Process It
Resume:
Multiresolution Sampling Procedure
for Analysis and Synthesis
of Texture Images
Jeremy S. De Bonet
Learning & Vision Group
Arti cial Intelligence Laboratory
Massachusetts Institute of Technology
E MAIL : ***@**.***.***
H OMEPAGE : http://www.ai.mit.edu/ jsd
Abstract for example.) The approach presented here uses these resulting psy-
chophysical models to provide constraints on a statistical sampling
This paper outlines a technique for treating input texture images procedure.
as probability density estimators from which new textures, with In a two-phase process, the input texture is rst analyzed by
similar appearance and structural properties, can be sampled. In a computing the joint occurrence, across multiple resolutions, of sev-
two-phase process, the input texture is rst analyzed by measuring eral of the features used in psychophysical models. In the second
the joint occurrence of texture discrimination features at multiple phase, a new texture is synthesized by sampling successive spatial
resolutions. In the second phase, a new texture is synthesized by frequency bands from the input texture, conditioned on the similar
sampling successive spatial frequency bands from the input texture, joint occurrence of features at all lower spatial frequencies.
conditioned on the similar joint occurrence of features at lower spa- The sampling methodology is based on the hypothesis that tex-
tial frequencies. Textures synthesized with this method more suc- ture images differ from typical images in that that there are regions
cessfully capture the characteristics of input textures than do previ- within the image which, to some set of feature detectors, are less
ous techniques. discriminable at certain resolutions than at others. By rearrang-
ing textural components at locations and resolutions where the dis-
criminability is below threshold, new texture samples are generated
1 Introduction which have similar visual characteristics.
Synthetic texture generation has been an increasingly active re-
search area in computer graphics. The primary approach has been
2 Motivation
to develop specialized procedural models which emulate the gen-
erative process of the texture they are trying to mimic. For ex-
The goal of probabilistic texture synthesis can be stated as follows:
ample, models based on reaction-diffusion interactions have been
to generate a new image, from an example texture, such that the new
developed to simulate seashells [15] or animal skins [14]. More
image is suf ciently different from the original yet still appears as
recently work has been done which considers textures as samples
though it was generated by the same underlying stochastic process
from probabilistic distributions. By determining the form of these
as was the original texture.
distributions and sampling from them, new textures that are simi-
If successful, the new image will differ from the original, yet
lar to the originals can, in principle, be generated. The success of
have perceptually identical texture characteristics. This can be mea-
these methods is dependent upon the structure of the probability
sured psychophysically in texture discrimination tests. To satisfy
density estimator used in the sampling procedure. Recently sev-
both criteria, a synthesized image should differ from the original in
eral attempts at developing such estimators have been successful in
the same way as the original differs from itself.
limited domains. Most notably Heeger and Bergen [10] iteratively
From an input texture patch, such as that shown in Figure 1,
resample random noise to coerce it into having particular multireso-
there are in nitely many possible distributions which could be in-
lution oriented energy histograms. Using a similar distribution, and
ferred as the generative process. Sampling from such distributions
a more rigorous resampling method Zhu and Mumford [16] have
results in different synthesized textures, depending on the priors as-
also achieved some success. In work by Luettgen, et al [12] mul-
sumed. Depending on the accuracy of these assumptions, the result-
tiresolution Markov random elds are used to model relationships
ing textures may, or may not, satisfy the above criteria for good
between spatial frequencies within texture images.
synthesis.
In human visual psychophysics research, the focus of texture per-
One possible prior over the distribution of pixels is that the orig-
ception studies has been on developing physiologically plausible
inal texture is the only sample in the distribution, and that no other
models of texture discrimination. These models involve determin-
images are texturally similar. From this assumption, simple tiling
ing to which measurements of textural variations humans are most
results, as shown in Figure 2. Clearly this fails the suf ciently
sensitive. Typically based on the responses of oriented lter banks,
different criteria stated above.
such models are capable of detecting variations across some patches
Another feasible though also clearly inadequate prior is to
perceived by humans to be different textures ([1, 2, 3, 4, 6, 9, 11],
assume that the pixels in the input texture are independently sam-
Research supported in part by DARPA under ONR contract No. pled from some distribution. Textures generated with this model do
not capture the non-random structure within the original. The re-
N00014-95-1-0600 and by the Of ce of Naval Research under contract No.
sult of such an operation is shown in Figure 3. As expected it fails
N00014-96-1-0311.
