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Symbol stability is the result of convergence towards a reciprocal-image attractor (continuous symbol)


A reciprocal-image attractor is stable if all of the images within some neighborhood are pulled towards it. Each attractor, which can be created from an arbitrary pair of images, acts like an "energy well". Just like a ball rolling towards the bottom of a valley, the pair of images converges towards the attractor and remains there like a ball coming to rest at the lowest point. If disturbed slightly from this "energy minimum", for example by introducing noise, it will roll back towards the point of minimum elevation. The system illustrated above has six stable attractors and is documented in this automatically generated experimental report. Because the reciprocal-image attractors are stable, they can serve as symbols in a cognitive system.


Color images are implemented as three-dimensional fields where the third dimension has a resolution of three, which corresponds to the red, green and blue color channels. The resolution can be used to specify any number of spectral bands and the fields can be of any dimension.  A multichannel (color) system with four attractors was tested and documented in the reports: montereytreelena and butterfly.  As described in the automata specification section of the report, the grid resolution has been defined as three-dimensional and the transmitter and receptor regions have been set so the width of the third, color channel, is "0.1". Since the total field size in each dimension is always defined as 1.0, this narrow color channel specification causes each color channel to act independently of the other colors. This causes the red, green and blue images to converge at different rates and generates intermediate images with only some of the some colors present. This effect can be clearly seen seen in some of the experimental results. If the third coordinate in the transmitter or receptor region specification is set to a larger value (greater than 1/3 or 2/3) then the recognition in the red, green and blue channels will reinforce each other and the convergence towards the attractor will be more uniform with respect to the color channels.

Transfer Function Characteristics

The behavior of the image association transformation depends on many parameters, including the resolution of the processing element grid, the size and shape of the receptor and transmitter functions, and the characteristics of the sigmoidal transfer function that controls how many input pixels must equal the predefined pattern in order for there to be an effective "match". The transfer function characteristics are documented in all of the reports in the automata specification section. They can be found in the <Transfer ...> XML element nested inside of the <Grid><Cell> element and contains three parameters, type, center and slope. The type "tanh" specifies a hyperbolic tangent function, which is shifted horizontally so the inflection point is located at the position defined by the center parameter and is compressed or expanded so that the slope at the center position equals the value specified. The center position defines the relative number of pixels that must match for recognition; for example, by setting the parameter center="0.20", when 20% of the pixels match, the transfer function will equal its "midpoint" value at the inflection point. Pattern matches occur when the transfer function is near its maximum value.

In these experimental reports, all of the parameters were set to the same value except the position of the inflection point, which was set at the values:  center=0.20center=0.23center=0.25center=0.27center=0.30 and center=0.35. When the  inflection-point center value is small, only a few pixels must match in order for the images to quickly converge towards the attractor. As the position of the inflection point is increased, the convergence becomes slower and slower, to the point where it no longer converges at all. At some values (e.g. center=0.27) the image associations have "collapsed" in some local areas but correctly formed in other areas. However, the overall, global image association becomes locked in a local minimum. Finally, as the inflection point of the transfer function increases even further (center=0.35) the associations collapse altogether and the images converge to all black with the pixel values nearly equal to zero. Similar behavior, but with slightly different convergence paths, can be observed near another attractor but with the same system parameters: center=0.20center=0.23center=0.25center=0.27center=0.30 and center=0.35. In this test suite, a pattern with solid square regions is used for the reciprocal image so the computed values are more visible. The complete collection of all inflection-point variation results can be seen sequentially in this combined animation.

General Manifolds Sapphire Software

This page contains links to several HTML experimental reports that were generated automatically by the General Manifolds Sapphire software system. An overview of this software is described in the Sapphire Implementation Notes. In order to further scientific research and allow these results to be independently verified, this sofware is being made available for educational and non-profit research purposes without a fee (See Sapphire Software).  An original set of full resolution (512x512) PNG color images used for testing is available in the Jar file: TestImages.jar.

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