1. 3D Mesh Representation

A typical 3D model represented in triangular mesh contains the following information: connectivity, geometry, photometry, and animation. A triangular mesh is a set of vertices and polygons, and a polygon (or triangle) surface is formed by three vertices. How to form faces with the given number of vertices is described as connectivity.

Three coordinate values in a Cartesian coordinate system form a vertex position. When represented in 32-bit floating point number, the size of coordinate values for a vertex position would require 96 bits. To render polygon surfaces, additional information such as surface normals, colors, and texture coordinates is needed.

In the presence of photometry information, the size of a 3D mesh object without compression may be huge in transmission and storage. Animation of object is done by transformation of objects over time (e.g., translation, rotation, and shape deformation). Many standard APIs specifies the animation properties of object separated from the object description itself.

Representation of 3D models in triangular mesh is not cheap for transmission or storage. Compression is,

therefore, an imperative process for 3D mesh objects. Compression of 3D mesh objects has been extensively studied in the last ten years [1]–[8]. Among the existing technologies, Taubin’s work [1][10] became a part of MPEG-4 graphics standard as 3D mesh coding (3DMC). 2.2. MPEG 3D Mesh Coding

MPEG 3DMC basically provides compression to IndexedFaceSet (IFS) representation of virtual reality modeling language (VRML). The encoding process in MPEG 3DMC for geometry and connectivity information is depicted in Figure 1. The connectivity information needs to be encoded prior to encoding of other properties including geometry. Topological analysis called topological surgery decomposes a 3D mesh object into two dual graphs called vertex graph and triangle tree, which is a critical process in order to achieve high compression on the connectivity information. Other properties such as colors go through the similar encoding process to geometry encoding.

Connectivity

Geometry

Photometry

Topological surgery

Vertex graph coding

Triangle graph coding

Quantization DPCMArithmetic

coding

Figure 1. 3DMC encoding process

The additional process in topological analysis is found to be a time-consuming process at the encoder side. We have conducted a computing time complexity analysis on the reference 3DMC software with Intel VTune™ Performance Analyzer. With some test models, the average computing time for topological surgery was about 36.7 percent of the entire encoding time as described in Table 1.

As a result of topological analysis, the order of vertices and/or faces in the input object may be altered during compression. Without additional information, it is not possible to recover the original order of input mesh object. Our early work was to efficiently compress the additional information with minimal overhead [7]. And the work is now a part of MPEG-4 3DMC extension.

The increased overhead in decoding time for vertex/face index reordering is also given in Table 1. Topological analysis was introduced for the efficient compression of the connectivity information. From the above analysis, we can conclude that the topological analysis process is costly not only at the encoder side, but also at the decoder side.

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Table 1: Encoding and decoding complexity analysis of VO/FO overhead

Test Models Number of vertices

Encoding overhead

(%)

Decoding overhead

(%) JUMPYMAN 79 21.1 59.1 ROOK 130 23.2 53.0 KING 314 20.0 58.6 FACE 472 30.3 51.7 HEART 1072 31.1 65.3 MARTINI 1401 21.6 56.8 AH64 1972 48.4 47.3 TRASHCAN 2022 48.0 53.0 HANDS 2504 27.1 59.0 SKYSCRPR 3020 52.3 57.5 POWERLNS 4091 27.8 68.9 STRATOCASTER 4500 30.9 66.3 MODEL_ENGINE 4875 25.9 70.3 DART2 5659 72.0 53.8 E1M2 6016 84.6 42.7 HELIX 6480 33.2 61.8 PICKUP 7037 31.2 73.2 KEYBOARD 8441 35.8 72.4 KITTYHAWK 10604 33.5 79.6 AVERAGE 3720 36.7 60.9 3. Proposed Method

The encoder block diagram of the proposed method is shown as Figure 2. Compared to Figure 1, the compression of connectivity no longer precedes the compression of geometry and photometry. This is a clear difference from MPEG-4 3DMC. All the 3D mesh data except connectivity are encoded by three stages of compression: quantization, DPCM, and entropy coding. Compression of connectivity should be encoded losslessly to preserve the topology of the model; therefore, quantization is not used.

