Geometric signal compression

Sept. 2004, 1/oi.19, No.5, pp.596-606 J. Comput. Sci. &: Tectmol.

Geometric Signal Compression

Kun Zhou, Hu-Jun Bao, Jiao-Ying Shi, and Qun-Sheng Peng

State Key Lab of CAD& CG, Zhejian 9 University, Hangzhou 3100~7, P.R. China

E-maih {kzhou, bao, jyshi, peng}@cad.zju.edu.cn

Revised June 7, 2004.

Abs t rac t Compression of mesh attributes becomes a challenging problem due to the great need for efficient storage and fast transmission. This paper presents a novel geometric signal compression framework for all mesh attributes, including position coordinates, normal, color, texture, etc. Within this framework, mesh attributes are regarded as geometric signals defined on mesh surfaces. A planar parameterization algorithm is first proposed to map 3D meshes to 2D parametric meshes. Geometric signals are then transformed into 2D signals, which are sampled into 2D regular signals using an adaptive sampling method. The JPEG2000 standard for still image compression is employed to effectively encode these regular signals into compact bit-streams with high rate/distortion ratios. Experimental results demonstrate the great application potentials of this framework.

Keywords mesh, geometry compression, parameterization, JPEG2000, wavelet transform

1 I n t r o d u c t i o n

With the development of 3D scanning tech- niques, to accurately acquire finely detailed geometry models is no longer a hard task. Such large models are awkward for storage, display and transmission. Therefore, it is desirable to compress meshes to reduce their storage, facilitate the rendering procedure, and save the transmission time, as what we have done for other types of digital media such as sounds, images and videos.

Since meshes are usually curved, irregular and without intrinsic parameterizations, mesh compression is more challenging than image compression, which has a long history and full-developed indus-

t ry standards. Concretely, a mesh consists of two components: connectivity and geometry information. The geometry information includes position coordinates, color, normal, texture, etc. While many geometry compression algorithms have been proposed to encode the connectivity and position coordinates, it still lacks a unified and efficient scheme to compress all kinds of mesh attributes. Although 3D geometry compression based on topological surgery has been included in the MPEG4 standard, this scheme does not look at 3D compression as a rate/distortion problem, which plays an important role in image compression standards.

Our goal in this paper is to design a unified compression framework for all mesh attributes. Within

(a) (b) (c) (d)

Fig.1. The synthesized color compression of the Bunny model. (a) Original model (b), (c) and (d) Compressed results with bit rates 16.11, 5.09 and 1.24 bits/vertex, respectively.

*Regular Paper Supported by the National Natural Science Foundation of China under Grant No.60333010, the National Natural

Science Foundation of China for Innovative Research Groups under Grant N0.60021201, the National Natural Science Foundation of China under Grant No.60133020, and the National Grand Fundamental Research 97"3 Program of China under Grant No.2002CB312102.

Kun Zhou ef al.: Geornetrfc Signal Compression 597

this framework, mesh attributes are transformed into 2D regularly sampled signals. The JPEG2000 standard for still image compression is employed to achieve high rate/distortion ratios. In order to describe our geometric signal compression framework, we first briefly review the previous work in this domain.

1.1 R e l a t e d W o r k

Existing geometry compression algorithms can be classified into two categories. One is based on vertex traversal and prediction, and the other is based on geometric signal processing.

Vertex Traversat and Prediction Scheme. This kind of algorithms has been described both for the single rate and progressive setting (for a detailed overview see [1]). Most of these algorithms concentrate on encoding the connectivity information, while the vertex positions are compressed by performing a quantization followed by a predictive coding induced by the traversal order of the connectivity encoding[ 2-9] . In the single rate schemes, the mesh connectivity can be encoded at around 2-6 bits/vertex, and the TG algorithm [9] is recognized as the best single rate encoder. Since the single rate scheme is not suitable for progressive transmission over network, many progressive compression algorithms have been proposed to address this issue [z~ Among these algorithms, the valence- driven conquest algorithm [1~ can achieve the best rate/distortion ratios at any time during transmission.

