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# Computer-Generated Watercolor

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## Abstract and Figures

This paper describes the various artistic effects of watercolor and shows how they can be simulated automatically. Our watercolor model is based on an ordered set of translucent glazes, which are created independently using a shallow-water fluid simulation. We use a Kubelka-Munk compositing model for simulating the optical effect of the superimposed glazes. We demonstrate how computergenerated watercolor can be used in three different applications: as part of an interactive watercolor paint system, as a method for automatic image "watercolorization," and as a mechanism for nonphotorealistic rendering of three-dimensional scenes.
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Computer-Generated Watercolor
Cassidy J. Curtis Sean E. Anderson Joshua E. Seims Kurt W. Fleischer David H. Salesin
University of Washington Stanford University Pixar Animation Studios
Abstract
This paper describes the various artistic effects of watercolor and
shows how they can be simulated automatically. Our watercolor
model is based on an ordered set of translucent glazes, which are
created independently using a shallow-water ﬂuid simulation. We
use a Kubelka-Munk compositing model for simulating the optical
effect of the superimposed glazes. We demonstrate how computer-
generated watercolor can be used in three different applications:
as part of an interactive watercolor paint system, as a method for
automatic image “watercolorization,” and as a mechanism for non-
photorealistic rendering of three-dimensional scenes.
CR Categories: I.3.3 [Computer Graphics]: Picture/Image Gener-
ation; I.6.3 [Simulation and Modeling]: Applications.
Additional Keywords: Fluid simulation, glazing, illustration,
Kubelka-Munk, non-photorealistic rendering, optical compositing,
painting, pigments, watercolor.
1 Introduction
Watercolor is like no other medium. It exhibits beautiful textures
and patterns that reveal the motion of water across paper, much as
the shape of a valley suggests the ﬂow of streams. Its vibrant colors
and spontaneous shapes give it a distinctive charm. And it can
be applied in delicate layers to achieve subtle variations in color,
giving even the most mundane subject a transparent, luminous
quality.
In this paper, we characterize the most important effects of wa-
tercolor and show how they can be simulated automatically. We
then demonstrate how computer-generated watercolor can be used
in three different applications: as part of an interactive watercolor
paint system (Figure 7),as a method for automatic image “watercol-
orization” (Figure 10), and as a mechanism for non-photorealistic
rendering of three-dimensional scenes (Figures 14 and 13).
The watercolor simulator we describe is empirically-based: while it
does incorporate some physically-based models, it is by no means
a strict physical simulation. Rather, our emphasis in this work has
been to re-create, synthetically, the most salient artistic features of
watercolor in a way that is both predictable and controllable.
1.1 Related work
This paper follows in a long line of important work on simulating
artists’ traditional media and tools. Most directly related is Small’s
groundbreaking work on simulating watercolor on a Connection
Machine [34]. Like Small, we use a cellular automaton to simulate
ﬂuid ﬂow and pigment dispersion. However, in order to achieve
even more realistic watercolor effects, we employ a more sophis-
ticated paper model, a more complex shallow water simulation, and
a more faithful rendering and optical compositing of pigmented
layers based on the Kubelka-Munk model. The combination of
these improvements enables our system to create many additional
watercolor effects such as edge-darkening, granulation, backruns,
separation of pigments, and glazing, as described in Section 2.
These effects produce a look that is closer to that of real watercol-
ors, and captures better the feeling of transparency and luminosity
that is characteristic of the medium.
In the commercial realm, certain watercolor effects are provided by
products such as Fractal Design Painter, although this product does
not appear to give as realistic watercolor results as the simulation
we describe. In other related work, Guo and Kunii have explored the
effects of “Sumie” painting [13], and Guo has continued to apply
that work to calligraphy [12]. Their model of ink diffusion through
paper resembles, to some extent, both Small’s and our own water
simulation techniques.
Other research work on modeling thick, shiny paint [2] and the
effects of bristle brushes on painting and calligraphy [30, 36] also
bears relation to the work described here, in providing a plausible
1
The work described here
also continues in a growing line of non-photorealistic rendering
research [5, 6, 9, 16, 22, 23, 26, 33, 39, 40], and it builds on previous
work on animating the ﬂuid dynamics of water [1, 10, 19] and the
effects of water ﬂow on the appearance of surfaces [7, 8, 28].
1.2 Overview
The next section describes the physical nature of the watercolor
medium, and then goes on to survey some of its most important
characteristics from an artist’s standpoint. Section 3 discusses how
these key characteristics can be created synthetically. Section 4
describes our physical simulation of the dispersion of water and
pigment in detail. Section 5 discusses how the resulting distribu-
tions of pigment are rendered. Section 6 presents three different
applications in which we have used our watercolor simulation
and provides examples of the results produced. Finally, Section 7
discusses some ideas for future research.
2 Properties of watercolor
For centuries, ground pigments have been combined with water-
soluble binding materials and used in painting. The earliest uses
of watercolor were as thin colored washes painstakingly applied to
detailed pen-and-ink or pencil illustrations. The modern traditionof
watercolor, however, dates back to the latter half of the eighteenth
century, when artists such as J. M. W. Turner (1775–1851), John
Constable (1776–1837), and David Cox (1783–1859) began to
experiment with new techniques such as wiping and scratching out,
and with the immediacy and spontaneity of the medium [35].
