Rate-Distortion Optimized Wavelet-based Irregular Mesh Coding

Jonas El Sayeh Khalil

, Adrian Munteanu

and Peter Lambert

ELIS Department, IDLab, Ghent University-iMinds, Sint-Pietersnieuwstraat 41, B-9000, Ghent, Belgium

Dept. of Electronics and Informatics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050, Brussels, Belgium

Keywords:

Quality-scalable Mesh Representation, Wavelet-based Mesh Coding, Rate-Distortion Optimization.

Abstract:

This work investigates the optimization of mesh quality at lossy rates for a lossless scalable wavelet-based

irregular mesh codec. Whereas previously proposed wavelet-based irregular mesh codecs offer coarse-grain

resolution scalability, in this paper we propose a coding scheme which enables ﬁne-grain quality scalability.

This is done by avoiding the use of geometric data in the encoding process, which reduces dependencies within

the data stream and allows for an unrestricted storage and transmission order of wavelet subband bitplanes

and connectivity information. This in turn allows us to perform rate-distortion optimization, whereby the

subband bitplanes to be encoded are determined by minimizing distortion subject to an overall target bitrate.

Experimental results show that the proposed coding approach offers ﬁne-grain quality scalability, achieves

optimality in rate-distortion sense and improves compression performance over the state of the art.

1 INTRODUCTION

As the amount and the quality of obtained media in-

creases, so does the need for efﬁcient compression in

multimedia domains. This need ignited advancements

in subsequently audio, image, and video compression

systems. Today, advanced compression solutions are

also required for three-dimensional data. While the

gaming industry tries to cope with increasing quality

requirements by making use of shader tricks, and the

movie industry makes use of huge rendering farms for

their rendering needs, we observe that, in general, a

scalable coding solution is required. With the rise of

virtual and augmented reality applications, the advent

of 3D printing and the increasing detail of 3D scan-

ners comes the need for lossless storage and transmis-

sion, while offering interactivity in a scalable way.

Wavelet-based techniques have been successful

in multimedia compression, including several 3D

codecs. A wavelet-based solution uses a set of high-

pass and low-pass ﬁlters to obtain a low-resolution

base mesh where all high-frequency data have been

removed, and a set of wavelet subbands containing

increasingly higher frequency information. By ex-

ploiting ﬁne-grain quality scalability, which allows

for scaling the quality of reconstructed data by decod-

ing per wavelet subband bitplane, data can be trans-

mitted such that the distortion in the reconstructed

mesh decreases optimally.

Contributions. In this work, we investigate our pre-

vious wavelet-based irregular mesh codec offering

resolution scalability and propose a coding scheme

based on it which offers quality scalability. This im-

proves over the state of the art in two ways: (1) the

coding performance at low bitrates is improved by our

proposed algorithm by performing rate-distortion op-

timization, and (2) functionally, an additional form of

scalability is offered without negatively impacting the

lossless coding rate.

The remainder of this work is structured as fol-

lows: related work is given in Section 2, our

resolution-scalable codec we improve upon is de-

scribed in Section 3, our contributions are detailed in

Sections 4 and 5 and we evaluate the proposed codec

in Section 6. Section 7 concludes this work.

2 RELATED WORK

Until today, triangle meshes have been the main rep-

resentation of 3D models for real-time rendering. A

mesh is often seen as the combination of both geome-

try information, i.e., the positions of the vertices, and

connectivity or topology, i.e., the edges (and conse-

quently the triangles) between these vertices. Mesh

coding has been an active topic of research for over

two decades. State-of-the-art single-rate coders, e.g.,

(Touma and Gotsman, 1998), proved to be insufﬁcient

212

El Sayeh Khalil J., Munteanu A. and Lambert P.

Rate-Distortion Optimized Wavelet-based Irregular Mesh Coding.

DOI: 10.5220/0006108602120219

In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), pages 212-219

ISBN: 978-989-758-224-0

despite good performance, as single-rate codecs only

allow for decoding an entire mesh. An ad-hoc solu-

tion is the use of Levels of Detail (LODs), obtained by

simplifying a mesh several times using a decreasing

amounts of vertices to approximate the mesh. This al-

lows for improved interactivity but increases the stor-

ing cost as each LOD is stored separately.

