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motion blur
  Eurographics Workshop on Natural Phenomena (2007)D. Ebert, S. Mérillou (Editors) Eulerian Motion Blur Doyub Kim † and Hyeong-Seok Ko ‡ Seoul National University Abstract This paper describes a motion blur technique which can be applied to rendering fluid simulations that are carried out in the Eulerian framework. Existing motion blur techniques can be applied to rigid bodies, deformable solids,clothes, and several other kinds of objects, and produce satisfactory results. As there is no specific reason todiscriminate fluids from the above objects, one may consider applying an existing motion blur technique to render  fluids. However, here we show that existing motion blur techniques are intended for simulations carried out in the Lagrangian framework, and are not suited to Eulerian simulations. Then, we propose a new motion blur techniquethat is suitable for rendering Eulerian simulations. Categories and Subject Descriptors  (according to ACM CCS) : I.3.7 [Computer Graphics]: Three-DimensionalGraphics and Realism 1. Introduction Motion blur is essential for producing high-quality anima-tions. The frame rate of most films and videos is either 24or 30 Hz, whereas human vision is reported to be sensitiveup to 60 Hz [Wan95,CJ02]. Due to the lower frame rate of  film and video, when each frame is drawn as a simple in-stantaneous sampling of the dynamic phenomena, artifactssuch as temporal strobing can occur. The graphics commu-nity has been aware of this problem, and several motion blurtechniques have been proposed to solve this problem.Fluids are often important elements of a dynamic scene,and for the artifact-free production of such a scene, fluidsneed to be rendered with motion blur. Since the graphicsfield already has an abundance of motion blur techniques,one may consider applying existing techniques to the mo-tion blur of fluids. Unfortunately, existing techniques do notproduce satisfactory results. This paper describes why theexisting solutions do not work for fluids and how to modifyexisting motion blur techniques to make them applicable tofluids.Motion blur techniques developed so far are intended for † kim@graphics.snu.ac.kr ‡ ko@graphics.snu.ac.kr Figure 1:  A motion blurred image (left) produced with thealgorithm presented in this paper and an unblurred image(right): A slice of water is falling along the wall, whichhits the logo and makes the splash. To factor out the effectscaused by the transparent material, we rendered the water as opaque. c  The Eurographics Association 2007.   Doyub Kim & Hyeong-Seok Ko / Eulerian Motion Blur  rendering simulations that are performed in the Lagrangianframework. † We will call this type of motion blur techniques  Lagrangian motion blur   (LMB). The majority of objects en-countered in 3D graphics scenes (including rigid bodies, ar-ticulated figures, deformable solids, and clothes) are simu-lated in the Lagrangian framework; thus their motion blurcan be readily rendered with LMB.Simulation of fluids, however, is often carried out in theEulerian framework. Considering the high quality and broadapplicability of LMB, and considering there is no specificreason to discriminate fluids from other 3D objects, one mayconsider employing LMB for rendering fluids. An interest-ing finding of this paper is that LMB is not suitable for ren-dering the results generated by an Eulerian simulation. Sofar no algorithm has been proposed that can properly rendermotion blur of fluids that are simulated using the Eulerianframework. In this paper, we explain why Lagrangian mo-tion blur should not be used for rendering Eulerian simula-tions. Insight obtained during this process led us to developa simple step that can be added to existing motion blur tech-niques to produce motion blur techniques that are applicableto Eulerian simulations (i.e.,  Eulerian motion blur   (EMB)). 