How to make a smoother Perlin noise generator?

Question

I try to use the Perlin noise generator to make map tiles, but I notice that my noise is too wide, I mean that it has too many elevations and there are no flat places, and they don't seem like mountains, islands, lakes or what something else; they seem too random and with lots of peaks.

At the end of the question there are changes necessary to correct it.

Important code for the question:

1D:

def Noise(self, x):     # I wrote this noise function but it seems too random
    random.seed(x)
    number = random.random()
    if number < 0.5:
        final = 0 - number * 2
    elif number > 0.5:
        final = number * 2
    return final

 def Noise(self, x):     # I found this noise function on the internet
    x = (x<<13) ^ x
    return ( 1.0 - ( (x * (x * x * 15731 + 789221) + 1376312589) & 0x7fffffff) / 1073741824.0)

2D:

def Noise(self, x, y):     # I wrote this noise function but it seems too random
    n = x + y
    random.seed(n)
    number = random.random()
    if number < 0.5:
        final = 0 - number * 2
    elif number > 0.5:
        final = number * 2
    return final

def Noise(self, x, y):     # I found this noise function on the internet
    n = x + y * 57
    n = (n<<13) ^ n
    return ( 1.0 - ( (x * (x * x * 15731 + 789221) + 1376312589) & 0x7fffffff) / 1073741824.0)

I left Perlin in my code for 1D and 2D noise because maybe someone is interested in it: (It took me a long time to find the code, so I think someone would be happy to find an example here). You do not need Matplotlib or NumPy to make noise; I use them only to create a graph and a better result.

import random
import matplotlib.pyplot as plt              # To make graphs
from mpl_toolkits.mplot3d import Axes3D      # To make 3D graphs
import numpy as np                           # To make graphs

class D():     # Base of classes D1 and D2
    def Cubic_Interpolate(self, v0, v1, v2, v3, x):
        P = (v3 - v2) - (v0 - v1)
        Q = (v0 - v1) - P
        R = v2 - v0
        S = v1
        return P * x**3 + Q * x**2 + R * x + S

class D1(D):
    def __init__(self, lenght, octaves):
        self.result = self.Perlin(lenght, octaves)

    def Noise(self, x):     # I wrote this noise function but it seems too random
        random.seed(x)
        number = random.random()
        if number < 0.5:
            final = 0 - number * 2
        elif number > 0.5:
            final = number * 2
        return final

    def Noise(self, x):     # I found this noise function on the internet
        x = (x<<13) ^ x
        return ( 1.0 - ( (x * (x * x * 15731 + 789221) + 1376312589) & 0x7fffffff) / 1073741824.0)

    def Perlin(self, lenght, octaves):
        result = []
        for x in range(lenght):
            value = 0
            for y in range(octaves):
                frequency = 2 ** y
                amplitude = 0.25 ** y            
                value += self.Interpolate_Noise(x * frequency) * amplitude
            result.append(value)
            print(f"{x} / {lenght} ({x/lenght*100:.2f}%): {round(x/lenght*10) * '#'} {(10-round(x/lenght*10)) * ' '}. Remaining {lenght-x}.")     # I don't use `os.system('cls')` because it slow down the code.
        return result

    def Smooth_Noise(self, x):
        return self.Noise(x) / 2 + self.Noise(x-1) / 4 + self.Noise(x+1) / 4

    def Interpolate_Noise(self, x):
        round_x = round(x)
        frac_x  = x - round_x
        v0 = self.Smooth_Noise(round_x - 1)
        v1 = self.Smooth_Noise(round_x)
        v2 = self.Smooth_Noise(round_x + 1)
        v3 = self.Smooth_Noise(round_x + 2)
        return self.Cubic_Interpolate(v0, v1, v2, v3, frac_x)

    def graph(self, *args):
        plt.plot(np.array(self.result), '-', label = "Line")
        for x in args:
            plt.axhline(y=x, color='r', linestyle='-')    
        plt.xlabel('X')
        plt.ylabel('Y')
        plt.title("Simple Plot")
        plt.legend()
        plt.show()

