Noname manuscript No.(will be inserted by the editor)

Automated Sampling and Control of Gaseous Simulations

Ruoguan Huang · John Keyser

Received: date / Accepted: date

Abstract In this work, we describe a method that au-

tomates the sampling and control of gaseous fluid sim-

ulations. Several recent approaches have provided tech-

niques for artists to generate high resolution simula-

tions based on a low resolution simulation. However, of-

ten in applications the overall flow in the low resolution

simulation that an animator observes and intends to

preserve is composed of even lower frequencies than the

low resolution itself. In such cases, attempting to match

the low resolution simulation precisely is unnecessar-

ily restrictive. We propose a new sampling technique

to efficiently capture the overall flow of a fluid simula-

tion, at the scale of user’s choice, in such a way that

the sampled information is sufficient to represent what

is virtually perceived and no more. Thus, by applying

control based on the sampled data, we ensure that inthe resulting high resolution simulation, the overall flow

is matched to the low resolution simulation and the fine

details on the high resolution are preserved. The sam-

ples we obtain have both spatial and temporal continu-

ity that allows smooth keyframe matching and direct

manipulation of visible elements such as smoke density

through temporal blending of samples. We demonstrate

that a user can easily configure a simulation with our

system to achieve desired results.

Keywords Eulerian simulation · Simulation control ·Poisson disk sampling · Bilateral sampling

Ruoguan HuangTexas A&M UniversityE-mail: hrg@cs.tamu.edu

John KeyserTexas A&M UniversityE-mail: keyser@cs.tamu.edu

1 Introduction

Fluid simulation has been an area of great interest within

the computer graphics community. As quality and ef-

ficiency have improved, fluid-based visual effects have

been widely applied in industry. This has led to a stronger

desire to have control over the simulation, while still

preserving a behavior that seems realistic.

Despite improvements in simulation techniques, large

fluid simulations are still notoriously slow to compute.

This makes the prospect of controlling and directing

such simulations problematic. Users can specify initial

conditions and other parameters, but it can take hours

to see the overall effect those conditions produce.

To improve the rate of feedback and thus conver-

gence to an acceptable result, artists often first run a

lower-resolution simulation (which will be much faster)

to adjust parameters and settings. Once an acceptable

simulation has been found, a higher resolution simula-

tion is produced. Simply using the same parameters as

the lower resolution can cause problems, since the small

differences in the simulation (e.g. from artificial viscos-

ity or object boundary effects) can quickly lead to much

larger and visually significant differences. Adding artifi-

cial noise and vortices provides detail, but not the real-

ism of a higher-resolution simulation. Keyframe match-

ing can provide good results, but it can be problematic

to determine good keyframes and ensure that the sim-

ulation is not overspecified, removing desired detail.

In this paper we present improvements for automat-

ically matching a high resolution gaseous fluid simula-

tion to a lower resolution simulation. Specifically, we

show that our automated system requires minimal user

editing and tweaking and a relatively low additional

computational overhead to provide necessary control

for achieving desired visual effects. Our approach fol-

2 Ruoguan Huang, John Keyser

lows the general framework of Huang et al. [9], where

control is achieved by sampling data from a low reso-

lution preview simulation at a set of “match points” in

a user-selected region and then matching a high reso-

lution to the match points. In many cases, the method

of Huang et al. requires that the user provide rather

specific placement (spatial and temporal) and detailed

properties of match points to produce desired results.

Even with the help of user direction, the statically po-

sitioned match points are usually inefficient and some-

times inaccurate in how well they capture the constantly

flowing motion of fluids.

To better and more efficiently capture interesting

fluid features, we propose a new method for dynam-

ically sampling arbitrary types of data from a simu-

lation with the goal of achieving spatially and tem-

porally more continuous sampling. With this dynamic

sampling, we can easily incorporate temporal blend-

ing in the matching stage to achieve much smoother

and hence more plausible control effects. This is key

to manipulating directly visible fluid components (e.g.

gaseous density), supporting discontinuous control (e.g.

keyframe matching), and allowing artistic direction.