The quantization process of geometry and photometry data is the same as MPEG-4 3DMC. The quantized value (Q(x)) of an input value (x) can be obtained as follows:

( ) 5.012minmax

min)( +−×−

−= txxQ (1)

where min and max are the minimum and maximum values of input values. The symbol (t) denotes the quantization parameter.

Connectivity

Geometry

PhotometryQuantization DPCM

Entropy coding

DPCM Entropy coding

Figure 2. The proposed 3D mesh encoding process

After quantization, a simple first-order differential

prediction is applied as follows:

0)()1()()(

=−−=

ieiXiXie

)0()0(

=>

ii

(2)

where e(i) is the differential value at i-th position and X(i) the value at i-th position. This simple prediction is even simpler than that used in 3DMC, where 3DMC utilized the second order prediction called parallelogram prediction.

In the case of connectivity information, the differential value, e(i), was calculated to reduce the sum of the absolute differences of a given triangle. For instance, assume that the current triangle is defined by the 7th, the 4th, and the 3rd vertices and the previous triangle is defined by the 2nd, the 3rd, and the 4th vertices. Then, the differential values of the current triangle can be computed as (2, 0, 3) instead of (5, 1, 1), which can be possible by rotating the order of vertices in the current triangle(e.g., 4-2=2, 3-3=0, 7-4=3). This rotation process reduces the size of differential values, which yields the better compression.

For entropy coding of the differential values after prediction, we have tested two entropy coding methods: 1) arithmetic coding and 2) bit-precision method. Arithmetic coding is also used in MPEG-4 3DMC for its superiority in compression efficiency. Bit-precision method, which was tested for image and video coding in our early work [12], is newly tried in this paper for the compression of 3D mesh. 3.1. Binary Arithmetic Coding

We applied binary arithmetic coding for each symbol, where this symbol is a differential value after prediction. The binary arithmetic coding method used in the proposed paper is conceptually identical to that used in MPEG-4 3DMC. For encoding each symbol, the symbol is divided into two parts: the number of zero run (ZR) from MSB to LSB and non-zero residual bits (NZB). If a symbol is represented in 10-bit representation, ZR can be described with four bits (24 > 10). And the size of NZB may vary from zero (when the symbol is zero) to 10 (when the symbol is greater

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than 511). The first bit of NZB is skipped from encoding, since the first bit of NZB is always one. Once ZR and NZB are formed, binary arithmetic coding of ZR and NZB is performed bit by bit. 3.2. Bit Precision Method

The main idea of bit precision (BP) method is to define the bit length of symbols. The size of a symbol in BP representation is a multiple of BP length (BPL) bits. A symbol in BP representation can be determined by a modulo operation with a pair of quotient and remainder as shown in Equation 3:

}22,,0{,)12( −=+−×= BPLii

BPLii VVNPDS … (3)

where DSi denotes the i-th symbol, NPi the number of prefixes (or quotient), and Vi the remaining value (or remainder) for the i-th symbol. To represent a symbol value in BP, the value of 2BPL-1 (all ‘1’s in BPL bits) is inserted NPi times and the remainder Vi follows.

Examples are provided in Table 2. In the table, the symbol ‘5’ can be represented as ‘11 10’ when BPL = 2 and represented as ‘101’ when BPL = 3. When BPL = 2, the quotient ‘11’ (=3) is preceded before the remainder ‘10’ (=2). No preceding quotients are necessary in the case of BPL = 3 because the symbol value is less than the dividend value ‘111’ (= 23-1). The more technical details can be found in [12].

One good feature of BP method is the simplicity of the decoding process as shown from Equation 3. To decode a symbol, only one addition and one multiplication is required. It was shown that BP is faster than Huffman coding several times, since BP does not require any search in finding the right symbol from the table [12]. 4. Experimental Results

We designed and conducted the experiments to evaluate the compression efficiency and the decoding complexity of the proposed method. MPEG-4 3DMC extension technology [13] is used for comparison.