To simulate watercolor effectively, it is important to study not only
the physical properties of the medium, but also the characteristic
phenomena that make watercolor so popular to artists. A simulation
is successful only if it can achieve many of the same effects. In the
1
This approach is essentially the same as the “minimal simulation”
approach taken by Cockshott et al. [2], whose “wet & sticky” paint model is
designed to behave like the real medium as far as the artist can tell, without
necessarily having a real physical basis.
ab cd e f
Figure 1 Real watercolor effects: drybrush (a), edge darkening (b), backruns (c), granulation (d), ﬂow effects (e), and glazing (f).
rest of this section, we therefore discuss the physical nature of wa-
tercolor, and then survey some of the most important characteristics
of watercolor from an artist’s standpoint.
2.1 Watercolor materials
Watercolor images are created by the application of watercolor
paint to paper. Watercolor paint (also called, simply, watercolor)is
a suspension of pigment particles in a solution of water, binder, and
surfactant [17, 25, 35]. The ingredients of watercolor are described
in more detail below.
from linen or cotton rags pounded into small ﬁbers. The paper itself
is mostly air, laced with a microscopic web of these tangled bers.
Such a substance is obviously extremely absorbent to liquids, and
so the paper is impregnated with sizing so that liquid paints may be
used on it without immediately soaking in and diffusing. Sizing is
usually made of cellulose. It forms a barrier that slows the rate of
water absorption and diffusion. For most watercolor papers, sizing
is applied sparingly and just coats the ﬁbers and ﬁlls some of the
pores, leaving the paper surface still rough.
A pigment isa solid material in the form of small,separate particles.
Watercolor pigments are typically ground in a milling process
into a powder made of grains ranging from about 0.05 to 0.5
microns. Pigments can penetrate into the paper, but once in the
paper they tend not to migrate far. Pigments vary in density, with
lighter pigments tending to stay suspended in water longer than
heavier ones, and thus spreading further across paper. Staining
power, an estimate of the pigment’s tendency to adhere to or
coat paper ﬁbers, also varies between pigments. Certain pigments
exhibit granulation, in which particles settle into the hollows of
rough paper. Others exhibit ﬂocculation, in which particles are
drawn together into clumps usually by electrical effects. (Since
ﬂocculation is similar in appearance to granulation, we discuss the
modeling of granulation only in this paper.)
The two remaining ingredients, binder and surfactant, both play
important roles. The binder enables the pigment to adhere to
the paper (known as “adsorption of the pigment by the paper”).
The surfactant allows water to soak into sized paper. A proper
proportion of pigment, binder, and surfactant is necessary in order
for the paint to exhibit the qualities desired by artists. (However, as
these proportions are controlled by the paint manufacturer and not
the artist, we have not made them part of our model.)
The nal appearance of watercolor derives from the interaction
between the movements of various pigments in a ﬂowing medium,
the adsorption of these pigments by the paper, the absorption of
water into the paper, and the eventual evaporation of the water
medium. While these interactions are quite complex in nature, they
can be used by a skilled artist to achieve a wide variety of effects,
as described in the next section.
2.2 Watercolor effects
Watercolor can be used in many different ways. To begin with,
there are two basic brushing techniques. In wet-in-wet painting, a
brush loaded with watercolor paint is applied to paper that is already
saturated with water, allowing the paint to spread freely. When the
brush is applied to dry paper, it is known as wet-on-dry painting.
These techniques give rise to a number of standard effects that can
be reliably employed by the watercolor expert, including:
Dry-brush effects (Figure 1a): A brush that is almost dry, applied
at the proper grazing angle, will apply paint only to the raised
areas of the rough paper, leaving a stroke with irregular gaps and
ragged edges.
Edge darkening (Figure 1b): In a wet-on-dry brushtroke, the
sizing in the paper, coupled with the surface tension of water,
process, the pigment migrates from the interior of the painted
region towards its edges as the paint begins to dry, leaving a dark
deposit at the edge. This key effect is one that watercolor artists
rely upon and that paint manufacturers take pains to ensure in
their watercolor paint formulations [17].
Intentional backruns (Figure 1c): Whena puddle of water spreads
back into a damp region of paint, as often happens when a
wash dries unevenly, the water tends to push pigment along as
it spreads, resulting in complex, branching shapes with severely
darkened edges.
Granulation and separation ofpigments (Figure 1d): Granulation
of pigments yields a kind of grainy texture that emphasizes
the peaks and valleys in the paper. Granulation varies from
pigment to pigment, and is strongest when the paper is very wet.
Separation refers to a splitting of colors that occurs when denser
pigments settle earlier than lighter ones.
Flowpatterns (Figure 1e): In wet-in-wet painting, the wet surface
allowsthe brushstrokes tospread freely, resulting in soft,feathery
shapes with delicate striations that follow the direction of water
ﬂow.
One other very important technique in watercolor is the process of
color glazing (Figure 1f). Glazing is the process of adding very
thin, pale layers, or washes, of watercolor, one over another, to
achieve a very clear and even effect. Each layer of watercolor is
added after the previous layer has dried. More expensive watercolor
paints are specially formulated to have a low resolubility,which
not only allows thin uniform washes to be overlaid, but in fact
allows any type of brushing technique to be employed over a dried
wash (including dry-brush and wet-on-wet) without disturbing the
underlying layers.
Glazing is different from ordinary painting in that the different
pigments are not mixed physically, but optically—in their super-
position on the paper. Glazes yield a pleasing effect that is often
described as “luminous, or as “glowing from within” [4, 32].
We suspect that this subjective impression arises from the edge-
darkening effect. The impression is intensiﬁed with multiple super-
abcd e f
Figure 2 Simulated watercolor effects created using our system.
imposed wet-on-dry washes.
Figure 1 shows scanned-in images of real watercolors. Figure 2
illustratessimilareffects obtained from our watercolor simulations.