Pioneering work in progressive mesh representa-

tions was done by Hoppe (Hoppe, 1996). In this work,

Hoppe describes how a mesh can be simpliﬁed ver-

tex per vertex, minimizing an energy function at ev-

ery step. This results in a so-called continuous LOD

chain: this progressive mesh representation generates

a nearly continuous spectrum of LODs where each

new level is obtained by splitting one vertex of the

previous level. Each obtained level is optimal given

a speciﬁc number of vertices. Further improvements

in compression performance was obtained by group-

ing the individual increments in batches, each batch

splitting half of the decoded vertices (Pajarola and

Rossignac, 2000). (Peng and Kuo, 2005) and (Valette

et al., 2009) present state-of-the-art codecs which al-

low for ﬁne-grain vertex-by-vertex resolution control.

The idea of constructing scalable representations

and compression systems is well-known in signal

processing. In this context, wavelets play a major

role, being used to generate multiresolution repre-

sentations of an input signal and to build scalable

codecs based on them. Wavelet-based scalable codecs

include well-known examples for images (Said and

Pearlman, 1996; Munteanu et al., 1999; Taubman

and Marcellin, 2001), for video (Taubman and Za-

khor, 1994; Ohm, 1994; Andreopoulos et al., 2004),

and were introduced for surfaces by Lounsbery et

al. (Lounsbery et al., 1997). Essentially, Lounsbery et

al. established the link between subdivision schemes

and multiresolution analysis for meshes. Subdivi-

sion schemes result in semi-regular meshes, however,

whereas models are most efﬁciently represented using

irregular meshes, allowing for adaptive sampling of a

surface. This avoids oversampling near areas lack-

ing high frequencies, and undersampling where high

frequencies are present. One solution is to convert ir-

regular meshes to semi-regular ones in a preprocess-

ing step (Payan et al., 2015). Semi-regular mesh cod-

ing allows for superior compression performance due

to the implicit connectivity knowledge (Khodakovsky

et al., 2000; Denis et al., 2010). However, not ev-

ery application allows for a lossy remeshing step,

and the inherent oversampling associated with semi-

regularity results in a larger number of vertices and,

consequently, a higher rendering cost.

Few wavelet transforms for irregular meshes have

been proposed in the literature. Bonneau describes a

generalization of Haar-wavelets in (Bonneau, 1998).

Valette et al. (Valette et al., 1999; Valette and Prost,

2004a) propose an extension of the work of Louns-

bery et al., describing a new subdivision approach

which is not restricted to 1-to-4 subdivision, but al-

lows creating one, two, three or four triangles from a

single triangle. In (Valette and Prost, 2004b), Valette

et al. describe how this wavelet transform can be

efﬁciently encoded, resulting in the state-of-the-art

Wavemesh codec. Recently we proposed a wavelet-

based irregular mesh codec (El Sayeh Khalil et al.,

2016) which we will refer to as EMD+16 in the

remainder of this paper. This codec produced im-

proved coding performance over Wavemesh in low

and midrange bitrates, and additionally tackled a

new untouched issue: whereas scalable mesh cod-

ing mainly focuses on minimizing the distortion for a

given amount of bits, efﬁcient rendering requires min-

imizing the distortion for a given triangle limit, which

is directly related to a given memory limit.

In this paper we build upon EMD+16. Observ-

ing that the decoding of higher-range bits only yields

marginal improvements in visual quality, we focus on

the low and midrange rates and examine the possibil-

ity and the effect of performing a rate-distortion op-

timized storage and transmission of wavelet subband

bitplanes. Rate-distortion performance is optimized

by identifying at every encoding step the wavelet sub-

band bitplane which maximally reduces distortion in

the reconstructed mesh at minimum cost in bits.

3 FEATURE-PRESERVING

IRREGULAR MESH CODING

In (El Sayeh Khalil et al., 2016), we describe

EMD+16, a wavelet-based irregular mesh coding sys-

tem with implicit preservation of important visual fea-

tures. We reiterate the most important aspects of this

codec, of which an overview is given in Figure 1.

3.1 Wavelet Transform

The wavelet transform is based on the lifting scheme

proposed by Sweldens (Sweldens, 1998). This

scheme is composed of three steps to perform the for-

ward wavelet transform.

Downsampling and Retriangulation. A low-pass

ﬁltering step splits vertices into even and odd ver-

tices. Each odd vertex is surrounded by even neigh-

bouring vertices, forming a patch. After downsam-

pling, the remaining polygonal patches are retriangu-

lated while implicitly preserving geometric features,

Rate-Distortion Optimized Wavelet-based Irregular Mesh Coding

213

Wavelet

Transform

Geometry Coder

Connectivity Coder

Base Mesh Coder

Wavelet Subband Coder

Figure 1: Overview of the original mesh coder. The transform step generates a base mesh and a sequence of wavelet subbands.

