Color Contoning for 3D Printing
Color Contoning for 3D Printing
VAHID BABAEI, Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology
KIRIL VIDIMČE, Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology
MICHAEL FOSHEY, Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology
ALEXANDRE KASPAR, Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology
PIOTR DIDYK, Saarland University, MMCI and MPI Informatik
WOJCIECH MATUSIK, Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology
Fig. 1. We introduce a novel color reproduction method for 3D printing. In contrast to halftoning techniques that create colors by spatial combination of inks
on the surface, our method creates colors by superposing inks inside the printed volume. This mitigates a variety of artifacts associated with halftoning. Parts
(a) and (b) of the figure show the preview of an image generated using clustered-dot rotated halftoning and error-diffusion halftoning, respectively. Part (c)
shows a photograph of the 3D printed image using our technique. Parts (d), (e) and (f) present three examples fabricated with our technique on a custom,
multi-material 3D printer.
Appearance reproduction is an important aspect of 3D printing. Current color
reproduction systems use halftoning methods that create colors through a
spatial combination of different inks at the object’s surface. This introduces
a variety of artifacts to the object, especially when viewed from a closer
distance. In this work, we propose an alternative color reproduction method
for 3D printing. Inspired by the inherent ability of 3D printers to layer
different materials on top of each other, 3D color contoning creates colors by
combining inks with various thicknesses inside the object’s volume. Since
inks are inside the volume, our technique results in a uniform color surface
with virtually invisible spatial patterns on the surface. For color prediction,
we introduce a simple and highly accurate spectral model that relies on
a weighted regression of spectral absorptions. We fully characterize the
proposed framework by addressing a number of problems, such as material
arrangement, calculation of ink concentration, and 3D dot gain. We use a
custom 3D printer to fabricate and validate our results.
CCS Concepts: • Computing methodologies → Reflectance modeling;
Image processing; Perception; • Hardware → Printers; Displays and imagers;
Additional Key Words and Phrases: Fabrication, 3D Printing, Color, Halftoning, Color reproduction
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https://doi.org/10.1145/3072959.3073605
ACM Reference format:
Vahid Babaei, Kiril Vidimče, Michael Foshey, Alexandre Kaspar, Piotr Didyk,
and Wojciech Matusik. 2017. Color Contoning for 3D Printing. ACM Trans.
Graph. 36, 4, Article 124 (July 2017), 15 pages.
https://doi.org/10.1145/3072959.3073605
1
INTRODUCTION
Additive manufacturing enables the design and reproduction of intricate geometries, and fabrication of novel multi-material composites
which cannot be created using existing manufacturing techniques.
A novel research area enabled by additive manufacturing is the
fabrication of objects with a desired appearance. For many applications, reproducing the diffuse color is sufficient to guarantee an
appearance reproduction without objectionable differences.
Color reproduction is a mature field in 2D printing. Almost all 2D
printers are restricted to depositing discrete choices of materials at
any given point on the surface. Therefore, halftoning techniques are
employed to create an illusion of continuous-tone colors. Halftoning
takes advantage of the human visual system that, due to its low-pass
filtering property, integrates a set of discrete dispersed dots across
a surface into a single color [Baqai et al. 2005]. Although halftoning
is a clever solution for devices with binary tone reproduction, it has
some drawbacks. Most importantly, the spatial halftone patterns can
introduce visual artifacts which are particularly noticeable when
the printed image is viewed from a close distance.
Similarly to 2D printers, 3D printers are also “discrete” devices
and share the same restrictions. Instead of adapting 2D color reproduction methods to 3D printing [Brunton et al. 2015; Reiner et al.
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
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V. Babaei et. al.
2014], we propose to utilize the inherent ability of 3D printers to
layer different materials on top of each other. We propose a “contoning” technique which prints layers of color inks around a white,
diffuse material. In contrast to halftoning, our technique synthesizes colors by combining the inks within the volume of the object.
Hence, regardless of the viewing distance, the decomposition of the
color into multiple base materials remains invisible to the eye of
the observer. Contoning mimics much better traditional materials
which are often built from layers, such as wood, skin, marble, etc.
We use a custom, inkjet 3D printer with UV curable inks to explore different problems and trade-offs in the design of our color
reproduction workflow, such as material design, and the layering
order. We show that, compared to the halftoning algorithms, our
method is free of spatial artifacts and has a much more scalable
color prediction. Furthermore, our technique can create plausible
color prints at very low resolutions. On the other hand, contoning
requires a larger number of color layers than halftoning. Therefore,
it is more prone to dot gain. We propose a simple and effective
method to counteract this effect.
The main contributions of this work are:
• A complete color 3D printing framework for printing halftoneless textures suitable for 2.5D and 3D printers,
• A color prediction model for 3D printing,
• A simple and elegant pipeline for color reproduction of
textured 3D objects,
• Fabrication of compelling prints at high and low printer
resolution and comparison with halftoning.
2
RELATED WORK
Appearance reproduction for 3D printing is an emerging research
area. Here, we review the research areas closely related to this paper.
Readers are referred to [Hullin et al. 2013] for an extensive review
of physical appearance reproduction literature.
Color matching. Color matching is a mature field, and the existing
methods are capable of high accuracy reproduction. The equations
introduced by Allen [1966; 1974] are the basis for many commercial
systems for color matching of paints, plastics, and textiles. These
equations benefit from the linear relationship between KubelkaMunk constants and colorant concentrations [Kubelka and Munk
1931]. We use the same principles to predict the color of a stack of
3D printed inks.
Halftoning. Halftoning creates an illusion of a continuous-tone
image by printing ink microdots with certain areas and structures
[Baqai et al. 2005]. Two fundamental halftoning techniques are rotated clustered-dot and error diffusion. Clustered-dot halftoning
[Sharma and Bala 2002] is designed such that individual printer
microdots are grouped into clusters. It is used by printing technologies that have difficulties in rendering single dots. Error-diffusion
halftoning [Ostromoukhov 2001] works within a local neighborhood
and produces halftone prints with less dot-visibility than clustereddot halftones. For creating color halftones, usually three or four
halftone layers are superposed on top of each other.
There are several difficulties associated with halftoning methods.
Clustered-dot halftones exhibit moiré [Amidror 2009] due to the
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
interference between periodic structures of different layers. Error
diffusion can suffer from divergent artifacts from worm-like structures to patterns with regular structures. These problems intensify
with an increased number of superposed inks and limit the application of halftoning in high-end 2.5D or 3D printing. A further
problem is the color predictability of halftones. It scales poorly with
the number of inks and/or number of halftone layers [Babaei and
Hersch 2016].
Continuous color printing. Collotype, a century-old technique, is
a continuous-tone photomechanical process that produces photographic quality prints [Defibaugh 1997]. Dye sublimation, mostly
used in printing textiles, is another printing process that transfers
the colorants continuously from the carrier into the substrate using
heat [Durbeck 2012]. Berns [1993] introduces a color prediction
model for such printers. Our current effort is in the same vein except that here we focus on producing continuous-tone colors on 3D
objects.
Appearance for 3D objects. There are several commercial color 3D
printers on the market including several powder-binder systems
from 3D Systems, photopolymer, phase-change printers from Stratasys and paper lamination printers from Mcor. Traditionally these
printers have not provided a formal, color-managed pipeline and
not much is known about their color reproduction strategy.