3 Computer-generated watercolor
Implementing all of these artistic effects automatically presents an
interesting challenge, particularly given the paucity of available
information on the physical processes involved.
2
In this section,
we propose a basic model for the physical and optical behavior of
watercolors. The details of this model are then elaborated in the
next two sections.
We represent a complete painting as an ordered set of washes over
a sheet of rough paper. Each wash may contain various pigments in
varying quantities over different parts of the image. We store these
quantities in a data structure called a “glaze.
Each glaze is created independently by running a ﬂuid simulation
that computes the ﬂow of paint across the paper. The simulation
takes, as input, parameters that control the physical properties of
the individual pigments, the paper, and the watercolor medium.
represent the areas of paper that have been touched by water. These
masks control where water is allowed to ﬂow by limiting the ﬂuid
ﬂow computation. The next section describes this ﬂuid simulation
in detail.
Once the glazes are computed, they are optically composited using
the Kubelka-Munk color model to provide the ﬁnal visual effect, as
described in Section 5.
4 The ﬂuid simulation
In our system, each individual wash is simulated using a three-layer
model (Figure 3). From top to bottom, these three layers include:
The shallow-water layer — where water and pigment ﬂow above
the surface of the paper.
The pigment-deposition layer where pigment is deposited
onto (“adsorbed by”) and lifted (“desorbed”) from the paper.
The capillarylayer where water that is absorbed into the paper
is diffused by capillary action. (This layer is only used when
simulating the backrun effect.)
2
Indeed, As Mayer points out in his 1991 handbook [25, p. 13]: “The
study of artists materials and techniques is hampered by the lack of
systematic data of an authentic nature based on modern scientiﬁc labora-
tory investigations with which to supplement our present knowledge—the
accumulation of the practical experience of past centuries, necessarily quite
full of principles which rest on the shaky foundations of conjecture and
consensus. . . . We await the day when a sustained activity, directed from the
viewpoint of the artists, will supply us with more of the beneﬁts of modern
science and technology.
In the shallow-water layer (Figure 3a), water ﬂows across the
surface in a way that is bounded by the wet-area mask. As the water
ﬂows,it lifts pigment from the paper, carries it along, and redeposits
it on the paper. The quantities involved in this simulation are:
The wet-area mask M, which is 1 if the paper is wet, and 0
otherwise.
The velocity u, v of the water in the x and y directions.
The pressure p of the water.
The concentration g
k
of each pigment k in the water.
The slope h of the rough paper surface, deﬁned as the gradient
of the paper’s height h.
The physical properties of the watercolor medium, including its
viscosity and viscous drag . (In all of our examples, we set
=0.1and = 0. 01.)
Each pigment k is transferred between the shallow-water layer
and the pigment-deposition layer by adsorption and desorption.
While pigment in the shallow-water layer is denoted by g
k
, we will
use d
k
for any deposited pigment. The physical properties of the
individual pigments, including their density , staining power ,
and granularity
—all affect the rates of adsorption and desorption
by the paper. (The values of these parameters for our examples are
shown in the caption for Figure 5.)
The function of the capillary layer is to allow for expansion of the
wet-area mask due to capillary ﬂow of water through the pores of
the paper. The relevant quantities in this layer are:
The water saturation s of the paper, deﬁned as the fraction of a
given volume of space occupied by water.
The ﬂuid-holding capacity c of the paper, which is the fraction of
volume not occupied by paper ﬁbers.
All of the above quantities are discretized over a two-dimensional
grid representing the plane of the paper.
We will refer to the value of each quantity, say p, at a particular cell
using subscripts, such as p
i, j
. We will use bold-italics (such as p)to
Shallow−water layer
(flow of water above paper)
Pigment−deposition layer
Capillary layer
(transport of water through pores)
Figure 3 The three-layer ﬂuid model for a watercolor wash.
Figure 4 Example paper textures.
denote the entire array of discretized values.
4.1 Paper generation
In real watercolor, the structure of the paper affects ﬂuid ﬂow,
backruns, and granulation. The mechanics underlying these effects
may be quite complex, and may depend on the precise connections
among the individual ﬁbers, as well as the exact slopes of the ne-
scale peaks and valleys of the paper. We use a much simpler model
in our system. Paper texture is modeled as a height ﬁeld and a
ﬂuid capacity ﬁeld. The height ﬁeld h is generated using one of
a selection of pseudo-random processes [29, 41], and scaled so
that 0
h 1. Some examples of our synthetic paper textures
can be seen in Figure 4. The slope of the height eld is used
to modify the uid velocity u, v in the dynamics simulation. In
addition, the ﬂuid capacity c is computed from the height ﬁeld h,as
c=h
(c
max
c
min
)+c
min
.
4.2 Main loop
The main loop of our simulation takes as input the initial wet-
area mask M; the initial velocity of the water u, v; the initial
water pressure p; the initial pigment concentrations g
k
;andthe
initial water saturation of the paper s. The main loop iterates over
a speciﬁed number of time steps, moving water and pigment in
the shallow-water layer, transferring pigment between the shallow-
water and pigment-deposition layers, and simulating capillary ﬂow:
proc
MainLoop
(M, u,v,p,g
1
, ,g
n
,d
1
,,d
n
,s):
for each time step do:
MoveWater
(M,u, v, p)
MovePigment
(M, u, v, g
1
, ,g
n
)
TransferPigment
(g
1
, ,g
n
,d
1
,,d
n
)
SimulateCapillaryFlow
(M, s)
end for
end proc
4.3 Moving water in the shallow water layer
For realism, the behavior of the water should satisfy the following
conditions:
1. The ﬂow must be constrained so that water remains within the
2. A surplus of water in one area should cause ﬂow outward from
that area into nearby regions.
