Computers & Graphics 35 (2011) 713–718

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Computers & Graphics

0097-84

doi:10.1

� Corr

E-m

ep6tri@

sungyul

journal homepage: www.elsevier.com/locate/cag

Short Communication to SMI 2011

Bayesian AD coder: Mesh-aware valence coding for multiresolution meshes

Junho Kim a, Changwoo Nam a, Sungyul Choe b,�

a School of Computer Science, Kookmin University, 861-1 Jeongneung-Dong, Seongbuk-Gu, Seoul 136-702, Republic of Koreab 1040-2 Yeongtong-Dong, Yeongtong-Gu, Suwon-Si 443-470, Republic of Korea

a r t i c l e i n f o

Article history:

Received 10 December 2010

Received in revised form

13 March 2011

Accepted 14 March 2011Available online 31 March 2011

Keywords:

Multiresolution mesh compression

Bayesian Alliez Desbrun (AD) coder

Mesh-aware valence coding

93/$ - see front matter & 2011 Elsevier Ltd. A

016/j.cag.2011.03.023

esponding author. Tel.: þ82 1085977684.

ail addresses: junho@kookmin.ac.kr (J. Kim),

kookmin.ac.kr (C. Nam), ggalssam@postech.a

.choe@gmail.com (S. Choe).

a b s t r a c t

The Alliez Desbrun (AD) coder has accomplished the best compression ratios for multiresolution

2-manifold meshes in the last decade. This paper presents a Bayesian AD coder which has better

compression ratios in connectivity coding than the original coder, based on a mesh-aware valence

coding scheme for multiresolution meshes. In contrast to the original AD coder, which directly encodes

a valence for each decimated vertex, our coder indirectly encodes the valence according to its rank in a

sorted list with respect to the mesh-aware scores of the possible valences. Experimental results show

that the Bayesian AD coder shows an improvement of 8.5–36.2% in connectivity coding compared to the

original AD coder despite of the fact that a simple coarse-to-fine step of the mesh-aware valence coding

is plugged into the original algorithm.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The common use of high-resolution meshes has provided themotivation for intensive studies of mesh compression techniques,including single-rate mesh compression [24,8,22,20,2,16,11],multiresolution mesh compression [21,5,17,1,6,19], out-of-coremesh compression [9,10], and random-accessible mesh compres-sion [15,27,4]. Particularly, multiresolution mesh compressionprovides a rate-distortion sense in shape transmissions via thenetwork by sending the shape of an object in a coarse-to-finemanner. Although much research has been done on multiresolu-tion mesh compression due to its usefulness, the Alliez Desbrun(AD) coder [1] has been the best coder for multiresolution2-manifold meshes over the last decade.

This paper presents a Bayesian AD coder, based on a mesh-aware valence coding approach, which has better connectivitycoding efficiency than the original AD coder [1]. The excellentcharacteristics of the AD coder stem from the observation that theconnectivity of a multiresolution mesh can be solely representedby a sequence of valences for the decimated vertices inducedfrom a rule-based simplification hierarchy. Our observation isthat the original AD coder [1] can be classified as a frequentistapproach toward valence coding such that it is only aware of thefrequencies of the previously decoded valences in each insertionof a new vertex in the decoder. In contrast, our coder can be

ll rights reserved.

c.kr,

classified as a Bayesian approach, as we are also aware ofthe current mesh in each valence coding. We indirectly encodethe valences in such a manner that for each gate we sort allpossible valences with respect to the geometric regularity scoresof their expected 1-ring patches and encode the rank of the truevalence in the sorted list. Moreover, our coder utilizes theconnectivity of the current mesh and enhances the compressionratios by excluding impossible symbols in each valence coding. Asa result, we obtain a skewed symbol distribution that much moresuitably obtains high compression ratios. Experimental resultsshow that the Bayesian AD coder shows improvement of8:5236:2% in terms of its connectivity coding compared to theoriginal coder, with a little computational overhead.

