K-L Reconstruction of the Counter Rotating Rings
This state consists of two concentric rings with six cells in
the outer ring and two cells in the inner ring. The rings counter rotate
with the outer ring moving at approximately 1 revolution/second and the
inner ring moving at approximately half of the speed.
(Click here (66K) to view an MPEG movie of the flame motion.
To perform the K-L analysis, videotape of the flame front was digitized at a rate of 30 frames/second.
300 images of dimension 64x64 were used.
The average of the image ensemble (denoted by I0) was subtracted from each image prior to the K-L decomposition.
The K-L analysis produced a set of eigenvectors that are denoted by
I1, I2, ... where I1
represents the K-L eigenvector containing the most energy.
The resulting K-L energy spectrum is shown at the left.
The spectrum indicates that the first 6 K-L eigenvectors contain 85% of the energy.
The ensemble average I0 and the 10 K-L eigenvectors are:
Modes I1, I2, I5 and I6
contribute to the outer ring, while modes I3, I4, I7
and I8 contribute to the inner ring.
The dominant outer ring coupling modes I1 and I2 have
D6 symmetry, while I5 and I6 have
D12 symmetry indicating that the later are higher harmonics
that serve to define cell shape. Similarly I3 and I4,
the dominant coupling pair that define the inner ring have D2
symmetry, while I7 and I8 have D4 symmetry
and correspond to a higher spatial harmonic of the dominant mode.
The images in the original data set are denoted by
U(t) for t = 1 .. 300.
The m-th order K-L reconstruction of image U(t) is:
Um(t) = I0 + a1(t)I1
+ a2(t) I2 + ... + am(t)
where ai(t) is the projection of U(t)
on Ii. The successive reconstructions for U(1)
are shown below (left to right top to bottom). The first
image is the average of the ensemble and the last image is
the original image from the dataset.
An animation showing a side-by-side comparison of
selected K-L reconstructions can be found at:
The constructions from left to right are order-2 K-L reconstruction, order-
4 K-L reconstruction, order-
6 K-L reconstruction,
and the original image.
This view of the Counter Rotating Rings state consists of 3 dimensional renderings
of selected K-L reconstructions. The upper left corner
is the order-1 K-L reconstruction, the upper right corner is the order-
2 K-L reconstruction, the lower left corner is the order-3 K-L reconstruction and the lower right corner is the original image. The animations are:
In order to emphasize the relative contributions of the
different K-L modes to the motion as a function of time,
this animation displays an instantaneous bar chart of the
projections of the images on the first
10 K-L eigenvectors.
The first bar represents the instantaneous size of
a1(t), the second bar represents the instantaneous
size of a2(t) and so on. The middle image is a
3-dimensional rendering of the K-L reconstruction using the
average and the first 10 K-L eigenvectors The rightmost image is a 3-dimensional
rendering of the original data. The animations are:
This animation shows a bar chart of the instantaneous
values of the projections of the images on the first
10 K-L eigenvectors. The resulting
reconstruction is presented as a grey scale image (middle figure)
and is compared with the original image (rightmost figure).
The animations are:
The following animation more clearly separates
the contributions of the modes. Each row corresponds to a
reconstruction. The leftmost image in each row is a bar chart
showing the instantaneous amplitudes. The right image shows the corresponding reconstructions.
The animations are:
These animation show the correlation between the 3D phase plot
of selected projections, the appearance of reconstruction
and the original image.
The phase space plot in the animation shows 1 vs 2 vs 3 The middle image shows the order-
3 KL reconstruction, while the rightmost image shows the original image.
The animations are:
This animation shows different views of
phase plane plots of the projections.
In each phase plane plot, all of the
projections are represented as black dots. The current position
in phase space is marked by a larger red dot. The
animations are laid out:
| a 1(t) vs a 2(t) |
a 3(t) vs a 4(t) |
a 1(t) vs a 3(t) |
| a 1(t) vs a 5(t) |
a 3(t) vs a 5(t) |
a 5(t) vs a 6(t) |
The movies of the phase plots indicate that the motion consists of two
incommensurate periodic motions.
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To perform the K-L analysis, videotape of the flame front was digitized at a rate of 30 frames/second.
300 images of dimension 64x64 were used.
The average of the image ensemble (denoted by I0) was subtracted from each image prior to the K-L decomposition.
The K-L analysis produced a set of eigenvectors that are denoted by
I1, I2, ... where I1
represents the K-L eigenvector containing the most energy.
The resulting K-L energy spectrum is shown at the left.
The spectrum indicates that the first 6 K-L eigenvectors contain 85% of the energy. 
