CONTROL POLYGON BASED TEXTURE SYNTHESIS ON

BIQUADRATIC BĒZIER RATIONAL SURFACES

Rupesh N. Shet, H. E. Bez and E. A. Edirisinghe

Department of Computer Science, Loughborough University, UK

Keywords: Texture synthesis Discrete Wavelet Transform (DWT), EZW, Rendering, Bézier surfaces.

Abstract: Existing texture synthesis algorithms fail to deliver effectively in application areas where progressive

rendering of texture is required. To provide a practical solution to this problem we propose a novel

algorithm for progressive-texture synthesis on surfaces, which makes use of the Embedded Zero-tree of

Wavelet (EZW) idea proposed by Shapiro et al., 1993 which is capable of prioritising the coefficients of a

DWT decomposed image according to their visual significance. We demonstrate the use of the proposed

algorithm on texturing a single biquadratic surface and two smoothly joined biquadratic surfaces. It is

further shown that the proposed texture synthesis approach on Bézier patches allows the algorithm's general

use in texture synthesis on many common surface topologies and can be generalised for arbitrarily shaped

surfaces. We provide experimental results to prove the effectiveness of the proposed approach, when

synthesising textures of regular, irregular and stochastic nature. Further experimental results are provided to

illustrate the practical use of the proposed texture synthesis algorithm in resource constrained application

domains.

1 INTRODUCTION

Texture synthesis provides a practical solution for

data acquisition and is often used to enhance realism

of artificially created scenes. As result, a number of

‘image based' texturing algorithms have been

proposed in the past decade. A typical texture

synthesis algorithm starts from a sample image and

attempts to produce a larger texture with a visual

appearance similar to the sample, by repeated

placement of micro patterns of texture elements. It

does this in a way that when perceived by an

observer, the synthesized texture appears to be

generated by the same underlying stochastic process.

However all texture synthesis algorithms are

challenged by the high statistical variability of

textures involved in synthesis. Thus a universal

solution to fast texture synthesis yet remains an open

problem. Texturing surfaces provides further

challenges and attracted much research interest in

the recent past due to applications in computer

graphics, animated movie production, computer

games, education, architecture, computer art and

virtual productions.

A major proportion of research in the area of

texture synthesis has focused on synthesizing texture

on planner surfaces. Recently a number of

approaches have been proposed for texture synthesis

on surfaces. These texture synthesis approaches can

be broadly classified into two groups, namely, pixel

based (Wei & Levoy, 2000, 2001; Turk, 2001;Ying

et al., 2001; Ashikhmin et al., 2001; Tong et al.,

2002; Shet et al., 2006; Lefebvre and Hoppe, 2006 )

and patch based ( Neyret and Cani, 1999; Praun et

al., 2000; Soler et al., 2002; Sebastian et al., 2003;

Wang et al., 2005; Wing Fu et al., 2005) approaches.

Pixel based approaches consider a pixel as the basic

unit in the synthesis process. Patch based approaches

are an alternative to pixel based approaches where

an attempt is made to synthesis texture by copying

selected regions of pixels from the sample texture

and stitching them together. This approach

overcomes the limitations of the pixel based

approaches, i.e. being limited to work with certain

types of textures and the lack of computational

speed. Neyret-Cani’s 1999, technique is based on

precomputed triangular texture samples which are

mapped non-periodically. However this method is

restricted to synthesising isotropic textures. In the

lapped texture technique proposed by Praun et al,

2000, the texture patches are first oriented and are

subsequently placed in an overlapping fashion on a

45

N. Shet R., E. Bez H. and A. Edirisinghe E. (2008).

CONTROL POLYGON BASED TEXTURE SYNTHESIS ON BIQUADRATIC BÄŠZIER RATIONAL SURFACES.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 45-52

