Two-stage Color Texture Synthesis using the Structure Tensor Field
Adib Akl
1,2
, Charles Yaacoub
2
, Marc Donias
1
, Jean-Pierre Da Costa
1
and Christian Germain
1
1
Bordeaux University, IMS Lab, UMR CNRS 5218, Talence cedex, France
2
Faculty of Engineering, Holy Spirit University of Kaslik (USEK), Jounieh, Lebanon
Keywords: Texture Synthesis, Color Structure Tensor, Color Texture, Multi-scale, Non Parametric Synthesis.
Abstract: Since reproducing the realism of the physical word is a major goal for computer graphics, color texture
synthesis is important for rendering synthetic images and animations. Most of the existing synthesis
techniques provide impressive results in many cases, but fail in difficult situations with large patterns, or
with long range directional variations. Based on a previously developed two-stage structure/texture
synthesis algorithm where the structure tensor is used to represent the structure layer, an extension to color
texture synthesis is proposed. Two different methods are used for the computation of the color structure
tensor field. An acceleration method for the proposed algorithm is also presented. Results show that the
proposed approach successfully synthesizes the output texture in many situations where traditional
algorithms fail to reproduce the exemplar’s patterns and dynamics. These promising results pave the way
towards 3D color textures synthesis showing multi-scale patterns.
1 INTRODUCTION
Texture synthesis has been particularly dynamic
with different applications in computer vision
including image extrapolation, restoration, editing
and compression. It has also been extended to video
completion/merging and animations, and the
description of the geometry of a surface (Bargteil et
al., 2006, Yamauchi et al., 2003, Bertalmio et al.,
2000, Winkenbach et al., 1994). Most of graphics
applications require color texture synthesis to
represent real word textures observed under different
lighting conditions. In rendering, textures can mimic
the surface details of real objects, ranging from
varying the surface’s color, perturbing the surface
normal, to actually deforming the surface geometry.
In pen and ink style illustrations, textures can
delineate the tone, shade, and pattern of objects.
With hand-drawn pictures, most scanned images are
of inadequate size and can lead to visible seams or
repetitions if they are directly used for texture
mapping (Winkenbach et al., 1994).
Several recent 2D texture synthesis algorithms
(Portilla & Simoncelli, 2000, Wei & Levoy, 2000,
2003, Paget & Longstaff, 1998, Kwatra et al., 2003,
Kopf et al., 2007, Vanhoey et al., 2013, Efros &
Freeman, 2001, Han et al., 2006) achieved success
in modeling a large panel of textures, including
stochastic and structured textures. For instance, a
Markov Random Field texture modeling method is
proposed in (Paget & Longstaff, 1998). It
mathematically captures the visual characteristics of
a texture into a unique statistical model that
describes the interactions between pixel values. A
synthesis algorithm based on copying patch regions
from the sample to the output is proposed in (Kwatra
et al., 2003). The method uses a graph cut technique
to determine the patch region without choosing its
size a-priori, in contrast to other existing methods. In
(Portilla & Simoncelli, 2000) an over complete
complex wavelet transform is used to parameterize
the model by a set of statistics, in the frequency
domain, corresponding to basic functions at adjacent
locations, orientations and scales. The 2D texture
synthesis method in (Wei & Levoy, 2000, 2003)
models the texture as a realization of a local and
stationary random process. The algorithm starts from
an input texture and an output image initialized by a
white random noise. The texture is synthesized in a
scan-line order. The neighborhood of each output
pixel is captured and the most similar neighborhood
is searched for in the exemplar based on the
Euclidian distance. Then the corresponding pixel is
copied to the target position in the output texture.
Many of the above 2D synthesis techniques were
extended to the 3D environment, i.e. solid texture
synthesis. Among such extensions, the non-
182
Akl A., Yaacoub C., Donias M., Da Costa J. and Germain C..
Two-stage Color Texture Synthesis using the Structure Tensor Field.
DOI: 10.5220/0005358301820188
In Proceedings of the 10th International Conference on Computer Graphics Theory and Applications (GRAPP-2015), pages 182-188
ISBN: 978-989-758-087-1
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
parametric approach of (Kopf et al., 2007), which
integrates histogram matching to help the global
statistics of the synthesized solid converge towards
those of the exemplar. Other parametric (Da Costa &
Germain, 2010) and non-parametric (Urs, 2013)
2D/3D extensions have also shown their efficiency
in many cases including simulation of atomic
structure of materials (Leyssale et al., 2009, 2012).
