Product Integral Binding Coefficients for High-order Wavelets
Nick Michiels, Jeroen Put and Philippe Bekaert
Hasselt University - tUL - iMinds, Expertise Centre for Digital Media Wetenschapspark 2, 3590 Diepenbeek, Belgium
Keywords:
Wavelets, Signal Processing, Product Integrals, Tensor, Rendering.
Abstract:
This paper provides an efficient algorithm to calculate product integral binding coefficients for a heterogeneous
mix of wavelet bases. These product integrals are ubiquitous in multiple applications such as signal processing
and rendering. Previous work has focused on simple Haar wavelets. Haar wavelets excel at encoding piecewise
constant signals, but are inadequate for compactly representing smooth signals for which high-order wavelets
are ideal. Our algorithm provides an efficient way to calculate the tensor of these binding coefficients. The
algorithm exploits both the hierarchical nature and vanishing moments of the wavelet bases, as well as the
sparsity and symmetry of the tensor. We demonstrate the effectiveness of high-order wavelets with a rendering
application. The smoother wavelets represent the signals more effectively and with less blockiness than the
Haar wavelets of previous techniques.
1 INTRODUCTION
Efficient rendering of complex scenes with detailed
lighting effects is an important application. To a-
chieve this goal, we need to solve the rendering
equation at every point in the scene (Kajiya, 1986).
The rendering equation calculates the radiance that
reaches the observer, depending on the incoming
light, the objects and the materials in the scene. Naive
sampling to solve this equation is impractical. To
make the calculations more tractable, light transport
is often precalculated in a factored form. The render-
ing equation is often evaluated at each point x, viewed
from direction ω
o
, by calculating the triple product in-
tegral over the hemisphere Ω (Ng et al., 2004):
B(x, ω
o
) =
∑
i
∑
j
∑
k
V
i
ρ
j
(ω
o
) L
k
C
ijk
(1)
with visibility V, reflectance ρ, environment lighting
L and binding coefficient C
ijk
defined as:
C
ijk
=
Z
Ω
Ψ
i
(ω) Ψ
j
(ω) Ψ
k
(ω) dω (2)
The factorization process in our rendering application
is demonstrated in Figure 7.
This paper focuses on calculating the binding co-
efficientsC
ijk
for an arbitrary mixture of wavelet basis
functions Ψ
i
, Ψ
j
and Ψ
k
. The position and dilation
factor are controlled by the basis function numbers i,
j and k. Choosing an appropriate basis to represent
the various factors becomes critical to achieve high
quality results with minimal computation time. This
paper recognizes that each factor in the triple prod-
uct has different signal characteristics. Therefore, it
would be efficient to encode each factor with a ba-
sis specifically tailored to it. While the Haar wavelet
basis Ψ
H
excels at encoding piecewise constant sig-
nals, like the visibility factor
V
, it fails at representing
smooth signals compactly such as reflectance ρ and
lighting L. For this task, smooth high-order wavelets,
e.g. the Daubechies-4 basis Ψ
D
, are generally con-
sidered more appropriate (the number 4 indicates the
support size of the basis functions with the smallest
dilation). These smooth bases often require an or-
der of magnitude less coefficients for smooth signals.
This paper devises a method to calculate product inte-
grals for arbitrary wavelet bases, focusing specifically
on triple product integrals for rendering applications.
Michiels et al. (Michiels et al., 2013) already showed
the applicative advantages of using smooth wavelets
for inverse rendering. Figure 1 shows two wavelet
bases: Ψ
H
on the left and Ψ
D
on the right
1
. These
bases are orthonormal:
Z
Ω
Ψ
Hi
Ψ
H j
dω = δ
ij
and
Z
Ω
Ψ
Di
Ψ
Dj
dω = δ
ij
, (3)
where δ
ij
is the Kronecker delta (Clapham and
1
Without loss of generality, the illustrations in this paper
show 1D wavelet functions. Separable wavelets in higher
dimensions can easily be obtained by combining two lower
dimensional basis functions.
17
Michiels N., Put J. and Bekaert P..
Product Integral Binding Coefficients for High-order Wavelets.
