Images and pictures (continued)
More fractal doodles. Recall the fractal simulation of the Queen Anne's Lace flower (See previous page.). By modifying its transformations only slightly, we obtain the following image:
Image SCruz1.jpg: Santa Cruz 1 (31.1 KB)
Image SCruz1.jpg: Santa Cruz 1 (End)
Template SCruz1T.jpg for Image SCruz1.jpg: Santa Cruz 1 (Start.) (19.4 KB)
Template SCruz1T.jpg for Image SCruz1.jpg: Santa Cruz 1. (End.)
Comparison of the floral image and its template of yesterday with the corresponding image and template above will illustrate how easy it is to generate a movie. The changes made to yesterday's template are as follows:
(a) The rectangle in white occupying the right-hand half of the template area (with a vertical white 'handle' hanging on to its left side) has been spun 5 degrees clockwise from that of yesterday.
(b) A similar rectangle, in fuchsia, has been spun counterclockwise by 5 degrees.
(c) The green vertical 'stem' has been expanded horizontally, and squashed vertically.
(d) Two rectangles -- one with a white handle, the other with a yellow one -- and both occupying the width of the template area, have been shifted vertically.
That's all it took to transform a flower into an ornate cross image. I almost feel a sermon coming on!
Recall that each of the five transformations comprising the iterated function system (ifs) of each image, is represented by six parameters. The total image, therefore, can be generated by 5 x 6 = 30 numbers. One can interpret two of these six as representing a translation of the original figure (the outer perimeter of the picture box (in green)) along horizontal and vertical directions. The remaining four parameters represent a combination of rotation and contraction (never expansion!). For the mathematically inclined, the determinant of the 4 x 4 transformation matrix represents the area of the ifs relative to that of the original perimeter rectangle. That area is used to calculate the number of points to plot (here, 207,579), given a particular plotting density (here, 200,000 points per 1,000 x 1,000 twips (1/72")). The outer perimeter rectangle has a size of 1,000 x 1,000 points (pixels).
Consider that the two images are only two of many frames in a movie. All we need to do to generate the intermediate frames between these two is to use a single 'movie-script parameter' by which we can modify all of the six parameters of each frame by a factor determined by the intermediate values of a single 'movie-script parameter'. Our movie script can be very simple. All we need do is specify the number of frames in the movie and the start and end images, then assume that the interpolations of intermediate frames is linear in time (in the simplest case), and we have a movie. All we need now is some popcorn! There is a bit more to it than that. We can also arrange it so that, for example, a background sunset also varies in color, from not only the top to the bottom of each frame, but from frame to frame.
In the present case, we could have made a movie in which a round ball (a seed) transforms (morphs) into a Queen Anne's Lace flower, then into an oak tree, then into a cross, then into any image we wish that can be generated from the same number (5) of transforms of the ifs. Another way of thinking of this is to consider that, with 30 numbers representing each image, each image can be represented by a single point in a 30-dimensional space. For a more complex transformation, in which, for example, one transforms the face of John F. Kennedy into that of Marilyn Monroe, the procedure is just the same, and just as simple -- except for one practical complication. To represent a human face with almost photographic quality one would need about 157 transformations, not just 5. So, JFK gets represented by a single point in a 157-dimensional space, Marilyn by another one, and a Volkswagen Beetle by another. (A young lady did her Master's thesis on this very topic, although it was apart from the study of fractals.) Police forces are now using such techniques to not only have computers automatically recognize bad guys, but to peek at people like us in public places to see where we are today, and maybe to guess why we are there.
In the present case, it would be just as easy to set up the movie as having 10 frames as it would be for 1,000 frames.
It might seem that each intermediate (interpolated) image would not offer much variety. Actually, there can be interesting surprises. Even more surprising (dare we say creative?) are images obtained when we extrapolate beyond the final frame or backward to before the first frame. I have seen some lovely -- and some ghastly -- images generated this way!
I may try to show an actual example of such a movie later, by showing perhaps just three frames, illustrating that all of the frames differ by the effect of the value of only a single 'movie-script parameter', no matter how complex the subject matter might be.
'See you at the movies!
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You can e-mail me at waynerp@sympatico.ca