Most of the existing synthesis algorithms are
appropriate for color images by considering the three
channels (Red, Green and Blue) of the color texture
in the RGB model (Wei & Levoy, 2000, 2003,
Kwatra et al., 2003).
For the synthesis of structured anisotropic
textures, most of existing approaches tend to
produce more regular textures than the exemplar
(Kopf et al., 2007, Wei & Levoy, 2000). They are
hardly able to reproduce long range orientation
variations, dealing badly with non-stationary
textures presenting undulating, circular or laminar
structures. In this case, the prior synthesis of a
geometric layer may help in the synthesis of the
texture layer (Peyré, 2009). A two-step synthesis
approach consisting in first producing a structure
layer from the analysis of the original exemplar, and
then using this structure layer to constrain the
synthesis of the texture itself, is presented in (Akl et
al., 2014). In this method, the structure layer is
represented by the structure tensor field which
captures the dominant orientations and the degree of
local anisotropy in the texture.
Based on this latter algorithm, this paper presents
an extension to color texture synthesis using two
different approaches for the computation of the
structure tensor on color images. The proposed
method leads to a same or better quality results than
those obtained using a standard approach, with a
significant computational load reduction.
The remainder of the paper is organized as
follows. The computation of the color structure
tensor field is presented in Section 2. The proposed
two-stage color texture synthesis algorithm is then
described in Section 3. Experimental results are
discussed in Section 4, and finally conclusions are
drawn in Section 5.
2 THE COLOR STRUCTURE
TENSOR FIELD
The structure tensor T of a gray-scale image M is the
covariance matrix of the first partial derivatives of
M, and built from previously estimated gradient
fields M = [M
x,
M
y
]with M
x
= M*G
x
and M
y
= M*G
y
where G
x
and G
y
are Gaussian derivative kernels,
and (*) denotes convolution.
At a point (x, y), the structure tensor T(x, y) is
symmetric and can be written as:
,,
,;
,,
xx xy
xy yy
TxyTxy
Txy
TxyTxy
(1)
the tensor components T
xx
, T
xy
, and T
yy
are given by:
**;;*,
xx x x xy x y yy y y
T S MM T S MM T S MM
(2)
where S is a weighting function used for gradient
field smoothing.
The structure tensor can be interpreted as an
ellipse (Toujas et al., 2010) characterized by a shape
(or coherence) indicator and an orientation factor.
The former is given by:
12 12
,/CT T T T T
(3)
where λ
1
(T) and λ
2
(T) are the tensor eigenvalues.
The latter is computed from the eigenvector [e
x
, e
y
]
associated with λ
1
(T) as:
,/
-1
yx
tan eTe
(4)
The first stage of the texture synthesis algorithm
in (Akl et al., 2014) consists of synthesizing the
texture’s structure layer represented by the structure
tensor field. Therefore, an important issue in the
color synthesis extension is the computation of the
color structure tensor. (Zenzo, 1986) proposes a
tensor formulation for the gradient of a multi-
component image with the extraction of a single
vector (direction and magnitude of maximum
variation). (Weijer & Gevers, 2004) propose to add
the three structure tensor components computed for
each color channel in an RGB image. The same
applies in (García et al., 2008) with an additional
Gaussian smoothing.
In this paper, two different approaches are used
for the computation of a color image tensor-field.
The first approach consists of extracting the
luminance component (Y) from the color input
texture, as defined in the ITU-R BT.601
recommendation (ITU-R, 2011):
0.299 0.587 0.114 ,YRGB
(5)
where R, G, and B represent the Red, Green and
Blue components of the input image, respectively.
The structure tensor is then computed from the
luminance component the same way it is computed
from the gray-scale images.
The second method, used in this paper, relies on
Two-stageColorTextureSynthesisusingtheStructureTensorField
183
the additivity of tensors for different channels as
(Weijer & Gevers, 2004):
,;
CC
xx xy
C
CC
xy yy
Txy
TT
TT
(6)
the color tensor components
C
x
x
T ,
C
x
y
T
and
C
yy
T are expressed as:
,,
,,,
CC
ab ab
CRGB
ab xx xy yyTT
(7)
where
R
ab
T
,
G
ab
T and
B
ab
T are the structure tensor
components obtained on the three channels R, G,
and B of the color texture respectively.