Among the key features of our method are:

– Automated sampling detects and captures the promi-

nent features of fluids and follows the flow, resulting

in smoother more accurate control without tedious

manual editing.

– Temporal sampling and blending allow seamless key-

frame matching and plausible density manipulation,

and handle various other sudden and drastic changes

in control.

– Our density control preserves edges presented in the

density field by using bilateral density sampling that

reduces the blurring of high frequency details in the

high resolution.

2 Previous Work

Our method is based on a Eulerian fluid simulation

framework. Interest in fluid dynamics simulations in

computer graphics increased significantly with the work

of Foster and Metaxas [5,6] and the subsequent Sta-

ble Fluids approach introduced by Stam [27]. Stam’s

method used an implicit solver, making the simula-

tion unconditionally stable, and thus allowing coarse

grid resolutions and large time steps. There has been

a tremendous amount of follow-on work which has pre-

sented improvements in various aspects of the simu-

lation, and a thorough discussion of this work is be-

yond the scope of this paper. While particle-based (La-

grangian) simulations have also become popular [16],

Fig. 1 Example of our controlled simulation with temporalblending. On each row from left to right are low resolutionpreview, controlled simulation, uncontrolled high resolutionsimulation. The top row shows images taken at the time westart to apply control. Before that simulation in the middleis uncontrolled, hence it looks exactly the same as the highresolution simulation at right. The bottom row shows imagestaken approximately 150 frames later when the applied con-trol is fully in effect. The middle now resembles the previewat left.

Eulerian simulations continue to be widely used, par-

ticularly for gaseous and incompressible situations.

Controlling fluid simulation was first investigated

by Foster and Metaxas [6] and later by Foster and Fed-

kiw [4], and has been addressed by a number of other

researchers. Rasmussen et al. [22] proposed a produc-

tion model using particles to control liquid simulations.

Shi and Yu have proposed methods for matching smoke

[25] and liquids [26] to changing target shapes. Pighin

et al. [21] have combined Eulerian and Lagrangian rep-

resentations to create a system for interactively manip-

ulating fluid flows. Their method parameterizes a fluid

simulation from advected particle paths, allowing fine-

grained control of the system by manipulating these

paths. Hong and Kim [8] use a geometric potential field

to generate forces to match target shapes. Schpok et

al. [24] control the simulation by manipulation of auto-

matically extracted simulation features. Angelidis et al.

[1] and Kim et al. [12] provide path control for smoke

simulations.

Automated Sampling and Control of Gaseous Simulations 3

A few recent methods approach the problem de-

scribed from a different perspective, namely that high

frequency “detail” is added into a coarse simulation,

thus giving the appearance of a higher-resolution sim-

ulation. Kim et al. [11] use a wavelet-based method to

generate fine turbulent details within a fluid simula-

tion. Schechter and Bridson [23] similarly upsample a

lower resolution simulation and add turbulent detail.

Narain et al. [17] also use a procedural method to add

in turbulence, but in addition they identify locations for

additional turbulence based on the energy computation

in the fluid simulation. Pfaff et al. [20] add the procedu-

ral turbulence by generating and advecting turbulence

particles based on the simulation. Finally, He et al. [7]

improve the Second Vorticity Confinement method [28]

to produce detailed turbulence effect on a coarse grid.

Closer to our approach, several algorithms have fo-

cused on the keyframe concept. Treuille et al. [31] pro-

posed a fluid control system based on user-defined smoke

density keyframes. The fluid system is controlled by

parametric wind and vortex fields that are manipulated

to make the simulation achieve the specified keyframes.

The amount of adjustment to the vector field is mini-

mized via a non-linear optimization. This optimization

is expensive, and dimensionality grows with the length

of the simulation. The approach was practical only for

2D simulation, but McNamara et al. [15] improved on it

by using a discrete adjoint method for gradient calcula-

tion. They achieve much faster convergence for control

parameters, thus enabling fine-grain keyframe control

on larger 3D fluid animations.

Instead of optimization, Fattal and Lischinksi [2] in-

troduce new terms into the standard flow equations, of-fering a closed form solution for the control parameters.