3. The ﬂow must be damped to minimize oscillating waves.
4. The ﬂow must be perturbed by the texture of the paper to cause
streaks parallel to ﬂow direction.
5. Local changes should have global effects. For example, adding
water in a local area should affect the entire simulation.
6. There should be outward ﬂow of the ﬂuid toward the edges to
produce the edge-darkening effect.
The ﬁrst two conditions are satisﬁed directly by the basic shallow-
water equations using appropriate boundary conditions [24, 38]:
u
t
=
u
2
x
2
+
uv
y
2
+
2
u
p
x
(1)
v
t
=
v
2
y
2
+
uv
x
2
+
2
v
p
y
(2)
These equations are implemented in the
UpdateVelocities
() sub-
routine. Conditions 3 and 4 are met by adding terms to the ﬂuid
ﬂow simulation involving the viscous drag
and the paper slope
h, as shown in the
UpdateVelocities
() pseudocode. Conditions 5
and 6 are accomplished by two additional subroutines,
RelaxDiver-
gence
() and
FlowOutward
(). All three of these routines are used to
implement the movement of water in the shallow-water layer:
proc
MoveWater
(M, u, v, p):
UpdateVelocities
(M, u,v,p)
RelaxDivergence
(M,u, v,p)
FlowOutward
(M, p)
end proc
4.3.1 Updating the water velocities
To update the water velocities, we discretize the equations (1) and
(2) spatially on a staggered grid (as in Foster [10]). An effect
of this discretization is that our solution is resolution-dependent.
Generalizing to a resolution-independent model is an important
goal for future work.
The staggered grid represention stores velocity values at grid cell
boundaries and all other values (pressure, pigment concentrations,
etc.) at grid cell centers. We use the standard notation for stag-
gered grids, referring to quantities on cell boundaries as having
“fractional” indices. For example, the velocity u at the boundary
between the grid cells centered at (i,j)and(i+1,j) is called
u
i+.5, j
. Furthermore, we will use the shorthand notation (uv)
i, j
to
denote u
i, j
v
i, j
. We will also use indices to denote quantities that
are not represented directly, but computed implicitly from their two
p
i+.5, j
(p
i, j
+ p
i+1, j
) 2
u
i, j
(u
i .5, j
+ u
i+.5, j
) 2
In the pseudocode below, we discretize equations (1) and (2) in time
and solve forward using Euler’s Method with an adaptive step size.
The step size t is set to ensure that velocities do not exceed one
pixel per time step:
proc
UpdateVelocities
(M, u, v, p):
(u,v)
(u,v) h
t1max
i, j
u , v
for t 0 to 1 by t do
for all cells (i,j) do
A
u
2
i, j
u
2
i+1, j
+(uv)
i+.5, j .5
(uv)
i+.5, j+.5
B (u
i+1.5, j
+ u
i .5, j
+ u
i+.5, j+1
+ u
i+.5, j 1
4u
i+.5, j
)
u
i+.5, j
u
i+.5, j
+ t (A B + p
i, j
p
i+1, j
u
i+.5, j
)
A
v
2
i, j
v
2
i, j+1
+(uv)
i .5, j+.5
(uv)
i+.5, j+.5
B (v
i+1, j+.5
+ v
i 1, j+.5
+ v
i, j+1.5
+ v
i, j .5
4v
i, j+.5
)
v
i, j+.5
v
i, j+.5
+ t (A B + p
i, j
p
i, j+1
v
i, j+.5
)
end for
(u, v)
(u , v )
EnforceBoundaryConditions
(M, u, v)
end for
end proc
The
EnforceBoundaryConditions
() procedure simply sets the ve-
locity at the boundary of any pixel not in the wet-area mask to zero.
4.3.2 Relaxation
Following Foster et al. [10], we also relax the divergence of the
velocity ﬁeld u x + v y after each time step until it is less
than some tolerance
by redistributing the ﬂuid into neighboring
grid cells. In our implementation of the following pseudocode, we
have used N = 50,
=0.01and =0.1:
proc
RelaxDivergence
(u, v, p):
t
0
repeat
(u
,v ) (u, v)
max
0
for all cells (i,j) do
(u
i+1 2, j
u
i 1 2, j
+ v
i, j+1 2
v
i, j 1 2
)
p
i, j
p
i, j
+
u
i+.5, j
u
i+.5, j
+
u
i .5, j
u
i .5, j
v
i, j+.5
v
i, j+.5
+
v
i, j .5
v
i, j .5
max
max( ,
max
)
end for
(u, v)
(u ,v )
t
t +1
until
max
or t N
end proc
4.3.3 Edge darkening
In a wet-on-dry brushstroke, pigment tends to migrate from the
interior towards the edges over time. This phenomenon occurs
in any evaporating suspension in which the contact line of a
drop is pinned in place by surface tension [3]. Because of this
geometric constraint, liquid evaporating near the boundary must be
replenished by liquid from the interior, resulting in outward ﬂow.
This ﬂow carries pigment with it, leading to edge darkening as the
water evaporates. In our model, we simulate this ﬂow by decreasing
the water pressure near the edges of the wet-area mask.
The
FlowOutward
() routine removes at each time step an amount
of water from each cell according to the cell’s distance from the
boundary of the wet-area mask, with more water removed from
cells closer to the boundary. The distance to the boundary is
approximated by ﬁrst performing a Gaussian blur with a K
K
kernel on the wet-area mask M. Then an amount of water is
removed from each cell according to the value of the resulting
Gaussian-blurred image M
:
p p (1 M )M (3)
In our examples, K = 10 and 0. 01
0. 05.
