Automated Sampling and Control of Gaseous Simulations

Transcription

1 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 automates the sampling and control of gaseous fluid simulations. Several recent approaches have provided techniques for artists to generate high resolution simulations based on a low resolution simulation. However, often 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 unnecessarily restrictive. We propose a new sampling technique to efficiently capture the overall flow of a fluid simulation, 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 in the 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 samples we obtain have both spatial and temporal continuity 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 Huang Texas A&M University hrg@cs.tamu.edu John Keyser Texas A&M University keyser@cs.tamu.edu 1 Introduction Fluid simulation has been an area of great interest within the computer graphics community. As quality and efficiency 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 convergence 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 simulation 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 viscosity or object boundary effects) can quickly lead to much larger and visually significant differences. Adding artificial noise and vortices provides detail, but not the realism of a higher-resolution simulation. Keyframe matching can provide good results, but it can be problematic to determine good keyframes and ensure that the simulation is not overspecified, removing desired detail. In this paper we present improvements for automatically matching a high resolution gaseous fluid simulation 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 2 Ruoguan Huang, John Keyser lows the general framework of Huang et al. [9], where control is achieved by sampling data from a low resolution preview simulation at a set of match points in a user-selected region and then matching a high resolution 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 positioned match points are usually inefficient and sometimes 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 dynamically sampling arbitrary types of data from a simulation with the goal of achieving spatially and temporally more continuous sampling. With this dynamic sampling, we can easily incorporate temporal blending 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 prominent features of fluids and follows the flow, resulting in smoother more accurate control without tedious manual editing. Temporal sampling and blending allow seamless keyframe 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 Stable Fluids approach introduced by Stam [27]. Stam s method used an implicit solver, making the simulation unconditionally stable, and thus allowing coarse grid resolutions and large time steps. There has been a tremendous amount of follow-on work which has presented improvements in various aspects of the simulation, and a thorough discussion of this work is beyond the scope of this paper. While particle-based (Lagrangian) simulations have also become popular [16], Fig. 1 Example of our controlled simulation with temporal blending. On each row from left to right are low resolution preview, controlled simulation, uncontrolled high resolution simulation. The top row shows images taken at the time we start to apply control. Before that simulation in the middle is uncontrolled, hence it looks exactly the same as the high resolution simulation at right. The bottom row shows images taken approximately 150 frames later when the applied control is fully in effect. The middle now resembles the preview at left. Eulerian simulations continue to be widely used, particularly for gaseous and incompressible situations. Controlling fluid simulation was first investigated by Foster and Metaxas [6] and later by Foster and Fedkiw [4], and has been addressed by a number of other researchers. Rasmussen et al. [22] proposed a production 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 representations to create a system for interactively manipulating fluid flows. Their method parameterizes a fluid simulation from advected particle paths, allowing finegrained 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 automatically extracted simulation features. Angelidis et al. [1] and Kim et al. [12] provide path control for smoke simulations.

3 Automated Sampling and Control of Gaseous Simulations 3 A few recent methods approach the problem described from a different perspective, namely that high frequency detail is added into a coarse simulation, thus giving the appearance of a higher-resolution simulation. Kim et al. [11] use a wavelet-based method to generate fine turbulent details within a fluid simulation. 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 procedural 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 focused on the keyframe concept. Treuille et al. [31] proposed 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 minimized 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 calculation. 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] introduce new terms into the standard flow equations, offering a closed form solution for the control parameters. Matching target densities using only advection is usually not enough, and so they propose smoke gathering, diffusing the error field. This allows complex smoke animations to be controlled with little additional expense. Later, Thürey 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 resolution simulation is created by incorporating an upsampled 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] extend the earlier work to support more efficient timevarying 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 provide 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 Thürey et al. in [29]. Control particles can be generated from a Lagrangian simulation and hence themselves already 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 insufficient 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 resolution 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 = f (u )u p t (2) and density d advected by the flow is defined as d t = S (u )d + k d 2 d (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 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 x i, a scale r i, and a matching function m r i. We will refer to a match point by its position x i. 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: d r i = 1 G r (x i, x)d(x) (4) Gr x D r i,1 = 1 G r (x i, x)d(x) (5) Gr x D r i,2 = 1 G r (x i, x)d(x) (x x i ) (6) Gr x G is the Gaussian weight function with radius r: G r (x i, x) = e r xi x 2 (7) Note that we obtain two pieces of information from D, direction D 1 and angular momentum around x i with unit mass D 2, to measure both linear and angular motion in the vector field. In the matching process, we sample the final (unmatched) 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 x i and scale(s) r. For scalar data we obtain the (weighted) current value d r i and for vector data determine Dr i,1 and D r i,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: δ r i = m r i (d r i d r i ) (8) r i,1 = m r i (D r i,1 D r i,1 ) (9) r i,2 = m r i (D r i,2 D r 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(x i, x) = G r (x i, x) x (G r(x i, x)) 2 (11) Thus, for a point x in space, we increase the density value at x by δi rg r(x i, x). A similar process is used to spread directional velocity control r i,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(x i, p)( r i,1 (x i p)) (12) 4 Improving Sampling The spatial and temporal placement and the weight m i r 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 invariant match points described by Huang et al. [9] raise several issues. The match points need to cover the specified 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 accurate alignment. The sample points are set up with static positions that are inefficient and inaccurate for capturing 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 matching a whole sequence, the weight m i r 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 improvements to the sampling methodology. We describe below 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