In this kind of algorithms, geometry coding is driven by connectivity coding, thus geometry code is not optimal. More effort should be invested in reducing geometry code.

Geometric Signal Processing Scheme. Bene- fit from the progress of digital geometric signal processing [15'16], geometry compression can be ef- [ectlvely performed in the frequency domain to ~chieve bet ter rate/distort ion ratios. Based on raubin 's signal processing framework [lz], Karni e~ aI. [ls] proposed an efficient spectral compression method to compress the mesh geometry by pro- iecting the mesh geometry onto an orthonormal basis derived from the mesh connectivity. In or- ier to accelerate the spectral decomposition pro- :ess, the input mesh is partit ioned into a number )f patches, and each patch is compressed indepen- iently, which leads to "edge-effects" along patch )oundaries and extra bits to mark these patches. Fhis algorithm is inherently a compression scheme

for coefficients produced by Discrete Fourier Trans- form. To achieve bet ter rate/dis tor t ion ratios, new orthogonal basis functions need to be sought. By re-sampling 3D models into semi-regular meshes, Khodakovsky et al. [19] proposed an efficient progressive geometry compression scheme by processing parameter and geometry information sepa- rately. In this algorithm, parameter information and connectivity information are virtually elimi- nated using the semi-regular scheme, and a wavelet transform is used for high order de-correlation and reconstruction. Although this algorithm can achieve the best geometry compression ratios by far, the original connectivity which is crucial to some applications is lost.

1.2 O v e r v i e w

In this paper, we only concern about the compression of mesh attributes. The main contribu- tion of our framework is the observation that the "bottleneck" of the compression of mesh attributes is the complexity of their definition domain, i.e., the surface of the mesh. The ready industry standards for images cannot be applied to geometric attributes of general meshes directly. Our basic idea is to construct planar parametric meshes for arbitrary meshes, and adopt JPEG2000 to compress those attributes defined on the parametric meshes.

Specifically, we make the following contribu- tions:

�9 A robust and fast parameterization algorithm is proposed for constructing a bijective map from an open 3D mesh to a planar mesh. The algorithm incorporates the progressive mesh representation, local parameterization and stretch minimization.

�9 An adaptive sampling scheme is designed to e~ciently sample the attributes defined on the planar parametric mesh into 2D regular signals. Dis- tinct sampling error metrics are introduced for different attributes.

�9 The JPEG2000 standard for still images is used to compress 2D regular signals, which allows the user to conveniently make a tradeoff between bit rate and distortion.

As this work is finished, we find a similar method proposed by Gu et al. [2~ called geometry image, also addressing this issue. They cut an arbitrary mesh automatically and parameterize the opened mesh onto a single chart, which is then sampled into a regular 2D array structure, and processed like images. To achieve a good parameterization, an [terative cut augmentation scheme

598 J. Compu~. Sci. ~z Technol., Sept. 2004, Vol.19, No.5

is designed by iteratively estimating the distortion of the parameterizat ion and optimizing the cut. In our work, we parti t ion a mesh into charts through a user-specified cut, and parameterize each chart over a square using a novel progressive parameterization scheme. An adaptive sampling method is designed to sample the non-uniform parametr ic mesh. Obviously, Gu's cutting algorithm can be directly integrated into our algorithm.

2 N o t a t i o n s

In this paper, only oriented 2-manifold triangle meshes are considered. A triangle mesh M is denoted as a triple M = (~w, KM, PM), where VM = {1, 2 , . . . , IMI} is the verte~ set (IMi denotes the number of vertices/, and KM is an abstract sim- plicial complex, i.e., a set of subsets of VM, which contains all connectivity information of the mesh. PM is a set of n-dimensional at t r ibute vectors defined on VM, i.e., PM = {OM(i)[i E VM}, where r = ( f l ( i ) , f 2 ( i ) , . . . , f ~ ( i ) ) is the at t r ibute vector of vertex {i} concerned with its position coordinates, normal, color, texture coordinates, etc. Obviously, PM contains all geometry information of mesh M.