Figure 1: An example texture image for input to a texture synthesis
process.
Figure 4: Sampling each spatial frequency band from the corre-
sponding band in the original does not capture the detail which is
characteristic of the input texture, indicating that relationships be-
tween frequencies is critical. The synthesized texture is different
from the superimposed original texture, which is clearly discrim-
inable.
Figure 2: Simple repetition of the image does not result in a texture
which appears to have come from the same stochastic distribution
as the original.
Figure 5: The objective is to generate a patch such as the one above
which is different from the original yet appears as though it could
have been generated by the same underlying stochastic process.
This texture, which was synthesized using the technique described
Figure 3: Textures that contain randomness not present in the orig-
in this paper, is perceptually very similar to the original, and the
inal are perceptually different textures. This texture was generated
superimposed original is not readily located.
by uniformly sampling the pixel values of the original. The original
texture superimposed on the synthetic one is easily identi ed.
to capture the character of the original and is perceptually differ- of this is given by the following induction. Consider a new im-
0
age, Iinput, which is generated from an image Iinput by removing its
ent. This is evidenced by the ease with which the original can be
located when superimposed on the synthesized texture. This effect, high frequencies by low pass ltering with a Gaussian kernel. With
0
just Iinput, and without knowledge of the additional information in
commonly known as popout ([3, 9, 11], e.g.), occurs because the
0
Iinput, one could still consider generating a new image Isynth which
textures are perceptually different and do not appear to have been
0 . Thus, the process of gen-
generated by the same process.
is similar in textural appearance to Iinput
0 0
The goal of texture synthesis is to generate a texture, such as that
erating Isynth from Iinput is independent of highest frequency band
shown in Figure 5, which is both random, and indiscriminable from
00
of Iinput . This argument can be repeated to show that Isynth can be
the original texture. Figure 5 satis es these criteria in that it differs
00 0
generated from Iinput without knowledge of Iinput, and so on. Fi is
signi cantly from the original yet appears to have been generated
by the same physical process. Because of the perceptual similarity then given by:
between this texture, which was synthesized by the procedure in
this paper, and the input texture (generated by some other process), 0:::0
Fi Iinput =
it is dif cult to locate the region which contains the superimposed
, , ,
original.
= Fi Li Iinput ; Li+1 Iinput ; ; Ln Iinput (4),
= Li Isynth
3 Functional synthesis framework,
where Li Isynth is the ith spatial frequency octave (or equiva-
Mathematically, the goal of texture synthesis is to develop a func-
lently the ith level of the Laplacian pyramid decomposition.) The
tion, F, which takes a texture image, Iinput, to a new texture sample,
Isynth, such that the difference between Iinput and Isynth is above original function, F, in equation (1) is then constructed by combin-
ing the spatial frequency bands generated by F0 through FN The
some measure of visual difference from the original, yet is textu-
rally similar. Formally, method presented here simpli es the dif culty of minimizing (ap-
proximate) D difference by initially synthesizing textures which
F Iinput = Isynth (1) are similar at low spatial frequencies, and then maintaining that
similarity as it progresses to higher frequencies. A new texture is
subject to the constraints that synthesized by generating each of its spatial frequency bands so
, that as higher frequency information is added textural similarity is
D Iinput ; Isynth Tmax disc (2) preserved.
and,
V Iinput ; Isynth Tmin diff 4 Texture generation procedure
(3)
where D is a perceptual measure of the perceived difference of
textural characteristics, and V a measure of the perceived visual
4.1 Hypothesis of texture structure
The sampling procedure used by this method is dependent upon the
difference between the input and synthesized images. To be accept-
accuracy of the following hypothesis. Images perceived as textures
able, the perceived difference in textural characteristics must fall
below a maximum texture discriminability threshold Tmax disc, and differ from other images in that below some resolution they contain
regions which differ by less than some discrimination threshold.
the perceived visual difference must be above a minimum visual
difference threshold, Tmin diff . Further, if the threshold is strict enough, randomization of these re-
gions does not change the perceived characteristics of the texture.