In this work, we modify the coding components to allow for an unrestricted storage and transmission order.

resulting in a lower-resolution approximation of the

higher-resolution mesh. Inversely, the upsampling

step adds a vertex per patch, and trivially retriangu-

lates each patch by connecting each border vertex

with the newly added odd vertex.

Prediction Step. The high-frequency data is pre-

dicted using the local patch information, determining

a prediction error per patch. To allow the same pre-

dictions during reconstruction, additional information

is required to determine the low-resolution patches.

Additional connectivity data is common to all irreg-

ular mesh codecs, but the higher cost per triangle is

compensated by requiring fewer triangles overall.

Update Step. The low-frequency data can ﬁnally

be post-processed to avoid aliasing artefacts. In the

given codec however, this step is chosen to be a null-

operation to avoid smoothing artefacts around geo-

metric features which should remain sharp.

3.2 Wavelet Subband Coder

The incremental data consists of two key components.

Firstly, the connectivity information has to be en-

coded to allow for locating the patches. For every

edge of the intermediate mesh, a single bit describes

if the edge was a new edge which was constructed

by retriangulation, or if the edge was present in the

higher-resolution mesh. Secondly, the wavelet coef-

ﬁcients are represented as vectors in global orthonor-

mal space and indicate the prediction errors of the odd

vertices. Due to Successive Approximation Quanti-

zation (SAQ), the wavelet coefﬁcients can be coded

bitplane by bitplane. The quantizers are deﬁned as:

(w) =

(

s(w)

|w|

∆

, if

|w|

∆

> 0

0, otherwise

(1)

with 2

∆ the quantization cell width, ξ the width of

the deadzone, and s() the sign function.

The connectivity information and wavelet coef-

ﬁcients are both encoded using an octree structure.

This octree structure decouples the encoding process

from any mesh traversal order and allows for exploit-

ing spatial locality. Within the octrees, the connectiv-

ity samples are encoded at the edge middles, while the

geometry samples are encoded at the predicted posi-

tions. For the encoder and decoder to operate in a

synchronized fashion, a template mesh is used to map

the samples. (El Sayeh Khalil et al., 2016) does not

give requirements for this, but indicates that the pre-

vious resolution is used for the template mesh.

4 GEOMETRY-INDEPENDENT

TEMPLATE MESHES

The original design of EMD+16 allows for resolution

scalability and only for a limited form of quality scal-

ability: at each resolution, the decoder can determine

the amount of quality bits for the newly added ver-

tices; however, a new resolution can only be decoded

after fully decoding the previous resolution. This is

a disadvantage of using this previous resolution as a

template mesh for coding the connectivity informa-

tion and wavelet coefﬁcients of the next resolution.

Dependencies between data blocks are shown in Fig-

ure 2(a), clearly showing that this approach allows

only one order in which data can be transmitted, de-

picted in Figure 2(b).

Instead, we explore quality-scalable encoding of

irregular meshes where the wavelet subbands are en-

coded at possibly different quality levels. Moreover,

the subband bitplanes are encoded in a rate-distortion

optimal manner, ensuring minimal distortion in the re-

constructed mesh for any target bitrate.

We propose to use the base mesh as a ﬁrst template

mesh again. However, instead of reconstructing each

resolution using its previous resolution, effectively

doing an identical inverse transform on the template

mesh as performed on the actual mesh, we propose to

perform a modiﬁed inverse transform which only uses

connectivity information. This results in a template

mesh which increases in resolution by only predict-

ing the added vertices without requiring the decoded

GRAPP 2017 - International Conference on Computer Graphics Theory and Applications

214

increasing resolution

increasing quality

(0)

(1)

(2)

(a) (b)

increasing resolution

increasing quality

(0)

(1)

(2)

Figure 2: Data dependencies and coding orders. Fig. 2(a)

shows dependencies in EMD+16, and Fig. 2(b) the only

possible coding order; Fig. 2(c) shows dependencies in the

proposed framework, Figs. 2(d), 2(e) and 2(f) possible cod-

ing orders. For resolution j, C

( j)

represents the connectivity

data and G

( j)

the geometry data at bitplane i.

wavelet coefﬁcients to reposition these vertices. The

result is that the connectivity and geometry of a reso-

lution can be decoded as soon as the connectivity of

the previous resolution is known. The resulting de-

pendencies are depicted in Figure 2(c): the encoder

is able to store the data blocks in an unrestricted or-

der as long as (a) the order of connectivity informa-

tion blocks is maintained, (b) geometry information

within each resolution is stored in the correct order,

and (c) connectivity information for a speciﬁc reso-

lution is encoded before geometry information. This

allows for storing and transmitting the blocks in any

order: resolution per resolution as before (Fig. 2(d)),

bitplane by bitplane or purely quality-scalable (Fig.