Matching target densities using only advection is usu-

ally not enough, and so they propose smoke gathering,

diffusing the error field. This allows complex smoke an-

imations to be controlled with little additional expense.

Later, Thurey et al. [29] in their particle based method,

also apply correcting forces through Lagragian particles

extracted from coarser simulations to control the low

frequencies of higher resolution simulations. Sharing the

same motivation, Nielsen et al. [18,19] create a direct

coupling between a low resolution preview simulation

and a higher resolution final simulation. The high reso-

lution simulation is created by incorporating an upsam-

pled velocity field from the low-resolution simulation as

additional constraints into pressure projection. In their

earlier paper, Nielsen et al. [18] introduce the method,

and present the multigrid framework used to solve it.

In their later paper, Nielsen and Christensen [19] ex-

tend the earlier work to support more efficient time-

varying computation, and use density to help determine

the level of control to provide. Though density is used

to help determine control, the actual adjustments are

made only to the vector field. Yuan et al. [32] also pro-

vide a guiding simulation that does not directly correct

density. Their method extracts flow patterns from a low

resolution simulation and uses a pattern-based guiding

force to force the higher resolution simulation to match.

Huang et al. [9] control a high resolution simulation

based on data sampled at user selected regions in a low

resolution simulation. The “match points” that they use

to cover and sample select regions, have finite support

and fairly scarce distribution which allow quick local

adjustment and leave room for high frequency features

to develop. In terms of sampling a simulation, match

points resemble control particles that are discussed by

Thurey et al. in [29]. Control particles can be generated

from a Lagrangian simulation and hence themselves al-

ready carry the nature of fluids, yet they rely on fully

simulated particle-based fluid dynamics. Match points

work in a more general way and are better for matching

scalar values such as density, yet they are nontrivial to

configure and their static positions are sometimes in-

sufficient for tracking fluid motions.

3 Review of Preview Based Control

We begin with an initial (low resolution) simulation

called the preview. Our goal is to produce a high res-

olution final simulation that is adjusted to match the

behavior of the preview at specified regions and times.

Our method follows the approach outlined in Huang et

al. [9] and we give an overview of that method here as

background.

In broad terms, we will run a low-resolution fluid

simulation, sample the vector and scalar data from that

simulation, and then run a high-resolution simulation

in which we match the sample data collected at lower

resolution.

3.1 Sampling the Flow Equations

The inviscid equations for incompressible flow u are

∇ · u = 0 (1)

∂u

∂t= f − (u · ∇)u−∇p (2)

and density d advected by the flow is defined as

∂d

∂t= S − (u · ∇)d+ kd∇2d (3)

As mentioned earlier, instead of precisely matching

all details of the low resolution simulation, we want to

be able to control the scale and locations at which the

4 Ruoguan Huang, John Keyser

final will match to the preview. Thus, we sample the

properties of the preview at specified locations, scales,

and times, and use these samples for matching, later.

We call these samples match points. Each match point

is defined by a position xi, a scale ri, and a matching

function mri . We will refer to a match point by its po-

sition xi. To sample the simulation in scale space, we

convolve the simulation data with a Gaussian kernel of

the given scale. We sample scalar d and vector data D

separately as follows:

dri =1∑Gr

∑x

Gr(xi,x)d(x) (4)

Dri,1 =

1∑Gr

∑x

Gr(xi,x)D(x) (5)

Dri,2 =

1∑Gr

∑x

Gr(xi,x)D(x)× (x− xi) (6)

G is the Gaussian weight function with radius r:

Gr(xi,x) = e−r|xi−x|2 (7)

Note that we obtain two pieces of information from

D, direction D1 and angular momentum around xi with

unit mass D2, to measure both linear and angular mo-

tion in the vector field.

In the matching process, we sample the final (un-

matched) simulation at the same points specified in

the keyframe. The difference between the final and the

keyframe will be used to modify (match) the final.