The success of a synthesis technique is measured by its ability to
minimize Tmax disc while maximizing Tmin diff . In other words, at some low resolution texture images contain re-
gions whose difference measured by D is small, and reorganizing
Human perception of texture differences, indicated by the hypo-
thetical function D , depends on our prior beliefs about how tex- these low frequency regions, while retaining their high frequency
detail will not change its textural (D ) characteristics yet will in-
crease its visual (V ) difference.
tures should vary. These beliefs incorporate much of human visual
experience; therefore, determining a computable metric, D, to ap-
proximate D , is a complex and often ill-de ned task. Devising In Figure 6, at each resolution examples of potentially inter-
a good approximation for V is an even more dif cult task. For changeable regions are highlighted. Rearranging the image at
these resolutions and locations, while retaining their high resolu-
texture synthesis purposes however, a poor approximation such as
tion structure, corresponds to moving whole textural units (which
direct correlation, is suf cient.
The dif culty of determining a function D, to approximate D , in Figure 6 are individual pebbles.)
depends on the structure and textual complexity of the two images.
Many psychophysically based approximations have been proposed 4.2 Analysis and Synthesis Pyramids
(e.g. [4, 6].)
Clearly, more complex textures can be represented in larger im- A new texture is synthesized by generating each of its spatial fre-
ages; therefore, determining a discrimination function, say Dsmall, quency bands so that as higher frequency information is added tex-
between images which have few pixels is less dif cult than deter- tural similarity is preserved. Each synthesized band is generated
mining a similar function Dlarge over larger images. by sampling from the corresponding band in the input texture, con-
Using a multiresolution approach, this work approximates D strained by the presence of local features. The general ow of this
process is outlined in Figure 7.
with a process which begins from low resolution small images.
By decomposing the function F into a set of functions Fi which In a rst phase the input image is decomposed into multiple res-
olutions. This is done using the standard Laplacian pyramid formu-
each generate a single spatial frequency band of the new texture,
lation where band pass information at the point x; y at level i, in
Isynth .
the image I, is given by:
The domain of the each function Fi is a subset of the domain of
F, as Fi s need only be a function of the information contained
Li I; x; y = Gi I , 2" Gi+1 I x; y
in the low spatial frequency bands of Iinput . An intuitive proof (5)
Figure 6: The synthesis procedure is based upon the hypothesis that at lower resolutions there are regions which are below some threshold of
discriminability and that the randomness within a texture is in the locations of these regions.
same textural characteristics as the original, yet vary from it in
global form, it is assumed that global structure within the input
texture is coincidental and should not constrain synthesis. Given
this assumption it is suf cient to use the responses of a set of local
texture measures as features which provide the basis for an approx-
imation to the human perceptual texture-discriminability function
D . A lter bank of oriented rst and second Gaussian derivatives
simple edge and line lters were used in addition to Laplacian
response. At each location x; y in the analysis pyramid level i,
the response of each feature j, is computed for use in constraining
the sampling procedure. When, at the lowest resolutions, the pyra-
mid layers are too small, the features cannot be computed, and a
constant value is used.
fj
GI fj x; y if size of Gi I
Fij I; x; y = i
0 otherwise
(7)
Figure 7: Multiple regions in the analysis pyramid can be candi- The constraints provided by these features are stronger than just
date values for a location in the synthesis pyramid (as shown in the parent value, because they capture some of the relationships
Figure 8). between pixels within a local neighborhood. This analysis pyra-
mid which contains the multiresolution band-pass and feature re-
sponse information, is directly computed from the input image.