3 THE PROPOSED TWO-STAGE
COLOR TEXTURE SYNTHESIS
The first stage of the proposed texture synthesis
algorithm consists in synthesizing the texture’s
structure layer represented by the color structure
tensor field. Hence, the non-parametric Wei and
Levoy (W&L) algorithm (Wei & Levoy, 2000),
which usually operates on scalar data, is adapted to
the specificities of tensor-valued images.
The algorithm starts by computing the input
structure tensor field from the color exemplar. Then
an output structure tensor field is initialized by
choosing randomly tensors from the input structure
tensor field. This field is modified in the synthesis
process to look like the input tensor field. Therefore,
the neighborhood of the output tensor (a vector of
tensors) is first captured, then the most similar
neighborhood is searched for in the input tensor
field, and the corresponding tensor is copied to the
output target position.
The same process is repeated for each output
tensor until all the tensors are determined (Akl et al.,
2014).
Due to its versatility, the metric in (8) is used to
measure the similarity between tensors
1
C
T
and
2
C
T
.
12
12 12
1
2
2
2
2
.
,
2
CC
xx xx y
CC CC
yy
CC
y
xy xy
TT T T T T
TT
(8)
The choice of the tensor neighborhood shape and the
scan type directly influences the resulting
synthesized tensor image. We either use a causal
neighborhood with a lexicographical scan type or a
square non-causal neighborhood with a completely
random walk (Wei & Levoy, 2000).
Since the neighborhood size has to be adequately
chosen in order to preserve texture structures, multi-
resolution image pyramids can be used to capture
the structures more compactly in lower resolution
pyramid levels (Wei & Levoy, 2000). This presents
an alternative method to the use of large
neighborhoods which makes the synthesis
computationally expensive while the number of
pyramid levels has as much influence as the
neighborhood size.
Figure 1: Illustration of the proposed two-stage color
texture synthesis algorithm.
The synthesized structure tensor field is used as
a constraint for the synthesis. Fig. 1 presents the
color texture synthesis method. The algorithm uses
as inputs the exemplar, its structure tensor field and
the synthesized structure tensor field. The algorithm
finds, for every output pixel, the pixel with the most
similar neighborhood in the input texture and copies
it to the target output position. In other words, each
neighborhood has two components: a pixel domain
neighborhood in the color texture image, and a
tensor domain neighborhood in the structure image
(Akl et al., 2014).
The neighborhood resemblance is measured by:
,1 ,,
in out in out
RpSSCDNN pSTDQQ
(9)
where p is a weight factor (0 ≤ p ≤ 1), N is the pixel
domain neighborhood component in the input (N
in
)
and output (N
out
) textures, and Q represents the
tensor domain component in the initial (Q
in
) and
synthesized (Q
out
) tensor field. SSCD is the Sum
Square Color Distance used for the pixel-domain
neighborhood:
Color Tensor Map
Structure-constrained Texture Synthesis
Tensor Synthesis
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2
12 1 2
1
22
12 12
,
,
N
b
RR
n
GG BB
SSCD N N N n N n
NnN n NnN n
(10)
where N
b
is the number of pixels within each
neighbourhood and
(),
C
i
Nn
1, 2 ,i
,,CRGB
represents the n
th
pixel within the neighborhood
C
i
N .
STD is the Sum of Tensors Dissimilarity used for the
tensor-domain component:
12 1 2
1
,,,
N
b
n
STD Q Q Q n Q n
(11)
where N
b
is the number of tensors within each
neighborhood and
1, 2,
i
Qn i represents the n
th
tensor within the neighborhood
i
Q .
In case of multi-scale synthesis, pyramids are
obtained by smoothing the tensor field then down-
sampling with a 2:1 factor for each additional scale.
The synthesis starts from the highest pyramid level
and ends at the bottom of the pyramid. To assure
that the added high-frequency details are consistent
with the already synthesized low-frequency
structures, the multi-resolution neighborhood of the
current tensor at level i contains its same-level
neighborhood as well as the neighborhood of the
corresponding tensor position at the previously
synthesized level (i+1).
