Exploiting Material Properties to Select a Suitable Wavelet Basis

for Efﬁcient Rendering

Jeroen Put, Nick Michiels and Philippe Bekaert

Hasselt University - tUL - iMinds, Expertise Centre for Digital Media, Wetenschapspark 2, 3590 Diepenbeek, Belgium

Keywords:

Rendering, Wavelets, Parameterisation, Materials.

Abstract:

Nearly-orthogonal spherical wavelet bases can be used to perform rendering at higher quality and with signif-

icantly less coefﬁcients for certain spherical functions, e.g. BRDF data. This basis avoids parameterisation

artifacts from previous 2D methods, while at the same time retaining high-frequency details in the light-

ing. This paper demonstrates the efﬁciency of this representation for rendering purposes. Regular 2D Haar

wavelets can still occasionally perform better, however. This is due to their property of being fully orthogonal.

An important novelty of this paper lies in the introduction of a technique to select an appropriate wavelet basis

on-the-ﬂy, by utilising prior knowledge of materials in the scene. To show the inﬂuence of different bases on

rendering quality, we perform a comparison of their parameterisation error and the compression performance.

1 INTRODUCTION

Research in computer graphics has led to mathemat-

ical models and algorithms to render photorealistic

images (Dutr

´

e et al., 2006). A challenging aspect

of rendering algorithms is how to parameterise the

functions that are used in the rendering equation (Ka-

jiya, 1986). For instance, in an all-frequency relight-

ing framework, it is particularly important to repre-

sent all the details in the visibility, BRDF and lighting

functions. Previous approaches use harmonic anal-

ysis to approximate the lighting, but fail to capture

sharp details like shadows and specular highlights. To

provide these, representations based on 2D wavelets

have been introduced. These representations suffer

from distortion artifacts, because a completely uni-

form one-to-one mapping of the spherical to the pla-

nar domain is not available. Their inherent compact-

ness is particulary important for inverse rendering ap-

plications, since the estimation of geometry, lighting

and material from a set of uncalibrated images is a

challenging problem and requires a lot of processing

power and storage.

Earlier work focused either on improving the efﬁ-

ciency of the underlying representation or enhancing

the performance of triple product calculations. Our

contribution in this paper is the combination of these

ideas. The nearly-orthogonal basis avoids parame-

terisation artifacts, while at the same time retaining

high-frequency details in the lighting. On the other

hand, Ng.’s triple product (Ng et al., 2004) provides

an efﬁcient manner to simulate complex lighting in-

teraction from precalculated datasets. It is through

the combination of precisely these ideas that we are

able to perform rendering with more complex spa-

tially varying BRDFs and more intricate lighting. An-

other contribution of this paper is the exploitation of

prior knowledge of scene materials to choose an ap-

propriate wavelet basis. We have observed during our

experiments that specular and diffuse BRDFs require

different bases for optimal approximation.

2 PREVIOUS WORK

Rendering requires the processing of vast quantities

of data. Often such data consists of visibility, BRDF

and lighting information. An efﬁcient representation

is required for rendering with high quality and per-

formance. We focus on inverse rendering as an ap-

plication in this paper, as the problem of large vol-

umes of data is even more pronounced for this cate-

gory of techniques. Earlier techniques used spherical

harmonics, extending Fourier decomposition to the

spherical domain (Okabe et al., 2004; Yu et al., 2006).

Spherical harmonics have the disadvantage that they

can only represent low frequencies. For high frequen-

cies, the number of coefﬁcients required grows expo-

nentially. This makes solving the triple product (Ng

et al., 2004) integral inefﬁcient. Okabe et al. (Ok-

218

Put J., Michiels N. and Bekaert P..

Exploiting Material Properties to Select a Suitable Wavelet Basis for Efﬁcient Rendering.

DOI: 10.5220/0004717202180224

In Proceedings of the 9th International Conference on Computer Graphics Theory and Applications (GRAPP-2014), pages 218-224

ISBN: 978-989-758-002-4

Copyright

c

2014 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Quality comparison for two different datasets. Leftmost column: environment map and meshes. Two central

columns: Rendering with optimal choice of basis (green) versus 2D parameterisation (red). Rightmost column: Zoomed

images to compare quality. Our method reduces parameterisation errors. See also the video accompanying this paper.

abe et al., 2004) showed an alternative representation

based on 2D Haar wavelets. Haber et al. (Haber et al.,

2009) use a 2D Haar wavelet basis to represent the

BRDF and visibility functions for every vertex in the

scene and to approximate the illumination environ-

ment map for every viewpoint. The main advantage

is that 2D Haar wavelets are able to capture high fre-

quency detail. However, they also suffer the disad-

vantage that there is no one-to-one mapping from the

planar domain to the spherical domain, resulting in

distortion when using 2D wavelets to represent spher-

ical functions. To minimise the distortion, octahedral

parametrisation (Praun and Hoppe, 2003a) is often

used.