DOI: 10.5220/0005013300170024
In Proceedings of the 11th International Conference on Signal Processing and Multimedia Applications (SIGMAP-2014), pages 17-24
ISBN: 978-989-758-046-8
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
Figure 1: Double and triple product of Haar wavelets (left) and Daubechies wavelets (right). (a), (b) and (c) are the different
basis functions Ψ
i
, Ψ
j
and Ψ
k
. Each column represents a permutation of i, j and k that create basis functions with different
dilation and/or position. The double product function is depicted in (d) with integral (f). The triple product functions is shown
in (e) with integral (g). Green represents the positive part of the basis functions, red the negative part.
Nicholson, 2009). The double product integral of
such orthonormal bases reduces to a convenient dot
product. This can be verified in Figure 1(f). Only ba-
sis functions with the same position and dilation result
in a non-zero double product integral.
Unfortunately, the triple product integral is sub-
stantially more difficult to calculate. An analytical ap-
proach was developed for the Haar product integral,
which iterates only over non-zero coefficients (Ng
et al., 2004). This is possible due to the strongly regu-
lar and hierarchical structure of Haar wavelets. Chil-
dren with a smaller dilation factor fall completely un-
der a constant part of their parent. Figure 1(g) (left)
shows that only specific and predictable combinations
result in non-zero product integrals.
High-order wavelets, on the other hand, have three
properties that prevent them from having simple bind-
ing coefficients:
1. Their larger support results in more overlap and
circular wrapping.
2. Children are no longer entirely contained by the
support of their parent.
3. The subset of the parent support that a child over-
laps is not necessarily constant.
It can be seen in Figure 1(g) (right) that the triple
product integral is more complex. If we iterate over
all combinations of i, j and k andput them in one large
tensor we get Figure 2(a) for Haar and Figure 2(b) for
Daubechies. With this tensor precalculated, only the
sparse non-zero coefficients need to be evaluated at
runtime.
The specific contributions of this paper are:
• An efficient algorithm to calculate the tensor with
binding coefficients of n-product integrals of a
wide range of wavelet basis functions. Mixing
of various basis types is supported. The paper
focuses on double and triple coefficients, but the
approach can easily be extended to quadruple or
higher product integrals.
• Analysis of the tensor characteristics such as spar-
seness and symmetry. We also study the effect of
mixing Haar and higher-order bases.
• We demonstrate the effectiveness of our approach
with a rendering application.
2 PREVIOUS WORK
Spherical harmonic basis functions were one of the
first functions proposed to approximate the factors in
the triple product integral (Kautz et al., 2002). Spher-
ical harmonics are an extension of the Fourier trans-
formation to the spherical domain. They have the
advantage of being a well studied mathematical tool
that can succinctly approximate low-frequency sig-
nals. Expressing detailed high-frequency effects with
spherical harmonics, however, requires exponentially
more coefficients. Another disadvantage is that no ef-
ficient analogue of the Fast Fourier Transform (Coo-
ley and Tukey, 1965) exists for harmonic analysis in
the spherical domain. A relatively fast alternative al-
gorithm is known, but its time complexity is still su-
perquadratic (Mohlenkamp, 1997). A triple product
integral theorem can also be formulated for Legen-
dre polynomials, but due to the global support of this
basis, it suffers from the same shortcomings as spher-
ical harmonics (Gupta and Narasimhan, 2007). Their
graphical representation of tensor slices inspired the
computational approach taken in this paper.
To accurately represent high frequency lighting
details, wavelet bases can be utilized (Daubechies,
1992). An alternative representation for spheri-
cal harmonics in the wavelet domain are spherical
SIGMAP2014-InternationalConferenceonSignalProcessingandMultimediaApplications
18
(a) (b)
Figure 2: Tensor of tripling binding coefficients for (a)
Haar wavelets and (b) high-order wavelets. The Haar ten-
sor clearly shows the three cases identified by Ng et al. (Ng
et al., 2004). High-order wavelets have a broader support,
resulting in more non-zero binding coefficients. However,
the tensor still remains sparse.
wavelets (Sweldens, 1998), which have demonstrated
to be a compact and efficient representation. While
the possibility to construct an orthogonal spherical
wavelet basis with compact support and symmetry has
been demonstrated (Lessig and Fiume, 2008), these
wavelets are considerably more difficult to construct
than their 2D counterparts. Therefore, functions are
often parameterized with an area-preserving parame-
terization (Praun and Hoppe, 2003), so that conven-
tional 2D wavelet analysis can be leveraged.
Ng. et al. were the first to solve a triple prod-
uct integral of three factors approximated in Haar
bases (Ng et al., 2004). They noticed that the product
integral tensor of wavelet functions is very sparse and
the calculations can be categorized in a small number
of cases, which they exhaustively list in their Haar
tripling coefficient theorem. They manually studied
the different possible wavelet combinations and their
outcome. Only a fraction of the wavelet combinations
on different levels resulted in a non-zero integral, so
that only these particular cases need to be treated to
evaluate the triple product integral.
Previous work has also developed a general tech-
nique for importance sampling products of complex
functions using wavelets (Clarberg et al., 2005). They
perform on-the-fly stochastic sampling of the wavelet
scaling coefficients to evaluate a double product in-
tegral of the BRDF and an environment map. They
base their sampling scheme on the characteristics of
the Haar wavelet basis and only double product inte-
grals are demonstrated. In addition, they preprocess
the 4D BRDFs, so that only sample points need to be
evaluated at runtime, but this means the sample pat-
tern does not adapt when either the environment map
or the BRDF is dynamic. Also, the random sampling
patterns they use introduce considerable noise in the
resulting images.
A generalized Haar integral coefficient theorem
was proposed for evaluating arbitrary dimensional
Haar product integral coefficients (Sun and Mukher-
jee, 2006). They extend the approach of Ng. et al.
and create an efficient sublinear algorithm to evaluate
these N-product integrals. As in previous work how-
ever, they are limited to simple Haar wavelet bases.
There exists also a geometry-dependent
basis for diffuse precomputed radiance trans-
fer (Nowrouzezahrai et al., 2007). Their basis is
derived from Principal Component Analysis of the
sampled transport functions at each vertex. They only
demonstrate double product integral capabilities and
the rendering results are diffuse only. Interpolation
artifacts also arise due to the dependency of the basis
on the geometric representation of the scene.
An affine double and triple product integral theory
was developed, enabling one of the product functions
to be scaled and translated (Sun and Ramamoorthi,
2009). They demonstrate that these operations are
very sparse and scale with linear complexity. This
sparsity enables them to add some of the first near-
field lighting effects. In their disposition, they give
specific attention to the common Haar wavelets and
rely on its non-overlapping property. They state that
an implementation for non-Haar wavelets is more ex-
pensive but that their general approach can be similar
applied to general wavelets. They do not, however,
provide a solution to this problem.
Previous work has tried to exploit the fact that
in areas where the visibility factor is constant, the
triple product integral reduces to a double product
integral (Inger et al., 2013). Nevertheless, in the ar-
eas where a full triple product evaluation is needed, a
fall back to an expensive pixel domain integral is still
required. In addition, mixing of arbitrary and high-
order wavelet bases is not supported.
This paper differs from previous work in that a
computational approach is taken to calculate the ten-
sor coefficients for triple product integrals. This al-
lows the use and mixing of a wide range of high-
order wavelets, as opposed to simple Haar wavelets.
High-order wavelets can compactly approximate the
smooth factors in the product. In the next section, the
use of these high-order wavelets will be motivated.
The rest of this paper will explain our computational
approach and evaluate the results.
3 WAVELET PRODUCT
INTEGRAL
Many functions are naturally expressed in the spher-
ical domain. Solving the rendering equation (see
Equation 2) at each point in space can be considered a
spherical convolution operation in signal processing.
Each term in the convolution, in the rendering case V,
ProductIntegralBindingCoefficientsforHigh-orderWavelets
19
Figure 3: Reconstruction of an environment map (left) with
a small number of coefficients. A Haar wavelet basis is used
in the top row on the right. The result is much more blocky
in comparison to the smoother Daubechies-6 reconstruction
in the bottom row.