2. Background

Here, we survey previous work related to mesh compressionand briefly review the original AD coder [1]. We also refer thereader to several excellent survey papers [23,7,3,18] for details.

2.1. Related work

Single-rate mesh compression encodes the entire graph struc-ture of a given mesh by conquering its primitives, such as verticesor faces, in a deterministic manner. Touma and Gotsman [24]presented the first work on valence-driven single-rate meshcompression, in which encoding is accomplished via mesh con-nectivity with the valences, each of which is the number ofassociated edges of a vertex. Alliez and Desbrun [2] noted thatthe valence-driven coder is closely related to the upper bound on

J. Kim et al. / Computers & Graphics 35 (2011) 713–718714

the cost of the graph structure coding [25]. Kalberer et al. [11]introduced an efficient valence-driven coder, called FreeLence.FreeLence [11] indirectly represents the valence information withthe numbers of non-conquered edges, so called free valences,which usually have less entropy compared to naive valencecoding approaches [24,2].

Multiresolution mesh compression provides an overview of acomplete mesh in a coarse-to-fine manner with a trade-offbetween bit rates and distortion. The early work on multiresolu-tion mesh compression [21,5,17,12] investigated efficient meth-ods of coding the detail data of a multiresolution irregular mesh.Unfortunately, the compression ratios in [21,5,17,12] were notcompared to those from single-rate coders. In contrast, Alliez andDesbrun [1] present a valence-driven multiresolution mesh coder,hereafter called the AD coder, that is based on their observation ofthe importance of the valence information [2]. In the AD coder [1],a multiresolution mesh hierarchy is constructed with valence-driven decimations such that the detail data of the multiresolu-tion mesh is solely represented with a sequence of the valences ofthe decimated vertices. As a result, its compression ratios arecomparable to those from single-rate coders.

For efficient geometry compression, there have been excellentapproaches based on spectral geometry analysis. Karni andGotsman [13] provided the spectral compression of mesh geo-metry in which the 3D positions of the vertices are projected ontothe low-frequency eigenvectors of the discrete Laplacian matrixinduced from the mesh connectivity. Khodakovsky et al. [14]presented the wavelet-based geometry compression for semi-regular meshes, in which the geometry of the mesh should beresampled by using remeshing, and thus it causes the originalconnectivity to be lost. We focus on connectivity coding formultiresolution irregular meshes, and the geometry coding isout of the scope of this paper.

There are two important studies of multiresolution non-manifold mesh compression. Gandoin and Devillers [6] presenteda multiresolution mesh compression scheme utilizing the geo-metry encoding based on kd-trees. Peng et al. [19] presented anoctree-based approach to encode multiresolution non-manifoldmeshes. These approaches are very effective when coding non-manifold meshes, such as trees and plants, in a coarse-to-finefashion. Unfortunately, however, their rate-distortion character-istics are not suitable for manifold meshes in general, especially ata coarse level.

2.2. Review of the original AD coder

In the AD coder [1], rule-based simplifications generate asequence of valences as the essential information of the con-nectivity for a multiresolution mesh. The levels of a mesh

valence 3 valence 4

enco

de

Fig. 1. Rules for the decimation conquests in the AD coder (note: � can be either � or

patch of the decimated vertex is based on rules, each of which is determined by the v

decoder, the patch is discovered based on the rules related to the valence code and the

the current input gate but will be related to the future input gates, can be gene

(For interpretation of the references to color in this figure legend, the reader is referre

hierarchy are generated with several applications of two inter-laced conquests: a decimation conquest followed by a cleaningconquest.

A decimation conquest decimates the vertices with valence-3/-4/-5/-6 in an independent set of patches and retriangulates the1-ring patches of the decimated vertices, while coding a sequenceof valences for the decimated vertices. In particular, each retrian-gulation is performed based on the fixed rules depicted in Fig. 1.The ingenious idea in [1] is that the decoder can conquer the sameindependent set of patches solely with a sequence of the valencesabout the decimated vertices. This is possible because theretriangulation rules not only utilize but also no-costly generatethe plus � or the minus � tags for the vertices, each of whichindicates the strategy of increasing or decreasing the vertexvalence, respectively. As a result, in the decoder it is possible todiscover the patch of a newly introduced vertex from a decodedvalence and the no-cost generated � or � tags on two vertices ofthe current gate, as depicted in Fig. 1.

