Automatic Dwell Limit

Robert P. Munafo, 1999 Feb 2.

Automatic Dwell Limit algorithms automate the decision of the dwell limit setting for an image. Usually the decision is made just before starting to generate the image, but it can also be done implicitly through successive increments.

The dwell limit is difficult for users to specify, because much experience in exploring the Mandelbrot Set is required to select an appropriate value. Too low and the image will be unacceptable; time will be wasted re-computing the image with a higher dwell limit. Too high, and excessive time will be wasted iterating Member Point. Therefore, it is desirable to have the computer select the parameter automatically, and the automatic dwell limit algorithm serves as a speed improvement.

There are different approaches which can be taken in computing a dwell limit:

Histogram Method

After a zoom, a set of points is selected (perhaps at random) and each is iterated up to a histogram dwell limit based on the dwell limit of the previous view (for example, 10 times the dwell limit used in the previous view). Once the dwells of these points are determined, a histogram is generated (frequency versus dwell value). An optimal dwell limit is selected in such a way that most of the points in the histogram fall below this dwell limit, not including those points which were equal to the histogram dwell limit (these are considered to be in the Mandelbrot Set.)

Known Statistics of Previous View

In the previous view before a zoom, there was a known dwell limit and many points with known dwells. This method looks to see how many of the points in the previous image had dwells which were close to the previous limit. If this number is small, the dwell limit was probably all right. If this number is large, the dwell limit should be increased. This method sometimes does not work very well, particularly when zooming on Embedded Julia Set or on features deep in Cusp.

Orbit Detection Method

A modification of the histogram method in which no histogram dwell limit is necessary. The points in the test set are iterated forever, until they escape or until a period is detected via one of the Orbit Detection methods. Those which converge on a period are discarded, and those which do not are used as the basis of the histogram as above. This method can become time-consuming when the magnification becomes large.

Incremental method ("Successive Dwell Limit")

The image is evaluated at an arbitrary low dwell limit. Any points that reach the dwell limit are remembered, along with their last iterate. Then, the image is rescanned, and these points are each iterated for an additional N iterations. The process continues indefinitely or until the user selects another zoom. This method is particularly attractive because it provides the fastest response and also handles arbitrarily large jumps in dwell limit. Like Successive Refinement, it is a Successive Tradeoff Methods which serves as a user interface optimization as well as an imaging optimization.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index

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