Mathematica in Action by Stan Wagon

Author:Stan Wagon
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

We can zoom in by choosing the range appropriately. It is usually necessary to increase the size of bailOut to get nice pictures.DensityPlot [MandelbrotIterations [x+i y, 100],

{x, −0.65, −0.45}, {y, 0.5, 0.71}, FrameTicks→

{Automatic, ComplexTicks[Range[0.5, 0.7, 0.1]], None, None},

PlotPoints →100, ColorFunction → (If [‡‡ == 1, Black, Hue [0.9 ‡‡3/4 ]] &) ]

Boundary Scanning

As described in [GG, Chap. 45], many interesting image processing operations can be achieved via convolution with a kernel. In particular we can use a technique called boundary scanning to highlight the fractal boundary of the Mandelbrot set. We’ll use the image we generate to make a nice dynamic Julia set generator.

Suppose we have a large two-dimensional matrix, which might represent color values for an image. A kernel is a typically much smaller two-dimensional matrix that is used to process the large matrix via convolution. The easiest way to describe convolution is to investigate the formula in a small, but arbitrary case.ListConvolve// MatrixForm

(a 22 k11 + a 21 k 12 + a 12 k 21 + a 11 k 22 a 23 k 11 + a 22 k 12 + a 13 k 21 + a 12 k 22) a32 k11 + a31 k12 + a22 k21 + a21 k22 a33 k11 + a32 k12 + a23 k21 + a22 k22)

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