Sweldens et al. (Schr

¨

oder and Sweldens, 1995)

proposed the construction of various wavelet basis

functions on the surface of the sphere. This construc-

tion is based on his earlier work on the lifting scheme.

He concluded that lifted spherical wavelets are par-

ticulary performant for the representation of BRDF

functions. This paper draws from these ideas, but con-

tributes by combining the triple product integration

from Ng. (Ng et al., 2004) with a nearly-orthogonal

wavelet basis.

A large body of work has been dedicated to the

analysis and representation of BRDFs (Ruiters and

Klein, 2010; Bilgili et al., 2011). However, these

representations are not particularly suitable for other

spherical functions, such as the illumination. Tsai et

al (Tsai et al., 2008) developed an importance sam-

pling strategy to sample products from illumination

and BRDF with spherical radial basis functions. This

technique is limited in its application, due to the

costly process of ﬁtting their basis functions. Their

method also focuses on double products, where we

require triple product integral calculations in our for-

ward rendering.

The remainder of this paper is organized as fol-

lows. Section 3 shows how a Haar wavelet basis can

be constructed on a sphere and explains which sub-

division scheme is required. Section 4 presents the

results, where a comparison is made between the new

spherical Haar wavelet basis and the original 2D Haar

wavelet basis. Finally we complete this paper by pre-

senting our conclusions and future work in Section 5.

ExploitingMaterialPropertiestoSelectaSuitableWaveletBasisforEfficientRendering

219

3 SPHERICAL HAAR WAVELETS

Many functions in graphics are naturally expressed in

the spherical domain. Like previous methods, in our

rendering application the light at every surface point

x is expressed as a triple product integral:

B(x,ω

0

) =

Z

Ω

L(x, ω

i

)V (x,ω

i

)ρ(x,ω

i

,ω

o

)(ω

i

·n(x))dω

i

(1)

In this equation, B is the radiance as a function of

position x and outgoing direction ω

o

. L and V are the

lighting and visibility functions respectively, ρ is the

BRDF and n is the surface normal. To avoid the need

for separate environment maps per vertex, we assume

furthermore that L is a distant illumination function

and incorporate the term (ω

i

·n(x)) into V, so equation

1 becomes:

B(x,ω

0

) =

Z

Ω

˜

L(ω

i

)V (x,ω

i

)ρ(x,ω

i

,ω

o

)dω

i

(2)

L, V and ρ are all functions on the spherical domain

Ω.

˜

L is the globally deﬁned environment map, rotated

into the local frame of V and ρ.

Current techniques use 2D Haar wavelets and are

able to represent details with relatively few coefﬁ-

cients. On the other hand, they introduce distortion

artifacts due to the parameterisation step of the spher-

ical domain to the planar domain. Often, the hemi-

octahedral parameterisation by Praun and Hoppe is

used to perform this mapping (Praun and Hoppe,

2003b):

V (ω) =

∑

i

V

i

Ψ

i

(ω),

˜

L(ω) =

∑

j

˜

L

j

Ψ

j

(ω),

ρ(~x,ω) =

∑

k

ρ

k

(~x)Ψ

k

(ω)

(3)

where Ψ is an appropriate basis on the octahedron.

In this paper, V , ρ and L are projected into a spherical

wavelet basis Ψ. This way, we can write equation 2

in terms of these basis functions:

B(x,ω

0

) =

∑

i

∑

j

∑

k

L

i

V

j

ρ

k

Z

Ω

Ψ

i

(ω)Ψ

j

(ω)Ψ

k

(ω)dω

(4)

Ψ

i

(ω), Ψ

j

(ω) and Ψ

k

(ω) are the tripling coefﬁcients,

as deﬁned by Ren et al. (Ng et al., 2004). More re-

cent work pointed out the existence of a generalized

wavelet product integral (Sun and Mukherjee, 2006).

3.1 Wavelet Basis and Subdivision

Scheme

To avoid artifacts, sampling of the spherical function

domain should happen as uniform as possible. In this

paper, an octahedral subdivision structure was cho-

sen, because of its favourable sampling characteristics

and because of symmetric sampling of hemispheric

functions.