The subsets in KM are called simplices and come in three types: vertices v = {i}, edges e = { i , j } and faces t = { i , j , k } . Two vertices {i} and {j} are neighbors if { i , j } e KM. The 1- ring neighborhood of a vertex v = {i} is denoted by

N(v) = { j I { i , j } C KM}. The 1-ring neighboring triangle set T(v) of vertex v consists of all the triangles incident to v. The valence of a vertex v is defined as the number of elements within its 1-ring neighborhood. The star of a vertex {i} is defined as the sub-mesh Star({i}) = [-Jies,seKM S.

The piecewise linear n-dimensional function interpolating all a t t r ibute vectors in PM is defined as the geometric signal FM on M. Consider a planar parameterizat ion of mesh M, there is a bijective map FM-D from M to a 2D parametr ic mesh D, which has the same connectivity as M. Therefore, the geometric signal FM on M can be transformed into a 2D signal FD on D by interpolating all at- t r ibute vectors in {@M(F~I_D(i'))Ii ' C VD}.

3 G e o m e t r i c S igna l C o m p r e s s i o n F r a m e w o r k

As illustrated in Fig.2, our geometric signal compression framework consists of the following components:

Planar Parameterization. A 2D parametric mesh D isomorphic to 7VI is constructed using a planar parameterization algorithm and the original geometric signal FM is transformed into a 2D signal FD.

Sampling. The 2D signal FD is adaptively sampled into a regular signal Fu under the user- specified precision threshold.

I Compressed [_~

Parameterization

| 2D regular / I " [ signal F~ i [ 2D mgnal F ;

JPEG2000 ~] Compressed I encoder / I datafile I

Fig.2. Block diagrams of our geometric signal compression framework. The top row is the encoding stage, and the bottom one is the decoding stage.

Kun Zhou etal.: Geometric Signal Compression 599

JPEG2000 Encoding and Decoding. The JPEG2000 encoder is used to compress the regular signal Fu, and the compressed data can be decoded into a regular signal F~.

Signal Reconstruction. Together with the regular signal F{/ and the parametric mesh D, a new signal F~I is reconstructed using linear interpolation and the inverse mapping of the parameterlzation.

It is necessary to point out that in our framework the encoding stage and decoding stage must share the same parametric mesh. Since the parametric mesh is constructed using both mesh connectivity and vertex positions, we have two ways to solve this problem. The first way is to provide the parametric mesh (connectivity and parametric coordinates) to the decoder explicitly. The alter- native way is to transfer the original mesh (con- neetivity and position coordinates) to the decoder, and the original mesh will be used to construct the same parametric mesh as that constructed at the encoder. Of course, both the parametric mesh and the original mesh mentioned above can be compressed using existing geometry compression schemes. In this paper, we adopt the first method. Position coordinates are compressed as signals and the parametric meshes are compressed using the TG algoritl~n [9] .

4 A l g o r i t h m C o m p o n e n t s

4.1 P l a n a r P a r a r n e t e r i z a t i o n

Parameterization is crucial to many applications such as surface modeling, texture mapping

and morphing. In the last two decades, man)' excellent parameterization algorithms [2i-23] have been presented to flatten open meshes. However, for large and complex shaped open meshes, it still lacks a fast and robust planar parameterization algorithm. By combining the we]J-known progressive mesh (PM) [i3] and a new local parameterization scheme, we introduce a novel planar parameterization algorithm, which can fast and progressively flatten arbitrary open meshes. If the input is a close mesh, it will be partitioned into two open sub- meshes along a user-specified close path, then the two sub-meshes can be processed independently.