DOI: 10.5220/0001096400450052

Copyright

c

SciTePress

surface with a predefined vector field. The method

works for a limited set of textures. Soler et al., 2002

introduces hierarchical texturing to overcome

drawbacks of previous algorithm. The method is

capable of capturing low-frequency pattern while

preserving high frequency randomness in the

texture. The synthesis time may vary from few

minutes to few tens of minutes. Sebastian et al.,

2003 separated the texture pre-processing from

synthesis and proposed two independent phases. Pre-

processed texture is stored on a disk and used when

needed. This process is very slow but only needs to

be performed once. The pre-processing time vary

from minutes to a few tens of minutes. Further

storing the texture on a disk is essential. Wang et al.,

2005 algorithm is mainly based on global conformal

parameterization of surfaces, where the textures are

preserved on surfaces without seams or cracks. This

algorithm is simple for texture synthesis but

parameterization process adopted is time consuming

thereby slowing down the overall performance.

Wing Fu et.al., 2005 introduced the concept of

Wang tiles. Initially a low distortion conformal map

is created from the input surface, which forms a

quad based geometry. The texture is then laid out on

quad surfaces, properly oriented and then mapped

back on to the surface. However this approach

inherits all drawbacks of the image quilting

algorithms.

All the above techniques are applied on irregular

shapes of triangular meshes, which results in seams

at edges. Size of triangles in the mesh also varies

which makes the visual artefacts on the surface,

prominent. Further to this, it will also use extensive

bandwidth in transmission media as triangular mesh

information and texture are in uncompressed format.

Further the animation of this triangular mesh is

difficult as they are rigid. To overcome many of the

above problems we have proposed to use NURBS, a

form of surface representation which helps to

compress a mesh and thus can be applied in

constrained bandwidth environments. NURBS also

provides additional facilities to animate the surface.

The inspiration of our work comes from the

present requirements for progressive texture

synthesis on surfaces, which results in extensive use

of transmission media with limited bandwidth for

modern application domains such as remote

visualisation, distributed/collaborative gaming etc.

Current texture synthesis algorithms on surfaces are

time consuming and fail to perform in

progressive/transform domain. To overcome this

problem we propose a progressive texture synthesis

algorithm using multiresolution DWT

decomposition, coefficient prioritisation using EZW

(embedded zero-tree wavelet) algorithm and surface

representation using biquadratic rational surfaces

which is falls under patch base category. We prove

the proposed novel algorithm is capable of creating

seamlessly varying quality levels of synthesized

texture on surfaces. According to the authors

knowledge it is a first attempt that demonstrates

progressive texture synthesis on meshes, which

utilises control polygons generated from the

biquadratic Bézier equations. We show that the

proposed work can be generalised to any type of

arbitrary mesh.

For clarity of presentation the paper is organised

as follows Section-2 introduces the reader to the

research background and fundamentals. Section-3

presents the proposed algorithm. Section-4 provides

experimental results and a detailed analysis. Finally,

Section-5 concludes, with an insight to possible

improvements and future variations.

2 RESEARCH BACKGROUND

For the purpose of clarity and ease of reference we

have summarized the fundamental techniques used

for multiresolution representation of texture (DWT)

(Wickramanayake et al., 2005), DWT coefficient

prioritization (EZW) (Shapiro et al., 1993) and

surface parameterization (biquadratic rational

surfaces) in this section. Hence readers who are

familiar with these concepts can forgo reading this

section.

2.1 DWT Representation of Texture

Image

Textured images contain a large amount of

perceptual data. Therefore the number of bits

required to represent/encode a texture image is high.

However typical images consist of a wide range of

frequency components spread throughout the human

visual frequency band. Some of theses frequency

components have a significant effect in human

perception while some others have very low

significance. Fortunately texture images are often of

this type. The Discrete Wavelet Transforms (DWT)

provide a compact multi resolution representation of

an image. It gives a signal representation in

correspondence to a narrow band, low frequency

range and some of the coefficients represent short

data lags corresponding to a wide band, high

frequency range. Using the concept of scale, data

representing a continuous trade off between space

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

46

and frequency can be made available for further

processing.

Figure 1: Transforming the sample texture into a multi-

resolution image representation. (a) Sample texture, (b)

single level decomposition, (c) two level decomposition,

(d) three level decomposition.