The wavelet basis is deﬁned over the triangles of

the subdivision scheme. Each of the octahedron oc-

tants is represented by a quadtree of wavelet coef-

ﬁcients. In each subdivision step, every triangle is

split into four children, using the geodesic bisector

criterium as deﬁned by Sweldens et al. (Schr

¨

oder and

Sweldens, 1995).

Let T

k

be such a triangle at depth k (k=0 for the

lowest level). T

k+1

0

, T

k+1

1

, T

k+1

2

and T

k+1

0

are the four

children of T

k

. Figure 2 shows an example of such a

triangle.

Figure 2: Decomposition of four pixel values into one

scaling coefﬁcient and three wavelet coefﬁcients (Bonneau,

1999).

Here, Φ

k

represents a scaling function. It is a con-

stant function with value 1 on T

k

and zero otherwise.

Ψ

k

1

, Ψ

k

2

and Ψ

k

3

are the three wavelet functions, with

T

k

as their support.

A fundamental property of the above triangular

Haar wavelets, is that every piecewise constant func-

tion over the four subtriangles (T

K+i

,i = 0, 1, 2, 3)

can be expressed as a linear combination of the con-

stant scaling function and the three wavelet func-

tions. From this, a local reconstruction can be de-

rived, which converts this combination of functions

to colour values again. The conversion of a scaling

function and three wavelet functions into colour val-

ues is called synthesis:

x

k+1

0

x

k+1

1

x

k+1

2

x

k+1

3

=

1 r

k

01

r

k

02

r

k

03

1 r

k

11

r

k

12

r

k

13

1 r

k

21

r

k

22

r

k

23

1 r

k

31

r

k

32

r

k

33

x

k

y

k

1

y

k

2

y

k

3

(5)

Here, r

k

i j

is the value of ψ

k

j

for the subtriangle T

k+1

i

.

Analysis can be performed by inverting the foregoing

synthesis matrix.

GRAPP2014-InternationalConferenceonComputerGraphicsTheoryandApplications

220

Figure 3: Compression with Bonneau spherical wavelets.

Left: original; Right: reconstruction with 5% of the coefﬁ-

cients.

3.2 Choosing a Suitable Basis

During our evaluation, we experimented with var-

ious Haar wavelet bases. The ﬁrst representation

were the Bio-Haar wavelets introduced by Sweldens

et al. (Schr

¨

oder and Sweldens, 1995). This basis has

the property of being semi-orthogonal:

Z

ψ

k

1

φ

k

=

Z

ψ

k

2

φ

k

=

Z

ψ

k

3

φ

k

= 0 (6)

Semi-orthogonality is a necessary condition for the

existence of a wavelet basis over a certain domain.

This paper has opted for the nearly-orthogonal Bon-

neau wavelet basis (Bonneau, 1999). This basis is,

in the limit of its subdivision, guaranteed to provide

good compression performance when truncating the

least signiﬁcant coefﬁcients. At the same time it is

a convenient basis for a triple product integral imple-

mentation. Figure 3 demonstrates the compression on

a spherical image.

More interesting bases, like SOHO

wavelets (Lessig, 2007), also provide orthogo-

nality:

Z

ψ

k

1

ψ

k

2

=

Z

ψ

k

1

ψ

k

3

=

Z

ψ

k

2

ψ

k

3

= 0 (7)

Orthogonality guarantees that the best approximation

of the L

2

-norm can be achieved when removing the

least signiﬁcant coefﬁcients. They also have the ad-

vantage of a fast matrix inversion for the analysis

stage. SOHO wavelets provide orthogonality, but do

so at the cost of increased computation for their subdi-

vision scheme to create triangles of equal area, while

not providing upper bounds on the distortion of image

quality of this scheme.

0

50

100

150

200

250

300

350

L1-norm

6500 BRDF slices; sorted worst to best approximated

2D Haar

Bio

Bonneau

Pseudo Haar

(a)

0

100

200

300

400

500

600

L1-norm

6500 BRDF slices; sorted worst to best approximated

2D Haar

Bio

Bonneau

Pseudo Haar

(b)

0

200

400

600

800

1000

1200

L1-norm

6500 BRDF slices; sorted worst to best approximated

2D Haar

Bio

Bonneau

Pseudo Haar

(c)

Figure 4: Comparison of compression performance in the

L

1

-norm of four wavelet bases on a database with (a) a

representative mixture of BRDF slices, (b) more specular

BRDF slices and (c) more diffuse BRDF slices. 5% of co-

efﬁcients were retained. A smaller norm is better. Our ap-

plication chooses the best basis on-the-ﬂy, by taking the ma-

terial properties into account. As a consequence we always

get the best possible representation.