Recently Brunton et al. [2015] proposed a 3D halftoning algorithm for creating 3D color prints. Their error-diffusion method
extends the capabilities of 2D halftoning to 3D objects and copes
well with challenges of 3D halftoning. To obtain a good color reproduction, they rely on a fully data-driven color prediction scheme
which requires per layer calibration that requires extensive sample
measurements. Although their technique provides faithful color
reproduction and reproduces well fine-detail structures, it suffers
from inherent limitations of halftoning techniques. In our work,
we follow the same goal, but we advocate for a fundamentally different strategy. We generate colors through a combination of our
materials inside the print and not on its surface. This allows us to
mitigate many halftoning artifacts at the cost of losing details in
regions of sharp color discontinuities. We show that this effect can
be compensated for using our pre-processing step.
Dong et al. [2010] and Hašan et al. [2010] fabricate slabs with
predefined subsurface scattering. In this work, we ignore the translucency and focus on the accurate reproduction of diffuse, spectral
color. Papas et al. [2013] design a framework for reproduction of
homogenous materials with a given color and translucency.
An alternative, two-step approach is to print the texture separately, and then transfer it to the object. Hydrographic printing
[Zhang et al. 2015] and thermoforming [Schüller et al. 2016; Zhang
et al. 2016] are two recent examples.
3
OVERVIEW
We propose a technique for 3D printing color objects using a layering approach. The core of our method is a color reproduction
workflow that for each color in an input color space, e.g., in sRGB,
computes its equivalent printer control values. In our representation,
the control values are layer-layouts: the number of layers of each
printing ink in a printed stack. The steps for this task are shown
Color Contoning for 3D Printing
Base Material
Design
Spectral
Measurements
Magenta, absorption
Magenta, scattering
Yellow, absorption
Yellow, scattering
0. 3
0.25
0. 2
0.15
0. 1
0.05
400
500
600
Wavelength (nm)
700
Fig. 3. The Kubelka-Munk absorption and scattering coefficients of our
cyan, magenta and yellow inks at 0.5% concentration.
Preview
Color
Prediction
Gamut
Mapping
Layer
Separation
Print
Fig. 2. Our color reproduction workflow. It takes an input color to its equivalent layer-layout.
For a fixed set of inks, the input-color to layer-layout workflow
is run once and saved in a table. It can then immediately be used
to generate flat prints. In Section 6, we show how these tables are
used to generate color 3D objects. After evaluating our method in
Section 7, we discuss the dot gain and propose a simple method to
estimate it and compensate for it (Section 8). Finally, we present a
comparison of halftoning and our method (Section 9).
4
Cyan, absorption
Cyan, scattering
Scattering / Absorption Coefficients
in Figure 2. First, we design a color prediction model that, for a
given layer-layout, predicts the resulting reflectance spectrum in
the visible range (Section 4). The color prediction model uses measured spectral reflectances of a set of prints made of primary inks
and their sparse combination. We use the color prediction model to
compute the concentrations of the primary inks used for printing
(Section 5.2). Since our printer is limited to 5 materials, we choose
to work with the cyan, magenta, yellow, black and white inks. Next
step is color gamut mapping where we map the input colors to the
color gamut of our printer (Appendix D). The gamut of the printer
is the range of all achievable colors by the printer and is predicted
using the color prediction model. Finally, we perform layer separation (equivalent to color separation in halftoning terminology)
to find a layer-layout that best estimates a given gamut mapped
color (Section 5.3). As our gamut is a set of discrete colors, the layer
separation is a nearest neighbor search in the color space with an appropriate color difference formula that plays the role of the distance
function.
• 124:3
COLOR PREDICTION MODEL
We propose a color prediction model that computes the resulting
spectral reflectance (and consequently the perceived color) of a 3D
printed stack as a function of its layer-layout. This model is the
main component of our color reproduction workflow (Figure 2). Our
prediction model uses a small set of measured prints to perform
a linear regression in the intrinsic absorption space. The result of
regression is a matrix that maps the layer-layout of any stack to its
intrinsic absorption. We show that going from spectral reflectance to
intrinsic absorption and vice versa is possible via a few closed-form
formulas. In this section, we explain how to construct this model.
A common practice in color modeling is to find a space where
there is a linear relationship between the "concentrations" of primary materials and the resulting color of their mixture [Berns 1997].
This explains the widespread use of the Kubelka-Munk (KM) equations [Kubelka and Munk 1931] in computational color science. The
KM scattering and absorption coefficients scale linearly with the
concentrations of coloring agents [Allen 1966, 1974]. The KM equations are the analytical solutions to the radiative transfer equation
[Chandrasekhar 1960], subject to some approximations [Mudgett
and Richards 1971].
Our material formulation avoids color inks with significant light
scattering. This assumption can be verified by computing the KM
scattering and absorption coefficients of our inks for all visible
wavelengths. We use the black-white method for calculating these
coefficients [Zhao and Berns 2009]. As we see in Figure 3, the absorption is the more significant event occurring within our inks.
Although, for some wavelengths, scattering reaches about 30% of
absorption, this occurs only for low-absorption and low-scattering
regions that do not contribute significantly to color perception. The
absorption assumption also is supported by the high accuracy of
our prediction model (Section 7). Consequently, we can use a simplified version of the KM theory (equivalent to Beer’s law) which
is based on absorption coefficients and build the regression using
absorptions only.
In the absence of scattering, the absorption of an ink layer ψ (λ)
printed on a diffusing substrate with reflectance spectrum Rд (λ)
has the following relationship with the reflectance spectrum R(λ)
of stack of ink and substrate [Berns 1993, 1997]:
R(λ) = Rд (λ) exp(−2ψ (λ)).
(1)
All above quantities are functions of the wavelength λ. As our materials do not show fluorescence, all wavelengths can be treated independently and the computations are done wavelength-by-wavelength.
The KM theory assumes no discontinuity in the refractive index
(neglecting the obvious air-ink interface). Therefore, we first apply
a correction on the measured spectral reflectance to obtain the intrinsic reflectance. The intrinsic reflectance is the reflectance of a
surface if there were no air-surface interface (Appendix A). We can
then calculate the intrinsic absorption coefficient ψs (λ) of a print
using Equation 1 as:
ρ (λ)
1
ψs (λ) = − ln(
),
2
ρд (λ)
(2)
where ρ (λ) and ρд (λ) are the intrinsic reflectances of the sample
R(λ) and the substrate Rд (λ), respectively.
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V. Babaei et. al.
According to the classic KM equations, the spectral absorption of
every printed slab can be computed through convex combination of
the spectral absorptions of its components. This gives large errors
because of inaccuracies in KM theory assumptions, slight scattering of our materials, and the stray specular light captured during
measurement. A compromise is to build a multilinear regression
in the absorption space based on a small training dataset [Babaei
et al. 2012]. The following relationship is built by concatenating the
intrinsic absorption coefficients ψs (λ) of the prints belonging to the
training dataset in matrix Ψs and their corresponding layer-layout
in matrix C.
Ψs = TC.
1
.
si j = ci − cj 2
(4)
The division by zero does not happen as we do not allow ci and cj
to be equal. In that case, the test print is in the training set and there
is no need for predicting it. For p test prints to be predicted using q
training prints, we compute a p × q similarity matrix S with entries
si j :
s 12
s 22
..