2(e)), or in any arbitrary order that preserves the de-

pendencies as indicated above (e.g., Fig. 2(f)).

5 RATE-DISTORTION

OPTIMIZATION

Rate-distortion optimization requires encoding data

blocks such that the distortion is minimal at all bit-

rates. Such distortion optimization depends on the

distortion measure used and as such does not yet have

an unambiguous solution. Our aim is to show that

such optimizations are indeed possible by proposing

a rate-distortion optimization algorithm, proving that

an optimized subband bitplane storage and transmis-

sion order is enabled by our codec architecture.

We construct our optimization algorithm by con-

sidering the distortions introduced by the wavelet

transform. With N

( j)

the number of wavelet coefﬁ-

cients for resolution j, the remaining distortion D

( j)

related to this j

resolution decoded up to bitplane p

is given by:

( j)

∑

i=1

( j)

i,p

, (2)

with d

( j)

i,p

the distortion on the odd vertex o

( j)

when

the most signiﬁcant bits of wavelet coefﬁcient i of res-

olution j are decoded up to the p

bitplane, i.e.,

( j)

i,p

= |x

( j)

−

( j)

i,p

= |x

( j)

− (

( j)

+ w

( j)

i,p

(3)

In Equation 3,

( j)

is the predicted position of odd

vertex o

( j)

and

( j)

i,p

is its reconstructed position when

decoding the most signiﬁcant bitplanes of the accom-

panying wavelet coefﬁcient w

( j)

up to bitplane p. To

simplify notations, we drop the superscript j and sub-

script i as the following paragraphs always handle a

speciﬁc odd sample i of a speciﬁc resolution j.

If we denote the total number of bitplanes by p

max

then w

max

= 0, i.e., when no bitplanes are encoded,

the wavelet coefﬁcient is zero and the prediction is not

corrected: d

max

= |x −

x|. The distortion becomes

exactly 0 as soon as the least-signiﬁcant bitplane is

decoded: w

= w = x −

x and d

= |x − (

x + w

)| =

|x − (

x + x −

x)| = 0.

The wavelet coefﬁcients are encoded using SAQ,

so let q

= Q

(w); the wavelet coefﬁcient up to bit-

plane p can be dequantized as w

= Q

−1

), with:

−1

) =

(

0, q

= 0

s(q

)



| −

+ δ



∆, q

6= 0

(4)

where 0 ≤ δ < 1 determines the placement of w

within the quantization cell. In the current implemen-

tation, it is chosen to be δ = 0.5.

To ﬁnd weights α for the wavelet coefﬁcients,

we ﬁrst assign weights β

to all vertices. These

weights indicate an estimation of the effect on the

full-resolution mesh of repositioning vertices. After

downsampling, the position of each removed odd ver-

tex o

( j)

is inﬂuenced by the positions of its neigh-

bours. Hence, the weights of these neighbour ver-

tices of o each increase by β

/d with d the amount

of neighbours. The weight of a wavelet coefﬁcient is

given by its accompanying odd vertex: α = β

At the highest resolution, each vertex (and conse-

quently each wavelet coefﬁcient) has a weight of 1/N.

If the downsampling terminates at a tetrahedral base

mesh counting four vertices, each vertex (and con-

sequently each wavelet coefﬁcient on average) has a

weight of 1/4. This indicates that a single wavelet

Rate-Distortion Optimized Wavelet-based Irregular Mesh Coding

215

( j)

p+1

( j)

p+1

( j)

Figure 3: Generic rate-distortion curve. With decreasing

bitplane p the rate increases and the distortion decreases.

coefﬁcient at a lower resolution is equally valuable as

a multitude of wavelet coefﬁcients at a higher reso-

lution. Note that the use of such weights is similar

to the scaling of wavelet coefﬁcients prior to encod-

ing – see e.g. (Said and Pearlman, 1996), rendering

biorthogonal transforms approximately unitary.

The complete algorithm for assigning weights is

presented in Algorithm 1.

Algorithm 1: Assigning weights to vertices.