As illustrated in Fig.3, our method involves two steps:

1. A progressive mesh representation with local parameterization information is generated. For each edge collapse, the two decimated vertices are parameterized over the resultant simplified mesh, and the local parameterization information is recorded in the corresponding vertex split operation.

2. A base planar mesh is generated by pro- jecting the base mesh onto a plane. The sequence of vertex split operations in the PM is performed progressively on the planar mesh. For each vertex split, the two split vertices are positioned using the recorded local parameterization information.

To construct the PM of a mesh is straightfor- ward. In this paper, we adopt Lindstrom's volume- preserving algorithm [24]. As illustrated in the top row of Fig.3, a sequence of edge collapse operations is performed until the current simplified mesh has

M = t~I,~

1 Edge collapses ( - - * ......

s p n , . . . ~ S P i

Vertex s p l l t s ~

D : D~ D{

~ E d g e collapses

\ s p i _ l , . . . , s p j

Vertex splits

Mj Mo

i Edge collapses ~:'~ ......

1 't]"~'~'7'F/~';,? " ~"-....-.i-- "-, .' !<,)' ~ : --'~-~;~j ;.,~ Vertex splits <. ' i ~,~, ,~: ~..-~ .... . .... ~

Dj Do

Fig.3. Progressive planar parameterization.

600 J. Comput . S c i . & Technoi., Sept. 2004, Voi.19, No.5

only four boundary vertices and one inner vertex. As a result, the original mesh M = M~ is repre- sented as a base mesh Mo followed by a sequence of vertex split operations { sp l , sp~, . . . , sp~}.

For each edge collapse operation, the local parameterization of the two decimated vertices is constructed using the conformal mapping technique [25] and the harmonic mapping technique [2t] . As Fig.4 illustrates, for an edge collapse operation (vl , v2) --+ v, we first construct a conformal map[ 251 to flatten S t a r ( v ) into a planar region U, then map Vk (k = 1, 2) into U using a harmonic map II [~t]. Let

I i , q l T~ = (ql,k, q2,k 3,k) be the triangle in U containing H(vk) (k = 1, 2) and Tk be its corresponding triangle incident to v, we can compute the barycentric coordinates (ak, ~3~, 1 - a~ - /3k) of II(v~) in 2"s

H ( v k ) ' ' = akq~,k + 13kq2,k + (1 -- ak --/~k)q~,k

Therefore, vk (k = 1, 2) can be locally parameterized with respect to its containing triangle Tk with the barycentric coordinates (ak,/3k). The parameterization information, together with other edge collapse information, is recorded in the vertex split operation for later reconstruction. In our scheme, two vertices are processed each time, and the vertices decimated at previous steps are no longer concerned.

" 2"1 U

~ d g e collapse

ConformaI map

Fig.4. Edge collapse with lecal parameterization.

The construction of the parametric mesh is equivalent to the recovery of a mesh from the PM, but with a base planar mesh and a novel vertex positioning strategy. The base planar mesh Do is generated by mapping M0 into a square region using the harmonic mapping technique[21]. Starting with ~I0 and Do, the split operations in { s p l , s p 2 , . . . , s p ~ } are then progressively performed to simultaneously recover the original mesh and construct the parametric mesh. Let Mi be the recovered mesh after performing the i-th split operation, and Di be its corresponding parametric mesh. Now the (i + 1)-th split operation is

performed. With the containing triangles Tk and barycentric coordinates (c~k,/~k) of the two split vertices v~ (k = 1, 2) retrieved from spi+l, two new vertices qk (k = 1, 2) can be inserted into De with the same connectivity as vk in/V/~+I and positioned at

qk = c~q~,k +/?kq2,~ + (1 - ~k - Pk)q3,k

where (ql ,k ,q2,k,q3,k) are the triangles in Di corresponding to Tk. Note that the two inserted vertices might cause triangle flipping, we further use the Laplacian operator (similar to the method of MAPS[ 2s]) to relax qk (k = 1, 2) and their 1-ring neighbors to keep triangles nnflipping. Once the n- th vertex split operation has been performed, the parametric mesh is generated accordingly.