.
sp2
s 13
s 23
..
.
sp3
Fig. 4. Halftoning synthesizes colors by surface mixing of inks; increasing
the amount of an ink is achieved by increasing its relative area. Contoning
synthesizes colors by volumetric mixing of inks; increasing the amount of
an ink is achieved by increasing its thickness.
(3)
Sampling the visible spectrum at n wavelength, using q prints in the
training set, and having N inks, Ψs , T, and C are n × q, n × N and
N × q matrices, respectively. Matrix T maps any print with known
layer-layout (thicknesses of different inks in matrix C) to its intrinsic
absorption coefficient ψs (λ). From intrinsic absorption coefficient,
we can use Equation 2 to obtain intrinsic reflectance ρ (λ). Finally,
we can add back the effect of discontinuity in refractive index to
the intrinsic reflectance to obtain a predicted reflectance that is
comparable to measured reflectances (Appendix A). Conceptually,
matrix T contains the spectral absorptance bases for all inks at all
wavelengths. Its size is therefore the number of wavelengths times
the number of materials (n × q). The training set is a set of spectral
measurements of prints in which thicknesses of different inks are
uniformly sampled from 0 to L layers, with a fixed interval, where
L is the maximum thickness.
One limitation of this approach is that matrix T is identical for all
test prints with different layer-layouts. To address this limitation,
we use a weighted regression where a distinct parameter matrix Ti
is tailored for the i-th test print. In the regression, we give higher
weights to training samples with more similarity to a given test
print. As a similarity metric, we use the inverse of the distance
between the ink layer-layout of the i-th test print ci and that of the
j-th train print cj .
s 11
s
21
S =  .
 ..
sp1
Contoning
•
Halftoning
124:4
...
...
..
.
...
s 1q 
s 2q 

..  .
. 
spq 
(5)
For the i-th test print, we define matrix Wi as a q ×q diagonal matrix
whose diagonal elements are the i-th row of S.
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
0
si2
..
.
...
si1
 0
Wi =  .
 ..
 0
...
...
..
.
0
0
0


 .

0 
siq 
(6)
We can then rewrite Equation 3 as a weighted multilinear regression for each test print separately:
Ψs Wi = Ti CWi .
(7)
In order to obtain the parameter matrix Ti , we need to perform
the following optimization:
argmin
Ti
∥Ψs Wi − Ti CWi ∥ F ,
(8)
where ∥.∥ F denotes the Frobenius norm. We need to solve this
equation to predict the spectrum of every test print. The least square
solution to this equation is the pseudo-inverse solution that can be
computed as:
Ti = Ψs Wi 2 Ct (CWi 2 Ct ) −1
(9)
Matrix Ti is then used, exactly like T as explained above, to calculate the intrinsic absorption of its corresponding 3D printed slab.
We then again compute the intrinsic reflectance from intrinsic absorption and, finally convert the intrinsic reflectance to spectral
reflectance. We further discuss the performance of our proposed
color prediction model when put in practice in Section 7.
5
COLOR CONTONING
Inspired by additive layered manufacturing, we propose to synthesize colors by layering inks on top of each other (a brief study of ink
layering is presented in Appendix B). Unlike halftoning that creates
the perception of continuous colors by changing the area of dots,
contoning superposes available inks with proper thickness at each
printing location. Figure 4 demonstrates the schematic difference
between these two methods. In this section, we present an algorithm
for computing the ink concentrations. Then we show that layers can
have a strict order of inks. This reduces the complexity of workflow
significantly. Finally, we show how to invert the color prediction
model in order to generate layer-layouts for any input color.
Color Contoning for 3D Printing
z
y
x
Fig. 5. Our printer grid. The layer thickness is equal in X and Y (XY-layer)
but is different in Z direction (Z-layer).
5.1
Material Design
Like any other subtractive color mixing, color 3D printing needs
the triple of cyan (C), magenta (M) and yellow (Y) to produce a
reasonable color gamut (refer to [Yule 1967] for an excellent explanation behind this choice). In addition, a white (W) reflecting
"background" is required, comparable to the role of the paper in
2D printing. We also add a black (K) ink to the palette of our 3D
printer to improve black tones. Here, we discuss how we compute
the optimal concentrations of these primary inks.
In our method, we generate colors by arranging color layers on
top of a white background. The color layers are formed using lowconcentration inks to avoid abrupt color changes and thus color
quantization. The trade off is that with lower ink concentrations,
more layers are needed to produce the same color which results
in larger minimum feature size and increased optical and physical
dot gain (Section 8). Minimum feature size is the smallest print
feature whose color can be faithfully reproduced. The feature size
corresponds to different number of layers in Z (Z-layers), or in X or
Y (XY-layers), as our grid is not uniform (Figure 5). In this work, our
strategy is to set a strict threshold on color quantization error and
allow for a maximum number of layers that results in a reasonable
feature size.
To calculate the ink concentrations, we begin with formulating
the white ink. We carefully selected the highest white pigment
concentration that was feasible to print on our system in order to
make the background slab as opaque as possible with the fewest
printed layers. We experimentally found that 5.0% white ink is
reliable enough for printing. Higher concentration inks are generally
less stable since the white pigments (typically made of TiO2 ) are
prone to settling and can easily clog the printhead. After measuring
the spectral reflectance of different number of white layers on a
black background, we observe that at 20 Z-layers of 5.0% white, the
change in the white ink reflectance becomes negligible.
For colored inks (CMYK), we seek a set of ink concentrations
that in the CIELAB color space [Morovic and Luo 2001] generate
a color-quantization error less than a predefined threshold. Fortunately, we find that in our color prediction model, concentrations
and thicknesses correlate well enough. Simply put, a color generated by doubling the concentration is reasonably close to a color
printed with half concentration using two layers. This exempts
us from making new inks for each iteration when searching for
optimized concentrations. Therefore, we print only one set of ink
concentrations, calibrate the model, and computationally change
the concentration by changing the thicknesses in the model. We
• 124:5
start by calibrating a color prediction model (see Section 4) with
1% pigment content for CMYK. Since the quantization is larger in
the thicker XY-layers compared to Z-layers, the model is calibrated
with training prints layered in XY.
After calibrating the model, we build the gamut of the printer
and perform the gamut mapping (Appendix D) and layer separation
(Section 5.3) for all sRGB colors. The color-quantization error is
defined as the color error difference between a layer-separated color
and its nearest neighbor in all gamut-mapped sRGB colors. As a
quantization-error threshold, we choose the quantile 0.99999 of
the layer separation error for all gamut-mapped sRGB colors. We
perform a grid search by changing the initial concentrations in
0.05% steps until we satisfy the considered accuracy. Setting the
quantization error to 3.0 CIEDE2000 ( ∆E 00 ) units [Sharma et al.
2005], we get c = 0.25%, m = 0.25%, y = 0.25% and k = 0.15%. After
printing with these concentrations, we verify that the quantization
error is 3.33 ∆E 00 for XY-layering and 1.57 for Z-layering.
5.2
Ink Order
When permuting N inks within 1 to L layers, the number of possible
layer-layouts ϒ grows according to the following formula (Appendix C):
N L+1 − 1
− 1.
