柏林噪声

Perlin Noise是Ken Perlin在1983年开发的一种梯度噪音，这是一种用于在计算机生成的表面上产生自然出现纹理的技术，使用Perlin噪声合成的纹理通常用于CGI，通过模仿自然界中纹理的受控随机外观，使计算机生成的视觉元素（如物体表面，火焰，烟雾或云）看起来更自然。–Wiki

理论知识链接

噪音 - Perlin
一篇文章搞懂柏林噪声算法，附代码讲解
http://libnoise.sourceforge.net/index.html
不只是噪声，更是数学美 —浅谈Perlin Noise（理论知识完美）
不只是噪音–知乎（写的超棒！！）

代码实现(Python)

由Ken Perlin 2002年原始JAVA代码改编而来

#-*- coding:utf8 -*-

# PYTHON REFERENCE IMPLEMENTATION OF IMPROVED NOISE - COPYRIGHT 2002 KEN PERLIN.
import math
import numpy as np

def PerlinNoise(x,y,z, octaves=6, persistence=0.5):
    # Sum of Noise Function = Perlin Noise
    # Each successive noise function you add is known as an octave
    total = 0
    p = persistence  # reference value:  1/4, 1/2 ,3/4 
    for i in range(octaves):
        frequency=2**i
        amplitude=p**i
        octave=ImprovedNoise(x * frequency, y * frequency, z * frequency) * amplitude
        total+=octave
    return total

def ImprovedNoise(x, y, z):
    # frequency=1/wavelength
    # It returns floating point numbers between -1.0 and 1.0
    # FIND UNIT CUBE THAT CONTAINS POINT.
    X = int(math.floor(x)) & 255
    Y = int(math.floor(y)) & 255
    Z = int(math.floor(z)) & 255

    # FIND RELATIVE X,Y,Z OF POINT IN CUBE.
    x -= math.floor(x)
    y -= math.floor(y)
    z -= math.floor(z)

    # COMPUTE FADE CURVES FOR EACH OF X,Y,Z.
    u,v,w = fade(x),fade(y),fade(z)

    # HASH COORDINATES OF THE 8 CUBE CORNERS
    # AND ADD BLENDED RESULTS FROM  8 CORNERS OF CUBE
    A = p[X  ]+Y; AA = p[A]+Z; AB = p[A+1]+Z;
    B = p[X+1]+Y; BA = p[B]+Z; BB = p[B+1]+Z;
    return lerp(w, lerp(v, lerp(u, grad(p[AA  ], x  , y  , z   ),
                                   grad(p[BA  ], x-1, y  , z   )),
                           lerp(u, grad(p[AB  ], x  , y-1, z   ),
                                   grad(p[BB  ], x-1, y-1, z   ))),
                   lerp(v, lerp(u, grad(p[AA+1], x  , y  , z-1 ),
                                   grad(p[BA+1], x-1, y  , z-1 )),
                           lerp(u, grad(p[AB+1], x  , y-1, z-1 ),
                                   grad(p[BB+1], x-1, y-1, z-1 ))))

def fade(t):
    return t * t * t * (t * (t * 6 - 15) + 10)

def lerp(t, a, b):
    return a + t * (b - a)

def grad(hash, x, y, z):
    # CONVERT LO 4 BITS OF HASH CODE INTO 12 GRADIENT DIRECTIONS.
    h = hash & 15
    u = x if h < 8 else y
    v = y if h < 4 else x if h==12 or h==14 else z

    return (u if (h&1)==0 else -u)+(v if (h&2)==0 else -v)


permutation = [ 151,160,137,91,90,15,
131,13,201,95,96,53,194,233,7,225,140,36,103,30,69,142,8,99,37,240,21,10,23,
190, 6,148,247,120,234,75,0,26,197,62,94,252,219,203,117,35,11,32,57,177,33,
88,237,149,56,87,174,20,125,136,171,168, 68,175,74,165,71,134,139,48,27,166,
77,146,158,231,83,111,229,122,60,211,133,230,220,105,92,41,55,46,245,40,244,
102,143,54, 65,25,63,161, 1,216,80,73,209,76,132,187,208, 89,18,169,200,196,
135,130,116,188,159,86,164,100,109,198,173,186, 3,64,52,217,226,250,124,123,
5,202,38,147,118,126,255,82,85,212,207,206,59,227,47,16,58,17,182,189,28,42,
223,183,170,213,119,248,152, 2,44,154,163, 70,221,153,101,155,167, 43,172,9,
129,22,39,253, 19,98,108,110,79,113,224,232,178,185, 112,104,218,246,97,228,
251,34,242,193,238,210,144,12,191,179,162,241, 81,51,145,235,249,14,239,107,
49,192,214, 31,181,199,106,157,184, 84,204,176,115,121,50,45,127, 4,150,254,
138,236,205,93,222,114,67,29,24,72,243,141,128,195,78,66,215,61,156,180
]
p=permutation*2