5 Automated Sampling and Control of Gaseous Simulations 5 rendered (e.g. as smoke) and in most simulations intended 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 element 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) > d min (13) d(x) d(x) > d min (14) u(x) > ω min (15) where d min, d min, ω min are pre-defined minimum values 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 { { dmax min d(x), } ω max u(x), r max }, r min (16) where d max and ω max 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, r max, as the ceiling and the size of the Eulerian grid cell of the low resolution simulation, r min, as the floor. 4.2 Modified Poisson Disk Distribution Given the above criterion for determining good positions 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 region R with match points. The traditional dart throwing algorithm generates uniformly distributed points, and rejects points that do not satisfy the minimum separation with already generated points [13]. This process continues until no more points can be added. In our case, complete separation of match points is unnecessary and we instead allow some overlapping (a discussion of how much overlapping is allowed between samples 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 satisfies Equations 13, 14, or 15, with radius r determined by Equation 16 if x x i < α min (r + r i ) α min [0, 1] (17) and the overlap is not so great as to prevent convergence (see [9]). In our experiments, we find that α min = 0.6 performs well in most cases. The resulting distribution is not uniform as produced by a traditional dart throwing algorithm. Instead, 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 therefore 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 advect these particles by running a costly complete Lagrangian particle simulation. Instead, we simply use first order Eulerian integration to update the position of each match point with the velocity field u. More importantly, match points that follow Lagrangian particle physics are not necessarily aligned with prominent features because they have a much more scarce distribution and larger radii. After advection, the match points do not necessarily 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 spatial 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 r i for each match point using Equation Obtain samples from the preview simulation and sort the match points (typically by the sampled density value d r i, though other prioritizations can be used). 3. Assign an integer value a i to every match point to track its age. a i is increased by one each time step.

6 6 Ruoguan Huang, John Keyser 4. Remove any match point x i if a i > a m and it does not satisfy any of Equations 13, 14, or 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 dartthrowing algorithm to fill the gaps in R. Age tracking is intended to avoid repeated activation 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 a m, even if it fails the density and vorticity tests. We will also use age to modify the control weight m r i 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 purposely placed to capture key features of fluids. Following 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 between the preview and the high resolution simulation. But in cases where drastic differences are presented in the high resolution simulation, the edge from the preview is completely lost. Inspired by bilateral filtering [30] in computer vision, we use a bilateral Gaussian weighted function to improve edge preservation for density matching. We obtain two density samples through a match point at position x i with radius r, one for each side of the edge, as follows: d r i,1 = 1 G r (x i, x)h(x i, x)d(x) (18) Gr x d r i,2 = 1 G r (x i, x)(1 H(x i, x))d(x) (19) Gr x H is a radially symmetric closeness function with geometric spread σ h : H(x i, x) = e d(xi) d(x) 2 σ 2 h (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 Impoved Matching The matching process modifies the high resolution simulation so that it agrees with the preview at the match points. In prior work [9], matching followed a fixed process 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 visibly 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 without 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, control may introduce a sudden density/force into the system. 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 expected 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 convergence guarantees would no longer apply. For these reasons, the control applied at earlier timesteps should be limited, for example by reducing the weight m r i at earlier timesteps and not iterating the control process to convergence. Our method circumvents these difficulties with an advected Poisson disk distribution. Match points are