We used 50 test models from the MPEG 3D graphics compression (3DGC) FTP site [11] and the list can be found in Table 3. The number of vertices in the test models varies from a few hundreds to more than 10,000. The machine used for the experiment is equipped with Windows XP, Intel Core 2 CPU 1.86GHz, and 2GB RAM. Table 2: Example of BP representation

Symbol BPL = 2 BPL = 3

0 00 000 1 01 001 2 10 010 3 11 00 011 4 11 01 100 5 11 10 101 6 11 11 00 110 7 11 11 01 111 000 … … …

All the test models are quantized by 10 bits per

sample regardless of compression methods, which makes the visual quality of compressed models identical for different codecs.

For the evaluation of the proposed method, we tested three different methods:

3DMC with VO/FO coding (3DMC) – in this mode, 3DMC preserves the original order of vertices and faces.

Proposed method with binary arithmetic coding (BAC)

Proposed method with bit precision (BP) Compression efficiency with the proposed methods

is evaluated with 3DMC in Figure 3 and Table 4. In Figure 3, the compressed file size using 3DMC is set to zero overhead. From the figure, we can see that the proposed methods (both BAC and BP) are not as efficient as 3DMC. Out of 50 test models, the proposed methods performed better than 3DMC in 18 models. Overall, the additional overhead using the proposed methods are 1 percent and 38 percent for BAC and BP, respectively. Therefore, BAC method performed equally with 3DMC, whereas BP method is slightly worse than BAC in compression efficiency.

In order to estimate the computational complexity of the proposed methods, we measured the encoding time and the decoding time compared with 3DMC. The encoding time performance of the proposed methods with 50 test models is depicted in Figure 4. From the figure, both BAC and BP outperformed 3DMC in all models. From Table 5, it is clearly shown that both BAC and BP outperformed 3DMC in encoding time about four times, respectively. Compared with the size increase in compression, the improved computational complexity using the proposed methods is phenomenal. This proves that topological surgery is one major source that causes high computational complexity. Table 3: Test models for experiments

Test Model Number of

Vertices

Number of Faces

Size in VRML

(B) JUMPYMAN 79 150 4810 ROOK 130 256 8369 PADLOCK 193 350 11106

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KNIGHT 214 419 14463 KING 314 624 21166 LINK 422 706 21347 TWISTEDCOLUMN 440 836 27472 OUTLET 444 594 23665 FACE 472 896 28667 HEART 1072 1712 52744 SKATE_TRUCK 1250 1652 69044 MARTINI 1401 2750 100352 AH64 1972 2709 111282 TRASHCAN 2022 1808 93026 TEMPLE 2048 3928 124639 FLWCOUCH 2055 3243 121503 WALKMAN 2106 4136 136721 COFFEE_TABLE 2229 4422 146563 ROSE 2346 3576 134655 HANDS 2504 4842 162072 WEATHERVANE 2618 5224 176533 F16 2696 4428 164032 VIOLIN 2880 1508 108175 AIRCAR 2900 4629 180190 GALLEON 2961 4698 177136 SKYSCRPR 3020 3692 156998 COW 3078 5804 204420 FLWCHAIR 3718 4806 203192 SGIMONITOR 3854 3592 193228 AL 3950 7124 257884 POWERLNS 4091 8964 291318 MICROSCOPE 4181 8240 279249 GUNSTAR 4468 8152 293479 STRATOCASTER 4500 8660 302158 BICYCLE 4545 5049 220177 FIN_ACT 4655 6532 251514 MODEL_ENGINE 4875 9344 315242 WOODMAN 5227 10010 345993 JAPAN 5228 5047 235978 DART2 5659 6014 306108 E1M2 6016 4006 184450 HELIX 6480 6400 315418 PICKUP 7037 13402 480496 KEYBOARD 8441 16697 576002 KITTYHAWK 10605 19188 717023 KITTYHAWK2 10605 19188 717023 JEEP3 11039 21450 759591 ROUND_TABLE 11868 20594 755571 CAMARO3 12784 24863 895006 POWER_DRILL 17756 32652 1204151

The decoding time performance of the proposed

methods with 50 test models is depicted in Figure 5. The overall performance of BAC and BP is better than 3DMC. The average decoding times with BAC and BP are roughly two and three times faster than that of 3DMC as shown in Table 6.