3.2 Controlling the Simulation

Matching is done in two passes at each timestep. In

the first pass, a step of the simulation is taken at high

resolution, and that simulation is sampled at the xiand scale(s) r. For scalar data we obtain the (weighted)

current value dri and for vector data determine Dri,1 and

Dri,2. The computations are performed in the same way

as for the samples from the preview (see equations 4, 5,

and 6).

The weighted errors:

δri = mri (dri − d

r

i ) (8)

∆ri,1 = mr

i (Dri,1 −Dr

i,1) (9)

∆ri,2 = mr

i (Dri,2 −Dr

i,2) (10)

will represents the amount of “control” that needs to be

applied to the high resolution simulation at the match

point.

The idea of matching is that we want to modify

the high resolution simulation by the control amounts.

Notice that in Equation 2 f accounts for external forces

and in Equation 3 S defines a density source (or sink).

The controls will have the same effect as introducing

additional f and S to the system. The matching will

be created by spreading the control amount over the

region around the match points with a spatial blending

function G′.

G′r(xi,x) =Gr(xi,x)∑x(Gr(xi,x))2

(11)

Thus, for a point x in space, we increase the density

value at x by δriG′r(xi,x). A similar process is used

to spread directional velocity control ∆ri,1, but simple

vector addition is used in the sum. Applying control on

the curl is slightly more complicated. For a point, p, the

change in the vector value due to curl control is:

G′r(xi, p)(∆ri,1 × (xi − p)) (12)

4 Improving Sampling

The spatial and temporal placement and the weight mir

of match points dictate the result of control as both the

sampling and matching processes rely heavily on these

match points. However, the temporally and spatially in-

variant match points described by Huang et al. [9] raise

several issues. The match points need to cover the spec-

ified regions and be aligned with prominent features.

Automatic regular placement of sample points tends to

provide poor overall placement, and generally requires

oversampling to cover all areas of interest. While it is

possible for a user to manually specify a configuration

of samples, it is often tedious to place every match point

by hand and provide sufficient coverage as well as accu-

rate alignment. The sample points are set up with static

positions that are inefficient and inaccurate for captur-

ing the constantly flowing motion of fluids. The user

can set up one group of match points for each period

of time or simply cover the whole domain with match

points, yet neither is ideal in term of efficiency. If the

user intends to hit certain keyframes instead of match-

ing a whole sequence, the weight mir for each individual

match point needs to be set to vary over time to avoid

abrupt changes in the result.

To address these issues, we propose four improve-

ments to the sampling methodology. We describe be-

low methods for automatically determining features to

focus samples on, for generating good distributions of

sample points, for allowing sample points to move over

time, and for using a different filter for the samples.

4.1 Detecting Prominent Features in Simulations

The most prominent visual component in a gaseous

fluid simulation tends to be density. Density is often

Automated Sampling and Control of Gaseous Simulations 5

rendered (e.g. as smoke) and in most simulations in-

tended for realism it affects the velocity field through

forces like gravity. Within vector data (usually the fluid

velocity field) vortices not only are a fundamental el-

ement of fluids, but also provide strong visual detail.

Thus, we look at density and vortices to determine

whether a location is suited for placing a match point.

We assume that the user has browsed through a low

resolution preview simulation and specified a general

region R for matching. In implementations, we allow

a user to define R, e.g. as the projected volume from

a selected region in screen space, but the user can also

simply set the whole domain to be of interest and let our

system detect all the features for a complete match. We

consider a position x ∈ R to be a candidate position to

place a match point if any of following three inequality

conditions is satisfied.

d(x) > dmin (13)

|d(x)− d(x)| > ∆dmin (14)

|∇ × u(x)| > ωmin (15)

where dmin, ∆dmin, ωmin are pre-defined minimum val-

ues close to 0. That is, we identify points with high

density, large difference in density from the final, or

high vorticity as candidates for placing samples.

Given a candidate position x, we next determine

the radius r of the match point based on the gradient

of density and vorticity. This ensures that the sample

will be taken at a scale that matches the scale of a

feature.

r = max

{min

{dmax

∇d(x),

ωmax

∇|∇ × u(x)|, rmax

}, rmin

}(16)

where dmax and ωmax are pre-defined maximum values

to cap the change, within the support of the match

point, in density and vorticity, respectively. We clip the

radius r using user defined maximum radius, rmax, as

the ceiling and the size of the Eulerian grid cell of the

low resolution simulation, rmin, as the floor.