An example of the edge-darkening effect is shown in Figure 2b.
4.4 Moving pigments
Pigments move within the shallow-water layer as speciﬁed by the
velocity ﬁeld u, v computed for the water above. In this part of the
simulation, we distribute pigment from each cell to its neighbors
according to the rate of ﬂuid movement out of the cell:
proc
MovePigment
(M, u, v, g
1
, ,g
n
):
t 1 max
i,j
u , v
for each pigment k do
for t
0 to 1 by t do
g
g g
k
forall cells (i,j) do
g
i+1, j
g
i+1, j
+ max(0, u
i+.5, j
g
i, j
)
g
i 1, j
g
i 1, j
+ max(0, u
i .5, j
g
i, j
)
g
i, j+1
g
i, j+1
+ max(0, v
i, j+.5
g
i, j
)
g
i, j 1
g
i, j 1
+ max(0, v
i, j .5
g
i, j
)
g
i, j
g
i, j
max (0, u
i+.5, j
g
i, j
)+max(0, u
i .5, j
g
i, j
)
+max (0, v
i, j+.5
g
i, j
)+max(0, v
i,j .5
g
i, j
)
end for
g
k
g
end for
end for
end proc
At each step of the simulation, pigment is also adsorbed by the
pigment-deposition layer at a certain rate, and desorbed back into
the ﬂuid at another rate (in a process similar to the one described by
Dorsey et al. [8] for weathering patterns due to ﬂuid ﬂow.) The
density
k
and staining power
k
are scalars that affect the rate
at which each pigment k is adsorbed and desorbed by the paper.
The granulation
k
determines how much the paper height h affects
proc
TransferPigment
(g
1
,...,g
n
,d
1
,...,d
n
):
for each pigment k do
for all cells (i,j) do
if M
i, j
=1then
down
g
k
i, j
(1 h
i, j
k
)
k
up
d
k
i, j
(1 + (h
i, j
1)
k
)
k k
if (d
k
i, j
+
down
) 1
then
down
max (0, 1 d
k
i, j
)
if (g
k
i, j
+
up
) 1
then
up
max(0, 1 g
k
i, j
)
d
k
i, j
d
k
i, j
+
down up
g
k
i, j
g
k
i, j
+
up down
end if
end for
end for
end proc
4.6 Backruns: diffusing water through the capillary layer
Backruns occur only when a puddle of water spreads slowly into
a region that is drying but still damp [37]. In a damp region, the
only water present is within the pores of the paper. In this situation,
ﬂow is dominated by capillary effects, not by momentum as in the
shallow water equations.
In the backrun simulation, water is absorbed from the shallow-
water layer above at the absorption rate
, and diffuses through the
capillary layer. Each cell transfers water to its four neighbors until
they are saturated to capacity c. If any cell’s saturation exceeds a
threshold
, then the wet-area mask is expanded to include that cell.
In this way, capillary action within the paper can enable a puddle
to spread. The variation in cell capacity from pixel to pixel results
in an irregular branching pattern. Other parameters affecting this
process are
, the minimum saturation a pixel must have before it
can diffuse to its neighbors, and
, a saturation value below which
a pixel will not receive diffusion.
proc
SimulateCapillaryFlow
(s, M):
forall cells (i, j) do
if (M
i, j
0) then
s
i, j
s
i, j
+ max(0, min ( , c
i, j
s
i, j
))
end for
s
s
for all cells (i, j) do
for each cell (k,
) neighbors(i, j) do
if s
i, j
and s
i, j
s
k,
and s
k,
then
s max (0, min (s
i, j
s
k,
, c
k,
s
k,
) 4)
s
i, j
s
i, j
s
s
k,
s
k,
+ s
end if
end for
end for
s
s
for all cells (i, j) do
if s
i, j
then
M
i, j
1
end for
end proc
4.7 Drybrush effects
The drybrush effect occurs when the brush is applied at the proper
angle and is dry enough to wet only the highest points on the paper
surface. We model this effect by excluding from the wet-area mask
any pixel whose height is less than a user-deﬁned threshold. An
example of simulated drybrush is shown in Figure 1a.
5 Rendering the pigmented layers
We use the Kubelka-Munk (KM) model [14, 20] to perform the
optical compositing of glazing layers. (The same model was also
used by Dorsey and Hanrahan to model the transmission of light
through layers of copper patina [7].)
In our use of the KM model, each pigment is assigned a set
of absorption coefﬁcients K and scattering coefﬁcients S.These
coefﬁcients are a function of wavelength, and control the fraction of
energy absorbed and scattered back, respectively, per unit distance
in the layer of pigment. In our implementation, we use three
coefﬁcients each for K and S, representing RGB components of
each quantity.
5.1 Specifying the optical properties of pigments
In typical applications of KM theory, the K and S coefﬁcients for a
given colorant layer are determined experimentally, using spectral
measurements from layers of known thicknesses. However, in our
application we have found it to be much more convenient to allow
a user to specify the K and S coefﬁcients interactively, by choosing
the desired appearance of a “unit thickness” of the pigment over
both a white and a black background. Given these two user-selected
RGB colors R
w
and R
b
, respectively, the K and S values can be
computed by a simple inversion of the KM equations:
S =
1
b
coth
1
b
2
(a R
w
)(a 1)
b(1 R
w
)
K = S (a 1)
where
a =
1
2
R
w
+
R
b
R
w
+1
R
b
, b = a
2
1
The above computations are applied to each color channel of S,
K, R
w
,andR
b
independently. In order to avoid any divisions by
zero, we require that 0 R
b
R
w
1 for each color channel.