ExploitingMaterialPropertiestoSelectaSuitableWaveletBasisforEfficientRendering

221

(a) (b) (c)

Figure 5: Comparison of a diffuse BRDF approximation.

2% of coefﬁcients retained. Smaller L

2

-norm is better. (a)

original; (b) spherical wavelets (L

2

-norm: 2,2739); (c) 2D

Haar-wavelets (L

2

-norm: 2,5831).

(a) (b) (c)

Figure 6: Comparison of specular BRDF approximation.

2% of coefﬁcients retained. Smaller L

2

-norm is better. (a)

original; (b) spherical wavelets (L

2

-norm: 0,9323); (c) 2D

Haar-wavelets (L

2

-norm: 0,8895).

4 RESULTS

This section will compare the performance and qual-

ity of the nearly-orthogonal spherical wavelet repre-

sentation in a rendering context. Good compression

performance guarantees the need for less coefﬁcients.

Therefore, this shall be the main focus in this section.

Figure 4 shows a comparison of compression per-

formance of ﬁve wavelet bases in the L

1

-norm on a

database of BRDF functions. In literature, it is argued

that the L

1

-norm often corresponds better to the vi-

sual perception by humans observers (Antonini et al.,

1992). The ﬁve wavelet bases are the lifted Bio-Haar

basis introduced by Sweldens et al. (Schr

¨

oder and

Sweldens, 1995), the regular 2D Haar wavelets used

by Haber et al. (Haber et al., 2009), the pseudo Haar

spherical wavelet basis suggested by Ma et al. (Ma

et al., 2006) and the two nearly-orthogonal Bonneau

spherical wavelets (Bonneau, 1999). We do not in-

clude a comparison with a spherical harmonics basis,

as its unsuitability for representing high frequencies

has already been proved in earlier literature (Ng et al.,

2004). Every representation performed compression

with a ﬁxed budget of coefﬁcients, where the least

signiﬁcant coefﬁcients were truncated. We used the

MERL BRDF database (Matusik et al., 2003; Ngan

et al., 2005) for rendering since it contains a repre-

sentative mixture of Lambertian, Cook-Torrance and

Lafortune BRDF slices. The chart sorts the BRDF

functions from worst to best approximated. It can be

observed that the Bio-Haar basis is signiﬁcantly out-

performed in nearly all cases by the other bases. Both

Bonneau and pseudo Haar bases have better compres-

sion performance than the 2D Haar basis. The pseudo

Haar basis performs identical to the Bonneau basis.

To conﬁrm our observation that spherical wavelets

compress diffuse BRDF slices better than very specu-

lar BRDF slices, we customised the content of the in-

ternal BRDF database. This results in two additional

graphs. The ﬁrst graph is displayed in Figure 4(b)

and shows a scenario with signiﬁcantly more specu-

lar BRDF slices. The graph shows that the spheri-

cal wavelets perform slightly worse than the 2D Haar

wavelets. The second graph is shown in Figure 4(c)

and demonstrates a scenario with predominantly dif-

fuse BRDF slices. The graph clearly demonstrates

that there are two groups of BRDF approximations.

The approximation yields large norms for specular

BRDFs and lower norms for more diffuse BRDFs.

The contents of the database has an inﬂuence on how

the two groups are distributed on the graph. This indi-

cates that the choice of wavelet basis can signiﬁcantly

enhance the compression performance if prior knowl-

ege of material properties in the scene is available.

Therefore the selection of an appropriate basis can be

guided by the location of the cut-off that separates the

two groups.

For the next part of our evaluation, we select some

noteworthy BRDF slices out of the BRDF databse. A

BRDF slice is deﬁned as a 2D subfunction of a 6D

BRDF deﬁned over the hemisphere of all incomming

directions In Figure 5, a comparison of diffuse BRDF

slice approximation is shown between the Bonneau

spherical wavelet basis and the 2D Haar wavelet ba-

sis. It can be seen that the spherical wavelets provide

13,59% better compression performance in this case.

Figure 6 shows another comparison, with a

strongly specular BRDF. This time, 2D Haar wavelets

outperform spherical wavelets, albeit marginally.