4.2 Modified Poisson Disk Distribution

Given the above criterion for determining good posi-

tions to place match points and what their radii should

be, we employ a modified dart-throwing Poisson disk

distribution algorithm [14,13] to fill the user specified

regionR with match points. The traditional dart throw-

ing algorithm generates uniformly distributed points,

and rejects points that do not satisfy the minimum sep-

aration with already generated points [13]. This process

continues until no more points can be added. In our

case, complete separation of match points is unneces-

sary and we instead allow some overlapping (a discus-

sion of how much overlapping is allowed between sam-

ples is given by Huang et al. [9]). Thus, we enforce a

smaller minimum distance between match points than

their actual radii combined. More specifically, we accept

a newly generated match point at position x, that sat-

isfies Equations 13, 14, or 15, with radius r determined

by Equation 16 if

|x− xi| < αmin(r + ri) αmin ∈ [0, 1] (17)

and the overlap is not so great as to prevent convergence

(see [9]). In our experiments, we find that αmin = 0.6

performs well in most cases.

The resulting distribution is not uniform as pro-

duced by a traditional dart throwing algorithm. In-

stead, the match points with differing radii are densely

distributed to cover regions with prominent features,

and yet moderately separated to guarantee convergence.

4.3 Advecting Sample Points

As mentioned before, statically positioned match points

do not follow the flow of a fluid simulation. We there-

fore advect the match points generated from the dart

throwing algorithm through the velocity field of the low

resolution preview simulation. We do not need to ad-

vect these particles by running a costly complete La-

grangian particle simulation. Instead, we simply use

first order Eulerian integration to update the position

of each match point with the velocity field u. More im-

portantly, match points that follow Lagrangian particle

physics are not necessarily aligned with prominent fea-

tures because they have a much more scarce distribu-

tion and larger radii.

After advection, the match points do not necessar-

ily meet the requirements of the modified Poisson disk

distribution or the properties of featured positions and

gradient based radii. We wish to avoid resampling the

region R because our goal is to maintain as much spa-

tial and temporal continuity as we can of the match

points, imitating a Lagrangian particle system. Thus,

we perform the following steps to fix the distribution

and properties of match points.

1. Recalculate the radius ri for each match point using

Equation 16.

2. Obtain samples from the preview simulation and

sort the match points (typically by the sampled den-

sity value dri , though other prioritizations can be

used).

3. Assign an integer value ai to every match point to

track its age. ai is increased by one each time step.

6 Ruoguan Huang, John Keyser

4. Remove any match point xi if ai > am and it does

not satisfy any of Equations 13, 14, or 15.

5. Remove any match point that violates the minimum

distance constraint described by Equation 17 or the

convergence limits [9], starting from the match point

with the lowest density sample.

6. Remove any match point that violates boundary

conditions, i.e. crossed a boundary.

7. Apply several more iterations of the modified dart-

throwing algorithm to fill the gaps in R.

Age tracking is intended to avoid repeated activa-

tion and deactivation of control in an area with density

that borders on the pre-defined threshold. We wish to

keep a match point that is under a mature age am, even

if it fails the density and vorticity tests. We will also use

age to modify the control weight mri to buffer sudden

activation of control (see Section 5.2). These steps will

bring back the desired distribution while preserving the

continuity needed for smooth control.

4.4 Bilateral Density Sampling

We sample the data from the low resolution preview

simulation through these match points that are pur-

posely placed to capture key features of fluids. Follow-

ing Huang et al. [9], we had used a Gaussian weighted

function for sampling, although any other function with

similar isotropic fall off should work as well. However,

when sampling gaseous densities, we encounter an issue

analogous to filtering in the area of computer vision—

the assumption of slow spatial variations fails at edges,

which are consequently blurred by low-pass filtering

[30]. When a match point is placed across a density edge

in the preview simulation, the adjustment made in the

high resolution is spread out within the support of the

match point in a Gaussian weighted fashion regardless

of the edge. Normally, we rely on the similarities be-

tween the preview and the high resolution simulation.