Due to the additional information provided by
the structure map, reproducing the exemplar's
patterns is feasible using a smaller texture
neighborhood than the one used for the structure
layer synthesis (Akl et al., 2014). However, the
outperformance of the proposed algorithm still
comes at the expense of additional computation of
the synthesized structure layer, as will be shown in
Section 4. Thus, an algorithm acceleration targeting
the structure layer synthesis stage is of our interest.
Practically, the additional computational burden
mostly lies in the synthesis of the lowest pyramid
level (highest resolution) during a multi-scale
synthesis process. In addition, for most of the
textures, the structure information in the lowest
pyramid level is slightly different than the one
existing in the higher level. Therefore, considering a
structure tensor pyramid of L levels, we propose to
synthesize the tensor fields at the high pyramid
levels of lower resolution (the coarse levels) up to
level L-1, and to construct the highest resolution
level L (the lowest level) from the already
synthesized level L-1 using a bilinear interpolation.
This can be used as an interesting alternative to the
algorithm in (Akl et al., 2014) which consists in
synthesizing at each level of the Gaussian pyramid.
4 RESULTS
This section deals with evaluating the proposed
algorithm using different input color textures from
Brodatz database (Brodatz, 1966). Due to space
limitations, only a subset of the results is presented.
Fig. 2 presents synthesis results on four different
color textures. For each result, the first row shows
(from left to right) the input texture and the
orientation of its structure tensor field computed
using the additivity of tensor channels and on the
luminance component, the synthesized texture using
W&L’s algorithm and its orientation image. The
second, third and fourth rows show the orientation of
the synthesized structure tensor, the resulting texture
and its orientation image, respectively, using the
additivity of tensors channels for the structure tensor
computation (first column), by computing the
structure tensor on the luminance component
(second column) and using the accelerated algorithm
with the structure tensor computation on the
luminance component (third column). The best
possible parameters are used with W&L and with
the proposed algorithm.
Note that no software acceleration (Tree-
structured Vector Quantization for example) has
been used to overcome any effect related to the sub-
optimality of such solutions (Wei & Levoy, 2000).
Three iterations are used to obtain the
synthesized structure tensor in result A, B and C, and
four iterations in result D. For all the results, the
textures obtained using the proposed approach are
shown after two iterations. For tensor synthesis, two-
scale Gaussian pyramids are used in results A, B and
D with a neighborhood size of 11×11, 13×13 and
21×21, respectively. Three-scale Gaussian pyramids
are used in result C with an 11×11 neighborhood
size. A mono-scale texture synthesis and a causal
neighborhood with a lexicographical scan are used
for all the results. The texture neighborhood size is
9×9 in results A and B, 11×11 and 21×21 in results C
and D, respectively. The palette used for orientation
images is shown in the upper center of Fig. 2.
It can be observed that the textures obtained with
the tensor-constrained synthesis, using both
approaches for color structure tensor computation in
results A and B, are similar to those obtained with
W&L. On the contrary, both approaches outperform
W&L in results C and D mainly in structure
Two-stageColorTextureSynthesisusingtheStructureTensorField
185
conservation. With W&L, the synthesized image in
result C is of acceptable quality, however it is more
regular than the exemplar. In result D, the texture
obtained with W&L presents some undesired
artifacts and the alternation and periodicity of the
exemplar’s patterns are not respected. On the
contrary, our approach leads to smooth and artefact-
free textures very similar to the input sample giving
the impression that they were produced by the same
process.
Figure 2: Synthesis results. For each result (A, B, C and D), the 1
st
row shows (from left to right) the input texture and the
orientation of its structure tensor field computed using the additivity of tensor channels and on the luminance component,
the synthesized texture using W&L’s algorithm and its orientation image. The 2
nd
, 3
rd
and 4
th
rows show the orientation of
the synthesized structure tensor, the resulting texture and its orientation image, respectively, using the additivity of tensors
channels for the structure tensor computation (1
st
column), by computing the structure tensor on the luminance component
(2
nd
column) and using the accelerated algorithm with the structure tensor computation on the luminance component (3
rd
column). The palette used for orientation images is shown in the upper center.