class D2(D):
    def __init__(self, lenght, octaves = 1):

        self.lenght_axes = round(lenght ** 0.5)
        self.lenght = self.lenght_axes ** 2

        self.result = self.Perlin(self.lenght, octaves)

    def Noise(self, x, y):     # I wrote this noise function but it seems too random
        n = x + y
        random.seed(n)
        number = random.random()
        if number < 0.5:
            final = 0 - number * 2
        elif number > 0.5:
            final = number * 2
        return final

    def Noise(self, x, y):     # I found this noise function on the internet
        n = x + y * 57
        n = (n<<13) ^ n
        return ( 1.0 - ( (x * (x * x * 15731 + 789221) + 1376312589) & 0x7fffffff) / 1073741824.0)

    def Smooth_Noise(self, x, y):
        corners = (self.Noise(x - 1, y - 1) + self.Noise(x + 1, y - 1) + self.Noise(x - 1, y + 1) + self.Noise(x + 1, y + 1) ) / 16
        sides   = (self.Noise(x - 1, y) + self.Noise(x + 1, y) + self.Noise(x, y - 1)  + self.Noise(x, y + 1) ) /  8
        center  =  self.Noise(x, y) / 4
        return corners + sides + center

    def Interpolate_Noise(self, x, y):

        round_x = round(x)
        frac_x  = x - round_x

        round_y = round(y)
        frac_y  = y - round_y

        v11 = self.Smooth_Noise(round_x - 1, round_y - 1)
        v12 = self.Smooth_Noise(round_x    , round_y - 1)
        v13 = self.Smooth_Noise(round_x + 1, round_y - 1)
        v14 = self.Smooth_Noise(round_x + 2, round_y - 1)
        i1 = self.Cubic_Interpolate(v11, v12, v13, v14, frac_x)

        v21 = self.Smooth_Noise(round_x - 1, round_y)
        v22 = self.Smooth_Noise(round_x    , round_y)
        v23 = self.Smooth_Noise(round_x + 1, round_y)
        v24 = self.Smooth_Noise(round_x + 2, round_y)
        i2 = self.Cubic_Interpolate(v21, v22, v23, v24, frac_x)

        v31 = self.Smooth_Noise(round_x - 1, round_y + 1)
        v32 = self.Smooth_Noise(round_x    , round_y + 1)
        v33 = self.Smooth_Noise(round_x + 1, round_y + 1)
        v34 = self.Smooth_Noise(round_x + 2, round_y + 1)
        i3 = self.Cubic_Interpolate(v31, v32, v33, v34, frac_x)

        v41 = self.Smooth_Noise(round_x - 1, round_y + 2)
        v42 = self.Smooth_Noise(round_x    , round_y + 2)
        v43 = self.Smooth_Noise(round_x + 1, round_y + 2)
        v44 = self.Smooth_Noise(round_x + 2, round_y + 2)
        i4 = self.Cubic_Interpolate(v41, v42, v43, v44, frac_x)

        return self.Cubic_Interpolate(i1, i2, i3, i4, frac_y)

    def Perlin(self, lenght, octaves):
        result = []
        for x in range(lenght):
            value = 0
            for y in range(octaves):
                frequency = 2 ** y
                amplitude = 0.25 ** y            
                value += self.Interpolate_Noise(x * frequency, x * frequency) * amplitude
            result.append(value)
            print(f"{x} / {lenght} ({x/lenght*100:.2f}%): {round(x/lenght*10) * '#'} {(10-round(x/lenght*10)) * ' '}. Remaining {lenght-x}.")     # I don't use `os.system('cls')` because it slow down the code.
        return result

    def graph(self, color = 'viridis'):
        # Other colors: https://matplotlib.org/examples/color/colormaps_reference.html
        fig = plt.figure()
        Z = np.array(self.result).reshape(self.lenght_axes, self.lenght_axes)

        ax = fig.add_subplot(1, 2, 1, projection='3d')
        X = np.arange(self.lenght_axes)
        Y = np.arange(self.lenght_axes)
        X, Y = np.meshgrid(X, Y)        
        d3 = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=color, linewidth=0, antialiased=False)
        fig.colorbar(d3)

        ax = fig.add_subplot(1, 2, 2)
        d2 = ax.imshow(Z, cmap=color, interpolation='none')
        fig.colorbar(d2)

        plt.show()

, .