1: for all v ∈ M do

2: v.β ← 1/N

3: end for

4: while downsampling do

5: for all v

∈ M

i,o

6: d ← valence(v

)

7: for all v ∈ neighbourhood(v

) do

8: v.β ← v.β + v

.β/d

9: end for

10: end for

11: end while

At each resolution, a rate-distortion curve such as

shown in Figure 3 can be found. Optimization comes

down to considering the rate-distortion curves for ev-

ery resolution, and coding at every step the informa-

tion which introduces the largest distortion decrease

at the lowest rate. With P

the last encoded bitplane

of resolution j, we would encode the next bitplane of

resolution j

with

= argmax

j:P

6=0

( j)

− D

( j)

−1

( j)

−1

− R

( j)

(5)

Up to this point, we did not handle connectiv-

ity blocks. To start decoding a speciﬁc resolution,

the connectivity information of all previous resolu-

tions has to be decoded. This decoding comes at

a rate but does not introduce a distortion decrease

in mean-squared-error sense. Furthermore, as the

most-signiﬁcant bitplanes are mostly zero, decoding

these highest bitplanes also requires rate often with-

out decreasing distortion either. Hence, Equation 5 is

adapted to consider multiple bitplanes together to ﬁnd

the most optimal slope. We generalize the deﬁnition

of P

to encompass any data block required to encode

resolution j. Consequently, P

will start at p

max

+1 as

it counts for all p

max

wavelet subband bitplanes and

an additional data block for the connectivity informa-

tion. With L the ﬁrst unencoded resolution, we will

encode k

bitplanes of resolution j

using:

( j

) = argmax

j∈[0,L],k∈[1,P

]

( j)

− D

( j)

−k

( j)

−k

− R

( j)

(6)

For this, we make two conventions:

• R

( j)

max

= 0;R

( j)

max

= R

( j)

conn

6= 0

• D

( j)

max

= D

( j)

max

This states that encoding the connectivity informa-

tion, i.e., the ﬁrst data block of a resolution, intro-

duces a rate of R

( j)

conn

while not decreasing the dis-

tortion; this corresponds to adding vertices to the de-

coded mesh without reﬁning their locations. The dis-

tortions D

( j)

,∀p ∈ [0, p

max

] are deﬁned in Equation 2.

6 EXPERIMENTAL EVALUATION

To evaluate our codec, we assume a raw storage of the

base mesh. In practice, the wavelet transform stops

at a resolution which is small enough to beneﬁt from

scalability, while remaining qualitative enough to be

used as a base resolution. To evaluate, however, the

transform is applied until no more downsampling can

be performed. The base mesh is included in the re-

sults by counting B = 3Qn

+3dlog

)en

bits, with

Q the amount of quantization bits per vertex, n

the

number of vertices in this base mesh and n

the num-

ber of triangles. The base meshes were clearly neg-

ligible, hence the produced coding numbers can be

entirely ascribed to the wavelet subband codec itself.

We have used the METRO tool (Cignoni et al.,

1998) to obtain the distortion values, reported as

root-mean-squared (RMS) errors, while rates are ex-

pressed in amount of bits per vertex (bpv). The result-

ing rate-distortion curves have been made convex by

removing non-convex ratepoints.

6.1 Accuracy of the Template Mesh

Using a template mesh which does not take into ac-

count geometry information, i.e., a template which

develops using only connectivity information, had an

effect on the implementation: possibly overlapping

vertices in the template mesh have to be handled prop-

erly, by repositioning these vertices equally by both

the encoder and decoder without relying on the en-

counter order. Note that these geometric modiﬁca-

tions do not alter the geometry of the decoded meshes

GRAPP 2017 - International Conference on Computer Graphics Theory and Applications

216

Table 1: Additional cost when using an accurate template.

All models have been encoded using 12 bit precision.

Model (#verts) Rate increase

teapot (1,292) +0.068bpv (+0.21%)

beethoven (2,521) +0.187bpv (+0.55%)

triceratops (2,832) −0.017bpv (-0.052%)

elk (5,194) +0.015bpv (+0.049%)

fandisk (6,475) −0.015bpv (-0.057%)

maxplanck (7,399) −0.013bpv (-0.044%)

venushead (8,268) +0.022bpv (+0.074%)

bimba (8,857) +0.094bpv (+0.31%)

horse (19,851) +0.022bpv (+0.087%)

screwdriver (65,538) +0.182bpv (+0.89%)

rabbit (67,039) +0.073bpv (+0.32%)

dino (129,026) +0.074bpv (+0.37%)

Average +0.23%

after inverse wavelet transformation; they can only al-

ter where samples are encoded in the octrees.