GRAPP2015-InternationalConferenceonComputerGraphicsTheoryandApplications
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In other words, due to the additional information
provided by the structure layer, the proposed
approach is able to successfully reproduce the
variations of orientations in the exemplar even when
W&L fails to sustain the structure.
In results B, C and D, the orientation images of
the input tensor fields are almost similar using both
approaches for color structure tensor computation,
which leads to nearly similar synthesized tensor
fields. This is not the case for texture A where each
approach gives a visually different orientation
image. However, synthetic textures obtained with
the four different exemplars, using both approaches
for color tensor computation, hardly differentiate
from each other, leading to eye-friendly synthesized
textures of satisfying quality.
It is clearly seen that in all the results, the images
obtained using the bilinear interpolation for the
structure tensor synthesis stage are roughly similar
to those generated by the unaccelerated approach.
The interpolated structure tensor remains of good
quality and the synthetic orientation images
resembles the texture orientation images in terms of
both structure and dynamics preservation.
Table 1 presents the simulation time of the
results in Fig. 2. The second row shows the running
time of W&L’s algorithm. The third and fourth rows
show the running time of our algorithm using the
classical all-levels synthesis method and using the
accelerated algorithm by bilinear interpolation,
respectively. All the presented timings are in
seconds and measured using an Intel Core i7-
2670QM CPU with a 2.20 GHz clock.
Table 1: Running time (seconds) for the textures in Fig. 2.
Texture: A B C D
W&L
28 35 860 756
Proposed Method
41 42 1398 1114
Accelerated
Algorithm
20 21 577 424
Unlike the unaccelerated two-stage synthesis
method, the proposed accelerated algorithm using
the structure tensor interpolation outperforms W&L
in terms of time consumption. For example, W&L
and the proposed classical approach took 860
seconds and 1398 seconds to generate the output
texture in result C respectively, while the accelerated
algorithm requires 577 seconds.
It is important to mention that even when the
pixel-based synthesis algorithm of W&L is able to
successfully synthesize the input texture, the
accelerated version of the proposed two-stage
synthesis is beneficial in simulation time reduction
without any loss in the output texture quality, as it is
the case in results A and B.
Figure 3: Synthesis results on irregular textures. For each
result from top-left to bottom-right; the exemplar, the
obtained texture using W&L, the synthesised textures with
the proposed unaccelerated method and using the
accelerated algorithm.
Fig. 3 shows two synthesis results on irregular long
range directional variations textures. Each result
shows from top-left to bottom-right; the input
texture, the synthesized texture using W&L’s
algorithm, the obtained results with the proposed
unaccelerated and accelerated algorithms.
It can be observed that the textures obtained with
the unaccelerated as well as the accelerated tensor-
constrained synthesis approaches, in the first result,
are similar to those obtained with W&L. On the
other hand, W&L fails to reproduce the exemplar’s
variation of orientations, in the second result, while
the textures obtained by the proposed tensor-
constrained synthesis are more realistic, better
respecting the orientations of the sample structures.
Let us recall that for all the results, the
acceleration by Tree-structured Vector Quantization
(TSVQ) is not used with W&L’s algorithm (Wei &
Levoy, 2000) and could be used as well for the
proposed structure/texture synthesis method. Thus,
future adaptation of such acceleration algorithms to
the tensors neighborhood, is of our interest.
5 CONCLUSIONS
This paper presented a non-parametric color texture
Two-stageColorTextureSynthesisusingtheStructureTensorField
187
synthesis algorithm based on two-stage
structure/texture processing. The structure layer,
represented by the color structure tensor field, is
synthesized in the first stage and used as a constraint
for texture synthesis in the second stage. Two
different methods for color structure tensor
computation were developed. An acceleration
technique for the proposed algorithm is also
presented. The obtained results are highly
encouraging, in terms of structure and dynamics
preservation, and proved that the proposed method is
advantageous for simulation time consumption and
for accurately reproducing the exemplar’s variations
of orientations even when traditional algorithms fail
to reproduce the exemplar’s patterns. As for future
work, we aim at reinforcing the use of the tensor
constraint for the synthesis of anisotropic and non-
stationary textures and to develop a 2D/3D color
texture synthesis algorithm.
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