A cleaning conquest cleans the several valence-3 verticeswhich are induced from the previous decimation conquest. Inthis conquest, the AD coder encodes the history of conquers onthe triangles which are generated by cleaning the valence-3vertices.

3. Bayesian AD coder

The basic idea of our Bayesian AD coder is that we can design alikelihood function that helps us to infer the true valence in a highchance among all possible valences by utilizing the connectivityand geometry of the current mesh.

Let us consider a scenario that the decoder reconstructs amesh in a coarse-to-fine manner. A level-wise refinement of themesh can be accomplished by applying an inverse cleaningconquest followed by an inverse decimation conquest, as shownin Fig. 2. An inverse cleaning conquest inserts the valence-3vertices, which then ruins the shape of the triangles in the currentmesh (see Fig. 2(a)). The subsequent inverse decimation conquestinserts vertices with valence-3/-4/-5/-6 into the current mesh.Notice that the shapes of the triangles in the current meshbecome regular due to the inverse decimation conquest (seeFig. 2(b)). This observation leads us to a reasonable likelihoodthat connectivity coding can be improved by utilizing the geo-metric property induced from the inverse decimation conquests.

Suppose we stand on a gate during the inverse decimationconquest in the decoder. Before we obtain the valence code of anewly introduced vertex, we can always consider five possiblepatches where only one of them is true and other four are false(see Fig. 3). We devise a reasonable assumption such that the true

valence 5 valence 6

deco

de

� without violating the other rules): in the encoder, the retriangulation about the

alence of the decimated vertex and the two tags related to the input gate. In the

two tags related to the input gate. Here, all of the red tags, which are not related to

rated at no cost based on the rules in both of the encoder and the decoder.

d to the web version of this article.)

(a) inverse cleaning conquest (b) inverse decimation conquest

Fig. 2. In the decoder, a mesh is reconstructed in a coarse-to-fine manner with

several interlaced applications of an inverse cleaning conquest followed by an

inverse decimation conquest. In an inverse cleaning conquest, several valence-3

vertices (red) are inserted into the current mesh. In an inverse decimation

conquest, several valence-3/4/5/6 vertices (yellow) are inserted into the current

mesh. (For interpretation of the references to color in this figure legend, the reader

is referred to the web version of this article.)

Fig. 3. Geometry-awareness: without the knowledge of the true valence code in

each gate, we can think of five possible patches each of which is induced from the

valence code 3, 4, 5, 6, or the null-patch. To infer the true valence, we measure the

geometric regularity sðvijnÞ by virtually inserting the new vertex vi using its local

coordinate and the Frenet frame related to each patch.

Fig. 4. Connectivity-awareness: we stand on the current gate after conquering the

shaded region, as depicted in (a). Then, (b) and (f) are the only two possible cases.

The valence-4/-5/-6 can be trivially rejected because the associated patches

violate the already conquered region, as depicted in (c), (d), and (e).

J. Kim et al. / Computers & Graphics 35 (2011) 713–718 715

valence will introduce a geometrically regular patch with a highprobability. To realize such an idea as our likelihood function, weplug the following processes into both of the encoder and thedecoder.

3.1. Geometry-aware valence coding

In the encoder, we first utilize the original AD encoderalgorithm to obtain the sequence of valences of the decimatedvertices. We then indirectly encode the valences with the follow-ing steps in a coarse-to-fine refinement of the mesh.