But in cases where drastic differences are presented in

the high resolution simulation, the edge from the pre-

view is completely lost.

Inspired by bilateral filtering [30] in computer vi-

sion, we use a bilateral Gaussian weighted function to

improve edge preservation for density matching. We ob-

tain two density samples through a match point at po-

sition xi with radius r, one for each side of the edge, as

follows:

dri,1 =1∑Gr

∑x

Gr(xi,x)H(xi,x)d(x) (18)

dri,2 =1∑Gr

∑x

Gr(xi,x)(1−H(xi,x))d(x) (19)

H is a radially symmetric closeness function with geo-

metric spread σh:

H(xi,x) = e− |d(xi)−d(x)|2

σ2h (20)

Note that the sum of Equation 18 and 19 is a standard

Gaussian function. We use these bilateral samples for

density matching in high resolution in Section 5.3.

5 Impoved Matching

The matching process modifies the high resolution sim-

ulation so that it agrees with the preview at the match

points. In prior work [9], matching followed a fixed pro-

cess that effectively forced a simulation to match the

preview at precisely the given frame. This could cause

significant problems if a match point was introduced

later in a simulation, in that the simulation would vis-

ibly “jump” to meet the sample. As a result, generally

the entire region of interest would need to be matched

through an entire simulation in order for the simulation

to match at the desired timestep.

We propose two improvements, temporal blending

and age weighting, that help support matching with-

out visible jumps in the simulation. Furthermore, we

describe matching with bilateral match points, giving

higher quality results.

5.1 Temporal Blending of Samples

When keyframes are specified several frames apart, con-

trol may introduce a sudden density/force into the sys-

tem. This could be extremely unrealistic and jarring,

and therefore we wish to apply control over a period

of time, effectively providing a temporal blend to the

control. If we wish to apply control at some point prior

to the keyframe time, we need to account for the ex-

pected change within the velocity field itself. We can

approximate the effect by advecting our match points

themselves backward through the velocity field. This

process effectively repositions all of the match points

at some earlier point in time, and the control process

is applied at those repositioned points. Note that this

accounts for only advection (when other forces will also

be active), and perhaps more importantly the conver-

gence guarantees would no longer apply. For these rea-

sons, the control applied at earlier timesteps should be

limited, for example by reducing the weight mri at ear-

lier timesteps and not iterating the control process to

convergence.

Our method circumvents these difficulties with an

advected Poisson disk distribution. Match points are

Automated Sampling and Control of Gaseous Simulations 7

advected through the velocity field and their positions

are continuous in space and time. These attributes elim-

inate the need for advecting the points backward. Con-

vergence is automatically guaranteed due to the well

maintained distribution. To match a keyframe at time

t1, we start to apply control at time t0 < t1. For a

frame at time t ∈ [t0, t1], we apply control through

match points with the weight mri = M(t), where M is a

blending function that increases monotonically from 0

to 1 as t approaches the keyframe time t1, e.g. a linear

version M(t) = t−t0t1−t0 .

5.2 Age-weighted Density Control

Compared to velocity, density is directly visible and the

control needs to be more subtle. For density control,

temporal blending of samples applies even for matching

a whole simulation sequence. To smoothly match the

sampled density from the preview simulation, we vary

the control weight of a match point based on its age

(introduced in Section 4.3). When the dart throwing

algorithm generates new match points xi, we do not

want it to apply maximum density control immediately.

Instead, we increase the match points’ weights over time

to the maximum value. Thus, we determine the weight

mri of match point xi with age ai as follows:

mri =

{ai/am if ai < am

1 otherwise(21)

Similar to the example we give for keyframe match-

ing, this is a linear blending function although it can

be replaced by other types of monotonically increasing

functions for different effects. We combine age weight

and temporal weight in a multiplicative fashion for tem-

porally varying density control.