(10)
N −1
Performing any computation for all possible layer-layouts would
be impractical given the very large number of configurations. For
a realistic scenario with maximum 40 layers, this number is almost 1.6×1024 . Given the complexity that free permutation of layers
brings, allowing for it makes sense primarily if it produces a larger
color gamut. Since our material formulation exhibits minimal scattering (Figure 3), we conjecture that the volume of the color gamut
is mostly a function of the thicknesses of different inks in the layers
and does not change considerably with changing their specific order.
Therefore, we opt for a stratified layering scheme: identical inks
are grouped together and are sorted in a strict order. This results
in a significant reduction in the number of possible layer-layouts
(Appendix C):
!
L+N
ϒ=
− 1.
(11)
N
For example, stratified layering of 4 inks allowing a maximum of 40
layers gives only 135,750 possible configurations.
Ideally, to compare the two layering modes (free-permutations
and stratified), we should sample both color gamuts and compute
their volumes. However, our color prediction model would need a
very large calibration set in order to predict stacks with arbitrary
order. Instead, we verified our assumption with a simple experiment
where we printed two inks (magenta and yellow) and allowed 7 layers at most. This choice kept the number of prints and measurements
manageable. We printed and measured all possible permutations of
these two inks on a white background resulting in 254 colors, out of
which 35 corresponded to the stratified layering category. Figure 6
shows the colors of these samples in a hue-angle, chroma diagram
(Appendix B). Observe that colors with the same amount of inks
but different orders form small clusters in this figure.
ϒ=
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•
V. Babaei et. al.
Perceivable color
60°
120°
80
30°
Ink stacks
40
0°
Black
maximization
Fig. 7. Completely different ink stacks can generate almost identical colors.
Our black maximization strategy chooses the stack with largest amount of
black ink. Therefore, we produce a smooth transition between layer-layouts
of neighboring printer pixels with similar colors.
330°
Fig. 6. The measured colors of a set of printed slabs in the hue-angle, chroma
diagram. Slabs have up to 7 layers of magenta and yellow inks, where they
freely permute. Among them, prints corresponding to stratified layering
are shown with blue circles (the gray circles represent colors of stacks with
either only magenta or only yellow inks from 1 to 7 layers, therefore they
have one possible permutation).
Since lightness is absent in the hue-chroma representation, we
also measured the color differences between all pairs within each individual cluster. The color difference formula accounts for luminance
and chrominance. The average of the maximum color differences of
members of each cluster is only 5.8 ∆E 00 . Therefore, we observe that
while the color of an ink stack is a function of its order, the color
change due to the inks’ order is not significant enough to make a
change in the gamut volume. This is a key observation that leads us
to adopt an ink grouping strategy.
5.3
Layer Separation
Now we show how to find a layer-layout that best represents an
input colors. Before that, we must ensure that these input colors
are reproducible by the printer and its available materials. For this
reason, in the gamut mapping stage, we map the input colors into
the color gamut of the printer (Appendix D).
In halftone color reproduction, the color separation step deduces
the amounts of inks that are needed to print a target color Ct . This
is usually formulated as the following [Pjanic and Hersch 2015]
argmin
∆(C (c, m, y, k ), Ct )
(12)
c,m,y,k
where C (c, m, y, k ) is a color computed using a prediction model and
the ∆ function is a color difference formula. Color separation is the
most computationally expensive part of a color reproduction workflow as most color difference formulas are highly non-differentiable
[Kauvar et al. 2015].
In contone color reproduction, we need to solve the similar inverse problem of going from a color to a layer-layout. We call it
layer separation. Color contoning allows for a very efficient layer
separation. The contone color gamut is made of a set of discrete 3dimensional CIELAB points corresponding to distinct layer-layouts.
The number and dimension of this set of points lend themselves
very well to the nearest neighbor search algorithms. Using a KD-tree
data structure, we can find the nearest neighbor in the color gamut
set to a query point very quickly. Then, we find the layer-layout
corresponding to that color. The Euclidean distance offered by KDtree nearest neighbor search is equivalent to the classic ∆E 76 color
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
Fig. 8. A top-down photograph of a semi-cylinder printed with color separation tables from Z-layered calibration set.
difference formula [Wyszecki and Stiles 1982]. This formula gives
reasonable results but is known to be perceptually non-uniform in
some regions of the color space. In order to solve this challenge,
we perform a rapid K nearest neighbor search for a relatively large
K. Having obtained the K nearest colors for the target color in a
∆E 76 sense, we refine them with a more advanced color difference
formula, i.e. ∆E 00 [Sharma et al. 2005].
The refinement stage gives us an opportunity to optimize the
layer-layouts. Many target colors have layer-layouts whose colors
are within a very close distance. These colors are usually a result of
the trade-off between black ink on one hand and the three chromatic
CMY inks on the other hand. We can reduce the amount of CMY
inks in the stack and replace it with black without any noticeable
change in the color. In our work, inspired by black maximizing color
separation strategies [Sharma and Bala 2002], we select the layouts
with the thickest black layer during layer separation. Apart from
classic benefits , this method has some advantages specific to 3D
color contoning. It enforces inter-layer smoothing by preventing
sudden changes in the layer-layout of neighboring pixels with close
colors. Figure 7 demonstrates this property.
6
3D CONTONING
Generating contone colors on 3D surfaces is straightforward. We use
a lookup table, precomputed with our color reproduction workflow,
to simplify and accelerate the color assignment process. The lookup
table exhaustively generates a mapping from all 24-bit sRGB color
space to layer stack.
During our experiments, we observed that it is sufficient to use the
color prediction model calibrated with only Z-layered patches. This
relieves us from calibrating the model in other directions and therefore reduces the required number of printed patterns significantly.
To verify this, we printed a semi-cylinder using a layer-separation
table resulting from the color prediction model calibrated with Zlayers only. As we see in Figure 8, this generates consistent colors
over the curvature.
Color Contoning for 3D Printing
a)
• 124:7
b)
Fig. 9. Ink assignment conflict. Areas shown within dashed lines are in
conflict. a) On a convex surface, if the color of the texture changes, the
ink stacks of different colors collide inside the color shell. b) On geometric
edges, even if there is a constant color, collisions are inevitable.
Naive layering
Given an input 3D model with its texture and surface parameterization, one naive approach is to voxelize it, store color and normal
information on the surface voxels, and then follow the normals to
assign the corresponding color stack to the internal voxels. While
careful sampling on the surface is required to avoid missing internal
voxels, two main problems arise: (1) convex regions lead to neighboring color stacks colliding into each others and (2) a given color
stack height may not be available, as illustrated in Figure 9.
Our approach is instead to do per-voxel color assignment using
distance queries to the surface. For every voxel, we query the closest
surface point, extract the corresponding color stack, and sample
it using the surface distance to get our material assignment. This
strategy avoids sampling issues of ray marching from the surface,
as well as conflicting color stacks. However this local approach
does not handle the available color stack budget which leads to
invalid colors because any stack whose height is too large gets cut
instead of getting shrunk. To solve this last budget problem, we
can precompute the maximum available stack height per voxel. In
a first pass, we accumulate the maximum distances to the queried
surface voxels and store it with the color information on the surface.
In the second pass, we compute the color as before, but use an
adaptive color stack fitting the stack budget (we precompute color
stacks for heights up to h = 48 and assume further voxels use white).