where Gi I is a low-pass down-sampling operation:
Gi I = 2 Gi,1 I g 4.3 Sampling procedure
(6)
" A synthesis pyramid is generated by sampling from the analysis
where 2 and 2 are the 2 up- and down-sampling operations
respectively; g is a two dimensional Gaussian kernel; and G0 I = pyramid conditioned on the joint occurrence of similar feature re-
I. sponse values at multiple resolutions. When the synthesized pyra-
mid has been completely generated, the band-pass information is
Each level of the Laplacian pyramid contains the information
combined to form the nal synthesized texture.
from a one octave spatial frequency band of the input For a com-
Initially the top level lowest resolution of the analysis pyra-
plete discussion of Laplacian and Gaussian pyramids, the reader is
mid, which is a single pixel, is copied directly into the synthesis
referred to [5].
pyramid. When synthesizing a texture larger than the original, the
From each level of this Laplacian pyramid a corresponding level
top level of the synthesis pyramid is larger that in the analysis pyra-
of a new pyramid is sampled. If this sampling is done independently
mid; in this case the analysis level is simply repeated to ll the
at each resolution, as shown in Figure 4, the synthesized image
synthesis level.
fails to capture the visual organization characteristic of the original,
Subsequent levels of the synthesis pyramid are sampled from the
indicating that the values chosen for a particular spatial frequency
corresponding level of the analysis pyramid. At each location in the
should depend on the values chosen at other spatial frequencies.
synthesis pyramid, the local parent structure is used to constrain
From the iterative proof, above, we can also infer that these values
~
sampling. The parent structure, Si, of a location, x; y , in image
only depend values at that and at lower spatial frequencies.
I, at resolution i, is a vector which contains the local response for
However, using only the Laplacian information in the lower fre-
features 1 through M, at every lower resolution from i + 1 to N :
quency bands to constrain selection is also insuf cient. Such a pro-
cedure which samples from a distribution conditioned exclusively
~
Si hI; x; y =
on lower resolutions only loosely constrains the relationship be-, , ,
Fi0+1 x ; y ; Fi1+1 x ; y ; ; FiM x ; y ;
+1, 2 2
tween the child nodes of different parents. Sampling from such
2 2 2 2 ,,
Fi0+2 x ; y ; Fi1+2 x ; y ; ; FiM x ; y ;
a distribution can result in high frequency artifacts which are not
+2 4 4
44 44
present in the intended distribution. To prevent this, constraints
;, y , y (8)
must be propagated across children of different parents; however,
0N 1N
FN 2x ; 2N ; FN 2x ; 2N ; ;
constraint propagation on a two dimensional network results in, x y iT
dependency cycles, from which sampling requires iterative proce- M
FN 2N ; 2N
dures, and which is not, in general, guaranteed to converge in nite
time. This technique constrains the selection process within a spa-
The parent structure of a location in a synthesis pyramid is de-
tial frequency band without creating cycles by using image features
picted in Figure 8; in this schematic, each cell represents the set of
to constrain sampling.
local feature responses.
Because the objective is to synthesize textures that contain the
Figure 8: The distribution from which pixels in the synthesis pyra- Figure 9: An input texture is decomposed to form an analysis pyra-
mid are sampled is conditioned on the parent structure of those mid, from which a new synthesis pyramid is sampled, conditioned
pixels. Each element of the parent structure contains a vector of the on local features within the pyramids. A lter bank of local texture
feature measurements at that location and scale. measures, based on psychophysical models, are used as features.
Two locations are considered indistinguishable if the square dif-
Variations between the analysis and synthesis pyramids occur
ference between every component of their parent structures is below
some threshold. For a given location x0 ; y 0 in the synthesis im- when multiple regions in the analysis pyramid satisfy the above cri-
age, Isynth, the set of all such locations in the input image can be terion. The parent structure of such a group of candidate locations
is depicted in Figure 9. As the thresholds increase, the number of
computed:
candidates from which the values in the synthesis pyramid will be
sampled, increases. The levels of the thresholds, Tij, mediate the
~,
Si,Isynth ; x0 ; y 0 ;,
x0 ; y 0 =
Ci Ti
~
x; y D rearrangement of spatial frequency information within the synthe-
~
(9)
Si Iinput ; x; y sized texture, and encapsulate a prior belief about the degree of ran-
domness in the true distribution from which the input texture was
D, between two parent structures u
Where the distance function generated.
and v, is given by: Algorithmically, this sampling procedure can be described with
u, v T u, v the pseudo-code:
D u; v = (10)
Z SynthesizePyramid
where Z is a normalization,constant which eliminates the effect of
P Loop i from top level-1 downto 0
Loop x0 ; y0 over Pyrsynth [level i]
~
contrast, equal to x;y Si Iinput ; x; y .
C=;
To be a member of set Ci x0 ; y 0 the distance between each
Loop x;S over Pyranalysis [level
y i]
component of the parent structures must be less than the corre-
C = C f x; y g
sponding component in a vector of thresholds for each resolution
Loop v from top level downto i + 1
and feature:
Loop j for each feature,
Pyranalysis [v][j] x=2v,i ; y=2v, ;
i
Ti0+1 Ti1+1 TiM
~
Ti =, 0 v,i 0 v,i
if D
+1 Pyrsynth [v][j] x =2 ; y =2
Ti0+2 Ti1+2 TiM
+2 threshold[level v][feature j]
then
C C,f g
(11)
x; y
M T
=
01
TN TN TN break to next x; y
Where each element Tij is a threshold for the j th lter response at kCk
selection = UniformRandom 0;
C
the ith resolution. x; y = [selection]
Pyrsynth [v] x0 ; y0 = Pyranalysis [v] x; y
The values for new locations in the synthesis pyramid are sam-
pled uniformly from among all regions in the analysis pyramid that
have a parent structure which satis es equation . This yields With more complex code, additional ef ciency can be obtained
a probability distribution over spatial frequency band values condi- by skipping whole regions which share a parent structure element
tioned on the joint occurrence of features at lower spatial frequen- that is above threshold difference.
cies: Upon the completion of this sampling process for each level
of the synthesis pyramid the synthesized band-pass informa-, ,
P Li Isynth ; x0 ; y 0 Li Iinput ; x; y x; y 2 Ci x0 ; y 0
tion is combined to form the new texture using a standard
CollapsePyramid procedure.
Though each band is sampled directly from the input image, the
= kCi x0 ; y 0 k
= 1 image which results from the recombination of each of these syn-
(12) thesized layers contains pixel values (i.e. RGB colors) not present
Figure 11: This series of 6 images (b-g) was generated from the
Figure 12: A series of synthesized textures for which the thresholds
original (a). For each a single threshold is used for all features and
are inversely proportional to the spatial frequency and proportional
resolutions. Thresholds increase from 0.05 to 0.3 from (b) to (g).
to 0.05 in (b) to 0.3 in (g).
in the original, because non-zero thresholds allow synthesized spa-
belief about the randomness implied by the original. Another syn-
tial frequency hierarchies which differ from those in the original.
thesis series for a different input image is shown in Figure 12. In
Because the Laplacian pyramid representation is over-complete,
i.e. the space spanned by Laplacian pyramids is 4=3 larger than that this case = 1, a varies over the same range, and the ideal thresh-
old is somewhere around = 0:25 (image f.)
spanned by images, it is possible to synthesize pyramids that are off
of the manifold of real-images. When this occurs, the pyramid is
projected onto the closest point on this manifold before reconstruc-
6 Discussion
tion. This is done by collapsing the pyramid using full precision
images, then replacing values above or below the range of legal
Because it uses only local constraints, the estimator presented here
pixel values with the closest legal value.
cannot model, texture images with complex visual structures. Such
structures include: re ective and rotational symmetry; progressive
5 Examples of texture synthesis variations in size, color, orientation, etc.; and visual elements with
internal semantic meaning (such as symbols) or which have mean-
ing in their relative positions (such as letters.)
