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Monday, February 20, 2012

Magnitude Patterns

Magnitude patterns arise from Fourier transforms.

Bob Lansdon and Tom Hedges showed me magnitude patterns back at Calma in 1978. Tom had just finished a driver for the large-bed Versatec raster plotter. The plot was probably about 3 feet square and it was computed by using a 1024x1024 2D FFT of seven points evenly distributed about a circle. Bob had the program shade the magnitude of the result. FFTs work in the complex domain and by this he meant plotting for a+bi the result sqrt(a*a+b*b). Here to right you see a magnitude pattern. It is very much like a sort of puffy shape with black rivers wending and undulating through it.
Perhaps we can see the convolutions of a human brain in it. When you notch higher frequencies, you get patterns that may include flowers, lines of beads, and really all sorts of symmetries. I got to thinking that I might be able to show the patterns more effectively, and so I decided to render the magnitude pattern to a texture and back to an FFT, moving to to the real channel of the data. Then I highpassed the result to remove the wide range of the bias. Once that  was done, I then transferred it back to a texture and annealed it to reveal all the patterns in a much higher-contrast format.
Lines, beads, and strange patterns kind of like those we see in the convection patterns of the solar surface begin to appear.

I really don't have any idea why these patterns are like this, save for one simple explanation. They are the magnitude patterns of a notch-filtered texture. This texture has really only one frequency of data in it, which makes it ring on only one wavelength. So the patterns are like waves on a pool with several disturbance points, all adding up to a chaotic, but highly band-limited pattern.

Here you see the real component of the magnitude pattern at the top of this page. The dots are all in the same place, but the dark spots and the light spots in this version are both represented by light spots in the magnitude pattern. This is because the FFT, with its negative and positive values, was normalized to the black-white tonal range for display. Thus the dark spots are actually negative values and the light spots are positive. And both the negative and positive values have positive magnitude.

So that does describe why the patterns have black rivers between them, but it doesn't say why there are higher-level patterns in the image: lines, rosettes, and chains.

I call it happy coincidence.

When your image has a particular kind of directionality to it, the magnitude patterns do as well.

Here is a pattern from an image with some directionality to it. You can almost feel the sculpted surface, like some H.R. Giger style image. Dark, organic shapes.

I was intrigued by magnitude. I took a hatching texture and annealed it into a sort of protoplasm you see to left here.

This had the effect of generating a high-contrast image with individual randomly-shaped elements. The hatching was a nice touch since it generated protuberances on each element, kind of like cilia on a paramecium.

Then I moved it to the FFT and did a highpass on it. The result was basically the same, but with the black and white moved to gray and the edges still visible as local contrast.

Then I converted it to magnitude to reveal the edges as black lines on soft areas. I guess I was getting the hang of magnitude patterns. The edges of the elements become tiny black worms inside a pleasant glow. A bit like meandering rivers. I have seem the Mississippi river from the air, and it does have the same kind of undulations to it, especially down near Louisiana.
I would like to show you some more slice-and-dice images now. I think a more proper term for these patterns is image in cell textures. Here a very large crossfade region is used for each cell, so that the cells tend to totally blend together. But of course I am also using z-buffering to merge the images together between the cells. This creates, as we have seen before, interpenetrating effects.
Your choice of a source image can be varied for different results. Here, I just used a different part of the same source texture to get a stickery-spiny result instead of interlocking arches.

The most interesting thing is to take these results and anneal them. Then you get some really interesting patterns. Each element can be similar to the others, but with a more organic placement and even some erosion. It becomes very natural for this reason.
This came from a pattern that was recursively edited using slice and dice. Maybe ten times. After a while, the complexity of the individual elements is a bit like yarn. This complexity translates to even better images when you render them to very large textures.

I think I will experiment with images that use the slice-and-dice technique using two image sources instead of only one in each cell. This will lead to areas that look one way and other areas that look in another way. Some sweaters are crocheted in patterns that vary the stitch in global ways, to create large-scale patterns. I think I will try this.

1 comment:

  1. I remember you guys printing out those big fried-egg-looking lobate things. I'm not remembering Bob Lansdon. I wonder if I could get you to write back off-blog. After staring at your various posts talking about erosions and dilations, FFTs and such, I thought I should dig through my archive to show you some stuff I did with 2D trig functions just to test precision and aliasing by-eye over the last 20 or so years. Not sure if I can post them as a blog post, and not sure if I want to. Let me know if you'd like to write back.