Both the use of this updated template mesh as well

as the occasional repositioning of overlapping sample

positions can have an inﬂuence on the exploitability

of the spatial correlations within the data. However,

the effects of using the new geometry-agnostic tem-

plate proved to be negligible.

This can also be seen in the ﬁnal bitrates. Table 1

lists the changes in lossless bitrates for several mod-

els. We observe that the bitrate increases on aver-

age by 0.23%, while in some cases the bitrate even

goes down. These results show that lowering the ac-

curacy of the template mesh does not hinder the ex-

ploitation of spatial correlations, on the contrary it

is clear that topological locality information is pre-

served. Vertices that are located closely together will

have mapped vertices in the template mesh which

are also located closely together due to topological

proximity, albeit possibly at another global position

(hence, falling within a different octree cell) due to

geometry information not being taken into account.

6.2 Rate-distortion Optimization

Rate-distortion optimization orders the data such that

the quality gains come at the lowest rates. The results

for the three models considered in (El Sayeh Khalil

et al., 2016) are given in Figure 4. For fandisk and

horse the improvements compared with a resolution-

scalable transmission order are clear at low bitrates,

obtaining similar qualities at a lower cost. At higher

bitrates the improvements are minimal. This indi-

cates that the resolution-scalable transmission at high

bitrates is in general already nearly optimal in rate-

distortion sense.

Visual results of our proposed codec are given in

Figure 5. As it was also observed in (El Sayeh Khalil

et al., 2016), the information in the lower resolution

wavelet subbands contributes the most to the shape

of such densely sampled models, while higher resolu-

tions only serve to increase the quality when render-

ing from very nearby: Figure 5(c) already resembles

very closely the original mesh depicted in Figure 5(d).

In theory, a rate-distortion optimization should al-

ways be on par or better; in practice this is the case

if the optimization algorithm calculates distortions in

the spatial domain. Our results show that an opti-

mization based on the proposed weighted wavelet co-

efﬁcients while ignoring topological information also

proves to give superior results. The coding overhead

is minimal: the storage increases only a fraction of a

bit per vertex, due to the need to identify the resolu-

tion of each subsequent data block.

6.3 State of the Art Comparison

For comparison, we mainly investigated the

overall improvement over EMD+16. As in

(El Sayeh Khalil et al., 2016), we also compare

with Wavemesh (Valette and Prost, 2004b) and

IPR (Valette et al., 2009). EMD+16 was imple-

mented using ratepoints at every resolution; this is

more coarse-grain than our proposed codec which

gives a ratepoint after every bitplane. For Wavemesh

we made use of the publicly available software with

wavelet geometrical criterion enabled. We also made

a comparison with IPR for the models for which we

received their decoded results.

Figure 6 shows a comparison with the state of the

art. It shows even more competitive results brought

by the proposed codec compared with EMD+16. In

the case of the feature-rich model fandisk, our results

after rate-distortion optimization improve over both

Wavemesh and IPR. In the case of horse, the results

are entirely on-par with or better than previous work,

even at the lowest rates. Finally, for the feature-poor

model rabbit, results remain nearly unchanged com-

pared with EMD+16; IPR remains the better solution.

Results over a small set of models are summa-

rized in Table 2. This table makes use of a measure

similar to the Bjøntegaard delta rate (Bjøntegaard,

2001), as was also described in (El Sayeh Khalil et al.,

2016). It interpolates the ratepoints within a limited

rate range, samples the distortion values and measures

the differences in rate at these samples. This way we

can ﬁnd a maximal, minimal and average value of

these differences, with a positive difference indicating

that a state-of-the-art codec requires more bits for the

same quality, and a negative difference indicating that

the state-of-the-art codec outperforms our proposed

codec. In this case, the limited rate range over which

Rate-Distortion Optimized Wavelet-based Irregular Mesh Coding

217

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5 6

distortion (RMS forward+backward)

bitrate (bpv)

with RDO

without RDO

(a) Fandisk

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5 6

distortion (RMS forward+backward)

bitrate (bpv)

with RDO

without RDO

(b) Horse

0.002

0.004

0.006

0.008

0.01

0 1 2 3 4 5 6

distortion (RMS forward+backward)

bitrate (bpv)

with RDO

without RDO

Figure 4: Rate-distortion optimization.