2. Previous Work Motion blur was first introduced to the computer graph-ics field by Korein and Badler [KB83], and Potmesil andChakravarty [PC83]. Korein and Badler proposed a methodthat works on an analytically parameterized motion and cre-ates a continuous motion blur. Potmesil and Chakravartyproposed another method that creates continuous motionblur by taking the image-space convolution of the objectwith the moving path. We will classify the above sort of mo-tion blur techniques as  analytic methods .The next class of motion blur we introduce is the  tempo-ral super-sampling methods . Korein and Badler [KB83] pro-posed another method that renders and accumulates whole(not partial) images of the object at several super-sampledinstants, resulting in a superimposed look of the object.The distributed ray tracing work of Cook et al. [CPC84]brought improved motion blur results. Their method suc-cessfully increased the continuity of the motion blur byretrieving pixel values from randomly sampled instants intime. Recently, Cammarano and Jensen [CJ02] extended thistemporal super-sampling method to simulate motion-blurred † We note the intrinsic differences of the physical quantities used inthe Lagrangian and Eulerian frameworks. In the Lagrangian frame-work, the simulator deals with the quantities  carried   by the movingobjects (e.g., the position, velocity, acceleration of the objects). Inthe Eulerian framework, on the other hand, the domain is discretizedinto grids and the simulator deals with the quantities  observed   fromfixed 3D positions (e.g., the velocity and density of the fluid at afixed grid point). globalilluminationandcausticsusingraytracingandphotonmapping.The third class of motion blur is known as  image-based methods . Max and Lerner [ML85] proposed an algorithmto achieve motion blur effect by considering the motion onthe image plane. Brostow and Essa [BE01] also proposedan entirely image-based method which can create motionblur from stop motion or raw video image sequences. Thesemethods are suited to cases where the 3D motion is not avail-able or the motion is already rendered. A more completesurvey of motion blur techniques can be found in Sung etal. [SPW02]We assume in this work that the 3D data of the fluid atevery frame are available, but the data are not given in a pa-rameterized forms. Therefore the temporal super-samplingmethod seems to fit to the situation. In this paper, we de-velop a motion blur technique based on the temporal super-sampling method.Realistic rendering of fluids has been studied as wellas the fluid simulation itself in the graphics community.Fedkiw et al. [FSJ01] visualized smoke simulation using aMonte Carlo ray-tracing algorithm with photon mapping,and Nguyen et al. [NFJ02] presented a technique based onMonte Carlo ray tracing for rendering fire simulations. Tech-niques for rendering liquids were also developed by Enrightet al. [EMF02]. However, motion blur was not considered inthose studies.Müller et al. [MCG03] used blobby style rendering for vi-sualizing water represented with particles, and their methodwas subsequently improved by Zhu and Bridson [ZB05]to have smoother surfaces. For the visualization of La-grangian particles, Guan and Mueller [GM04] proposedpoint-based surface rendering with motion blur. Geundel-man et al. [GSLF05] and Lossaso et al. [LIG06] attempted to include the rendering of the escaped level-set particles tocreate the impression of water sprays.Motion blur of Eulerian simulation has rarely men-tioned/practiced before; To our knowledge, there have beenonly two reports on motion blur of Eulerian simulations incomputer graphics thus far. In rendering water simulation,Enright et al. [EMF02] mentioned that a simple interpolationbetweentwosigneddistancevolumescanbeappliedinorderto find ray and water surface intersection. A few years later,Zhu and Bridson [ZB05] mentioned that the method will de-stroy surface features that move further than their width inone frame. 3. Computing Motion Blur The basic principle of motion blur is to add up the radiancecontributions over time, which can be expressed as  L  p  =   t  s    A  L (  x  ,  ω , t  ) s (  x  ,  ω , t  ) g (  x  ) dA (  x  ) dt  ,  (1) c  The Eurographics Association 2007.   