Shaded & Power Normalized Rendering

Die Wiedergabe der Mandelbrot-Menge kann verbessert werden, indem eine normalisierte Neuzählung verwendet wird, die mit einer leistungsnormalisierten Farbabbildung (Gamma = 0,3) verbunden ist. Die Wiedergabe kann dank Schattierung weiter verbessert werden.

Das maxitergibt die Genauigkeit der Berechnung an. maxiter=200sollte auf den meisten modernen Laptops einige Sekunden dauern.

import numpy as np


def mandelbrot_set(xmin, xmax, ymin, ymax, xn, yn, maxiter, horizon=2.0):
    X = np.linspace(xmin, xmax, xn).astype(np.float32)
    Y = np.linspace(ymin, ymax, yn).astype(np.float32)
    C = X + Y[:, None] * 1j
    N = np.zeros_like(C, dtype=int)
    Z = np.zeros_like(C)
    for n in range(maxiter):
        I = abs(Z) < horizon
        N[I] = n
        Z[I] = Z[I]**2 + C[I]
    N[N == maxiter-1] = 0
    return Z, N


if __name__ == '__main__':
    import time
    import matplotlib
    from matplotlib import colors
    import matplotlib.pyplot as plt

    xmin, xmax, xn = -2.25, +0.75, 3000 // 2
    ymin, ymax, yn = -1.25, +1.25, 2500 // 2
    maxiter = 200
    horizon = 2.0 ** 40
    log_horizon = np.log2(np.log(horizon))
    Z, N = mandelbrot_set(xmin, xmax, ymin, ymax, xn, yn, maxiter, horizon)

    # Normalized recount as explained in:
    # https://linas.org/art-gallery/escape/smooth.html
    # https://web.archive.org/web/20160331171238/https://www.ibm.com/developerworks/community/blogs/jfp/entry/My_Christmas_Gift?lang=en

    # This line will generate warnings for null values but it is faster to
    # process them afterwards using the nan_to_num
    with np.errstate(invalid='ignore'):
        M = np.nan_to_num(N + 1 - np.log2(np.log(abs(Z))) + log_horizon)

    dpi = 72
    width = 10
    height = 10*yn/xn
    fig = plt.figure(figsize=(width, height), dpi=dpi)
    ax = fig.add_axes([0, 0, 1, 1], frameon=False, aspect=1)

    # Shaded rendering
    light = colors.LightSource(azdeg=315, altdeg=10)
    M = light.shade(M, cmap=plt.cm.hot, vert_exag=1.5,
                    norm=colors.PowerNorm(0.3), blend_mode='hsv')
    ax.imshow(M, extent=[xmin, xmax, ymin, ymax], interpolation="bicubic")
    ax.set_xticks([])
    ax.set_yticks([])

    # Some advertisement for matplotlib
    year = time.strftime("%Y")
    text = ("The Mandelbrot fractal set\n"
            "Rendered with matplotlib %s, %s - https://matplotlib.org"
            % (matplotlib.__version__, year))
    ax.text(xmin+.025, ymin+.025, text, color="white", fontsize=12, alpha=0.5)

    plt.show()