For each active gate in the inverse decimation conquest, weidentify the five possible patches of the new vertex vi, as shown inFig. 3. Next, we virtually construct all possible patches of vi withtheir own Frenet frames and the quantized local coordinate of vi.Next, we measure the geometric regularities of each case. Wesimply define the geometric regularity of a patch as the averagecompactness of the 1-ring triangles, as shown below:

sðvijnÞ ¼1

#Nf ðvijnÞ

XtANf ðvijnÞ

areaðtÞ

perimeter2ðtÞ

0@

1A:

Here, n denotes a possible valence code for the newly introducedvertex vi and Nf ðvijnÞ indicates the 1-ring faces of the vertex vi inthe patch defined from the valence code n. Next, we sort the fivescalars from sðvij3Þ, sðvij4Þ, sðvij5Þ, sðvij6Þ, and sðvijnull-patchÞ in adescending order. Finally, we encode the rank of the regularityvalue about the true valence in the sorted list instead of the truevalence itself.

In the decoder, for an active gate in the inverse decimationconquest, we utilize a sorting process that is identical to that ofthe encoder and identify the true valence code for the newlyintroduced vertex by decoding its rank in the sorted list.

3.2. Connectivity-aware valence coding

In several gates, we do not need to consider all of the fivepossible patches, as we can trivially reject some patches whichviolate the already conquered region in an independent set ofpatches.

Suppose we stand on a gate in an inverse decimation conquest,as shown in Fig. 4. In this case, the valence codes 4, 5, and 6 mustbe false, because their patches violate the already conqueredregion. Recall that the encoder decimates vertices in an indepen-dent set of patches where any two patches cannot overlap.Therefore, the valence codes 4, 5, and 6 can be trivially rejectedin our connectivity-awareness scheme.

The connectivity-aware approach can be simply used in con-junction with the geometry-awareness in both of the encoder andthe decoder, as follows. For each active gate, we test whether ornot each possible valence generates a trivially rejected patch. Wethen sort the geometric regularity scores only from the non-rejected valences and code the rank of the true valence in thesorted list.

Table 1 shows the statistics pertaining to the trivial rejectioncases in the decimation conquests for several models. As shown inTable 1, in nearly 50% of the gates, we can reject severalimpossible valences to encode our symbols.

Fig. 5 shows the distributions of the symbol frequencies forseveral models. As depicted in Fig. 5, our Bayesian AD coder hasmore skewed symbol distributions compared to the original ADcoder, which is better to obtain high compression ratios.

J. Kim et al. / Computers & Graphics 35 (2011) 713–718716

3.3. Priors

We further minimize the entropy of our symbols with priorknowledge about the valences in our Bayesian AD coder.

The simplest type of prior knowledge is the distributions of thepreviously decoded valences, which can be obtained at no cost in the

Table 1Statistics of the trivial rejection cases in the decimation conquests: each value

related to k (i.e., k¼0, 1, 2, 3) indicates the number of gates where we can ignore

#k different valences and where we must consider only #(5�k) possible valences.

Model # Total gates # Gates with #k rejected valences

k¼0 k¼1 k¼2 k¼3

Happy Buddha 126,993 51,887 27,494 11,757 35,855

Dragon 123,699 50,595 27,044 11,108 34,952

Mannequin 21,353 9,630 4207 1801 5715

Horse 48,665 19,827 10,869 4305 13,664

Tiger 3108 1787 466 230 625

0

2000

4000

6000

8000

10000

12000

14000

16000

0 1 2 3 40

5000

10000

15000

20000

25000

30000

35000

Fig. 5. Symbol distribution in the decimation conquests: (a) the original AD coder, w

respectively, and (b) our Bayesian AD coder.

1

1

2

2

3

3

Fig. 6. (a) Even with a highly regular mesh (i.e., the red mesh), several triangle seams (i.

decimation conquest followed by a cleaning conquest. This occurs because an indepen

vertex valences associated with the triangle seams in the simplified mesh are not reg

valence code distributions in consecutive decimation conquests. (For interpretation of th

of this article.)

decoder. However, we found that this simple approach is noteffective because the valence distributions differ from the level of amesh hierarchy, as shown in Fig. 6. This arises because, in general, anindependent set of patches used for the decimations cannot cover allof the faces in each level of the mesh hierarchy. Therefore, even witha highly regular model, several seams of non-decimated triangles canappear after a decimation conquest followed by a cleaning conquest,as shown in Fig. 6(a). In such a case, the seams of non-decimatedtriangles cause a change in the valence code distributions inconsecutive decimation conquests, as shown in Fig. 6(b).