To further improve the parameterization, the parametric mesh is optimized by minimizing the stretch using Sander's algorithm [23]. Fig.5 shows the parameterization of the Bunny model with 2000 triangles (partitioned into two patches).

(a)

I

(c)

(b)

(d)

Fig.5, Parameterlzation of Bunny model. (a) and (b) Two

partitioned patches. (c) and (d) Planar parametric meshes of (a) and (b), respectively.

Therefore, the original geometric signal can be transformed into a 2D signal F D defined on the planar parametric mesh D, and the 2D signal is ready to be sampled.

Kun Zhou et aI.: Geometric Signal Compression 601

4.2 S a m p l i n g a n d E r r o r M e t r i c s

Since the 2D signal defined on the planar parametric mesh is irregularly sampled, it is necessary to sample it into a regular signal before using the JPEG2000 encoder to compress it. Then we need to decide how many regular samples are needed to recover the irregular 2D signal. Although the Nyquist sampling theorem has told us that 4B 2 sample points are enough for signals with band- width B, it is difficult to calculate the bandwidths directly for those piecewise linear signals. Our solution is to increase the sampling rate step by step until the user-specified precision is satisfied. In the following we first describe error metrics for different geometric signals, then an adaptive sampling scheme is introduced to sample the signal with less sampling rate than the uniform sampling method under the same sampling error.

4.2.i Error Metrics

To measure errors is crucial to lossy compression algorithms. Since the attributes of a mesh ~I are completely different from each other, it is necessary to define a metric for each attribute. In this section, we present three metrics for position coordinates, color and normal, respectively. Metrics for other attributes can be defined similarly.

For a given sampling rat% the 2D signal FD is sampled into a regular signal Fu using linear interpolation. From this signal, a new signal F~) can be reconstructed also using linear interpolation. The L 2 difference between FD and F~) is defined as the sampling error.

The position metric is defined as the distance between the original mesh and the reconstructed mesh, which can be computed using the METRO too1126].

The normal metric is defined as

V/E,Ev arccos2 �9 e(?~o~l~a~) IDI

where n(i) and n'(i) represent the normalized normal components of FD and F~) at the i-th vertex, respectively.

The RGB color metric is defined as the distortion of standard intensity Y = 0.3R+0.59G+0. IIB by PSNR:

1 PSNR = 20 log10 e(Y)

where

IIY( ) - Y'(i)fl e(Y)

In[

For any attribute, the sampling rate will re- main increasing until the error metric is less than a user-specified precision. In our current implementation, the initial sampling rate for all at t r ibutes is set as 2v/]MIAera(D)/Area(M) • 2~/liVIIdera(D)/dera(M), where dera(D) and Area(M) are the surface areas of D and M, respectively.

4.2,2 Adaptive Sampling

The uniform sampling scheme (the sampling points are shown in Fig.6(a)) has the drawback that in order to satisfy a user-specified precision, a very high sampling rate may be required, especially for those parametric meshes which have quite non- uniform vertex distribution. The higher the sam- piing rate is, the larger the data size will be. Since large data size will reduce the efficiency of the algorithm, we design an adaptive sampling scheme (the sampling points are shown in Fig.6(b)) to achieve lower sampling rate than the tmi[orm sam- piing method under the same user-specified precision. For example, Fig.6(c) is the reconstructed Btmny model (color signal) using the uniform sampling scheme with 512 x 512 sampling rate, while Fig.6(d) is another reconstructed model using our adaptive sampling scheme with only 480 x 480. sampling rate. One can see the lat ter is less distor ted than the former, especially in the ear regions. The reason is that the vertex distribution in the ear regions of the parametric mesh is so dense (Fig.5(c)) that fine details cannot be sampled in the specified sampling rate. Contrarily, the adaptive sampling scheme tries to distribute the sample points in terms of the vertex distribution of the parametric mesh. As a result, it can sample details bet ter even with lower sampling rate.