5.3 Bilateral Spread of Density Control

In this section, we derive the spatial blending func-

tion to apply density control using the bilateral samples

mentioned in Section 4.4. Assume we have differences

Q1 = δri,1 and Q2 = δri,2 between the bilateral samples

of the preview and the final at match point xi. In order

to make the final match the preview, the sampled final

values must therefore increase by Q1 and Q2, respec-

tively. That is, we need to have

Q1 =∑x

Gr(xi,x)H(xi,x)Q′x (22)

Q2 =∑x

Gr(xi,x)(1−H(xi,x))Q′x (23)

where Q′x is the increase in density at point x. We wish

to find the amount to apply via a bilateral Gaussian

spread to result in the correct overall change. However,

because there are two constraints to meet, we instead

formulate the problem with two unknowns, Q′′1 and Q′′2 .

Q′x = Q′′1G1(x) +Q′′2G2(x) (24)

where G1(x) = Gr(xi,x)H(xi,x) and

G2(x) = Gr(xi,x)(1−H(xi,x)) for the sake of conve-

nience and simplicity. The idea behind Equation 24 is

that we wish spread two separated amounts, Q′′1 and

Q′′2 , such that a point x gets more contribution from

the amount that represents the same side of the edge

it resides on. Substituting into Equation 22 and 23, we

have:

Q1 = AQ′′1 +BQ′′2 (25)

Q2 = BQ′′1 + CQ′′2 (26)

where A =∑

x(G1(x))2, B =∑

xG1(x)G2(x) and C =∑x(G2(x))2. Solving for Q′′1 and Q′′2 , we get:

Q′′1 =BQ2 − CQ1

B2 −AC(27)

Q′′2 =BQ1 −AQ2

B2 −AC(28)

Thus, for each point in space, x, we would increase the

density value at x by Q′′1G1(x) +Q′′2G2(x).

In practice, if match point xi covers an area with rel-

atively smooth density distribution, the closeness func-

tionH evaluates close to 1. This brings the denominator

B2 − AC close to 0. In such cases, we revert back to

Equation 11 for a standard Gaussian spread.

6 Results

We have implemented our automated sampling and con-

trol system within an Eulerian fluid smoke simulator.

All simulations were run on a PC with Intel Core i7 Pro-

cessor at 4.5GHz. We employ BFECC [10] and mono-

tonic cubic interpolation [3] for advection. Identical time

steps were used in low and high resolution. We have de-

fined a few constants as minima for selecting regions in

Section 4.1. For the thresholds in Equation 13, 14 and

15, we have found that even small amount of density

(0.01 when density is normalized to [0,1]) is perceivable

after rendering. Thus, in our examples we use small val-

ues (0.01) for dmin, ∆dmin and ωmin, to avoid floating

point error and ensure all visible effects are matched.

We illustrate our method with the following exam-

ples. Please refer to the accompanying video for more

examples, and more complete version of the simula-

tions. In all of our examples except the user interface

demo, no manual configuration of match points is needed.

All match points are automatically generated by our

sampling method described in Section 4.

8 Ruoguan Huang, John Keyser

Fig. 2 On the left is the low resolution preview simulation. The second column is the results of [9]. 2a is without densitycontrol and 2b is with. The third column is our results. 3a is without density control, 3b and 3c are with. 3b uses a standardGaussian kernel while 3c uses a bilateral kernel. On the right is the uncontrolled high resolution simulation.

Figure 2 shows an example of simulation using our

automated sampling method with bilateral density sam-

pling and compares it to the method of Huang et al. [9]

that covers the domain with uniformly placed match

points with fixed positions. Arrows indicate one place

where differences are more obvious. The comparison be-

tween the two methods (Figure 2.2a and Figure 2.3a)

without density control shows that our advected match

points cause the flow to more closely match desired be-

haviors instead of waiting for the differences to appear

and then correcting them. Feature detection based sam-

pling significantly reduces the number of match points

used, and therefore requires much less computational

overhead. The number of match points used in this ex-

ample by our method is approximately one fourth of

that used in [9], which brings down the extra time (rel-

ative to an uncontrolled simulation) spent on the con-

trol process from 80% to 20%. The density control of

Huang et al. [9] (Figure 2.2b) shows some areas with

unmatched density while the result of our method (Fig-

ure 2.3c) with age-based density control and temporal

blending smoothly matches the preview.

