1. Keyframe Control of Smoke Simulations SIGGRAPH 2003

2. Overview ► Introduction ► Basis equation ► Proposed method ► Results ► Future work

3. Introduction ► Goal:  Control of smoke simulation ► Difficulties  Complexity  Non-linearity ► Proposed method:  Control the simulation by given density and velocity

4. Basis Equations ► Navier-Stoke Equation: Velocity diffusion Velocity advection External forces Smoke density advection

5. General procedure Add forceAdvect Diffuse Project

6. Framework ► State consists of  of densities  of velocity vector ► Initial state: ► State at time t: ► Simulation:

7. Control ► A set of keyframes that the smoke should achieve  Specifies the density should match at time t  Specifies the constraint on ► A set of parameterized forces  Amount/direction

8. Matching Keyframes ► Goal  Match the user-specified keyframe  Use as little force as possible Solve for the equation

9. Computing Derivatives ► Calculating derivatives by simulating the entire process in a space consisting of  A density and velocity field  Their derivatives ► Initial state: ► State at time t:

10. Computing Derivatives ► Standard solver process:  Mass preservation step  Advects the smoke density  Projects the resulting field  Performs diffusion  Advects the velocity  External forces Calculating S

11. Computing Derivatives ► Calculating  Each operation induces a operation  Ex:  And similarly for  Therefore,

12. Derivatives ► Projection ► Diffusion

13. Derivatives ► Advection

14. Derivatives ► Mass Preservation ► Forces

15. Control Parameters ► Two types: Wind forces Vortex forces

16. Wind forces ► A single control vector scaled by a Gaussian falloff function ► Derivative

17. Vortex Forces ► Using Gaussian falloff approach ► Derivatives

18. Objective Function ► Smoothness  Derivatives

19. Objective function ► Keyframe-matching  Straightforward method  Proposed method  Derivatives

20. Results

21. Future Work ► Drawbacks:  Computationally prohibitive with fine-grained control  Optimization might be caught in local minimum ► To paradigms other than keyframes