ρ and L, is expanded in an appropriate basis Ψ:
V(ω) =
∑
i
V
i
Ψ
i
(ω) (4)
ρ(ω) =
∑
j
ρ
j
Ψ
j
(ω) (5)
L(ω) =
∑
k
L
k
Ψ
k
(ω) (6)
As described before, previous work either used
harmonic analysis to model the functions compactly
on the sphere, spherical wavelets or regular 2D Haar
wavelets. Harmonic analysis has the advantage of be-
ing a natural choice of representation for spherical
functions, but requires a high number of coefficients
to represent high-frequency detail. Haar wavelets on
the other hand are better suited to compactly repre-
sent these local details, because the basis functions
have a small support. Haar functions lack smooth-
ness, however, which makes them less suitable for the
approximation of smooth signals. On the other hand,
high-order wavelet bases (e.g. Daubechies) are bet-
ter tailored to representing smooth signals. Figure 3
shows a comparison between the Haar wavelet basis
and the Daubechies-4 wavelet basis. A reconstruc-
tion with few coefficients leads to a noticeably more
blocky result when using the Haar basis. In this par-
ticular example, the original signal requires roughly 5
times less coefficients in the Daubechies basis. This
motivates our approach to perform the product inte-
gral calculation on this sparser representation.
3.1 The Haar Tripling Coefficient
Theorem
The Haar tripling coefficient theorem binds appropri-
ate weighting factors to each of the wavelet coeffi-
cients that combine to form the product integral. They
use an analytical approach to formulate the very lim-
ited set of cases that arise when calculating the triple
product integral of simple Haar basis functions. A
recursive algorithm is given with sublinear complex-
ity, which calculates only the relevant coefficients.
Their manual approach does not scale well when us-
ing smoother wavelets. This paper therefore takes a
computational approach to the problem and calculates
the tripling coefficients for a wide range of wavelets
automatically.
Ng et al. realized that the different binding coeffi-
cients are very redundant for a simple Haar basis (Ng
et al., 2004). Their final tripling coefficient theo-
rem has three cases, with variations for scaling and
wavelet combinations. Furthermore, Haar wavelets
with smaller dilation factors are always fully con-
tained in a subset of the parents support with constant
value. These characteristics allow for a fast sublinear
time algorithm, that iterates over all non-zero coeffi-
cients.
3.2 General Tripling Coefficient
Theorem
In contrast to the Haar wavelets, high-order wavelets
are not piecewise constant functions anymore. Since
high-order wavelets have a larger support, the number
of non-zero combinations will increase rapidly. It is
no longer feasible to manually indentify and enumer-
ate all the various cases. Figure 4 visually compares
the overlappingproperties for Haar and the high-order
Daubechies wavelets. It can be seen that the convo-
lution of Haar wavelets yields a much sparser tensor
than the convolution of high-order wavelets. Haar
wavelets with smaller dilation never overlap more
than one coarser Haar wavelet. This is not true for
high-order wavelets. The support of the high-order
wavelets remains compact, however, only being en-
larged by a constant factor. Because of this, the tensor
of binding coefficients remains sparse.
3.2.1 Naive Approach
It is easy to envision a naive approach to calculate
the tensor, based on the permutation of all i, j and
k and calculating the binding coefficients C
ijk
(Equa-
tion 2). This quickly becomes intractable, due to its
time complexity O(n
D
∗ r
2
) with D the dimensional-
ity of the product integral (double, triple, quadruple,
...), n the total number of translations and dilations
of the basis function and r
2
the resolution of the 2D
wavelet functions that are integrated. If the original
signal contains many high-frequency perturbations, n
can grow profoundly large. As a result the computa-
tion time increases drastically. For only a resolution
of 512x512, a triple product integral for 2D wavelets
requires an order of 9.5 × 10
14
calculations. Clearly
more sophisticated methods are required.
SIGMAP2014-InternationalConferenceonSignalProcessingandMultimediaApplications
20
Figure 4: Hierarchically overlapping wavelet basis func-
tions. Each wavelet basis function Ψ
i
will have a fixed
overlap with a set of other wavelet functions Ψ
j
. The three
columns on the left represent the overlap in case of Haar
wavelets, specifically for Ψ
i=4,12,21
. The three rightmost
columns show the analogous cases for the Daubechies-4
wavelets.
3.2.2 Hierarchical Approach
A second approach exploits the characteristics of
higher-order wavelet bases. By design, each ba-
sis function has a fixed support. For example, a
Daubechies-4 wavelet starts with a dilation factor of
size 4 and will further dilate accordingly (4, 10, 22,
46, ...). As a result, a wavelet basis function Ψ
i
will
overlap only with a fixed amount of other wavelet
basis functions Ψ
j
, each having a different position
and dilation factor. Figure 4 illustrates the overlap
of several basis functions Ψ
i
with other basis func-
tions Ψ
j
. The figure displays that a basis function
with a narrow dilation factor overlaps only with a
small number of other functions. Only these specific
combinations will possibly result in non-zero bind-
ing coefficients. All other combinations can safely
be skipped. The time complexity is now reduced to
O(n∗C ∗ log
D−1
(n) ∗ r
2
) with C a constant related to
the enlargement of support for high-orderwavelets (in
case of Daubechies-4, C ≃ 3). The reconstruction of
a 512 × 512 resolution will now take approximately
6.5 × 10
10
permutations, which is several orders of
magnitude less than the naive approach.
(a) (b)
Figure 5: Tensor mirroring. (a) shows the overlap regions
for a double product of Daubechies-8 wavelets Ψ
Di
and
Haar wavelets Ψ
H j
. (b) shows the overlap regions for a
double product where both Ψ
Di
and Ψ
Dj
are Daubechies-
8 wavelets functions. In the case of two identical wavelet
bases, the tensor is fully symmetric. The values in the upper
triangle, indicated in blue, can be mirrored into the lower
triangle. Note that in all cases the tensor is sparse.
3.2.3 Tensor Mirroring
When the mix of wavelet bases in a product integral is
homogeneous, symmetry can be observed in the plot-
ted tensor. For the double product case, the tensor is
depicted in Figure 5(b)). Since
R
Ψ
i
Ψ
j
=
R
Ψ
j
Ψ
i
is
true, each combination of basis functions where j ≥ i
can be mirrored around the diagonal, eliminating half
the work. In the case of a triple product integral with
i, j and k or a general product integral, taking advan-
tage of the symmetry is done in an analogous manner.
3.2.4 Wavelet Sliding
Previous sections have discussed improvements in
computational complexity by reducing the number of
wavelet permutations. The bottleneck for the overall
tensor calculation is not the amount of wavelet per-
mutations (e.g. 10
5
for 512× 512), but rather the ac-
tual work performed in each permutation (r
2
or 512
2
multiplications for integration). A lot of these calcu-
lations are redundant. This section will explain how
they can be avoided.
Let us observe two basis functions Ψ
i=x
and Ψ
i=y
and their respective sets of overlapping basis func-
tions O
i=x
and O
i=y
for which holds:
O
i
= {Ψ
j
: (
Z
(Ψ
i
Ψ
j
) 6= 0, j ≥ i} (7)
In Figure 6, Ψ
i=x
is circled in red and O
i=x
is marked
in red. Ψ
i=y
and O
i=y
are marked green analogously.
In the case of Ψ
i=x
and Ψ
i=y
having the same dila-
tion factor, the amount of overlapping functions will
be equal (#O
i=x
= #O
i=y
). However, the overlapping
functions of Ψ
i=y
are shifted with a sliding factor s:
s = (y− x) × support(Ψ
i=x
) (8)
Using the observation above, only the overlapping
functions of one specific translation need to be cal-
ProductIntegralBindingCoefficientsforHigh-orderWavelets
21
Figure 6: Wavelet sliding. Given a wavelet basis function Ψ
i=x
(circled in red), there is a branch in the wavelet tree of
overlapping basis functions (red). The precalculated binding coefficient values of this branch can be reused for every translated
Ψ
i=y
(circled in green) with the same dilation. The branch is translated with slide factor s (see Equation 8).
culated, e.g. the red area in Figure 6. We call
this a branch. The precalculated branch of binding
coefficients can then be reused for all other trans-
lated basis functions with that specific dilation fac-
tor. This will reduce the amount of work drastically
to O (C ∗ log
D
(n) ∗ r
2
). The 512 × 512 example will
now take approximately 3.1× 10
8
calculations.
3.2.5 Vanishing Moments
An additional advantage of high-order Daubechies
wavelets is that they provide the maximum amount of
vanishing moments for wavelets of that specific sup-
port size (Daubechies, 1992). In general, smoother
wavelets generate more vanishing moments. The in-
teresting property of wavelets with vanishing mo-
ments is that their product integral is not only zero
when the basis functions do not overlap, but even for
certain translations within their support. This yields
extra sparsity in the tensor. The position of these van-
ishing moments are predictable and their determina-
tion is incorporated in the wavelet sliding algorithm
of the previous section. Afterwards, the data entries
of vanishing moments can be removed from the pre-
calculated branch.