From this observation, we define the level-wise prior knowl-edge as regards the valences by investigating the frequencies ofthe valences at each level, as follows:

plðnÞ ¼freqlðnÞ

freqlðnull-patchÞþP6

i ¼ 3 freqlðiÞ:

Here, freqlðnÞ indicates the number of valence-n codes in the level l.Our Bayesian AD coder is finalized by utilizing the mesh-aware

valence coding as a likelihood function and the probabilitydefined from the frequencies of the valence symbols as a prior,

0 1 2 3 4

(a)

(b)

here 0, 1, 2, 3 and 4 represent the null-patch and the valence-6/-5/-4/-3 codes,

0

500

000

500

000

500

000

500

1st decimationconquest

2nd decimationconquest

e., red triangles) appear in a simplified mesh (i.e., the yellow mesh) obtained from a

dent set of patches cannot cover all of the faces of the mesh in general. Here, the

ular. (b) The vertices related to the triangle seams cause dramatic changes in the

e references to color in this figure legend, the reader is referred to the web version

J. Kim et al. / Computers & Graphics 35 (2011) 713–718 717

in this case sðnjviÞ ¼ sðvijnÞplðnÞ:

sðnjviÞ ¼plðnÞ

#Nf ðvijnÞ

XtANf ðvijnÞ

areaðtÞ

perimeter2ðtÞ

0@

1A:

In each gate, we compute the scores sðnjviÞ only for the possiblevalence codes n with connectivity-awareness. We then sort thescores sðnjviÞ in a descending order and code the rank of the scoreabout the true valence in the sorted list as a symbol for indirectlyrepresenting the valence.

Since our prior knowledge plðnÞ cannot be obtained at no costin the decoder, it is necessary to record the values of plðnÞ in a fileas additional information. We quantize each probability valuewith six bits. Although extra spaces are mandatory in ourmethod to preserve the prior knowledge of each level, they arenegligible compared to the gains of the compression ratio in ourexperiments.

4. Optimal symbol codings

Here we present an efficient approach to encode our symbolsfor the connectivity of a multiresolution mesh. To encode thesymbols with an entropy coding technique, we use the adaptivearithmetic coder [26]. The adaptive arithmetic coder manages afrequency table of possible input symbols while processingthe symbols, and utilizes it to improve the coding efficiency forthe next incoming symbol. Originally, the frequency table in thearithmetic coder is automatically managed based on the historyof the processed symbols. To optimize the bit rates of coding oursymbols further, we propose two strategies to manage thefrequency table in the adaptive arithmetic coder, as outlinedbelow.

4.1. Usages of connectivity-awareness

When we encounter trivial rejection cases in our connectivity-aware condition at a gate, the probabilities of the rejectedsymbols must be set to zero. To utilize this fact, we temporarilyremove the frequencies of the rejected symbols by insertingzeroes. Next, we encode our symbol with the adaptive arithmeticcoder. Finally, we recover the frequencies of the rejected symbolswith their original values for the next incoming symbols. In thismanner, we temporarily reduce the entropy of the symbols, inwhich improves the coding efficiency.

Fig. 7. Test models: the (#v, #f) values are listed in a clockwise order. Happy B

mannequin¼(11,703, 23,402), and tiger¼(2738, 5472).

4.2. Use of preset frequency tables

The initial state of the frequency table can influence theperformance of the adaptive arithmetic coder, although the adap-tive arithmetic coder can adjust the probabilities of the symbols.In particular, the performance gains from carefully selectedinitials are significant when the patterns of the symbols arenot fully randomized throughout the symbol sequence, as notedin [11].

