K-L Reconstruction of the 2 Rotating Cells - Experimental
The rotating two cell state shown to the right consists of two asymmetric
cells that are rotating counter clockwise.
(Click here (58K) to view an MPEG
movie of the flame motion.) A similar state in which the cells rotate clockwise
also exists. The state is the result of a symmetry-breaking bifurcation.
To perform the K-L analysis, videotape of the flame front was digitized at a rate of 30 frames/
/second. 64 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 4 K-L eigenvectors contain 83% of the energy.
The ensemble average I0 and the 10 K-L eigenvectors are:
The average I0 shows circular symmetry. I1 and
I2 form a coupling pair with D2 symmetry. This
pair forms a traveling wave that results in the uniform rotation.
I3 and I4 form a coupling pair with
D4 symmetry. As shown by phase plots below, this
the time projection of the data on this pair oscillates at half of
the temporal frequency as the projection on the dominant pair. The symmetry
relationship indicates that I3 and I4 are
higher spatial harmonics of the dominant pair and refine the cell
shape. (I3 and I4 only have to rotate
half as far as the dominant pair to reestablish the original
pattern since they have a higher degree of symmetry.)
The
rotating 1 cell and the rotating two cell are similar in structure in that
they are produced by a dominant coupling pair with a secondary pair
providing additional shape information. The difference is that the
rotating 1 cell dominant pair has D1 symmetry, while the
rotating 2 cell dominant pair has D2symmetry.
The images in the original data set are denoted by
U(t) for t = 1 .. 64.
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 2 Rotating Cells - Experimental state consists of 3 dimensional renderings
of selected K-L reconstructions. The upper left corner
is the order-2 K-L reconstruction, the upper right corner is the order-
4 K-L reconstruction, the lower left corner is the order-6 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) |
| a 1(t) vs a 7(t) |
a 3(t) vs a 7(t) |
a 7(t) vs a 8(t) |
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To perform the K-L analysis, videotape of the flame front was digitized at a rate of 30 frames/
/second. 64 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 4 K-L eigenvectors contain 83% of the energy. 