For a specified sampling rate, our adaptive sam- piing scheme first uniformly divides the square region and constructs a regular mesh U (Fig.6(a)). Then the vertices of U are redistributed to match the vertex distribution of the parametric mesh D by introducing the notion of the vertex density (Fig.6(b)). The vertex density at vertex v on mesh D is defined as:

[N(v)l + 1

dD(v) = ErecT(v) Area(A)/Area(D)

602

where Area(D) is the total surface area of mesh D.

J. Comput. Sci. & Technol., Sept. 2004, Vol.19, No.5

nal Fu defined on U (Fig.6(b)) can be calculated from FD using linear interpolation. Similarly, the resulting signal Fu can be inversely sampled into a new signal F~) defined on D. The sampling rate remains increasing until the error between FD and F~ satisfies the precision threshold.

The signal Fu is geometrically irregular, but logically regular. Therefore, it is ready to be compressed by the JPEG2000 encoder.

(# (b)

(~) (d)

Fig.6. Uniform sampling and adaptive sampling. (a) Reg- ular grid. (b) Redistributed grid using our vertex redistri- bution scheme. (c) Uniform sampling result. (d) Adaptive sampling result.

A relaxation procedm'e is used to redistribute all vertices of U_ Each vertex v of U is relaxed by:

_ 1

viEN(v)

w h e r e - d'( d du(vi) is the relative vertex den-

sity, du(vi) is the current vertex density at vi on U, d'(vi) = dn(v 0 is the vertex density at vi on D calculated by linear interpolation. Once v is updated, d'(v) and dv(vi) (vi E N(v)) are updated accordingly. The relaxation process is performed a user-specified number of times.

Once the relaxation process is completed, a sig-

4.3 J P E G 2 0 0 0 E n c o d e r a n d D e c o d e r

Encoding 2D regular signals is much straight- forward since these signals can be processed like images. All existing image compression algorithms can be used for our purpose. Therefore, our geometric compression framework can be conveniently embedded into various compression standards. As we know, many compression standards such as JPEG, JPEG2000, MPEG2 and MPEG4 have been proposed for encoding images and videos in the last decade, which makes digital media very popular in the modern society. In our current implementation, JPEG2000 is adopted for its nice features.

JPEG2000 [271 is a new industry standard for still image compression with many nice features, such as superior low bit-rate performance, lossless and lossy compression, progressive transmission, random code-stream access and processing, open architecture, etc. This standard is a transform- based encoding scheme, and works on image tiles. The block diagram of the JPEG2000 encoder and decoder is illustrated in Fig.7. In the standard, the discrete wavelet transform (DWT) is first applied to analyze each tile of the input regular signal, and the transform coefficients are then quantized and entropy encoded (arithmetic coding) before form- ing the output bit-stream (the top row of Fig.7). The decoder is the reverse of the encoder (the b o b tom row of Fig.7). The bibstream is first entropy decoded, de-quantized and inversely transformed, the regular signal is then recovered.

j Forward wavelet' ~ Quantization ~ Entr~ ] t C~ I regular encode I data

I Compressed l-- data

I ] Dequantizationl I Inverse wavelet I /Reconstructed t Entropydecode I---I " I--] transform i-- regutar signal !1

Fig.7. Block diagrams of JPEG2000. The top row is encoder and the bottom row is decoder.