The comparison between Gaussian (Figure 2.3b) and

bilateral (Figure 2.3c) density sampling and distribu-

tion methods shows that the result using bilateral ker-

nels produces a closer match to the shape of the smoke

in the preview simulation than the one using Gaus-

sians. The effect of Gaussian kernels seems to reduce

the intensity of the smoke to match the preview, yet

makes no change to its overall shape. The effectiveness

of bilateral density sampling is further demonstrated

in Figure 3. In this example, we set up a simulation

with constant u = 0 everywhere and refresh the distri-

bution of match points every frame. The results after

just one iteration demonstrate the preservation of edges

by bilateral kernels compared to the blurring effect of

Gaussians. With further iterations, the advantage of bi-

lateral kernels is even clearer. To show each individual

match point more clearly, the match points generated

in (only) this example are restricted to have a larger

Automated Sampling and Control of Gaseous Simulations 9

Fig. 3 Bilateral density sampling and matching on a staticdensity field. Each row presents from left to right: the preview,the result using Bilateral kernels, the result using Gaussiankernels. The first and second row show the results after oneiteration with match points displayed along the first. Thethird is after 50 iterations.

minimum distance than used in other examples. In the

accompanying video, we slow down the matching pro-

cess so that the faster convergence rate will be obvious.

Figure 1 shows that the user can easily configure the

control of a simulation in selected regions and times.

The control is set up to start in the middle of the simu-

lation and is effective only in the projected volume from

a selected region in screen space. The user can also mod-

ify settings on a time line to determine the interval and

curve at which the control weights mri ramp up as the

simulation approaches the first fully controlled frame.

The result, as set up, is uncontrolled for the first part of

the simulation, and then starts to shift towards the pre-

view, and after a short transition, becomes completely

matched to the preview.

Figure 4 shows the animation of a walking charac-

ter of smoke. We apply modified Poisson disk distribu-

tion to fill the hull of the character with match points,

and sample the density and velocity by discretizing the

space on a coarse grid. We then use the sampled data

to drive the smoke animation in higher resolution.

We provide the timings and the number of match

points used in Table 1.

7 Conclusion

We have presented a collection of techniques that allow

preview-based control of gaseous fluid simulations to be

Fig. 4 On the left is the animated character of smoke. On theright is the corresponding frame from the motion sequence.

handled in a more automated and more accurate man-

ner. Methods presented for identifying key features in

a simulation, automatic Poisson sampling, and advec-

tion of match points allow simulations to be matched

more accurately and with fewer match points than with

uniform sampling, and with far more ease of use than

manual placement. Methods for temporal blending and

age weighting allow for smooth transitions into a con-

trolled region/time of the simulation. Finally, a bilateral

approach to sampling and matching is presented that

allows sharp gradients within the simulation to be cal-

culated much more accurately. Together, these methods

allow a user to have a much more usable, efficient, and

accurate control method.

There are several avenues that would be good to

explore in the future:

1. Automatically determining control weights to han-

dle very drastic change in the preview simulation

without involving the user.

2. Extending our sampling and matching methods to

an arbitrary style of control such as artistic direc-

tion.

3. Adapting our method to Lagrangian frameworks,

and perhaps liquid simulations.

Acknowledgements

This publication is based in part on work supported by

NSF Grant IIS-0917286 and by Award Number KUS-

CI-016-04, made by King Abdullah University of Sci-

ence and Technology (KAUST).

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10 Ruoguan Huang, John Keyser

Crossing Bar Rotating Bar Smoke Column StairsHuang et al. N/A N/A 1.20 1.44Ours 0.45 0.71 0.76 0.36Simulation 2.23 0.86 4.16 1.83

Number of Match Points Huang et al. N/A N/A 1536 3072Number of Match Points Ours 104 218 215 906

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