Doyub Kim & Hyeong-Seok Ko / Eulerian Motion Blur  t  1  t  2  t  3  t  4 t  1 t  2 t  3  t  4(a)(b)(c)(d)(e)(f) Figure 2:  Motion blur with temporal super-sampling where  g ()  is the filter function,  s ()  represents the shut-ter exposure, and  L ()  is the radiance contribution from theray [CJ02]. The above principle applies to both Lagrangianand Eulerian motion blurs. In the equation,  x  is the placewhere the movement of objects jumps into the motion blur;for the evaluation of   x  , the locations of the objects at arbi-trary (super-sampled) moments need to be estimated, whichforms a core part of motion blur.For the development of a motion blur technique based ontemporal super-sampling, we use Monte Carlo integration.It computes the integral in Equation (1) by accumulating theevaluations of the integrand at super-sampled instants.More specifically, imagine the situation shown in Figure 2(a) in which a ball is moving horizontally. Suppose that wehave to create a blurred image for frame  t  n . Let  η  be theshutter speed. For each pixel, we associate a time samplepicked within the interval  [ t  n − η / 2 , t  n + η / 2 ] ; the samplesare taken from both past and future. Figure 2 (b) shows that,for example, the time samples  t  1 ,  t  2 ,  t  3 , and  t  4  (which do notneed to be in chronological order) are associated with thefour pixels in a row.For each pixel, we now shoot the ray at the associatedtime, test for intersection, and estimate the radiance contri-bution. Shooting a ray at a certain time and testing for inter-section implies that the location of the objects at that timeshould be estimated. Figure 2 (c) shows the object locationsat  t  1 ,  t  2 ,  t  3 , and  t  4 . In this particular example, only the rayshot at  t  3  hits the moving object. Figure 2 (d) shows the finalresult. Figure 2 (e) shows an image produced with an ac-tual ray tracer. Usually multiple rays are shot for each pixelfor better results (Figure 2 (f)), which can be easily done byassociating multiple time samples to a pixel. 4. Lagrangian Motion Blur LMB is used for rendering objects that have explicit sur-faces such as rigid bodies, deformable solids, and clothes.The core part of the LMB approach is to compute, from thegiven 3D data of each frame, ray–object intersection at arbi-trary super-sampled instants. In order to do this, the locationof the surface at an arbitrary moment has to be estimated. InLMB,theestimationisdonebytakingthe  time-interpolation of the vertices of the two involved frames; When the po-sitions  x ( t  n )  and  x ( t  n + 1 )  of the vertices at  t  n and  t  n + 1 aregiven, the estimated position  x  L ( τ )  at super-sampled time  τ is calculated by x  L ( τ ) =  τ − t  n t  n + 1 − t  n x ( t  n + 1 )+  t  n + 1 − τ t  n + 1 − t  n x ( t  n ) .  (2)We now briefly consider the physical meaning of the esti-mation given by rearranging Equation (2) into the form x  L ( τ ) =  x ( t  n )+( τ − t  n ) x ( t  n + 1 ) − x ( t  n ) t  n + 1 − t  n  .  (3)This equation shows that the estimation is the result of as-suming the movement was made with a constant velocity ( x ( t  n + 1 ) − x ( t  n )) / ( t  n + 1 − t  n ) . However, how valid is this as-sumption? Movement of any object with non-zero mass hasthe tendency to continue its motion, and thus has an iner-tial component. When specific information is not available,calculation of the object position based on the inertial move-ment turns out to give quite a good estimation in many cases, judging from images rendered using LMB. The error of theestimation is proportional to the acceleration. 5. Eulerian Motion Blur In developing Eulerian motion blur, we assume that the sim-ulation result for each frame is given in the form of 3Dgrid data. The grid data consists of the level-set (or density)and velocity fields. As in the Lagrangian motion blur, it isnecessary to know how a ray traverses the fluid at an ar-bitrary super-sampled instant. However, rendering Euleriansimulations needs a different type of information: insteadof the ray-surface intersection, the required information isthe level-set (in the case of water) or density (in the caseof smoke) values at the cell corners of all the cells the raypasses. ‡ ‡ When the fluid has a clear boundary, as is the case for water,the surface can be extracted from a Eulerian simulation using themarching cube algorithm [LC87]. In such a case, rendering can bedone with ray–surface intersections. However, this approach is notapplicable to surfaceless fluids such as smoke which do not have c  The Eurographics Association 2007.   Doyub Kim & Hyeong-Seok Ko / Eulerian Motion Blur  t  n = t  n +0.4 t  n +1   = 0.58   = 2.54 1.36 τ τ t  n t  n +1 0 -0.29 φ  φ φ  (a)(b)(c)(d) TI  φ φ  Figure 3:  Characterization of the level-set change in a sim- ple example: (a) the snapshot at t  n  , (b) the snapshot at t  n + 1  ,(c) the situation at t  n + 0 . 