7 Automated Sampling and Control of Gaseous Simulations 7 advected through the velocity field and their positions are continuous in space and time. These attributes eliminate the need for advecting the points backward. Convergence is automatically guaranteed due to the well maintained distribution. To match a keyframe at time t 1, we start to apply control at time t 0 < t 1. For a frame at time t [t 0, t 1 ], we apply control through match points with the weight m r i = M(t), where M is a blending function that increases monotonically from 0 to 1 as t approaches the keyframe time t 1, e.g. a linear version M(t) = t t0 t 1 t 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 x i, 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 m r i of match point x i with age a i as follows: { m r ai /a i = m if a i < a m (21) 1 otherwise Similar to the example we give for keyframe matching, 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 temporally varying density control. 5.3 Bilateral Spread of Density Control In this section, we derive the spatial blending function to apply density control using the bilateral samples mentioned in Section 4.4. Assume we have differences Q 1 = δi,1 r and Q 2 = δi,2 r between the bilateral samples of the preview and the final at match point x i. In order to make the final match the preview, the sampled final values must therefore increase by Q 1 and Q 2, respectively. That is, we need to have Q 2 = x Q 1 = x G r (x i, x)h(x i, x)q x (22) G r (x i, x)(1 H(x i, 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 1G 1 (x) + Q 2G 2 (x) (24) where G 1 (x) = G r (x i, x)h(x i, x) and G 2 (x) = G r (x i, x)(1 H(x i, x)) for the sake of convenience 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: Q 1 = AQ 1 + BQ 2 (25) Q 2 = BQ 1 + CQ 2 (26) where A = x (G 1(x)) 2, B = x G 1(x)G 2 (x) and C = x (G 2(x)) 2. Solving for Q 1 and Q 2, we get: Q 1 = BQ 2 CQ 1 B 2 AC Q 2 = BQ 1 AQ 2 B 2 AC (27) (28) Thus, for each point in space, x, we would increase the density value at x by Q 1G 1 (x) + Q 2G 2 (x). In practice, if match point x i covers an area with relatively smooth density distribution, the closeness function H evaluates close to 1. This brings the denominator B 2 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 control system within an Eulerian fluid smoke simulator. All simulations were run on a PC with Intel Core i7 Processor at 4.5GHz. We employ BFECC [10] and monotonic cubic interpolation [3] for advection. Identical time steps were used in low and high resolution. We have defined 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 values (0.01) for d min, d min and ω min, to avoid floating point error and ensure all visible effects are matched. We illustrate our method with the following examples. Please refer to the accompanying video for more examples, and more complete version of the simulations. 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 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 density control and 2b is with. The third column is our results. 3a is without density control, 3b and 3c are with. 3b uses a standard Gaussian 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 sampling 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 between 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 behaviors instead of waiting for the differences to appear and then correcting them. Feature detection based sampling significantly reduces the number of match points used, and therefore requires much less computational overhead. The number of match points used in this example by our method is approximately one fourth of that used in [9], which brings down the extra time (relative to an uncontrolled simulation) spent on the control 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 (Figure 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 distribution methods shows that the result using bilateral kernels produces a closer match to the shape of the smoke in the preview simulation than the one using Gaussians. 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 distribution 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 bilateral 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

9 Automated Sampling and Control of Gaseous Simulations 9 Fig. 4 On the left is the animated character of smoke. On the right is the corresponding frame from the motion sequence. Fig. 3 Bilateral density sampling and matching on a static density field. Each row presents from left to right: the preview, the result using Bilateral kernels, the result using Gaussian kernels. The first and second row show the results after one iteration with match points displayed along the first. The third is after 50 iterations. minimum distance than used in other examples. In the accompanying video, we slow down the matching process 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 simulation and is effective only in the projected volume from a selected region in screen space. The user can also modify settings on a time line to determine the interval and curve at which the control weights m r i 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 preview, and after a short transition, becomes completely matched to the preview. Figure 4 shows the animation of a walking character of smoke. We apply modified Poisson disk distribution 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 handled in a more automated and more accurate manner. Methods presented for identifying key features in a simulation, automatic Poisson sampling, and advection 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 controlled region/time of the simulation. Finally, a bilateral approach to sampling and matching is presented that allows sharp gradients within the simulation to be calculated 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 handle 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 direction. 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 and by Award Number KUS- CI , made by King Abdullah University of Science and Technology (KAUST). References 1. Angelidis, A., Neyret, F., Singh, K., Nowrouzezahrai, D.: A controllable, fast and stable basis for vortex based

10 10 Ruoguan Huang, John Keyser Crossing Bar Rotating Bar Smoke Column Stairs Huang et al. N/A N/A Ours Simulation Number of Match Points Huang et al. N/A N/A Number of Match Points Ours Table 1 The top three rows give timing information in average seconds per frame for the method of Huang et al. [9], our new approach presented here, and the high resolution simulation without any control. The bottom rows present the average number of match points used per frame in the two methods. smoke simulation. In: Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Computer Animation, pp (2006) 2. Fattal, R., Lischinski, D.: Target-driven smoke animation. ACM Trans. Graph. 23(3), (2004) 3. Fedkiw, R., Stam, J., Jensen, H.W.: Visual simulation of smoke. Proc. of ACM SIGGRAPH 01 pp (2001) 4. Foster, N., Fedkiw, R.: Practical animation of liquids. Proceedings of ACM SIGGRAPH 01 pp (2001) 5. Foster, N., Metaxas, D.: Realistic animation of liquids. 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