-40

-20

0

20

40

60

80

100

120

4810 28667 124639 177136 251514 315418 895006

Size of original VRML file (byte)

Siz

e C

om

parison (

%)

BAC

BP

3DMC

Figure 3. Evaluation of Compression Efficiency

Table 4: Average overhead in compression efficiency

Method 3DMC BAC BP Overhead 0 % 1 % 38 %

0

10

20

30

40

50

60

70

80

90

100

4810 28667 124639 177136 251514 315418 895006

Size of original VRML file (byte)

Encodin

g t

ime

Com

pariso

n (

%)

BAC

BP

3DMC

Figure 4. Evaluation of the encoding time

Table 5: Average encoding time (in seconds)

Method 3DMC BAC BP Encoding time 0.174 0.051 0.041

Ratio 100% 29.20% 23.72%

0

10

20

30

40

50

60

70

80

90

100

4810 28667 124639 177136 251514 315418 895006

Size of original VRML file (byte)

Decodin

g tim

e C

om

parison (

%)

BAC

BP

3DMC

Figure 5. Evaluation of the decoding time

Table 6: Average decoding time (in seconds)

Method 3DMC BAC BP Decoding time 0.153 0.069 0.049

Ratio 100% 44.83% 31.94%

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From the given experimental results, the proposed

method with BAC showed equally good performance in compression compared with 3DMC. The proposed method with BP was inferior to 3DMC in compression efficiency. However, the superior performance in encoding and decoding complexity with the proposed methods seems to overcome the slight overhead in compression efficiency. 5. Conclusion

We proposed simple, fast and efficient methods to compress 3D mesh data. In comparison with MPEG-4 3DMC, the proposed methods preserve the vertex and face order of the input 3D mesh. The simplicity of the proposed methods mainly comes from the first order prediction and simple entropy coding method. From the experimental results, the proposed methods with the slight overhead outperformed 3DMC in both encoding and decoding complexity. 6. References

1. G. Taubin and J. Rossignac, “Geometric Compression through Topological Surgery,” ACM Transactions on Graphics, 1998. 2. M. Isenburg and J.Snoeyink, “Mesh Collapse Compression,” In Proceedings of SIBGRAPI’99 - 12th Brazilian Symposium on Computer Graphics and Image Processing, 1999. 3. Fernando Pereira and Touradj Ebrahimi, The MPEG-4 Book, Prentice Hall, 2002. 4. Aaron E. Walsh and Mikael Bourges-Sevenier, MPEG-4 Jump-Start, Prentice Hall, 2002. 5. R. Bar-Yehuda and C. Gotsman, “Time/space tradeoffs for polygon mesh rendering,” ACM Transactions on Graphics, 15(2):141–152, April 1996. 6. S. Gumhold and W. Strasser, “Real time compression of triangle mesh connectivity,” in SIGGRAPH 98, pp. 133 – 140, 1998. 7. E.-Y. Chang, D. Kim, B. Min, S. Lee, N. Hur, S. I. Lee, E. S. Jang, “Vertex and face permutation order compression for efficient animation support,” in Proceedings of SPIE -- Volume 6061, Internet Imaging VII, Jan. 16, 2006. 8. M. Deering, “Geometry Compression,” in Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, pp. 13 – 20, 1995. 9. D. Hearn and M. Pauline Baker, Computer Graphics with OpenGL, 3rd Ed., Pearson Prentice Hall, 2004. 10. G. Taubin and J. Rossignac, “Geometry coding and VRML,” IBM Research Report, RC-20925, 1997.

11. The MPEG 3D Graphics Compression (3DGC) FTP site, http://www.gti.ssr.upm.es./~mpeg/3dgc/3Dmodels/. 12. Jong Woo Won, Hyun Soo Ahn, Wook Joong Kim, and Euee S. Jang, “Bit-Precision Method for Low Complex Lossless Image Coding”, ICIC 2005. 13. MPEG 3DMC extension reference software, http://mpeg.nist.gov/cvsweb/

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