(a) 0.25bpv (0.16% tris) (b) 0.5bpv (0.73% tris) (c) 3bpv (6.8% tris) (d) 33.3bpv (241 607 tris)

Figure 5: Visual results at speciﬁc rates. Figures 5(a), 5(b) and 5(c) show the model at increasing rates. Observe that 3bpv is

still in the low bitrate range considering a lossless rate of 33.3bpv for this 18 bit quantized fertility model shown in 5(d).

Table 2: Rate savings w.r.t. the state of the art. To obtain

the same quality at rates up to 3bpv, the numbers indicate in

bpv (∆

avg

) the average rate savings, (∆

max

) the largest rate

savings, and (∆

min

) the smallest rate savings. A positive

value means more rate is required than the proposed codec,

a negative value means the proposed codec performs worse.

Model Coder ∆

avg

∆

max

∆

min

teapot

EMD+16 +0.22 +1.00 0.0

(1,292)

beethoven

EMD+16 +0.32 +0.81 0.0

(2,521)

triceratops EMD+16 +0.07 +0.59 0.0

(2,832) Wavemesh −0.72 +0.43 −1.30

elk EMD+16 +0.16 +0.60 0.0

(5,194) Wavemesh +0.10 +1.20 −0.28

fandisk EMD+16 +0.08 +0.73 0.0

(6,475) Wavemesh +0.25 +0.93 +0.04

IPR +0.43 +2.90 −0.03

maxplanck EMD+16 +0.09 +0.39 0.0

(7,399) Wavemesh −0.25 +0.04 −0.37

venushead

EMD+16 +0.06 +0.73 −0.04

(8,268)

bimba EMD+16 +0.10 +0.39 0.0

(8,857) Wavemesh +0.28 +1.1 +0.14

horse EMD+16 +0.10 +0.39 0.0

(19,851) Wavemesh +0.15 +0.84 −0.17

IPR +2.40 +7.60 +0.55

screwdriver EMD+16 +0.01 +0.78 0.0

(65,538) Wavemesh −0.04 +0.01 −2.1

rabbit EMD+16 +0.011 +0.29 0.0

(67,039) Wavemesh +0.06 +1.3 0.0

IPR −0.15 +0.40 −0.80

dino EMD+16 +0.02 +0.94 0.0

(129,026) Wavemesh −0.08 0.0 −2.50

we measure is taken up to 3bpv for the proposed

codec. Furthermore, our rate-distortion optimization

and quality-scalable decoding improves all results of

EMD+16. At such low bitrates, relatively high gains

of up to 1bpv are obtained. Observe that the minimal

rate difference is zero as both the proposed codec and

EMD+16 start at the same base mesh, resulting in the

same distortion at the same rate. This conﬁrms that

our rate-distortion optimized codec never performs

worse than EMD+16, as the minimal differences are

never negative. The comparison with Wavemesh and

IPR shows that in most cases the proposed codec is

also more favourable at these low bitrates, except for

dino where we are on par at best.

7 CONCLUSIONS

We have shown how to achieve quality scalabil-

ity and optimized rate-distortion performance for the

wavelet-based irregular mesh codec EMD+16 which

employs octree-based encoding of both connectivity

and geometry information. Gains up to 1bpv are ob-

tained at low bitrates, while at higher bitrates the orig-

inal codec design is nearly optimal in rate-distortion

sense. Furthermore, the results and working imple-

mentation serve as a proof-of-concept that an unre-

stricted storage and transmission of subband bitplanes

can be provided using the proposed framework.

ACKNOWLEDGEMENTS

The research activities as described in this paper were

funded by Ghent University, iMinds, Flanders Inno-

GRAPP 2017 - International Conference on Computer Graphics Theory and Applications

218

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5 6

distortion (RMS forward+backward)

bitrate (bpv)

Proposed

EMD+16

Wavemesh

IPR

(a) Fandisk

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5 6

distortion (RMS forward+backward)

bitrate (bpv)

Proposed

EMD+16

Wavemesh

IPR

(b) Horse

0.002

0.004

0.006

0.008

0.01

0 1 2 3 4 5 6

distortion (RMS forward+backward)

bitrate (bpv)

Proposed

EMD+16

Wavemesh

IPR

Figure 6: Comparison with the state of the art.

vation & Entrepreneurship (VLAIO), the Fund for

Scientiﬁc Research-Flanders (FWO-Flanders), and

the European Union.

REFERENCES

Andreopoulos, Y., Munteanu, A., Barbarien, J., van der

Schaar, M., Cornelis, J., and Schelkens, P. (2004). In-

band motion compensated temporal ﬁltering. Signal

Process. Image, 19(7):653–673. Special Issue on Sub-

band/Wavelet Interframe Video Coding.