4 APPLICATIONS
Our main motivation for using high-order wavelets is
that they provide a compact and high-quality approx-
imation of smooth functions. This is particularly in-
teresting for triple product integration to evaluate the
rendering equation (see Figure 7). The visibility fac-
tor is a piecewise constant function, for which Haar
(a)
(b)
Figure 7: Triple product integration in our rendering appli-
cation. Solving the rendering equation requires taking an
integral over the hemisphere at each point in the scene. The
three factors of the product integral are visibility, BRDF
(bidirectional reflectance distribution function) and light-
ing. The output of all the integrals for the entire image
results in a rendered viewpoint. (a) and (b) show an ex-
ample of two rendered views, both with different lighting
conditions.
wavelets are ideally suited. We argue that the light-
ing environment map and certainly the BRDF (bidi-
rectional reflectance distribution function) exhibit, in
general, profoundly more smoothness. These factors
are better represented with a smoother wavelet, for ex-
ample Daubechies. In contrast to previous methods,
our algorithm is able to calculate the triple product
integral with a heterogeneous mix of wavelet bases,
where each factor is coded in a basis specifically tai-
lored to the signal characteristics.
Figures 8 and 9 compare the visual quality of
blocky Haar wavelets and smoother Daubechies-6
wavelets for various models and compression rates.
The ground truth rendering at full quality is included
for reference, alongside zoomed pictures on areas
with the largest differences. The L
2
norm is pro-
vided with each rendering as a quantitative compar-
SIGMAP2014-InternationalConferenceonSignalProcessingandMultimediaApplications
22
(a) Haar
(b) Daubechies-6
Figure 8: Quality comparison of the Elch dataset. The envi-
ronment map is compressed with: (a) the Haar wavelet ba-
sis and (b) the Daubechies-6 wavelet basis. The three right
columns show the quality differences with respectively 16,
64 and 128 coefficients retained. In (a), themesh is rendered
with the Haar triple product integral (Ng et al., 2004). (b)
is rendered with our tensor calculation. The results in (b)
have smaller L
2
norms and converge faster to the ground
truth (depicted in the left column).
ison measure. A rendering with a smaller norm is
closer to the ground truth. It can be seen that smooth
high-order wavelets outperform Haar wavelets both
qualitatively and quantitatively.
Figure 10 demonstrates the ability of our applica-
tion to render with different kinds of wavelets. It is
possible to mix and match wavelets to optimally rep-
resent the signals of all factors in the product integral
calculation. In general, the smoothness characteristics
of the wavelet basis should match those of the signal.
In that case the wavelets will be able to represent the
signal with a minimum of coefficients.
(a) Haar
(b) Daubechies-6
Figure 9: Quality comparison of the Lucy dataset. The envi-
ronment map is compressed with: (a) the Haar wavelet ba-
sis and (b) the Daubechies-6 wavelet basis. The three right
columns show the quality differences with respectively 16,
32 and 128 coefficients retained. In (a), the mesh is rendered
with the Haar triple product integral (Ng et al., 2004). (b)
is rendered with our tensor calculation. The results in (b)
have smaller L
2
norms and converge faster to the ground
truth (depicted in the left column).
5 CONCLUSIONS
This paper has provided a method to calculate the
binding coefficients of general product integrals. Our
method is able to cope with a mix of heterogeneous
bases. This allows the representation of factors in the
product integral with piecewise constant or smooth
basis functions, depending on the signal properties.
The algorithm exploits both the hierarchical nature
and vanishing moments of the wavelets basis, as well
as the sparsity and symmetry of the tensor. Our ren-
dering application demonstrates that the tensor-based
product integral leads to less blockiness in the results.
ProductIntegralBindingCoefficientsforHigh-orderWavelets
23
Figure 10: Product integral calculation with different wavelet basis functions. In this case, a compression with Haar, Coiflet-5
and Symmlet-5 is executed on the environment map. The result of the triple product integral on the Elch dataset is shown
respectively.
ACKNOWLEDGEMENTS
The authors acknowledge financial support by the Eu-
ropean Commission (FP7 IP SCENE), the European
Regional Development Fund (ERDF), the Flemish
Government, iMinds and IWT.
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