Effectively this shrinks individual color stacks to fit the voxel budget
while being spatially coherent.
Figure 10 illustrates the effect of adaptive stack height for a model
with edges. The left-hand example shows dramatic change in color
of four rows along the thick to the very thin triangles, whereas the
adaptive strategy achieves much more consistent colors even for
very thin triangles. The top-edge artifact (unprinted or unevenly
printed top layers), shared by both examples, are printer artifacts.
This is a typical problem of current printers which have difficulties
in reproducing such small geometric features.
7
EVALUATION OF CONTONING
We use a custom, inkjet 3D printer (Appendix E.1). As the printhead
has only 5 channels allotted to CMYKW, we cannot also print support material and thus our printed examples are all height fields.
The printer achieves resolution of 720 dpi in X and Y and between
1,200 and 3,600 dpi in Z. The main limiting factor of our 3D printer
is the print speed and jetting inconsistency.
Adaptive layering
Model side view and camera direction
Fig. 10. Contoning at sharp edges. The naive implementation allows inks to
collide and results in wrong color at edges. With adaptive ink assignment,
we are able to change the color gradually towards the tip, such that the color
change becomes less noticeable. A side-view illustration of color assignment
of the profile section of the triangles is provided (top row). Photographs of
the models (middle row) show that, due to printer difficulty, the heights
of narrower triangles are not fully printed. The schematic drawing of the
model, i.e., a number of triangles having fixed height and variable base, and
the direction of camera for photographs are also shown (bottom row).
Print speed is limited due to the small number of nozzles per ink
(only 60 for CMY) and the inefficient per-material nozzle arrangement (CMY inks are lined up along the Y axis which is perpendicular
to our print motion, thus requiring multiple passes over the same
area).
The inconsistency comes from jetting our highly viscous photopolymer inks (see Appendix E.2) using a consumer-level printhead
originally designed for low-viscosity aqueous inks. The printhead is
also mounted on an inexpensive gantry that has not been optimized
for precision placement.
Our material formulation results in perfectly clear, mechanically
rigid material to which we can add CMYKW pigments. The surface finish of our prints is highly specular. In order to remove the
specularity from the prints, we print all flat prints upside-down on
the printer metal platform that has fine microstructures which adds
some microgeometry to the surface of our prints.
In this work, we allow for color stacks that are 48-layer deep. This
generates a reasonable color gamut and corresponds to a feature
size of about 1 mm. When using 48 color layers, our table includes
270,725 printer colors and thus ink stack entries. Figure 11 and
Figure 12 show two images printed with our method. We start with
sRGB source images that represent different colors and textures. We
first generate the previews using our color reproduction framework.
There is a considerable difference between original and preview
images since the sRGB gamut is typically about two times larger
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
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•
V. Babaei et. al.
than most printer gamuts. We also show the photographs of our
fabricated results. Note that part of the color difference between
the preview and the print photograph is inherent to the challenges
of print soft proofing. Proper soft proofing needs to be seen using
a calibrated, colorimetric displays (e.g. Eizo CG245W). That said,
the prints do not show significant artifacts, preserve the edges,
and reproduce the colors well. On the other hand, the prints are
somewhat blurry, largely because of dot gain. We analyze dot gain
related issues in depth in Section 8.
Figure 1 e and f, and Figure 13 are 3D models printed using 3D
contoning. Since 3D prints need a large number of layers, we limit
their size in the favor of print time. Although the prints are small
(maximum 1 inch wide), the details are pronounced and texture
appears high definition.
Color prediction accuracy. In order to examine the prediction accuracy of our proposed color prediction model, we calibrate the
model by printing all layer-layouts of the CMYK inks in 12 Z-layer
intervals, while allowing a maximum of 48 layers. This yields 70
calibration (train) patches to print. For the test set, we print two
test patches per each possible ink-combinations. Since there are
15 possible combinations for four inks, the size of our test set is
30 patches. The thicknesses of individual inks for each test sample
are chosen randomly. This way we ensure that the test samples are
within different regions of the color space.
We run the same experiment for prints with XY-layering. The
obtained accuracy for the color prediction model calibrated in this
mode is lower. There are significantly more inconsistencies in the
prints because it is difficult for the printer to generate uniform
vertical patches. This is not however an important problem since
we use the more accurate color prediction model calibrated for Zlayering to print all results. The XY model is only used for estimation
of the ink concentrations.
Following standards in color reproduction, we summarize the
prediction accuracies in Table 1, with and without the weighting
strategy. The prediction accuracy is expressed using the ∆E 00 color
difference metric as well as the RMS% difference between the measured and predicted spectral reflectances. The model accuracy is
excellent for Z-layered prints and comparable to the accuracy of
state of the art CMYK models for 2D printing. Both train and test
patches are printed simultaneously and with the same batch of inks,
thus repeatability errors are minimized in this experiment to show
the full capacity of the color model. The weighting scheme is proved
to be quite helpful by reducing the prediction error. Furthermore,
Figure 14 shows a very good match between the measured and
predicted spectral reflectance curves of some randomly selected test
patches.
Contone color gamut. The gamut of our 48 Z-layer contone printer
is 325 kilo CIELAB, comparable with the gamut volume for a typical
inkjet CMYK printer [Rossier 2013]. The color gamut is directly
influenced by the layer thickness. We expect that after a certain
number of layers the gamut of color contoning saturates. The gamut
saturates when adding a layer to the ink stack does not change the
reflectance. In the literature, this is sometimes referred to as the
infinite reflectance (R ∞ ) [Hébert et al. 2008].
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
Prediction mode
normal Z
weighted Z
normal XY
weighted XY
mean ∆E 00
1.79
1.24
4.11
2.90
max ∆E 00
3.61
2.54
8.33
7.53
mean RMS%
1.71
1.19
3.04
2.41
Table 1. The prediction accuracy of our color prediction model under the
D65 illuminant for a test set of 30 samples. The results are listed with and
without the weighted regression scheme.
Performance. Our Matlab implementation takes less than 5 minutes to generate an exhaustive lookup table that maps every entry
in the sRGB gamut to layer-layouts. The ray-tracing in the gamut
mapping step was accelerated using a native C implementation.
Generating print data is instant for our 2D results. For 3D prints,
our current implementation is bound by the signed distance function queries but is sufficiently fast that print data generation can
be done upfront or interleaved with the printing process. We have
also implemented a simple preview procedure based on ray casting
and integrating through the color stacks. A real-time 3D preview is
feasible and left for future work.
8
DOT GAIN
Dot gain is a well-known problem in 2D printing. In most dropon-demand printing processes, the dot that gets deposited spreads
within the substrate resulting in physical or mechanical dot gain.
Furthermore, light can penetrate in an inked area, travel in the
translucent paper, and exit from non-inked areas; this is called optical dot gain. Separating these two effects is an unsolved problem.
Instead, in 2D printing the total dot gain is measured and compensated for through a set of color measurements [Hersch and Crété
2005].
Mechanical dot gain in our system results from inks merging in
the substrate before they undergo phase change to a solid material
and because of imprecisions in the inexpensive gantry, that introduces mechanical dot-gain as it cannot precisely register different
layers on top of each other.
