VISUALIZATION, ANIMATION AND K-L DECOMPOSITION OF SPATIOTEMPORAL DYNAMICS IN A PATTERN-FORMING SYSTEM

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1 VISUALIZATION, ANIMATION AND K-L DECOMPOSITION OF SPATIOTEMPORAL DYNAMICS IN A PATTERN-FORMING SYSTEM KAY A. ROBBINS Division of Computer Science, University of Texas at San Antonio San Antonio, TX and ANTONIO PALACIOS and MICHAEL GORMAN Department of Physics, University of Houston Houston, TX Abstract Cellular flames form patterns of concentric rings of cells that bifurcate to states exhibiting complex spatiotemporal dynamics. We have integrated image processing and computer animation with K-L decomposition to create visualizations of the dynamics. These techniques are illustrated by comparing two relatively simple states each consisting of two cells. Direct K-L decomposition of images taken from experimental data shows that, for each state, only four K-L eigenvectors are needed to reproduce the behavior. Understanding the interaction of the spatial and temporal dynamics is then enhanced by constructing different animations consisting of a side-by-side comparison of the original data with successive low-order K-L reconstructions. 1 Introduction Karhunun-Loeve (K-L) decomposition is a well-known technique for finding an optimal basis for a dataset of vectors [4]. The approach is particularly useful when a significant part of the dataset is restricted to a subspace spanned by a few vectors of the K-L basis. Individual vectors in the original dataset can then be characterized by a small number of scalars that are the projections on the K-L subspace. These projections can be used for classification, analysis or compression of the original dataset. In the application described here, K-L decomposition is applied to cellular flames exhibiting spatiotemporal dynamics. The dataset consists of digitized images of videotape of the experiment. Time series that describe the dynamics are constructed by projecting images from video of the experiment on the K-L basis. The approach separates the temporal and spatial behavior and clarifies the bifurcation structure of the system. This paper focuses on techniques for visualizing the spatiotemporal interactions from the K-L decomposition by comparing and contrasting the results for two similar states. The next section presents an overview of the experimental setup and reviews K-L analysis. Section 3 illustrates how the K-L projections and successive reconstructions can be animated to provide valuable insight into the dynamics.

2 2 The Experimental Procedure In the combustion experiments a uniform circular flame front is stabilized over a porous burner and videotaped from above [2]. The controlling parameters are the flow rate and the ratio of fuel (isobutane) to oxidizer (oxygen). The pressure of the combustion chamber is held constant at 1/3 atm. Videotape of the flame experiment is digitized using 64 by 64 pixel resolution at a frame rate of 30 frames per second. The images are converted to vectors by storing the images in row-major form. The software package KLTOOL [1] was used to perform the K-L decomposition. Approximately 128 images were used for each calculation, and the average of the images was subtracted from each image prior to performing the K-L decomposition. A cutoff of 75% of the energy was used to determine how many eigenvectors to keep [3]. Once the K-L eigenvectors were obtained, the original video frames (minus the average) were projected onto the most significant eigenvectors and successive K-L reconstructions were performed. For the purposes of notation, the K-L eigenvectors are denoted by Ii and the corresponding projections are denoted by ai(t) where t = 1; 2; : : : is the video frame number. The average value of the images is denoted by I 0. The eigenvectors are ordered so that I 1 is the most significant followed by I 2 and so on. K-L reconstructions are of the form: U (x; y; t) = I 0 + a 1 (t)i 1 + a 2 (t)i 2 + : : : + am(t)im where m is the order of the K-L reconstruction. This paper focuses on two relatively simple states to illustrate the visualization techniques. The simplest (State A) is a two-cell rotating mode consisting of two large asymmetric cells in uniform rotation around the center of the burner as shown in Figure 1. Each row of Figure 1 compares successive K-L reconstructions for a single frame of video. The leftmost column shows I 0 + a 1 (t)i 1, the second column shows I 0 + a 1 (t)i 1 + a 2 (t)i 2 and so on. The rightmost column contains the original images from the experimental videotape. The second (State B) oscillates between two configurations as shown in Figure 2. Each configuration consists of two cells that are separated by an axis of symmetry. The axes of symmetry for the two configurations are oriented at 90 degrees relative to each other. Visually the flame front appears to form two cells each of which breaks into two parts while maintaining the configuration s line of symmetry. Then each part combines with a part from the other pair to form two cells at approximately right angles to the original orientation. The last column of Figure 2 shows four consecutive frames of experimental video in which the original cells split apart and the parts join to form a pair of cells in the other orientation. 3 Visualization The purpose of this paper is to demonstrate the use of visualization and animation techniques in improving the understanding of spatial and temporal interactions. One goal of this research is an on-line K-L analysis that would allow the experimenter to better identify 2

