Fractals

Tasks

• Large images are taking a while to generate (8K image at 10000 iterations took 20 minutes). Need to look into optimization
• Need to add color smoothing. There is very noticable banding. Need to implement logarithmic smoothing to mandlebrot function

Color Smoothing

Added Color smoothing by normalizing the iteration count. Done by taking the log of the log of the magnitude of the imagninary number at escape value. See below;

if(abs(z) > 2)
{
  return iterations + 1.0 - log(log2(abs(z)));
  break;
}

The normalized iteration count is now calculated as follows;

return iterations + 1.0 - log(log(abs(z))) / log(ESCAPE_RADIUS);

This is more generalized as I may choose to increase the escape radius (bailout radius to some) later.

This produces images like those below; 50 iterations to produce the image below. Zoomed in regions shown below need a higher iteration count Julia sets look like this;

Estimating distance to nearest mandelbrot surface

I'll be adding in the ability to draw the mandelbrot set using a distance estimation algorithm. Currently I have the following;

double Mandelbrot::getIterations(double x, double y)
{
    complex<double> c(x, y);
    complex<double> z = 0;
    complex<double> dz = 0;

    double iterations = 0;

    while (iterations < MAX_ITERATIONS)
    {
        dz = 2.0 * dz * z + 1.0;
        z = z * z + c;

        if (abs(z) > 10000)
        {
            return abs(abs(z) * log(abs(z)) / abs(dz));
        }
        iterations++;
    }
    return 0;
}

This produces a blurred plot but this can be alleviated using a large bailout radius and also more iterations

Julia sets

Also working on implementing julia set generation. Currently have the following;

double Mandelbrot::getJulia(double x, double y)
{
    complex<double> z(x,y);
    complex<double> c(-0.7269, 0.1889);

    double iterations = 0;

    while (iterations < MAX_ITERATIONS)
    {
        z = z * z + c;
        if (abs(z) > 2) {
            return iterations + 1.0 - log(log2(abs(z)));
            break;
        }
        iterations++;
    }
    return double(MAX_ITERATIONS);
}

And also the distance estimation

double Mandelbrot::getIterations(double x, double y)
{
  complex<double> c(x, y);
  complex<double> z = 0;
  complex<double> dz = 0;

  double iterations = 0;

  while(iterations < MAX_ITERATIONS)
  {
      dz = 2.0 * dz * z + 1.0;
      z = z * z + c;

      if(abs(z) > 2)
      {
        break;
      }
      iterations++;
  }
  return 0.5*log(abs(z)*abs(z)/abs(dz);
}

Sample images

These images are genearated using the frequence of the number of iteration e.g. if at pixle x,y it took n iterations, and the total number of pixels with n iterations is m, normalize m and use taht to color in the plot.

Hue[0,60], saturation 100%, value 100%

Wide RGB range in zoomed in region Hue[0,360], saturation 100%, value 100%

Hue[220,260], saturation 100%, value 100%. Gradually zooming in