In our algorithm we use two-dimensional DWT.

To begin with, the texture image is subdivided into

four sub-bands using horizontal and vertical DWT

filters over the image pixels. The resulting sub bands

labeled LH1, HL1 and HH1 represent the finest

scale wavelet coefficient whereas the sub-band

labeled LL1 represents low resolution coefficients.

In order to obtain the next level of wavelet sub-

bands, the sub band labeled LL1 is further

decomposed and sampled using the vertical and

horizontal DWT filters. This process is repeated

until the required final decomposition is reached (see

Figure 1). The coefficients of the subbands are then

used for speeding up searching process and

prioritized using the EZW algorithm presented next.

2.2 Embedded Zerotree Wavelet

(EZW) Algorithm

The Zerotrees of wavelet coefficient concept was

originally introduced by Shapiro et al., 1993 in

progressive encoding of images. It is based on two

important observations:

1. Natural images in general have a low pass

spectrum. Therefore when an image is

wavelet transformed the energy in the

subbands decreases as the scale decreases

(low scale means high resolution), so the

wavelet coefficients will, on average be

smaller in the higher subbands than in the

lower subbands.

2. Large wavelet coefficients are visually more

important than smaller wavelet coefficients.

EZW provides a compact representation of

perceptually significant coefficients and multi

resolution construction capability of an image. The

idea is to organize DWT coefficients of an image

(see Figure 1) in a prioritized order of visual

significance, depending on their position and

magnitude in the DWT decomposition and to

subsequently encode the ordered list of coefficients

following an embedded coding algorithm. In an

embedded coding algorithm the encoder can

terminate the encoding at any point thereby allowing

a target bit rate or target distortion metric to be met

exactly. On the other hand, given a bit stream, a

decoder can cease decoding at any point in the bit

stream. Thus it is capable of producing exactly the

same image that would have been encoded at the bit

rate corresponding to the truncated bit stream.

In this paper we use the EZW algorithm’s initial

coefficient prioritization procedure to prioritize their

use within texture synthesis algorithm. Due to space

limitation we refer readers interested in the detail of

the EZW coefficient prioritization algorithm to

Shapiro et al., 1993. We show that the visually

prioritized availability of coefficients enables

seamless progressive texture synthesis capability on

biquadratic rational surface using control polygon.

This is the main contribution of our present work.

2.3 Biquadratics Bézier Surfaces

This section will briefly introduce biquadratic

surface patches and construction of simple surface

using them. For more details the readers are referred

to Bez H.E, 2006.

Rational parametrisation is a de-facto standard

representation in computer graphics and geometric

modelling software, allowing portability across

applications and systems. In addition to possessing

desirable geometric properties, rational

parametrisation

♦ requires the evaluation of only polynomial

functions,

♦ gives rise to a compact data-structure,

♦ facilitates interactive control of shape,

♦ is complete in the sense that approximation

of any shape to a specified tolerance δ can

CONTROL POLYGON BASED TEXTURE SYNTHESIS ON BIQUADRATIC BÉZIER RATIONAL SURFACES

47

be achieved, and exact parametrisation (i.e.

δ= 0) is often possible.

Rational parametrisations of surfaces comprise

local atlases, or patches, of the form:

()

(

)

() ()

(2.1)1,ts,0,

st

st

t)τ(s,

n.m

0jk,

jk,m.jn.k

n.m

0jk,

*

jk,m.jn.k

ωbb

vbb

<<=

∑

∑

=

=

where

ω

jk ,

are the weights and

v

jk ,

are the

Bernstein vectors. If all the weights are non-zero this

may be expressed as

()

()

()

()

)2.2(1t,s0,)t,s(

m.n

0j,k

j,kj.mk.n

m.n

0j,k

j.mk.n

st

st

bb

bb

j,k

j,k

<<=

∑

∑

=

=

ω

ω

ν

τ

where

ω

jk

jk

jk

v

v

,

*

,

,

=

are the Bézier vertices. The values of

n and m determine the degree of the

parametrisation; if n = m = 2 the patch is said to be

biquadratic and if n = m = 3 it is bi-cubic.