This restriction is reasonable even for opaque pigments, since the
user is specifying reﬂected colors through just a thin layer, which
should still be at least partially transparent. While for most valid
combinations of speciﬁed colors the computed K and S values fall
in the legal range of 0 to 1, for certain very saturated input colors
the absorption or scattering coefﬁcients computed by this method
may actually exceed the value of 1 in some color channels. Though
such a large value of K or S is clearly not possible for any physical
pigment, we have not noticed any ill effects in our simulation from
allowing such “out-of-range” values. The situation is somewhat
analogous to allowing an “alpha” opacity to lie outside the range
0 to 1, another non-physical effect that is sometimes useful [15].
We have found this method of specifying pigments to be quite
adequate for creating a wide range of realistic paints (see Figure 5).
In addition, the method is much easier than taking the kind of
extremely careful measurements that would otherwise be required.
By specifying the colors over black and white, the user can easily
create different types of pigments. As examples:
Opaque paints, such as Indian Red, exhibit a similar color
on both white and black. Such paints have high scattering in
the same wavelengths as their color, and high absorption in
complementary wavelengths.
Transparent paints, such as Quinacridone Rose, appear colored
on white, and nearly black on black. Such paints have low
scattering in all wavelengths, and high absorption in wavelengths
complementary to their color.
Interference paints, such as Interference Lilac, appear white (or
transparent) on white, and colored on black. Such paints have
high scattering in the same wavelengths as their color, and low
absorption in all wavelengths. Such pigments actually get their
color from interference effects involving the phase of light waves,
which have been modeled accurately by Gondek et al. [11].
While our simple model does not simulate phase effects, it
nevertheless manages to produce colors similar in appearance to
the interference paints used in watercolor painting.
Our method also makes it easy to simulate real paints that exhibit
slightly different hues over black than white, such as Hansa yellow.
Figure 5(i) shows a simulated swatch of this pigment over both
black and white backgrounds.
5.2 Optical compositing of layers
Given scattering and absorption coefﬁcients S and K for a pig-
mented layer of given thickness x, the KM model allows us to
compute reﬂectance R and transmittance T through the layer [20]:
R =sinhbSx
c
T = b
c
where c = a sinh bSx + bcosh bSx
We can then use Kubelka’s optical compositing equations [20, 21]
to determine the overall reﬂectance R and transmittance T of two
abutting layers with reﬂectances R
1
,R
2
and T
1
, T
2
, respectively:
R = R
1
+
T
2
1
R
2
1 R
1
R
2
T =
T
1
T
2
1 R
1
R
2
This computation is repeated for each additional glaze. The overall
reﬂectance R is then used to render the pixel.
For individual layers containing more than one pigment of thick-
nesses x
1
, ,x
n
,theSand K coefﬁcients of each pigment k are
weighted in proportion to that pigments relative thickness x
k
.The
overall thickness of the layer x is taken to be the sum of the
thicknesses of the individual pigments.
In our ﬂuid simulation (see Section 4), we use g
k
to denote the
concentration of pigment in the shallow-water layer, and d
k
for
the concentration of pigment deposited on the paper. These values
are summed to compute the thickness parameter x
k
used by the
Kubelka-Munk equations.
5.3 Pigment examples
Figure 5 shows the palette of colors used in the examples, with
each pigment shown as a swatch painted over a black stripe. The
colors we chose for these pigments were based on fairly casual
observations of the colors of the actual paints over black and white
backgrounds. The K and S coefﬁcients were then derivedfrom these
colors by the procedure outlined in Section 5.1. As the thickness
of a layer of pigment increases, its color traces a complex curve
through color space. For example, Figure 6 shows the range of
colors obtainable by glazing “Hansa Yellow” over both white and
black backgrounds. Note the difference in hue between the two
curves, and the change in both hue and saturation along each curve.
This complexity is one of the qualities that gives these pigments
their rich appearance.
5.4 Discussion of Kubelka-Munk model
The KM model appears to give very plausible and intuitive results
in all the cases we have tried. On the one hand, these results are not
very surprising, considering that the KM model was speciﬁcally
designed for situations akin to watercolor in which there are mul-
tiple pigmented layers that scatter and absorb light. It is, however,
a
g
b
h
c
i
d
j
e
k
f
l
Figure 5 Various synthetic pigments.
The swatches were all created using
identical initial conditions, with thicker
pigment in the top half, and extra water
in the upper left and lower right corners.
The only changes from swatch to swatch
are the pigments’ optical and physical
parameters, shown at right. The swatches
are painted over a black stripe to distin-
guish the more opaque pigments such as
“Indian Red” (b) from the more transpar-
ent ones such as “Brilliant Orange” (h).
PIGMENT K
r
K
g
K
b
S
r
S
g
S
b
a “Quinacridone Rose” 0.22 1.47 0.57 0.05 0.003 0.03 0.02 5.5 0.81
b “Indian Red” 0.46 1.07 1.50 1.28 0.38 0.21 0.05 7.0 0.40
c “Cadmium Yellow” 0.10 0.36 3.45 0.97 0.65 0.007 0.05 3.4 0.81
d “Hookers Green” 1.62 0.61 1.64 0.01 0.012 0.003 0.09 1.0 0.41
e “Cerulean Blue” 1.52 0.32 0.25 0.06 0.26 0.40 0.01 1.0 0.31
f Burnt Umber” 0.74 1.54 2.10 0.09 0.09 0.004 0.09 9.3 0.90
g “Cadmium Red” 0.14 1.08 1.68 0.77 0.015 0.018 0.02 1.0 0.63
h “Brilliant Orange” 0.13 0.81 3.45 0.005 0.009 0.007 0.01 1.0 0.14
i “Hansa Yellow” 0.06 0.21 1.78 0.50 0.88 0.009 0.06 1.0 0.08
j “Phthalo Green” 1.55 0.47 0.63 0.01 0.05 0.035 0.02 1.0 0.12
k “French Ultramarine” 0.86 0.86 0.06 0.005 0.005 0.09 0.01 3.1 0.91
l “Interference Lilac” 0.08 0.11 0.07 1.25 0.42 1.43 0.06 1.0 0.08
Black
R
G
White
R
G
B
B
Figure 6 The range of colors obtainable by compositing varying
thicknesses of “Hansa Yellow” over black (solid curve) and over
white (dashed curve). The point where the two curves meet is R ,
the color of an inﬁnitely thick layer. At left, the RGB cube is viewed
in perspective; at right, we look directly down the luminance axis,
showing the difference in hue between the two curves.
