Op Art and More 

Fractal representation with artificial neural networks

This page contains the representation of fractals, in which the pixels of the individual video images are formed by a logical linkage with the pixels of the preceding image. This results in a separate optical time series for each pixel, which is influenced by the logical linkage as well as by the respective image section and the respective calculation of the fractal. So that a pixel does not change into an unchangeable extreme value (e.g. black or white) by the iterative logical connection, extreme values are cyclically reset to a starting value, similarly as with a modulo operation with natural numbers. The chosen optical iterations lead to the fact that in the overall context of the pixels apparent vistas or apparent three-dimensional formations or also iridescent surfaces or points arise. The actual fractal then remains only partially visible.

The following Boolean functions were used as logical operators in the iteration of the pixels for each of the videos:

f_BOOL: AND / XOR / OR NOT / MAX / OR / AND NOT

Here the following iterations were performed:

a) for the calculation of the fractals:      

f _k+1 := f (o) f_k   

with 50 iterations (k) at each pixel per video frame; (o) corresponds to one iteration step
f: see above in the caption of each video;
c: variable in time as c(t) with an expiration of the time axis t := t+1 per video frame; c is here a given sequence of complex numbers by which the visible structures of the fractal f can be changed dynamically.

The calculated fractal value f_k is then converted to the RGB color model using a custom color model with a color depth of only 780 color steps at most.

b) subsequently for further iterative color processing of the pixel colors RGB:

RGB_i+1 := f_BOOL ( RGB_i , RGB(f_k+1) )

with a sequence (i) of 25 frames per second at each pixel. RGB(f_k) represents the value of the color corresponding to the original color value of the original fractal after the final k-th iteration;

The flow of the time axis t is harmonized with the iteration i in the circuit, i.e. t == i and corresponds to a clock pulse.

The Boolean functions were repeatedly applied individually for all 24 bits Pⁿ (n:=1,..24) of each pixel and linked with the respective current bit Pⁿ_i of the RGB color value (RGB_i) of the fractal. Subsequently, the newly determined bits (Qⁿ_i+1) were displayed on the screen as the new RGB color value RGB_i+1.

Since some Boolean operations in the iteration process end at white (all 24 bits Qⁿ == "1") or black (all 24 bits Qⁿ == "0"), a SET or RESET of the circuit (here in the example with a logical NAND) was performed for all 24 bits in each case for such events. The circuit for a complete pixel with the AND NOT function has therefore the following structure:

At a screen resolution of 500x500 pixels, 250,000 circuits of this type are thus emulated 25 times per second for each individually calculated fractal image, whereby a sequence of 25 new fractal images with 50 iterations each was previously calculated per second as an input variable for the circuits. The original fractal value calculated after 50 iterations for each pixel position is converted into an RGB value, which is then processed by the circuitry before finally being output to the display.

The circuits correspond to an artificial neural network that is single-layered and has a recurrent form with a direct feedback loop.

Through changing the coordinate window (enlargements and reductions), and the navigation in the fractal as well as through changes of the term c in the fractal functions jumps or movements of partial images and also apparent three-dimensional movements or apparent transparency of partial images in the course of the of the video become visible during the video calculation with the circuit incl. the automated removal of extreme pixel values (SET and RESET). All this results in the artistic representation of this kind of calculations. Some of the Boolean operations (e.g. XOR) lead to rather unpleasant iridescent optical impressions. For the sake of completeness, however, some videos are presented here anyway.

A comparison between a conventional representation and  the representation alienated by the circuit is shown in the following video as an example. Colors colors appear inverse (negation of the input colors in the circuit) and background colors are partially preserved due to the iterative feedback whereby new background structures can become visible:

Left Image output via circuit; right image original representation: