Modified Inverse Iteration Method
Robert P. Munafo, 2008 March 5.
The Inverse-Iteration Method for Julia sets draws points on the boundary of the Julia set much more quickly than an ordinary grid scan.
At each step in inverse iteration, there are two choices: Z' = sqrt(Z-C) and Z' = - sqrt(Z-C). If these are chosen equally often, then most of the points actually plotted will be on the outer ends of the various "lobes" or "branches" of the Julia set.
This can be improved slightly by choosing a value that hasn't been plotted yet (whenever possible). Another technique, about equally effective, is to choose the square root that is closer in value to C which ensures that the two values for Z'' (the choice we make at the next iteration) will be further from each other.
Such a technique, called MIIM or Modified Inverse Iteration Method, is presented (without source code) in The Science of Fractal Images, page 178. This image by Adam Majewski shows the type of result one can expect, and includes Maxima source code.
However, in order to make a really good image we have to look several or many steps ahead, planning the choices so that we will eventually reach pixels that haven't been plotted yet. Alternatively, one could use the Distance Estimator method.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2012. Mu-ency index