worth noting that there are a number of ﬁne points in the basic KM
assumptions thatare satisﬁed, at best, only partially in our situation:
1. All colorant layers are immersed in mediums of the same
refractive index. This assumption is in fact violated at both the
“air to pigment-layer” and “pigment-layer to paper” boundaries
(although a fairly simple correction term has been proposed [18],
that could be used to increase accuracy).
2. The pigment particles are oriented randomly. This assumption
is satisﬁed for most watercolor paints, although not all. For
example, metallic paint pigments have mostly horizontal ﬂakes.
3. The illumination is diffuse. Our simulated watercolors will
obviously not look entirely correct underall lighting andviewing
conditions. Duntley [20] has a more general theory with four
parameters instead of two that can account for more general
lighting conditions.
4. The KM equations apply only to one wavelength at a time.
Fluorescent paints violate this assumption.
5. There is no chemical or electrical interaction between different
pigments, or between the pigment and medium, which would
cause clumping of pigment grains and a non-uniform particle
size. These assumptions are violated for most watercolor pig-
ments.
In summary, the fact that the KM model appears to work so well
could actually be considered quite surprising, given the number of
basic assumptions of the model violated by watercolor. We suspect
that while the results of the model are probably not very physically
accurate, they at least provide very plausible physical approxima-
tions, which appear quite adequate for many applications.
6 Applications
In this section, we brieﬂy discuss three different applications
of computer-generated watercolor: interactive painting, automatic
image “watercolorization, and 3D non-photorealistic rendering.
6.1 Interactive painting with watercolors
We have written an interactive application that allows a user to
paint the initial conditions for the watercolor simulator. The user
sets up one or more glazes for the simulator, where each glaze has
sub-layers for pigments, water, and a wet-area mask. Common to
all glazes are a reference image and a shaded paper texture. There
are slider controls to adjust the physical parameters for each glaze
(including viscous drag
, edge darkening and kernel size K)as
well as the number of times to iterate the simulation.
The glaze’s pigment channels are represented by colored images in
the glaze. Each pigment is painted independently using a circular
brush with a Gaussian intensity drop-off. The brush size, penum-
bra, and overall intensity parameters are adjustable. A palette of
pigments associated with a glaze may be deﬁned by specifying the
color of each pigment over black and over white, as described in
pigments from les. For each pigment, the density
, stainingpower
, and granulation may also be controlled using sliders.
The wet-area mask can be painted directly using a similar brush,
or by selecting regions from the reference image using “intelligent
scissors” [27]. The user can achieve drybrush effects by setting the
Figure 7 An interactive painting application. At top center are the
initial conditions painted by the user; at top right, a watercolor
simulation in progress, showing two of the painting’s ﬁve glazes.
The large image is the ﬁnished painting.
depth to which the brush is allowed to touch the paper.
Although our watercolor simulator runs too slowly for interactive
painting, compositing many glazes of pigments using the KM
model is feasible in real-time, and provides the user with valuable
feedback about the colors resulting from the simulation. The ref-
erence image, paper texture, and set of glazes and their sublayers
can be independently toggled on and off, and displayed in any
combination. Another helpful feature is a rendering window that
displays the progress of running the simulator on a (possibly scaled-
down) set of glazes. Lower-resolution simulations are enlarged for
display in the viewing window, and each frame of the simulator’s
animation is typically displayed in a fraction of a second to a few
minutes. A screenshot from the application appears in Figure 7,
showing several simulations at different stages of completion.
6.2 Automatic image “watercolorization”
Another application we have built allows a color image to be
automatically converted into a watercolor illustration, once mattes
for the key elements have been extracted and an ordered set of
pigments has been chosen, with one pigment per glaze. In our
tests, we generated these regions quickly using a commercial
paint program. We have also chosen the pigments by hand in our
examples, although for further automation, the choice of pigments
could instead be computed through an optimization process [31].
The conversion is executed in two stages: color separation (Fig-
ure 8), in which the ideal distribution of pigment in each glaze is
calculated to produce the desired image; and brushstroke planning
(Figure 9), in which each glaze is painted in an attempt to re-create
the desired pigment distributions by adding brushstrokes of water
and pigment, taking into account the behavior of the medium. The
result is an image that approximates the original but has the ﬂow
patterns and texture of a watercolor painting.
Color separation. Color separations are calculated using a brute-
force search over a discrete set of thicknesses for each pigment.