Bjøntegaard, G. (2001). Calculation of average PSNR

differences between RD-curves. Technical Report

VCEG-M33, ITU-T SG16/Q6, Austin, TX, USA.

Bonneau, G.-P. (1998). Multiresolution analysis on irregu-

lar surface meshes. IEEE Trans. Vis. Comput. Graph-

ics, 4(4):365–378.

Cignoni, P., Rocchini, C., and Scopigno, R. (1998).

METRO: measuring error on simpliﬁed surfaces.

Comput. Graph. Forum, 17(2):167–174.

Denis, L., Satti, S. M., Munteanu, A., Cornelis, J., and

Schelkens, P. (2010). Scalable intraband and compos-

ite wavelet-based coding of semiregular meshes. IEEE

Trans. Multimedia, 12(8):773–789.

El Sayeh Khalil, J., Munteanu, A., Denis, L., Lambert,

P., and Van de Walle, R. (2016). Scalable feature-

preserving irregular mesh coding. Comput. Graph.

Forum.

Hoppe, H. (1996). Progressive meshes. In Proc. 23rd SIG-

GRAPH Conf. Computer Graphics, pages 99–108.

Khodakovsky, A., Schr

oder, P., and Sweldens, W. (2000).

Progressive geometry compression. In Proc. 27th

SIGGRAPH Conf. Computer Graphics and Interactive

Techniques, pages 271–278.

Lounsbery, M., DeRose, T. D., and Warren, J. D. (1997).

Multiresolution analysis for surfaces of arbitrary topo-

logical type. ACM Trans. Graph., 16(1):34–73.

Munteanu, A., Cornelis, J., Van der Auwera, G., and

Cristea, P. (1999). Wavelet-based lossless compres-

sion scheme with progressive transmission capability.

Int. J. Imag. Syst. Tech., 10(1):76–85.

Ohm, J. R. (1994). Three-dimensional subband coding with

motion compensation. IEEE Trans. Image Process.,

3(5):559–571.

Pajarola, R. and Rossignac, J. R. (2000). Compressed pro-

gressive meshes. IEEE Trans. Vis. Comput. Graphics,

6(1):79–93.

Payan, F., Roudet, C., and Sauvage, B. (2015). Semi-regular

triangle remeshing: A comprehensive study. Comput.

Graph. Forum, 34(1):86–102.

Peng, J. and Kuo, C.-C. J. (2005). Geometry-guided pro-

gressive lossless 3D mesh coding with octree (OT)

decomposition. In Proc. 32nd SIGGRAPH Internat.

Conf. Computer Graphics and Interactive Techniques,

pages 609–616.

Said, A. and Pearlman, W. A. (1996). A new, fast, and

efﬁcient image codec based on set partitioning in hier-

archical trees. IEEE Trans. Circuits Syst. Video Tech-

nol., 6(3):243–250.

Sweldens, W. (1998). The lifting scheme: A construction

of second generation wavelets. SIAM J. Math. Anal.,

29(2):511–546.

Taubman, D. and Zakhor, A. (1994). Multirate 3-D sub-

band coding of video. IEEE Trans. Image Process.,

3(5):572–588.

Taubman, D. S. and Marcellin, M. W. (2001). JPEG

2000: Image Compression Fundamentals, Standards

and Practice. Kluwer Academic Publishers, Norwell,

MA, USA.

Touma, C. and Gotsman, C. (1998). Triangle mesh com-

pression. In Proc. Graphics Interface Conf., pages

26–34.

Valette, S., Chaine, R., and Prost, R. (2009). Progressive

lossless mesh compression via incremental paramet-

ric reﬁnement. Comput. Graph. Forum, 28(5):1301–

1310.

Valette, S., Kim, Y.-S., Jung, H.-Y., Magnin, I., and Prost,

R. (1999). A multiresolution wavelet scheme for ir-

regularly subdivided 3d triangular mesh. In Proc. In-

ternat. Conf. Image Processing, pages 171–174.

Valette, S. and Prost, R. (2004a). Wavelet-based multires-

olution analysis of irregular surface meshes. IEEE

Trans. Vis. Comput. Graphics, 10(2):113–122.

Valette, S. and Prost, R. (2004b). Wavelet-based progressive

compression scheme for triangle meshes: Wavemesh.

IEEE Trans. Vis. Comput. Graphics, 10(2):123–129.

Rate-Distortion Optimized Wavelet-based Irregular Mesh Coding

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