Kun Zhou et al.: Geometric Signal Compression 603

5 R e s u l t s

We have implemented the described geometric signal compression framework using C++ language on a 600MHz Pentium Ill machine. Several models with different attributes have been tested. The Bunny, Horse and Venus head models are partitioned into two sub-meshes, which are compressed independently and merged back at the decoder. Table 1 shows the numerical values for the compression results. The parametric meshes axe corn-

pressed using the commerc ia l compress ion sof tware of V i r tue Ltd . (www.v i r tue3d .eom) , which incor- po ra t e s the T G a lgo r i t hm [9]. Since the p a r a m e t r i c coord ina tes make no con t r i bu t i on to the compres - s ion error, t hey are encoded wi th 9-12 bi ts quan t i - za t ion precisions. T h e color signals are quan t i zed w i th 8 b i t s prec is ion and o the r signals are quan- t i zed wi th 16 b i t s precision. T h e pos i t ion er rors r e p o r t e d here are re la t ive wi th respec t ive to t he b o u n d i n g box d iagona l on a scale of 10-4.

Figs.8, 9, 10, 11 are the compress ion resul t s for

Table I. Compression Results for Typical Meshes (All errors are measured between the reconstructed models and the original models.) Sampling Sampling Bit rate/distortion for (b), (c) and (d) images Parametric mesh

Models Vertex Attributes rate error (b) (c) (d) Quant. Bit rate Venus 2793 position 256 x 256 3.0 16.72/5.5 9.09/9.3 4.40/22.3 9 11.15 Manne. 6769 position 512 x 512 2.1 14.74/4.0 7.38/7.1 3.63/14.4 10 9.32 Horse 20002 position 640 • 640 1.7 11.26/3.6 4.59/7.8 1.74/18.7 I0 17.04

position 1.9 9.40/3.3 3.88/6.0 1.15/20.9 syn.color 73.74 16.11/70.33 5.09/66.06 1.24/62.67

Bunny 25120 ren. color 480 x 480 89.18 5.28/85.77 1.66/80.01 0.32/69.88 12 16.37

normal 0.015 7.34/0.033 2.64/0.060 0.91/0.105 Venus 10002 normal 512 • 512 0.029 23.42/0.054 7.78/0.098 1.47/0.219 12 14.33 head

(a) (b) (c) (d)

Fig.8. Position coordinates compression of the Venus model. (a) Original model. (b), (c) and (d) Compressed results with hit rates 16.72, 9.09 and 4,40 hits/vertex, respectively.

(a) (b) (c) (d)

Fig.9. Position coordinates compression of the IVIannequin model. (a) Original model. (h), (c) and (d) Compressed results with hit rates 14.76, 7.38 and 3.63 hits/vertex, respectively.

604 d. Comput. Sci .& Technol., Sept. 2004, Vol.I9, No.5

the position signals of the Venus, Mannequin, Horse and Bunny models. Fig.1 and Fig.12 are the compression results for different color signals of the Bunny model. The colored model shown in Fig.1 is generated using texture synthesis algorithm [2s] , while that shown in Fig.12 is produced using Phong shading technique. Fig.13 and Fig.14 show the compression results for the normal signals of the Bunny and Venus head models. The (a) images show the original models and the (b), (c) and (d)

images are the compressed results with different bit rates. Note that the (b) images look almost the same as the (a) images and the (c) images preserve most of the features of the (a) images.

For the position signals, our compression algorithm can achieve an average bit rate of 11 bits/vertex without any visual degeneracy and models can be reconstructed from a bit-stream of 2 bits/vertex with minor degeneracy. For the normal signals, much lower bit rates can be achieved for

(a) (b) (c) (d)

Fig.10. Position coordinates compression of the Horse model. (a) Original model. (b), (c) and (d) Compressed results with

bit rates 11.26, 4.59 and 1.74 bits/vertex, respectively.

(~) (b) (c) (d)

Fig. l l . Position coordinates compression of the Bunny model. (a) Original model. (b), (c) and (d) Compressed results with bit rates 9.40, 3.88 and 1.15 bits/vertex, respeetiveiy.