This is a trend that was observed during our exper-

iments. We believe this is due to three main fac-

tors. First, spherical wavelets approximate the origi-

nal signal with the triangles of the underlying octahe-

dral subdivision scheme, while 2D Haar wavelets use

rectangular areas (pixels). A large portion of specular

BRDFs tend to have a circular shape, due to the dis-

tribution of light around the specular lobe. It was ob-

served in our implementation that the circular shape

can be approximated by fewer coefﬁcients with rect-

angular areas.

Second, specular BRDFs with a large specular ex-

ponent subtend only a small solid angle on the hemi-

GRAPP2014-InternationalConferenceonComputerGraphicsTheoryandApplications

222

sphere. It is therefore less likely that the specular lobe

will cross areas of non-uniform sampling due to the

parameterisation method. In the cases such specular

BRDFs do overlap such areas, they are expected to

undergo severe distortion.

Finally, in the aforementioned case where very

specular BRDFs avoid parameterisation artifacts, the

2D Haar wavelets are expected to slightly outperform

the spherical wavelets. This can be attributed to the

fact that the 2D Haar basis is a fully orthogonal basis,

while the nearly-orthogonal wavelet bases are only or-

thogonal in the limit of the subdivision. As demon-

strated by Sweldens et al. (Schr

¨

oder and Sweldens,

1995), BRDF functions can be represented with very

few coefﬁcients and the approximation therefore of-

ten takes place on a low number of subdivisions of

the underlying octahedral structure.

To have a more clear measure of the parameterisa-

tion error, independent from different wavelet repre-

sentations, we performed another experiment shown

in Figure 8. Here, we ﬁrst sampled the BRDF

slices uniformly on the unit sphere with a statistical

method (Dutr

´

e et al., 2006) as a ground truth. We

then sampled the same slices with the octahedron as

well as with the 2D parameterisation method. Fig-

ure 8 shows the accumulated error for both parame-

terisations in comparison with the ground truth. We

can conclude that avoiding parameterisation artifacts

yields better quality approximation.

Figure 7 demonstrates the effectiveness of the

spherical wavelets for different compression ratios.

This graph can be used to read the reduction in coefﬁ-

cients for a certain norm. For example in Figure 7(a),

to reach an L

1

-norm of 20, the sperical wavelets use

9% less coefﬁcients. A ﬁrst observation is that for a

relatively diffuse BRDF, as shown in (a), the spher-

ical wavelet outperforms the 2D Haar wavelet over

all compression ratios. Second, for a more specu-

lar BRDF, as shown in (b), both wavelets perform

equally well. If we perform this experiment on the

entire BRDF database, spherical wavelets require on

average 20% less coefﬁcients. In the best case we ob-

served 40% reduction in coefﬁcients. A comparison

between our basis selection algorithm and regular 2D

parameterisation is shown in Figure 1.

5 CONCLUSIONS

This paper has introduced the use of nearly-

orthogonal spherical wavelets to perform rendering

at higher quality and lower computational cost. Our

evaluation shows that spherical wavelets often outper-

form 2D Haar wavelets, because they avoid inherent

0

20

40

60

80

100

120

140

50

45

40

35

30

25

20

15

10

5

L1-norm

% of coefficients retained

2D Haar

Bonneau

(a)

0

20

40

60

80

100

120

140

50

45

40

35

30

25

20

15

10

5

L1-norm

% of coefficients retained

2D Haar

Bonneau

(b)

Figure 7: Wavelet performance for different compression

ratios: (a) almost diffuse BRDF; (b) specular BRDF. It is

shown that material properties are a good indicator for an

on-the-ﬂy choice of compression ration.

parameterisation errors. By utilising prior knowledge

of scene materials, we can adaptively choose an ap-

propriate wavelet based on a specularity criterium.

Separation of materials in diffuse and specular pro-

vides only a rough categorisation, but in practise we

observed that compression performance of spherical

versus 2D Haar wavelets is strongly linked with these

properties.

ExploitingMaterialPropertiestoSelectaSuitableWaveletBasisforEfficientRendering

223

0

20

40

60

80

100

120

140

160

180

cumulative error

6500 BRDF slices

Octahedron

2D

Figure 8: Cumulative parameterisation error compared to the ground truth. Octahedron (spherical) parameterisation outper-

forms regular 2D parameterisation.

ACKNOWLEDGEMENTS

This work has been made possible with the help of

a PhD specialization bursary from the IWT. The au-

thors acknowledge ﬁnancial support from the Euro-

pean Commission (FP7 IP SCENE).

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