For 800 full color input textures, we synthesized new textures, each
Simply adding additional complex features to attempt to capture
four times larger than the original. Some typical results are shown
these sorts of visual structures over conditions the sampling pro-
in Figure 10. The results from these examples are indicative of the
cedure, and simple tiling results. If appropriate thresholds could
synthesis performance on the entire set and were chosen only be-
be determined through additional analysis of the input image, the
cause they reproduce well on paper. The results of all 800 textures
effects of complex features could be mediated, and they might pro-
are available on the world wide web via the URL:
vide useful constraints.
http://www.ai.mit.edu/ jsd/Research/TextureSynthesis Because it samples exclusively from the input image, this model
assumes that the true distributions from which each spatial fre-
In the synthesis examples through out this paper thresholds of the
quency band in the input was generated, can be accurately approx-
form:
Tij = =i imated by only those values present in that image. If there were
(13)
a model for the probability of values not present in the original,
2 2f g
0; 0:4 and 0; 1 . The parameter synthesized textures could possibly be generated which contain ad-
were used with es-
tablishes the prior belief about the sensitivity of D , the threshold ditional variation from the original which does not increase texture
(D ) difference yet increases the visual (V ) difference.
Tmax disc in equation (2); larger incorporates the belief that the
true distribution which generated the input texture is spatially ho-
mogeneous, and that the low frequency structure within the input
7 Conclusion
image should not be an in uential factor in region discrimination.
Shown in Figure 11 are a series of synthesized textures for = 0
f g
and = 0:05; 0:10; 0:15; 0:20; 0:25; 0:30 . As the threshold in- We have presented a method for synthesis of a novel image from
creases, progressively more locations in the original become in- an input texture by generating and sampling from a distribution.
distinguishable, and the amount of variation from the original in- This multiresolution technique is capable of capturing much of the
creases. For this texture, the synthesized image which balances suf- important visual structure in the perceptual characteristics of many
cient difference from the original with perceptual similarity, lies texture images; including arti cial (man-made) textures and more
somewhere between = :15 and = :20 (images d-e.) For dif- natural ones, as shown in Figure 13. The input texture is treated as
ferent images, the ideal threshold is different, re ecting our prior probability density estimator by using the joint occurrence of fea-
Figure 10: Texture synthesis results. The smaller patches are the input textures, and to their right are synthesized images which are 4 or 9
times larger.
Figure 14: An input texture (a) which is beyond the limitations of
Heeger and Bergen (1995) model (b), can be used successfully by
this techniques to synthesize many new images. Four such synthe-
sized images, using the same set of thresholds, are shown in (c) -
Figure 13: The characteristics of both arti cial / man-made and
(f).
natural textures can be captured and replicated with this process.
[8] A. Gagalowicz and S. D. Ma. Model driven synthesis of
tures across multiple resolutions to constrain sampling. Prior be-
natural textures for 3 D scenes. Computers and Graphics,
liefs about the true randomness in the input are incorporated into
10:161 170, 1986.
the model through the settings of thresholds which control the level
of constraint provided by each feature. Many of the textures gener- [9] N. Graham, A. Sutter, and C. Venkatesan. Spatial-frequency
ated by sampling from this estimator can simultaneously satisfy two and orientation-selectivity of simple and complex channels in
the two criteria of successful texture synthesis: the synthesized tex- region segregation. Vision Research, 33:1893 1911, 1993.
tures are suf ciently different from the original, and appear to have
been created by the same underlying generative process. These tex- [10] D. J. Heeger and J. R. Bergen. Pyramid based texture anal-
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produce textures which appear more akin to the originals, than those SIGGRAPH, 1995.
produced by earlier techniques (Figure 14.)
[11] B. Julesz. Visual pattern discrimination. IRE Transactions on
Information Theory, IT 8:84 92, 1962.
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