(~) (b) (c) (d)

Fig.12. Rendering color compression of the Bunny model. (a) OriginM model. (b), (c) and (d) Compressed results with bit rates 5.28, 1.66 and 0.32 bits/vertex, respectively.

Kun Zhou et al.: Geometric Signal Compression 605

(a) (b) (c) (d)

Fig.13. Normal compression of the Bunny model. (a) Original model. (b), (c) and (d) Compressed results with bit rates 7.34, 2.64 and 0.91 bits/vertex, respectively.

(a) (b) (c) (d)

Fig.14. Normal compression of the Venus head model. (a) Original model. (b), (c) and (d) Compressed results with bit rates 23.42, 7.78 and 1.74 bits/vertex, respectively.

the Bunny model than the Venus head model. The reason is that the Venus head model has finer details than the Bunny model. For the color signals, color richness and distribution affect the compression ratios greatly, which is common in image compression. That is why the compression ratios of the model shown in Fig.12 are much higher than that of the model shown in Fig.l.

6 C o n c l u s i o n a n d F u t u r e W o r k

In this paper we describe a lossy compression a lgor i thm to deal wi th all mesh at t r ibutes based on signal processing. All mesh attributes including position coordinates, normal, color, texture coordinates, etc. are regarded as geometric signals defined on mesh surfaces. A robust and fast parameterization algorithm is used to transform the original geometric signals into 2D signals defined on planar parametric meshes. Then the 2D signals are sampled into regular signals, which can be compressed by the JPEG2000 encoder, using an adaptive sampling scheme. The superiority of our geometric signal compression framework includes

the following aspects: consistent compression of all mesh attributes, a better tradeoff between bit rate and distortion and easy integration with existing compression standards.

There are several directions for future work: 1. The parametr ic mesh occupies a significant

fract ion of the bit budget. To reduce this information, a possible solution is to el iminate the con- nect ivi ty as Khodakovsky ' s a lgor i thm [i9] or only use topology information to cons t ruc t parametr ic meshes.

2. Efficient sampling schemes should be pur- sued to sample features of signals more accurately. For example, many CAD models contain sharp edges and folders, which must be sampled densely-.

3. More experiments should be carried out on the parameter set t ing of the J P E G 2 0 0 0 encoder for different kinds of at tr ibutes. We believe this will lead to higher compression ratios.

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K u n Z h o u received his : ~ Ph.D. degree from the Depart-

ment of Computer Science and Engineering of Zhejiang Univer- sity in 2002. He works in the areas of digital geometry processing, texture synthesis/analysis and real time rendering. He is currently an associated re- searcher at Microsoft Research Asia.

Hu-Jun Bao is a professor of the State Key Labo- ratory of CADACG at Zhejiang University, P.R. China. He received his M.Sc. and Ph.D. degrees in applied mathematics from Zhejiang University. His research interests include computer graphics, geometric modeling and virtual reality. More information can be found at http://www.cad.zju.edu.cn/home/bao.

J i a o - Y l n g Shi is a professor of the Department of Computer Science and Engineering at Zhejiamg Uni- versity. He is currently Deputy Chairman of China Image and Graphics Association, and Deputy Chair- man of China CAD and Graphics Society under China Computer Federation. He is the representative of Asia in Education Committee of ACM SIGGRAPH. Since 1990 his work is concentrated on computer graphics, visualization in scientific computing and vir~u~l en- vironment. He has published more than i00 papers and four books, More information can be found at http://www.cad.zju.edu.cn/home/jyshi.

Qun-Sheng Peng is a professor of the Depart- ment of Applied Mathematics at Zhejigng Univer- sity. He received his Ph.D. degree from Univer- sity of East Anglia, UK in 1983. He is currently Vice Chairman of Academic Committee of State Key Lab of CAD&CG at Zhejiang University. His research interests include realistic image synthesis, in- frared image synthesis, computer animation and scientific visualization. More information can be found at http:/ /www.cad.zju.edu.cn/home/peng.

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Geometric signal compression