With given nine control points we compute and

draw the biquadratic surface patch defined by them.

(see Figure 2)

(a) (b)

Figure 2: (a) Green Colour: Biquadratic control polygon

point, Red Colour: Smooth surface mesh generated using

control polygon (b) Control polygon mesh (for 9 control

points generate 4 faces).

Many of the desirable geometric properties of

rational representation, e.g. the convex hull property

and the existence of bézier vertices, are lost if

negative or zero weights occur - hence, in computer

graphics and geometric modelling applications,

positive weight parametrisations are always

preferred. For computational efficiency, low degree

parametrisations are desirable.

3 PROPOSED METHODOLOGY

In this section we provide the design details of the

proposed texture synthesis algorithm.

3.1 Texture Synthesis on Control

Polygon

Figure 3 illustrates the basic block diagram of the

proposed algorithm. The texture synthesis process

starts by applying a n-level (n=3 used in our

experiments) 2D DWT (e.g. Haar Transform) on

sample texture image, which is denoted as I

sample

,

The application of single level 2D DWT on the

sample texture results in a set of component images

i.e. sub-bands, as follows:

(

)

(

)

)1.3(DWT,,,

IIIII

sample1HH1LH1HL1LL

−

=

Figure 3: Proposed block diagram for texture synthesis on

biquadratic surfaces.

Where

IIII

HHLHHLLL 1111

,,,

are the texture image

sub-bands corresponding respectively to low-

resolution approximation, vertical details, horizontal

details and diagonal details of sample texture.

Similarly 2

nd

level and 3

rd

level decomposition are

obtained by applying DWT to the low-resolution

sub-bands of previous decomposition level. This can

be mathematically represented as follows.

(

)

(

)

)2.3(,,,

12222

−=

IIIII

LLHHLHHLLL

DWT

Sample

LL3 and HH3

3

rd

Levelled DWT

LL3 and HH3 bands

Paste block of shape and

size of face on control

polygon /

EZWIDWT

Best Match

Location

Find next best match

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

48

(

)

(

)

)3.3(,,,

23333

−=

IIIII

LLHHLHHLLL

DWT

(

)

(

)

)4.3(,,,

1111

−=

++++

IIIII

LLnHHnLHnHLnLLn

DWT

from n-level we extract low-resolution and diagonal

detail bands i.e.

I

LLn

and

I

HHn

.

We generalise the notation used in equation

(3.1)-(3.6)as

I

pl

where

}{

HHLHHLLLp ,,,

∈

and

}{

n,..2,1,0l ∈ where p represent the sub-bands

within each decomposition level (LL-low resolution,

LH-horizontal, HL- vertical, HH diagonal) and l

represent the decomposition level. DWT represent

the forward discrete wavelet transform.

The basic idea of proposed algorithm is to

synthesise texture on control polygons of a given

surface. Let

B

yxLLn ),(

represent a general polygonal

block of decomposed sample image located at

position (x, y) relative to the sub-bands (p, l)’s

origin.

Initially we randomly pick block

B

yxLLn ),(

from

the sample texture image with identical size and

shape to that of the control polygon face. This

randomly created texture block is mapped to the

control polygon surface shown in figure 3. [Note:

the details of the mapping process are described in

section 3.2.]

In locating the next block to be synthesized, we

cut a 8 pixel wide template of pixels along the edge

of already synthesized, neighbouring blocks, apply a

3 level DWT decomposition on the template block

and extract low-resolution and diagonal detail bands

which are used for searching in sample LL3 and

HH3 bands. Best matching block can be found by

minimizing the L2 norm. EZW algorithm can be

used if coefficient prioritisation is used to further

reduce complexity.

