2.4 Related Work In Graphics
2.4.3 Pattern Formation
While the specific goal of this dissertation is the modeling of ice formations, in a more general sense, the problem of solidification is one ofpattern formation. Formally, physicists define pattern formation as when “nonlinearities conspire to form spatial patterns that sometimes are stationary, travelling or disordered in space and time.”
(Bodenschatz et al., 2003). In this sense, the work in this dissertation is one of many pattern formation algorithms in computer graphics. The patterns that form in fluid simulation, for example, fall into this category due to the non-linear advection term. Visual phenomena that arise as a consequence of flow, such as sand dunes and rust patterns (Dorsey et al., 1996; Chen et al., 2005) meet this criteria as well.
Closely related to the phase field equations to be presented in Chapter 3 are reaction- diffusion systems. Pattern formation in reaction-diffusion systems was first described by Turing (Turing, 1952). The notion is counter-intuitive, because diffusion is usually thought of as a physical mechanism that smears out detail, not one that gives rise to it. However, Turing showed that when coupled with the appropriate reaction terms, sharp, standing wave solutions could be obtained. The formation mechanism has since been dubbed ‘Turing instability’ in physics. Reaction-diffusion was introduced to graphics by two articles that were published concurrently: (Witkin and Kass, 1991) and (Turk, 1991). The first article (Witkin and Kass, 1991) described how to simulate reaction- diffusion over a rectilinear grid. However, when this grid is stretched over a model, it experiences significant distortion, so a distortion correction technique was described as well. The second article (Turk, 1991) circumvented the distortion problem by generat- ing a Voronoi diagram on the surface of the model in lieu of a rectilinear grid and then solved the reaction-diffusion equations over the diagram instead. Reaction-diffusion equations have the form:
∂A
∂t =DA∇
2A+R(A).
The symbol A denotes some chemical, DA is its diffusion constant, and R(A) is
some reaction equation. A reaction-diffusion system is obtained when two chemicals are coupled via their reaction terms. For example:
∂A ∂t = DA∇ 2A+R(A, B) ∂B ∂t = DB∇ 2B +R(A, B).
In fact, the phase field equations can be viewed as a special case of reaction diffusion. In both cases, non-trivial patterns form from initially homogeneous concentrations due to non-linearities in the reaction equations.
DLA, the algorithm that will be presented in Chapter 4 has also been applied to pattern formation in other phenomena, such as lichen growth (Sumner, 2001; Desbenoit et al., 2004). DBM, which also be described in Chapter 4 has been used to simulate lightning (Kim and Lin, 2004). In both of these cases, the non-linearity arises from the implicit presence of Eqn. 1.2 in the simulation.
Algorithms similar to DLA have also been developed in graphics. The non-linearities in these algorithms are not as readily apparent as in DLA, but given the algorithmic and visual similarities, they seem likely. Ballistic deposition, a simplified version of DLA, has been used to model the formation of patinas (Dorsey and Hanrahan, 1996), and a novel venation algorithm that bears resemblance to DLA has been proposed (Runions et al., 2005).
The Phase Field Method
In this chapter, I describe one method of solving the Stefan problem, the phase field
method. The general approach of the phase field method was derived independent of the Stefan problem, using an approach that appeals more directly to thermodynamics. A summary of this free energy approach is available in the previous chapter. I will provide an overview of the phase field equations and describe how they intuitively map to the Stefan problem.
Additionally, I present techniques to simplify the phase field computation and make the problem of simulating ice crystal growth more tractable. I also show how the phase field method allows a user parameterization that a visual effects artist can use to manip- ulate the ice crystal growth. The phase field method often has smoothing artifacts as a result of its implicit representation, and it can only compute the outermost ice/water boundary. Therefore, a novel intermediate geometric processing step is introduced to add sharp edges and medial ridges to the interior of the ice. Finally, the simulated images are rendered using photon mapping (Jensen, 2001).
The basic simulation and rendering framework has been applied to several different scenarios. Fig. 3.1 shows an example image generated by the described method.
Figure 3.1: Detail of ice grown on a stained glass window. The inset shows the full window.
3.1
Overview
I will give a brief overview of the overall computational framework and the basic design of each step involved.
I use a simple and powerful implicit simulation technique from the crystal growth literature, known as the phase field method. This method can takeO(N3) time, where
N is the resolution of a single grid dimension. To obtain reasonable accuracy, N must be fairly large, making the computation quite expensive. I reduce the computation time significantly by using two acceleration techniques. The first is based on the observation that most ice crystals are very thin. I can simulate growth in 2D and add 3D detail later, reducing the computation time from O(N3) to O(N2). Second, I further improve the
performance of the simulation by performing banded computation around the “front” of the ice and water interface, instead of over the entire grid.
I then adapt the phase field method to include aesthetic controls for a visual effects artist to manipulate. This is achieved by user control of the seed crystal and freezing temperatures input into the phase field simulation.
The visually salient features of our target object are used for the seed crystal. The features are extracted with edge detection and used to set the initial conditions of the simulation. In addition to seeding the simulation, I also influence the simulation throughout by manipulating the freezing temperature.
Due to the smoothing artifacts of the phase field method and the lack of internal detail given by the evolving interface, a novel intermediate geometric processing step is introduced to add sharp features prior to rendering. This is performed by first computing the border and medial axis of the ice with morphological operators. Given the resulting medial axis and boundary edges, I generate a constrained conforming Delaunay triangulation upon which a subdivision step is performed to introduce creases and edges (DeRose et al., 1998). Finally, the triangles are rendered using photon mapping (Jensen, 2001).
Fig. 3.2 shows the overall system pipeline of our computational framework.
Figure 3.2: The overall system pipeline.