Optical dot gain in contoning is somewhat different from 2D
optical dot gain. Transparent CMYK inks are stacked on a diffuse
white. The light travels in the color stack and is reflected back at all
directions. Therefore optical mixing between adjacent ink columns
does occur.
Both physical and optical dot gain reduce the resolution and
blur the results. Similarly to 2D printing, we can compensate for
the total dot gain in 3D contoning without trying to distinguish
between physical and optical dot gain. If we consider the total dot
gain as a Gaussian filter, we can estimate the kernel parameters
using a calibration print. This print consists of representative color
rectangles that generates horizontal and vertical edges. Then, we
take a photo of the blur calibration chart, convert it to linear RGB
by an inverse gamma correction, and combine all color channels
to produce a grayscale image. We fit a gaussian blur kernel that
best reproduces the effect of dot gain on all edges. Having found
the blur kernel, we apply the Richardson-Lucy algorithm [Lucy
1974; Richardson 1972] to produce a deconvoluted texture which
Color Contoning for 3D Printing
• 124:9
Fig. 11. An image printed with our contoning technique at 720 dpi. From left to right: sRGB original image, color preview after gamut mapping and layer
separation, and fabricated result, respectively. The prints have 68 Z-layers, at least 20 of which are white backing layers. (print size (xyz): 2108×2560×68; print
time: 9hrs).
Fig. 12. An image printed with our contoning technique at 720 dpi. From left to right: sRGB original image, color preview after gamut mapping and layer
separation, and fabricated result, respectively. The prints have 68 Z-layers, at least 20 of which are white backing layers. (print size (xyz): 3825×2550×68; print
time: 15hrs).
Fig. 13. The preview of our 3D printed models (respective left side of each model) and a photograph of the final print (respective right side of each model). The
preview includes the gamut mapping and the layer separation steps. Note that our preview shows an orthogonal top-down view of the 3D printed models,
without including the ambient occlusions and/or the blur kernel. Adding these effects for 3D prints can improve the preview significantly.
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
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V. Babaei et. al.
1
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400
500
600
700
0
400
500
600
700
0
400
0.1
500
600
700
0.05
400
Wavelength (nm)
Fig. 14. The measured (solid-line) and predicted (dashed-line) spectral reflectance curves of a number of randomly selected test prints.
compensates for the blur effect. We assume that the dot gain kernel
is the same regardless of the print orientation. Therefore for 3D
objects, we use the physical size of the kernel obtained for flat
prints. Figure 15 shows two image printed using contoning, with and
without compensating for dot gain. Applying the kernel successfully
recovers the lost details in the print due to the dot gain.
9
COMPARISON WITH HALFTONING
Halftoning is the only solution for most 2D printers as they print
only a few layers. Contoning is an alternative solution when one
can use depth for color mixing. The two methods can be thought as
two extreme color reproduction techniques. We found it challenging
to have a fair comparison of contoning with color 3D halftoning
[Brunton et al. 2015]. One particular challenge is to determine the
balance between the number of layers and inks concentrations for
3D halftoning. One can use fewer layers of more concentrated inks
or more layers of less concentrated inks, with different consequences
on the print quality. In the future, we would like to perform a comprehensive comparison involving either finding the optimized set
of parameters (number of layers and ink concentrations) or showing examples that span them. Therefore, we only print flat images
for comparison of halftoning and contoning and employ standard,
state-of-the-art error diffusion halftoning (with only 4 CMYK layers). The only exception where we use multi-layer halftoning is the
comparison of gamut volumes.
Figure 16 shows an image printed with contoning and halftoning.
The halftone image has been printed using a state of the art color
reproduction workflow [Rossier 2013]. The framework compensates
for the total dot gain occurring in the halftone prints. To find the ink
concentration for halftoning, we try to compute them such that they
generate the same gamut volume as the gamut of contoning. We
use a simple Neugebauer model [Hersch and Crété 2005] because
it requires a small number of calibration samples. The number of
measurements would become unwieldy, if we wanted to use a full
ink spreading Yule-Nielsen and for each concentration. This is the
reason of gamut mismatch between contone and halftone prints in
Figure 16.
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
Color gamut. In order to have a fair comparison between the color
gamut of contoning and halftoning, we use identical concentrations
for CMY inks (c = m = y = 0.25%), and identical number of layers
(L = 24). For halftone color prediction, we use the ink-spreading
Yule-Nielsen spectral Neugebauer model [Hersch and Crété 2005]
calibrated for 24 layers. We use only CMY inks and up to 24 layers
because with this setup the halftone prediction model shows a
sufficient accuracy (∆E 00 = 4.9) for gamut volume computation.
Higher number of layers would result in unacceptable halftone
model accuracy because halftone color prediction models have not
been designed to account for the ink spreading only in 2D and not
in 3D. For this identical setup, the gamut of color contoning is 10%
larger than the color gamut obtained with color halftoning. As we
discuss in Appendix B, layer superposition is known to generate
more saturated colors than spatial mixing.
Print at low resolution. Figure 17 shows an image printed at 180
dpi printer resolution, using our contoning method and also with
the error diffusion algorithm. The contone prints does not show any
signs of spatial subsampling or other artifacts whereas the same
image printed with error diffusion halftoning exhibits unacceptable
dot patterns.
Color predictability. There are established models to accurately
predict the color of CMYK halftones [Hébert and Hersch 2015].
However, if more inks are involved [Babaei and Hersch 2016] or
if multi-layer halftones are required [Brunton et al. 2015], color
prediction becomes very challenging. The prior work in color prediction of multi-layer 3D halftone [Brunton et al. 2015] uses a fully
data-driven color prediction scheme. Such an empirical approach
requires a large number of measurements and is only applicable to
a fixed number of layers.
Our proposed color prediction model is the first model for predicting the color of a 3D printed stack of different inks. For CMYK
prints, it needs a small number of calibration samples, and can predict the color of any thickness from 1 to L layers once calibrated
for L layers. However, our sampling of the training samples of our
color prediction model, has an exponential growth with the number
of inks.
10
DISCUSSION AND FUTURE WORK
While advantages of contoning are due to the use of depth for
creating colors, advantages of halftoning come as a consequence of
it using the surface (and less depth) for color synthesis. Halftoning
can reproduce colors with the significantly lower number of layers.
This will guarantee a smaller feature size and less dot gain. A possible
extension is to combine both methods in a complementary manner.
One way is to use halftoning for high spatial frequency areas, and
contoning for uniform regions where the halftone dots are most
objectionable.
Future advances in 3D printers that enable printing with a larger
number of materials can release the full potential of contoning
by allowing it to create colors with a minimal height of the ink
stack. This will guarantee a smaller feature size and less dot gain.
With a larger number of inks, contoning generates a larger color
gamut while keeping the ink stack height low. With the current
trends in the market, such 3D printers are becoming more and more
Color Contoning for 3D Printing
•
124:11
Fig. 15. Left: The original image (gamut mapped). Middle: Photograph of the print of the original image with standard contoning. Right: Photograph of
the print of the presharpened image. The estimated blur kernel is a Gaussian with filter size 30 XY-voxels and standard deviation 10, estimated from a blur
calibration print.
accessible. The most recent industrial inkjet 3D printer can print
with 7 materials [Stratasys 2016]. With a larger number of inks,
we should strive for a more physical and scalable color prediction
model.