In general, if

)1,1( yxpl

B

and

)2y,2x(pl

B

are two

randomly shaped blocks to be matched, we

say

)1,1( yxpl

B

is the best match for

)2,2( yxpl

B

if

),(

)2,2()1,1( yxplyxpl

BBd

is minimum for all

possible

pl

B blocks, which is calculated as,

)5.3(

}])i(B)i(B[

])i(B)i(B[{

)B,B(d

Bi

2

)2y,2x(3HH)1y,1x(3HH

2

)2y,2x(3LL)1y,1x(3LL

)1y,1x()2y,2x(

−

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

∂−∂+

∂−∂

=

∑

∂∈

Where

pl

B∂ an edge is zone of block

),( yxpl

B

and

i is an element (coefficient) within the edge

zone.

Finally the overlap area of the best matching

edge is blended with the overlap area on the original

block using alpha bending. The non-overlapping

area of the block is picked from the sample texture

and subsequently appended to the synthesized

texture. This process will continue till all the faces

of control polygon are mapped. In some cases we

have considered two or more overlapping areas for

finding the best match.

3.2 Texture Mapping on Control

Polygons

In our implementation initially we create

propagating seed vertex directions, which are then

used to smooth the surface vector field. However

alternatively a number of other surface vector field

techniques (Wei-Levoy, 2001; Turk 2002; Ying etal,

2001) can be used to replace the approach we have

selected above. Once vector fields are assigned to

all control polygon faces, we then rotate all the faces

according to tangential vector field and surface

normal, thus placing all faces in the same 2D plane.

Using a modified version of Soucy et al., 1996,

approach (Note: modified from using triangle to

using polygon) a texture map. T is created. For each

face of the control polygon, we map it to a

corresponding face in T in compact form, i.e. with

no space being wasted. The faces in T are textured

using the corresponding best matching block. The

faces in T that we use are of non-uniform size that

are a better fit to the shape and size. It is noted that

the resulting texture can be rendered on the control

polygon surfaces at interactive rates. The images

illustrated in Figure 4 were rendered in this manner

using 256 x 256 textures. The models used in our

experiments are composed of smooth surfaces

having between 100 to 1000 faces, whereas the

control polygons used consisted of 4 faces to 8 faces

(It can be further increase to n faces). We have

observed that these surfaces render at real-time rates.

3.3 Projection of Texture from Control

Polygons to Biquadratic Rational

Surfaces

Firstly we calculate the distance between control

points of the control polygon using the standard

distance formula between two points in 3D space.

Depending on these distances we calculate relative

location of projections of these points on the rational

surface, parameterised by

0,1

s

t<<

. Using the

correspondence between points we then decide on

the area projection from control polygonal mesh to

the smooth surface. Figure 4(a) illustrates the texture

synthesized onto the control polygon using the

CONTROL POLYGON BASED TEXTURE SYNTHESIS ON BIQUADRATIC BÉZIER RATIONAL SURFACES

49

proposed algorithm and figure 4(b) illustrates the

mapped texture onto the smooth rational surface..

Figure 4: (a) Texture on control polygon (b) Projection

from control Polygon to smooth surface.

Note that closer the control polygon to the smooth

surface representation, lesser distortion in projection

will occur and vice –versa.

3.4 Progressive Texture Using EZW

In order to achieve progressive texture synthesis on

surfaces, we adopt Shapiro’s EZW idea in which

coefficient values with magnitudes above a given

threshold are considered significant. This threshold

(t) is calculated using equation (3.6) based on the

magnitude of wavelet coefficients of decomposed

sample image.

()

)6.3(Kt

2

y,xMAX

LL

log

n

2

−=

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

where MAX() means the maximum coefficient

value, K is a constant and

()

yx

LL

n

, denotes a

general coefficient in

LL

n

sub-band. By only

considering the coefficients of sub-bands, which are

larger than the threshold and ignoring all others (i.e.

setting to zero), an inverse DWT is calculated to

produce the texture at a given progressive texture

quality. This can be expressed generally as:

()

)7.3(

thershold

,,

,

EZWIDWT

BB

BB

B

HHnHLn

LHn,LLn

1LLn

−

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

=

−

The above equation can produce discrete quality

levels of texture depending on threshold or number

of coefficients need to be considered. Image quality

can be increased by decreasing the threshold and

vies-versa. Note that the function EZWIDWT above

represent an EZW constrained inverse discrete

wavelet transform. When progressive texture

synthesis is required we replace the normal texture

mapping process with the above EZW based

approach (see figure 5 & 6).