To optimize the bit rates of our coder, we reserve the level-wise symbol frequencies into a compressed file in a mannersimilar to that in Section 3.3. For the symbol frequencies obtainedfrom the level of the mesh hierarchy, we quantize each symbolfrequency with six bits such that the sum of the quantizedfrequencies becomes (26-1). Since each level requires two fre-quency tables, one for an inverse cleaning conquest and the otherfor an inverse decimation conquest, 30 bits are required as anextra cost to initialize the frequency tables of each level.

We tested two strategies for the initialization of the frequencytables. The simplest approach was one in which we initialized thefrequency tables at each level of a mesh hierarchy by investigat-ing the frequencies of the symbols in each level. The otherapproach was one in which we split the overall levels of a meshhierarchy into three parts: coarse, medium, and fine levels. Wethen reserve the frequency tables for the three parts by investi-gating the frequencies of the symbols in each part. Eitherapproach was feasible in our experiments, but we used thesecond approach for the results presented in this paper. Weempirically divide the overall levels into three parts where eachpart roughly has the similar number of symbols.

5. Experimental results

We tested our Bayesian connectivity coder with severalmodels, as depicted in Fig. 7. Table 2 shows the bit rates of ourconnectivity coder including the overhead from our prior knowl-edge and the preset frequency tables. We obtain improvement ina range of 8:5236:2% compared with the original AD coder [1].For geometry codings, we adopt the algorithm in [1], in which the3D position of a decimated vertex is encoded with a quantizedlocal coordinate of the Frenet frame related to its patch.

As shown in Table 2, our coder encodes highly regular meshes,such as the mannequin mesh and the tiger mesh more compactlycompared to that in [1]. One reason is that our symbol distribu-tions are much skewed, as shown in Fig. 5, as we indirectly code

uddha¼(49,990, 100,000), dragon¼(50,000, 100,000), horse¼(19,851, 39,698),

Table 2Connectivity compression ratios.

Model AD [1](bpv) Ours (bpv) Gains (%) Max level

Bayesian only Bayesianþ

Opt. coding

(incl. overhead)

Happy Buddha 4.74 4.54 4.34 8.5 17

Dragon 4.69 4.44 4.29 8.6 23

Mannequin 3.99 3.33 3.03 24.1 17

Horse 4.64 4.35 4.18 9.9 20

Tiger 3.02 2.39 1.93 36.2 13

J. Kim et al. / Computers & Graphics 35 (2011) 713–718718

the valences with geometric regularity scores. Another reason isthat our null-patch coding is highly efficient, as illustrated inSection 3.3. For irregular meshes, our Bayesian AD coder alsoworks very well. The mesh-awareness of our likelihood functionis still reasonable, as we utilize the fact that inverse decimationconquests tend to regularize the current mesh in terms of itsconnectivity as well as its geometry, as illustrated in Fig. 2.

6. Conclusion

In this paper, we present a Bayesian AD coder which shows aperformance improvement ranging from 8.5% to 36.2% comparedto the original AD coder [1] in terms of connectivity coding. Incontrast to [1], we indirectly code the valences in a multiresolu-tion mesh with a mesh-aware likelihood function, with which wecan identify the true valence with its rank in the sorted list of allpossible valences for a gate. Although our Bayesian approachrequires a little computational overhead, the high compressionratios compensate for the computational overheads.

We believe that there is still room for improvement in the ADcoder. In particular, additional investigations are required into amethod to remove the seams of triangles in the cleaning con-quests at a low cost. Additionally, efficient geometry coding for amultiresolution mesh is an important future research.

Acknowledgments

We would like to thank Seungyong Lee for helpful discussionand the anonymous reviewers for their insightful comments.Models are courtesy of Stanford Graphics Laboratory, CyberwareInc., Takashi Kanai, and Pierre Alliez. This work was supported bythe Korea Research Foundation Grant funded by the KoreanGovernment (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-D00510), the National Research Foundation of KoreaGrant funded by the Korean Government (NRF-2010-0015858),and the research program 2011 of Kookmin University in Korea.

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