Automatic Dwell Limit

Robert P. Munafo, 2010 Sep 7.

Automatic Dwell Limit methods are one type of speed improvement (see that article for more).

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 points. 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: a special dwell limit used just for making a histogram. (If this is a zoom from a previous view, the histogram dwell limit might be a constant times the dwell limit in that 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 either below this optimum value, or did not escape at all. In addition, the histogram is typically used to automatically generate a color map.

Known Statistics of Previous View

A less compute-intensive alternative to the histogram method when the current view is a zoomed-in portion of a 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 sets or on features deep in cusps.

Orbit Detection Method

A modification of the histogram method in which no histogram dwell limit is necessary. A set of sample points is chosen, and iterated forever, until they escape or until a period is detected via one of the orbit detection methods (see infinite dwell limit). Those which converge on a period are proven member points; 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 and/or when the filament density is very high (for one example of the latter, with pictures, see Feigenbaum point).

Incremental method ("Successive Dwell Limit")

The image is evaluated at an arbitrary low dwell limit. Any points that reach the dwell limit without escaping 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, this is a successive tradeoff method which serves as a user interface optimization as well as an imaging optimization.

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