Error-diffusion
Our contoning
Fig. 16. An image printed on our printer with error-diffusion halftoning,
and our contoning method.
Fig. 17. An image printed with halftoning (top) and contoning (below) at
180 dpi printer resolution (print size (xy) = 2280×1300).
A main simplifying assumption in our model is that we treat each
ink column independently of its neighbors. In reality, these columns
interact with each other due to both optical and mechanical mixing. The interaction is more complex for more difficult geometries,
where factors such as interreflection come into play. Our blur kernel
estimation and compensation method is a first solution carried out
in the texture domain. A more promising approach is to modify
the material arrangement, and account for the mechanical mixing
and the non-local light transport. Instead of finding the ink composition of a single column, optimizing for the materials within a
neighborhood, while considering its geometry, is most interesting.
One can predict these interactions by precomputing them within
small neighborhoods through a combination of measurement and
simulation. Then, similar to structure preserving halftoning [Pang
et al. 2008], the difference between the original appearance and
the print appearance can be minimized. Another exciting solution
to deal with this problem is to design techniques that compensate
for the unwanted light transport or ink spreading within printed
volume. For example, it could be addressed by printing thin, opaque
walls at color edges.
A user study of contone prints in order to understand their aesthetic aspects and to compare them with halftone prints is left for
future work. For sensitive applications, such as fine-art reproduction, this would be necessary. We believe that the current workflow
lends itself well to reproduction of other appearance attributes, such
as gloss and translucency. Finally, it would be exciting to print with
our workflow on an industrial 3D printer. The only requirement is
the ability to control the ink concentrations.
Concluding remarks. The method presented in this article is simple. While we believe this to be one of its main virtues, it is largely
thanks to eliminating the need for halftoning. The new color reproduction system proposed in this paper paves the way for highdefinition color reproduction for 2.5D and 3D printers.
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
124:12
V. Babaei et. al.
SAUNDERSON CORRECTION
The Kubelka-Munk theory assumes no discontinuity in the refractive index. In reality, when light enters the material from the air
side, a portion r s is reflected back in the specular direction. When
light exits the material, a portion r i of light is reflected back into the
material and bounces infinitely between the ink side of material-air
interface and the substrate. Based on these observations, Saunderson
[1942] proposed a correction on the measured reflectance spectrum
Rm (λ), in order to infer the intrinsic reflectance ρ (λ) as if there was
no interface:
Rm (λ) − Kr s
,
(13)
1 + (1 − K )r i r s + r i Rm (λ) − r i − r s
where K is set to 1 if the device measures the specular reflection,
otherwise it is set to 0. r i and r s are computed using the Fresnel equations and their values depend on the angle of incident illumination θ
and the relative index of refraction between ink and air. For our measuring geometry with hemispherical light (see Appendix E.3) and
assuming uniform illumination, r s is computed by integrating the
Fresnel component F (θ ) over the illumination hemisphere [Simonot
et al. 2006]:
ρ (λ) =
Z 2π Z π /2
F (θ )
cos θ sin θdθdϕ,
π
0
0
which simplifies to
rs =
rs =
Z π /2
F (θ ) sin(2θ )dθ .
(14)
0
3D PRINTED GLAZE
In painting, glaze is a technique where transparent layers of paint
are applied on top of each other. Each layer is made of pigments
scattered in a thin transparent medium. The artist creates different
colors by varying the number and color of glaze layers. Before
applying a new layer, the bottom layer should be dry. Glazes are
known to produce most saturated colors [Elias and Simonot 2006].
A very similar process also occurs in our prints; every time a layer
is deposited, it is instantly dried by UV-radiation and then the next
layer is applied.
Here, we study the color and spectral changes when increasing
the layers of an ink. Figure 18 (left) shows the spectral reflectance
factor of 1 to 50 layers of a yellow ink printed with our 3D printer.
As expected, the reflectance decreases with increasing the number
of layers. In theory, if we continue adding more layers, there will
be a point after which the reflectance does not change [Hébert et al.
2008].
For showing the colorimetric changes, we use hue-angle, chroma
diagram computed in CIECH color space [Schanda 2007]. From
CIELAB a ∗ and b ∗ , we compute the hue-angel as arctan(b ∗ /a ∗ ), and
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
135°
0.8
0.7
100
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0.5
90
180°
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80
70
Hue
an
gle
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60
0°
a
om
Chr
0.3
0.2
0.1
0
400
225°
450
500
550
600
650
315°
700
Wavelength (nm)
270°
Fig. 18. Spectral and color changes due to contoning; Left: the measured
spectral reflectance factor of 1 to 50 layers of a yellow ink with 0.5% ink
concentration. Right: The hue-angle, chroma diagram of 1 to 50 layers of a
yellow ink with 0.5% pigment concentration.
√
chroma as a ∗2 + b ∗2 . In this polar diagram, different angles represent different hues and larger radii correspond to more saturated
colors. Figure 18 (right) shows the effect of increasing the number
of layers in this graph. By adding more layers, the chroma first
increases and then decreases with a shift in the hue-angle. This
phenomenon has already been studied in glazes [Elias and Simonot
2006] and also in piling nonscattering colored sheets [Hébert et al.
2008]. These two processes are very similar to UV-ink color 3D
printing.
C
(15)
The same equation is used to compute r i assuming the light coming
from the substrate (and hitting the non-air side of the interface)
is Lambertian. The only difference is that the refractive index is
inverted as the light goes from denser media to the media with
lower index of refraction. Having set the refractive index of our
polymer carrier to 1.5, we obtain r s = 0.092 and r i = 0.596.
B
90°
1
0.9
Reflectance factor
A
•
COUNTING LAYER-LAYOUTS
First we compute the number of all possible layer-layouts with N
inks and maximum L layers. For 1 layer, there are N , for 2 layers
there are N 2 and for L layers there are N L possible layer-layouts.
This leads to a geometric series:
N L+1 − 1
− 1.
(16)
N −1
For stratified layering, number of all possible layer-layouts with N
inks and maximum L layers is equal to the number of non-negative
solutions of the following equation:
N + N 2 + ... + N L =
I 1 + I 2 + ... + I N + I Φ = L.
(17)
where Ii is the number of layers of the i t h ink and I Φ is the number
of void layers. The number of non-negative solutions to Equation 17
is know to be [Rosen and Krithivasan 2007]:
!
L+N +1−1
(18)
N +1−1
which after excluding the single all-void layer from our counting
gives:
ϒ=
!
L+N
ϒ=
− 1.