4 EXPERIMENTAL RESULTS &

ANALYSIS

In order to analyse the performance of proposed

algorithm and to show that surfaces can be rendered

effectively, we have implemented the proposed

algorithms in OpenGL, C++.

Experiment were performed on a diverse range

of texture samples that include regular, near-regular,

irregular and stochastic (Lin et al., 2004) textures.

Results illustrated in figure 5 indicate the ability of

proposed technique to efficiently map and

synthesized texture on surfaces, with minimal

artifacts. Further as matching and searching is

performed in wavelet domain, the texture synthesis

is fast. Textures illustrated in Figure 5 (a), (b), (c)

respectively belong to near-regular, regular, and

stochastic categories. Similar synthesized quality

levels are demonstrated for all three texture

categories. Further analysis revealed that the time

required to synthesize these texture is in the range of

few milliseconds.

To further extend the functionality of the

proposed method, we have extended our work to

progressive texture synthesis on surfaces. We have

preformed a wide range of experiments (see figure 6

& 7) to show that texture can be synthesized at

seamlessly different levels of quality on surfaces,

without consuming noticeable processing time.

Figure 6 illustrates the synthesis of a stochastic

texture of a flower. It is evident from the results that

only 10% of information from sample texture is

sufficient to create a texture with sufficiently rough

quality. By increasing the percentage of coefficients

further, the quality of the synthesized texture can be

seamlessly improved. Further experiments revealed

that for this texture, 20% of coefficients was

sufficient to synthesize a texture visually equal to

the texture that can be synthesized when all

coefficients are utilized. Progressive texture

synthesis gives the added advantage of being able to

truncate a bit stream representing the sample texture

at any intermediate stage, still being able to

synthesize texture at some intermediate quality level.

To further illustrate the application of the

proposed idea, we have extended our approach to

synthesizing texture on two smoothly joined

biquadratic rational surfaces, shown in figure 7.

Figure 7(c) shows two smoothly join biquadratic

patches. Figure 7(d) to 7(k) illustrates progressive

texture synthesis on this surface. This proves that

our technique can be extended to the many

geometric topologies. Results in figure 7 further

illustrates using regular and near-regular texture

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

50

samples that texture variations across patch

boundaries are smooth.

Figure 5: Texture synthesis on biquadratic surface.

5 CONCLUSIONS

We have introduced a novel DWT based approach to

synthesizing, texture and progressive texture, on

biquadratic surfaces. We have presented the methods

and algorithms in detail along with possible

applications and advantages. The proposed method

has the capability of synthesizing texture at

seamlessly different quality settings, a functionality

which is not possible via existing state-of –the art

techniques.

The use of visual prioritisation of information in

the sample image during texture synthesis allows the

task to be carried out at a higher speed but at an

equivalent visual quality level. We show that the

proposed approach is computationally efficient,

results in good quality texture synthesis, and is

applicable in bandwidth-adaptive/compressed-

domain applications such as remote visualization.

We have shown that the control polygon strategy

used can be extended to cover synthesizing texture

on many 3D objects with arbitrary surface topology.

We are currently in the process of generalizing the

proposed algorithm to address this issue.

Figure 6: Stochastic progressive texture on biquadratic

surface.

a1-3% coefficients

a2-5% coefficients

a3-10% coefficients

a4- ALL coefficients

(a) near-regular

(b) regular

(c) stochastic

CONTROL POLYGON BASED TEXTURE SYNTHESIS ON BIQUADRATIC BÉZIER RATIONAL SURFACES

51

Figure 7: Progressive texture on two-joined biquadratic

surface.

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(a) near-regular

(b) regular

(c) smoothly joint mesh

(d) – 1% coefficients (e) – 2% coefficients

(f) – 6% coefficients

(g) ALL coefficients

(h) – 2% coefficients (i) – 4% coefficients

(j) – 6% coefficients (k) ALL coefficients

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