Generating Various Flow Fields using Principal Component Analysis
Syuhei Sato∗
Hokkaido University

Yoshinori Dobashi
Hokkaido University/JST CREST

Hiroyuki Ochiai
Kyushu University/JST CREST

1

Tsuyoshi Yamamoto
Hokkaido University

Our Method

First, our method precomputes basis functions from the input velocity fields. We compute basis functions by applying PCA to input
velocity fields. After constructing a matrix consisting of input velocity fields V = [v0 , v1 , · · · , vM −1 ], principal components ofthe input velocity fields P = [Φ0 , Φ1 , · · · , ΦN −1 ] are calculated,where vm (m = 0, 1, · · · , M − 1) is an input velocity field at
m-th frame, M is the number of frames of the input dataset, Φis principal components, N is the number of basis functions with
M > N . Next, we represent the input flow field vm by a linear
combination of principal components Φi . The coefficient wim forthe i-th principal component is calculated using the following equation, wim = vm · Φi , where · is the dot product between two vectors. To generate various flow fields um , our method modulates wim
N −1for each principal component as follows, um = i=0gi wim Φi ,mwhere gi represents the gain with which wi is modulated.

3Tomoyuki Nishita
UEI Research/Hiroshima Shudo University

Introduction

The visual simulation of fluids has become an important element inmany applications, such as movies and computer games. In theseapplications, large-scale fluid scenes, such as fire in a village, areoften simulated by repeatedly rendering multiple fluid flows. In
these cases, animators are requested to generate many variations ofa fluid flow. Previously, we developed a method to help animators
meet such requirements [Sato et al. 2013]. However, the method
was limited to 2D fluid simulation. In the previous method [Satoet al. 2013], we use Laplacian eigenfunctions as the basis functions.
However, Laplacian eigenfunctions are too expensive in computation and storage costs for 3D fluid simulation. Furthermore, Laplacian eignfunctions force us to use slip boundary conditions and thismakes the method less practical. In order to address these problems, we introduce Principal Component Analysis (PCA) for thebasis functions. In generating the principal components of the input velocity fields, we use a subspace approach [Treuille et al. 2006;
Kim and Delaney 2013]. The variations are generated by modulating the coefficient of each principal component.

2Kei Iwasaki
Wakayama University/UEI Research

Results and Future Work

Fig. 1 shows examples created using our method. The number ofgrid points for the velocity fields is 128 × 128 × 192, the number∗ e-mail:sato@ime.ist.hokudai.ac.jp(a) original(b) decreasing
lower-order coefficients(c) increasing
higher-order coefficients(d) increasing
lower-order coefficients

Figure 1: Modulating coefficients of a fire animation by ourmethod. (a) shows the input fire animation. (b) through (d) show theresults generated by modulating coefficients of original animations.

of frames of the input velocity fields is 200, and the number of theprincipal components is 64. The flow field is visualized by advecting the fire temperature. As shown in these examples, our method
successfully generates similar but slightly different fire. The videos
corresponding to these examples and other results (smoke examples) can be found in the supplementary material.
One of the limitations of our method is the fact that the flow fields
generated by our method might not conform to the laws of fluid
flow, if the degree of modulation by gi is too large. To address
this problem, we calculate the Navier-Stokes equations for velocity
fields generated by our method, then compare two velocity fields.
In future work, we will make experiments to evaluate these flow
fields.

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