, :

test = D2(1000, 3)
test.graph()

- .

, 2D- , , 1D :

test = D1(1000, 3)
test.graph()

, . - .

- :

:

P.S: .

EDIT:

Pikalek:

/ .

geza:

geza :

def Perlin(self, lenght_axes, octaves, zoom = 0.01, amplitude_base = 0.5):
    result = []

    for y in range(lenght_axes):
        line = []
        for x in range(lenght_axes):
            value = 0
            for o in range(octaves):
                frequency = 2 ** o
                amplitude = amplitude_base ** o
                value += self.Interpolate_Noise(x * frequency * zoom, y * frequency * zoom) * amplitude
            line.append(value)
        result.append(line)
        print(f"{y} / {lenght_axes} ({y/lenght_axes*100:.2f}%): {round(y/lenght_axes*20) * '#'} {(20-round(y/lenght_axes*20)) * ' '}. Remaining {lenght_axes-y}.")
    return result

:

• Z = np.array(self.result) `Z = np.array(self.result).reshape(self.lenght_axes, self.lenght_axes)` ` .
• math.floor() ( import math) round() Interpolate_Noise round_x round_y.
• return Noise () return ( 1.0 - ( (n * (n * n * 15731 + 789221) + 1376312589) & 0x7fffffff) / 1073741824.0). D2(10000, 10) , , , () , , Noise.

Answer 1

:

• Interpolate_Noise "" (, x 0.01). 1D, ,
• 3 (3 )
• 0.5 ^ , 0.25 ^ ( , 0.25 , )
• 2D- 2 ( . , , ). "" , x x.
• . Perlin .
• 2D- : x n
• round math.floor.

, (++) Perlin-like ( perlin) : fooobar.com/questions/1021094/...

Answer 2

. - Matrix Convolution. , , "", , , . :

0.1 0.1 0.1
0.1 0.2 0.1
0.1 0.1 0.1

, :

2 4 1 3 5
3 5 1 2 3
4 9 2 1 2
3 4 9 5 2
1 1 3 6 7

, 2, :

. . . . .
. 5 1 2 .       0.1 0.1 0.1       0.5 0.1 0.2
. 9 2 1 .   x   0.1 0.2 0.1   =   0.9 0.4 0.1
. 4 9 5 .       0.1 0.1 0.1       0.4 0.9 0.5
. . . . .

, , 0.5+0.1+0.2+0.9+0.4+0.1+0.4+0.9+0.5 = 4, :

? ? ? ? ?
? ? ? ? ?
? ? 4 ? ?
? ? ? ? ?
? ? ? ? ?

... , , . , .

, . , ​​ 5x5 9x9, .

: ​​ , 1, ( , , a > 1, , ). 5x5 :

0.010 0.024 0.050 0.024 0.010
0.024 0.050 0.062 0.050 0.024
0.050 0.062 0.120 0.062 0.050
0.024 0.050 0.062 0.050 0.024
0.010 0.024 0.050 0.024 0.010

- ; . :.

1  4  16 4  1                    0.002808989    0.011235955 0.04494382  0.011235955 0.002808989
4  16 32 16 4                    0.011235955    0.04494382  0.08988764  0.04494382  0.011235955
16 32 64 32 16  (sum = 356) -->  0.04494382     0.08988764  0.179775281 0.08988764  0.04494382
4  16 32 16 4                    0.011235955    0.04494382  0.08988764  0.04494382  0.011235955
1  4  16 4  1                    0.002808989    0.011235955 0.04494382  0.011235955 0.002808989