3 Figure 1: Comparison of successive K-L reconstructions (columns 1 through 4) with the original image (column 5) using four consecutive frames of video of State A. Figure 2: Comparison of successive K-L reconstructions (columns 1 through 4) with the original image (column 5) using four consecutive frames of video of State B. 3

4 and characterize the dynamics during the conduct of an experimental run. The results of animations and visualizations are difficult to display in a paper. The two principal methods of video presentation are videotape and the world wide web. Figures 1 and 2 each contain four successive images from a video comparing different levels of K-L reconstruction. By playing a movie of these successive reconstructions, the viewer can get a sense of how each K-L eigenvector contributes to the motion. A movie of the successive reconstructions was made by converting the images into consecutively numbered PBM files. These files were then converted to an MPEG movie which can be accessed and played via the world wide web. The individual frames were also converted to consecutively numbered JPEG files and imported into Adobe Premiere for construction of a videotape. Finally the frames were also converted to consecutively numbered GIF files in order to construct an animated GIF. The movies of the successive reconstructions for State A reveal a simple traveling wave structure. The average intensity I 0 is circularly symmetric. The two principal K-L eigenvectors I 1 and I 2 (not shown) have identical structure, but are oriented 90 degrees with respect to each other. The second-order reconstruction for State A (second column of Figure 1) produces the state s characteristic rotation in much the same way as a linear combination of sin and cos would. Higher order reconstructions merely enhance the asymmetry and shape of the resulting cell structure. The movies of the successive reconstruction for State B reveal that two eigenvectors are sufficient to reproduce the characteristic sloshing motion from one configuration to the other. The relative contributions of the K-L eigenvectors I 1 and I 2 are not obvious from the side-by-side comparison of the successive reconstructions. The visualization in Figure 3 provides a clearer demonstration of the motion of State B. Each row of images corresponds to a time. The bar chart on the left gives the instantaneous values of the projections on the first four K-L eigenvectors. The middle image shows the resulting K-L reconstruction rendered as a 3-dimensional density plot. The rightmost image is a 3-dimensional rendering of the original data. The 3-dimensional rendering enhances the contrast in regions of low intensity and makes the internal structure of the cells more apparent. These images were produced in Mathematica [5] and transferred to individual PBM files for the construction of animations. The average intensity I 0 and the first K-L eigenvector I 1 for State B both have fourfold symmetry. The first K-L coefficient is always positive, resulting in a first-order K-L reconstruction with an oscillation of a pattern of lobes with four-fold symmetry. The sign of a 2 (t) determines which of the adjacent lobes are paired to form the two large single cells. The magnitude of a 3 (t) determines the shape of these two cells. Figure 4 shows an animation that gives the viewer a global view of the dynamics. The left image in each row is a phase space plot of K-L projections (a 1 (t); a 2 (t); a 3 (t)). The right image is the corresponding reconstruction. As the animation runs, a movie of the reconstruction plays on the right while its position in phase space moves on the left. In this way the viewer can make a correlation between the low-dimensional temporal behavior and the visual appearance of the experiment. Notice the more complicated structure of State B. 4

5 Figure 3: Animations of successive frames of State B. The bar chart in each row gives the values of the K-L projections, the middle image is the reconstruction based on the first four K-L eigenvectors and the rightmost image is a rendering of the original data. 4 Conclusions Appropriate visualization is an essential component in understanding spatiotemporal systems. We have demonstrated how multiple animations of K-L reconstructions enhance understanding of the dynamics of two apparently simple states. We are applying visualization of K-L decomposition to successively more complex states in order to understand how the flame front changes as the experimental parameters are varied. A multimedia version of this paper and additional animations can be found at: Acknowledgments: This research was supported by a grant N00014-K-0613 from the Office of Naval Research. The authors would like to thank Daniel Brown, Deepak Kumar, Jennifer Olveda and 5

6 Figure 4: Animation of the phase space plot and the corresponding pattern. The top row corresponds to one frame for State A, while the bottom row corresponds to one frame for State B. The right image in each frame is the corresponding third order K-L reconstruction. Joe Rodriguez for their assistance in the production of the animations. The authors would also like to acknowledge Mohamed el-hamdi for his work in developing the experiment and Gemunu Gunaratne and Martin Golubitsky for helpful discussions. References [1] D. Armbruster, R. Heiland and E. Kostelich, Chaos 4, no. 2, 421 (1993). [2] M. Gorman, C. F. Hamill, M. el-hamdi and K. A. Robbins, Combust. Sci. and Tech. 98, 25 (1994). [3] A. Palacios, G. H. Gunaratne, M. Gorman and K. A. Robbins, Chaos to appear [4] L. Sirovich, Q. Appl. Math. 5, 561 (1987). [5] S. Wolfram, Mathematica 3rd edition, Wolfram Media, Inc.,