N
D
(19)
GAMUT MAPPING
The gamut mapping takes place in a device-independent color space
which is usually the CIELAB [Morovic and Luo 2001]. We therefore convert colors from different spaces into the CIELAB color
Color Contoning for 3D Printing
a)
b)
180 180
Mapping region
Y
60
*
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W
L*=lh
K
M
60
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60
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3
2
1
4
3
2
1
4
3 y
2
1
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x
*
L =ll
L*min
C
*
Fig. 19. Lightness-chroma plot of the sRGB gamut (union of white, light
gray and dark gray areas) and a printer gamut (union of two light and
dark gray areas) in a constant hue-angle plane. In this figure, the lightness
mapping has already taken place such that the minimum and maximum
lightnesses of both gamuts coincide. In this step, the colors with chroma
outside the printer gamut are mapped into the mapping region (light gray
region) of the printer gamut . Colors with L ∗ ≤ ll or L ∗ ≥ lh are mapped
along the line segment made of those colors and ll or lh . Other colors are
mapped perpendicularly to the L ∗ axis. The choice of ll , lh and the mapping
region is not deterministic and can change from gamut to gamut.
space. This is particularly straightforward for input colors in standard RGB (sRGB) [Morovic and Luo 2001]. In order to generate
the CIELAB gamut of the printer, the resulting reflectance spectra
from all possible stratified layer-layouts are computed using our
developed color prediction model. Each spectrum is then converted
to CIEXYZ tristimulus values with a D65 illuminant and in respect
to the 2°standard observer [Wyszecki and Stiles 1982]. The CIE-XYZ
values are then converted into CIELAB color coordinates using the
reflectance spectrum of our 3D printed white material as a white reference. The non-convex gamut boundary is obtained by performing
a ball pivoting surface reconstruction on the CIELAB color points
[Bernardini et al. 1999].
For the gamut mapping, we follow the standard methods in the
color reproduction field [Morovic and Luo 2001]. As a first step, we
map the lightness range of the sRGB gamut into the lightness range
of the printer gamut. The lightness-mapped gamut is then chroma
mapped into the gamut of the printer for each hue-angle slice. See
Figure 19 for more details [Rossier 2013].
E EXPERIMENTAL SETUP
E.1 3D Printer
We use MultiFab [2015], a laboratory-scale, custom-built, multimaterial 3D printer that uses an inkjet-based, phase-shift, photopolymer printing process. The system is similar to commercial printers
developed by Stratasys [2016] and 3D Systems [2013]. We use a
consumer-level, drop-on-demand (DoD) Epson Workforce 30 piezo
inkjet printhead that consists of several nozzles. Each nozzle is provided with a piezo-actuator, which on application of a voltage pulse
(or waveform) generates pressure oscillations inside the ink channel.
These pressure oscillations push the ink drop out of the nozzle [Khalate et al. 2012]. The printhead accommodates 5 channels, allowing
Fig. 20. a) The schematic orientation of the nozzles in Epson Workforce 30
printhead. There are three lines of nozzles. The first and second lines are
connected to two ink channels and they have 180 nozzles each, while the
third line is connected to three channels and each channel has 60 nozzles. b)
The printhead moves with the maximum speed in X direction and traverses
the Y direction in four passes.
5 materials to be printed at the same time. As we use 5 materials
(CMYKW), we cannot have support material and our 3D prints are
height field examples. The inks are heated and kept at elevated
operational temperature to reduce their viscosity, improve ink flow
and jetting. We print with droplet volume of between 10 and 15
picoliters and achieve resolution of 720 dpi in X and Y and about
2000 dpi in Z, subject to the individual material characteristics. The
printer additively builds objects by printing one layer at a time; a UV
LED module is affixed immediately to the printhead and is activated
concurrently while the printhead is printing in either direction. The
emitted UV radiation pins and solidifies the jetted droplets.
The amount of the overlapping is primarily a function of the drop
volume which itself is a function of the waveform applied to the
piezo-electronic actuators at each nozzle channel. We observed that
the default waveform of the printhead results in very large drop
volumes. This causes sever overlapping of the adjacent contone
columns. Therefore, we experimentally designed a new waveform
for our materials that corresponds to a smaller drop volume.
The main limiting factor of our 3D printer is the print speed. The
slow process is due to the small print swathe of the printhead (1 inch)
and the limited number of nozzles (540). There are three lines of
180 nozzles on the printhead one of which is connected to three ink
channels and therefore three out of five channels are 1/3" wide and
have only 60 nozzles (Figure 20 a). Thus when printing with more
than two inks, our speed is reduced. The second factor influencing
the print speed is the nozzle density on the printhead. The rows of
the printhead are not fully populated with nozzles. The printhead
needs to traverse the same the Y direction multiple times in order to
fire at all designated pixels and fulfill the resolution requirement. In
our case, the printhead needs 4 passes (Figure 20 b). The other factor
is the speed of the printhead movement in X direction which itself
depends on the firing frequency of the printhead. With higher firing
frequency, we can move faster in X and increase the overall printing
speed. Finally, the employed inkjet printhead was not designed to
work with highly viscous photopolymer ink such as ours; the jetting
consistency is such that it requires continuous nozzle rotation which
can further slow down printing of certain geometries.
Industrial printheads have significantly larger swathes (2.55" is
considered standard), and are designed to work in one material
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
124:14
•
V. Babaei et. al.
per printhead configurations. Further, their firing frequency can be
several times faster than a typical consumer-level printhead such
the one employed in our system.
E.2
Materials
All materials that we use are UV-curable photopolymers. We mix
two monomers and one olygomer to make our base carrier. We add
a small amount of photoinitiator that triggers the polymerization
when exposed to the UV light. Before adding color pigments to
these materials, we measure and test various properties including
density, surface tension, viscosity, thermal stability, shelf-life, and
phase change speed. Additionally, we optimize the jetting of each
ink by adjusting the waveform applied to the printhead piezos.
We expect our formulation to result in a perfectly clear, mechanically rigid print. However, this depends on the careful selection of
the photoinitiator and its concentration. We observe that some photoinitiators introduce an objectionable yellowing tint to prints. This
is because the unreacted photoinitiators inside the polymer tend to
reduce the intensity of the blue region of the spectrum. Photoinitiators have their peak absorption in the UV range. The location of this
peak varies in different photoinitiators and usually shifts toward
the blue region when increasing the photoinitiator concentration.
Yellowing occurs when this peak is close to the visible blue range.
Formulating an optimal ink involves selecting a photoinitiator that
works well with the given UV curing unit and allows curing to occur
in low enough concentrations that limit the yellowing effect while
still promoting full polymerization.
E.3
Spectral Measurements
For calibration of our color prediction model, we measure all train
and test samples using a sphere spectrophotometer, namely the XRite Color i7 instrument. This instrument provides a diffuse incident
illumination through an integrating sphere. The diffuse incident
light falls on the measured sample at all possible angles and the
sensor captures the reflected light at 8°from the normal incidence.
The response is in the form of reflectance factor sampled at 10 nm
intervals from 350 nm to 740 nm. We work with the visible range
from 400 nm to 700 nm, therefore every measurement is a 31-vector
of reflectance factors. The device gives the possibility to include or
exclude the specular component reflected off the sample. We use
the device in the specular component excluded mode.
ACKNOWLEDGMENTS
We would like to thank our reviewers for the helpful suggestions, XRite (especially Peter Ehbets) for lending us the spectrophotometer,
David Kim and Wenshou Wang for help with materials, and Subra
Subramanian, Javier Ramos, and Pitchaya Sitthi-amorn for help
with printing. We thank Sergiu Gaman, Romain Rossier and Roger
Hersch from EPFL for the gamut mapping software. We are grateful
to Sylvain Paris for his feedback on the manuscript. The 3D models
and images of physical materials were obtained from TurboSquid,
GrabCAD, Wikimedia and ISO. Vahid Babaei acknowledges the
Swiss National Science Foundation (SNSF) support, under grant
number P2ELP2_161843.
ACM Transactions on Graphics, Vol. 36, No. 4, Article 124. Publication date: July 2017.
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