Givenanorderedlistofnpigments, the thickness range for each
pigment is ﬁrst divided into m steps, using a binary subdivision
by Manhattan distance in the six-dimensional space of the R
i
and
T
i
Kubelka-Munk parameters. Using the KM optical compositing
model, a composite color is then computed for each of the m
n
combinations, and the color is stored in a 3d-tree according to its
RGB color values. (The tree is pruned so that the difference between
Regions and Pigment
Choices
Target Image
Color Separations
+
{
Figure 8 Overview of the color separation process.
a
b
c
d
Figure 9 Brushstroke planning. At given intervals, the planner
identiﬁes regions containing too much pigment (a) and thins them
out by adding plain water (b). The planner can also compensate for
a lack of pigment (c) by adding a pigmented wash (d).
colors is less than 1/255.) For the separations used for Figures 10
and 14, m = 20, and n = 3. Color separations are computed by
searching the tree for each pixel to ﬁnd the pigment combination
yielding the closest color to the desired color. These colors are
stored in an image called the target glaze.
Brushstroke planning. A painter can somewhat control the con-
centration and ﬂow of pigment in a wash by carefully monitoring
the relative wetness of brush and paper, knowing that spreading
water carries pigment with it and tends to thin it out. Similarly, we
control a glaze by adding incremental brushstrokes of pigment, and
we control the direction of ﬂow by increasing or decreasing water
The overall process works by repeatedly querying and manipulating
the state of the glaze at a user-speciﬁed interval (30 to 100
steps in our examples) during the simulation, using user-controlled
parameters
g
,
g
,and
p
. In our examples, the steps below were
repeated between 2 and 5 times, and we used the following values:
0.01
g
0. 2,
g
=
g
,and
p
= 1. 0. At each step,
the current pigment distribution is compared to the target glaze,
ignoring high frequency details, by performing a low-pass ﬁlter
on the difference between the two. Then one of two actions is
performed:
1. In areas where the current glaze does not have enough pigment
(by more than
g
), increment g by
g
, and increment p by
g
.
2. In areas where the current glaze has too much pigment (by more
than
g
), increment p by
p
.
As a ﬁnal step, highlights are created by removing paint from
areas deﬁned by the user in the form of mattes. This step is
analogous to the “lifting out” technique used by artists for similar
effects. Figure 10 shows the ﬁnal results, and Figure 11 shows the
appearance of the painting in progress as glazes are added.
Figure 10 An automatic watercolorization (left)
of a low resolution image captured using a poor-
quality video camera (above). The ﬁnished paint-
ing consists of 11 glazes, using a total of 2750
iterations of the simulator, rendered at a resolution
of 640 by 480 pixels in 7 hours on a 133 MHz SGI
R4600 processor.
6.3 Non-photorealistic rendering of 3D models
A straightforward extension of the automatic watercolorization of
the previous section is to perform non-photorealistic watercolor
rendering directly from 3D models.
Given a 3D geometric scene, we automatically generate mattes
isolating each object. These mattes are used as input to the water-
colorization process, along with a more traditional “photorealistic”
rendering of the scene as the target image. The pigment choices and
brushstroke planning parameters are supplied by the user. As shown
in Figure 12, even a very primitive “photorealistic” image can thus
be converted into a richly textured painting (seen in Figure 14).
Figure 13 shows several frames from a painterly animation of
clouds generated using only a few dozen spheres.
7 Future Work
Other effects. There are several techniques we do not model, such
as spattering and some aspects of the drybrush technique. One way
to simulate the appearance of bristle patterns in drybrush would
be to integrate hairy brushes [36] with the watercolor simulator.
The integration of watercolor with other media such as pen-and-ink
would also be interesting.
Automatic rendering. We would like to explore further the idea
of automatic watercolorization of images in a more general sense.
An algorithm to automatically specify wet areas so that hard edges
are placed properly would be especially useful, as well as a color-
separation algorithm to calculate the optimal palette of pigments to
use for various regions of theimage [31]. Other possibilities include
automatic recognition and generation of textures using drybrush,
spattering, scraping, and other techniques.
Generalization. Our model treats backruns and wet-in-wet ﬂow
patterns as two separate processes. In real watercolor, however,
they are just two extremes of a continuum of effects, the difference
between them being simply the degree of wetness of the paper.
A model that could integrate these two effects, parametrized by
wetness, would be a signiﬁcant improvement.
Animation issues. When an animated sequence is converted to
watercolor one frame at a time, the resulting animation exhibits
certain temporal artifacts, such as the “shower door” effect [26]. In
the future we would like to develop a system that takes into account
the issue of coherency over time and allows the user to control these
artifacts.
Acknowledgements
The 3D target animation for Figures 12–14 was created by Siang
Lin Loo. We would also like to thank Ron Harmon of Daniel Smith
Artists’ Materials for connecting us to reality; John Hughes, Alan
Barr, Randy Leveque, Michael Wong, and Adam Finkelstein for
for assistance with the images.
This work was supported by an Alfred P. Sloan Research Fellow-
ship (BR-3495), an NSF Presidential Faculty Fellow award (CCR-
9553199), an ONR Young Investigator award (N00014-95-1-0728)
and Augmentation award (N00014-90-J-P00002), and an industrial
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... Tedious efforts or high art skills are often required in the typical semi-manual/manual colorization methods [10]. Computer graphics (CG) methods, e.g., physics-based simulation [11] and non-photorealistic rendering (NPR) [2], often need to craft a specific technique or require an expensive computational cost, in order to generate the desired result. Current learning-based methods for colorization, e.g., style-transfer with examples [14], are mainly used for transferring general visual features, such as texture and color. ...
... Visually realistic colorization results can be achieved using physical simulation-based methods, such as for the effects of oil painting and watercolor. Curtis et al. [11] performed a fluid simulation to model the water and pigment moving processes. Chu and Tai [8] as well as Van Laerhoven and Van Reeth [40] utilized the GPU computation for accelerating and generating realistic effects of watercolor. ...
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