4  , (d) the level-set changes. 5.1. Why Time-Interpolation Does Not Work Since the grid data are available only for the frames, we mustsomehow  estimate  the level-set values at arbitrary time sam-ple  τ . For the estimation, Enright et al. [EMF02] presenteda method which interpolates the level set data between twoframes. Note that this is just as same as LMB-style estima-tion. An LMB-style solution would be to make the estima-tion with φ TI  ( τ ) =  φ ( t  n )+( τ − t  n ) φ ( t  n + 1 ) − φ ( t  n ) t  n + 1 − t  n  .  (4)Contrary to expectation, the above estimation gives incor-rect results. Imagine the simple case shown in Figure 3, inwhich a spherical ball of water is making a pure translationalmovement along the horizontal direction at a constant veloc-ity. Figures 3 (a) and 3 (b) show two snapshots taken at  t  n and t  n + 1 , respectively. At the marked grid point, the level-setvalues are  φ ( t  n ) =  0 . 58 and  φ ( t  n + 1 ) =  2 . 54. The question iswhat would be the level-set value  φ ( τ )  at  τ  =  t  n + 0 . 4 at thatposition? Since the fluid movement is analytically known inthis example, we can find out the exact location of the waterball at  τ , as shown in Figure 3(c). At  τ , the marked positioncomes within the body of fluid; therefore  φ ( τ )  has a negativevalue. In fact, we can find out the trajectory of   φ ( t  )  for theduration  [ t  n , t  n + 1 ] , which is plotted as a solid curve in Fig-ure 3(d). On the other hand, the time-interpolated result is φ TI  ( τ ) =  1 . 36, which is far from what has happened. Varia-tion of   φ TI  ( t  )  within the duration follows a straight line andis plotted with a dashed line in Figure 3(d). Here, we notethat (1) the time-interpolation gives an incorrect result eveninsuchasimple,non-violent,analyticallyverifiablecase;(2) a distinct boundary. Even for cases where the surface extraction ispossible (as in the water), when the topology changes over frames,LMB is difficult because finding the vertex correspondence is a non-trivial process.  ( τ   t  n )  u ( τ  ) τ t  n ???? Figure 4:  Estimation of the level-set values for Eulerian mo-tion blur. The grid points marked with ?s are the locationswhose level-set values must be estimated. The short solid ar-rows at those points represent the estimated velocity  u ( x , τ ) . the error is remarkable; and (3) the error is not related to thegrid resolution.We now investigate why the time-interpolation gives suchan incorrect result. When specific information about themovement is not available, exploiting the inertial compo-nent of the movement works quite well. The reason the LMBmethod works so well for Lagrangian simulations can be at-tributed to the fact that the LMB-estimation of object loca-tion exploits the inertia. We can adopt this idea of   exploitinginertia  in the development of Eulerian motion blur. A ques-tion that arises here is whether the time-interpolation  φ TI   isexploiting the inertia.It is critical to understand that it cannot be assumed thatthe level-set/density change at a grid point will continue tohappen at the current rate. The space in which the fluid ex-periences inertia in the conventional sense is the 3D space.The inertial movement of the fluid in 3D space is  reflected   tothe level-set field by updating the level-set according to theequation ∂φ∂ t   + u ·∇ φ  =  0 .  (5)This equation states that the level-set should be advected inthe direction  u  at the rate | u | . 5.2. Proposed Method FortheEulerianmotionblurtoexploittheinertialmovementof fluids, therefore, we propose that the estimation of thelevel-set values at arbitrary super-sampled instants be basedon the level-set advection, rather than the time-interpolationof the level set values. More specifically, we propose to es-timate  φ  E  ( x , τ )  of the level set value at a 3D position  x ,at a super-sampled time  τ  with the semi-Lagrangian advec-tion [Sta99,SC91] φ  E  ( x , τ ) =  φ ( x − ( τ − t  n ) · u ( x , τ ) , t  n ) .  (6)This equation states that  φ  E  ( x , τ )  takes the level-set value of  t  n at the back-tracked position  x − ( τ − t  n ) · u ( x , τ ) . In the c  The Eurographics Association 2007.
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