Perlin noise library for Python

该包旨在为您提供简单易用的快速函数，用于在Python程序中生成Perlin噪声。

安装

安装前先下载微软开发环境
https://visualstudio.microsoft.com/downloads/
找到 Other Tools and Frameworks 点开，
下载 Microsoft Visual C++ Redistributable for Visual Studio 2017安装即可。

pip安装

pip install noise

本地安装
GitHub下载源文件，然后运行

python setup.py install

安装错误： Failed building wheel for noise
https://www.lfd.uci.edu/~gohlke/pythonlibs/#noise
下载对应版本，安装解析环境，cp后面是Python的版本号

noise-1.2.2-cp36-cp36m-win32.whl
noise-1.2.2-cp36-cp36m-win_amd64.whl
noise-1.2.2-cp37-cp37m-win32.whl
noise-1.2.2-cp37-cp37m-win_amd64.whl

pip install C:\Users\Admin\Anaconda3\Scripts\noise-1.2.2-cp36-cp36m-win_amd64.whl

示例代码

#-*- coding:utf8 -*-
from noise import pnoise2,pnoise3
import numpy  as np
from  numpy import sin,cos,arccos,arcsin,sqrt
import pandas as pd
import random

def timer(func):
    import datetime
    from functools import wraps

    @wraps(func)
    def decorated(*args, **kwargs):
        starttime = datetime.datetime.now()
        res=func(*args, **kwargs)
        endtime = datetime.datetime.now()
        print('time used {} sec'.format((endtime - starttime).seconds))
        return res
    return decorated

class Sphere:

    def __init__(self,radius):
        self.radius=radius

    def coord_trans(self,longitude,latitude,radius=None):
        '''
        coordinate transformation
        Cartesian coordinates
        '''
        lat,lon=latitude,longitude
        if radius is None:
            r=self.radius
        else:
            r=radius

        x = r * cos(lat) * cos(lon)
        y = r * cos(lat) * sin(lon)
        z = r * sin(lat)
        return x,y,z

    def distance(self,lon,lat,angle='degrees',radius=None):
        if angle=='degrees':
            lon,lat=np.radians([lon,lat])

        a1,a2=lat
        b1,b2=lon
        if radius is None:
            r1=r2 =self.radius
        elif isinstance(radius,(int,float)):
            r1=r2 = radius
        else:
            r1,r2,*_=radius

        'Cartesian distance'
        tmp = cos(a1) * cos(a2) * cos(b1 - b2) + sin(a1) * sin(a2)
        L = sqrt(r1 ** 2 + r2 ** 2 - 2 * r1 * r2 * tmp)

        if r1==r2:
            'spherical distance'
            S = r1 * arccos(tmp)
            return L,S
        else:
            return L,None

    @timer
    def create_sphere(self,unit=0.1,multiplier=1.0,stretch=1.0,
                      seed=None,*args,**kwargs):
        '''
        Spherical coordinate system (lon,lat,r)
        0 <= r < math.inf
        0 <= lon <= PI * 2
        -PI / 2 <= lat <= PI / 2
        '''

        lon=np.arange(-180,180,unit)
        lat=np.arange(-90,90+unit,unit)

        'coordinate transformation'
        # coord = pd.MultiIndex.from_product([lon, lat], names=['lon', 'lat'])
        # coord = pd.DataFrame(index=coord).reset_index()
        lon1,lat1=np.meshgrid(lon,lat)
        x, y, z = self.coord_trans(np.radians(lon1), np.radians(lat1), radius=stretch)

        'default arguments'
        if seed is None:
            seed=random.randint(0,256)
        else:
            seed=int(seed)

        'Define numpy ufunc(universal function)'
        ufunc_pnoise3=lambda x,y,z:pnoise3(x, y, z,base=seed,*args,**kwargs)
        self.ufunc_pnoise3=np.frompyfunc(ufunc_pnoise3, 3, 1)

        h = self.ufunc_pnoise3(x,y,z)
        # h=pd.pivot(coord.lat, coord.lon, h)

        print('seed={}'.format(seed))
        return lon,lat,h

    def draw_sphere(self,lon,lat,h,map='ellipse'):
        import matplotlib.pyplot as plt
        h1=h.copy()
        h2=h.copy()
        h1[h<0]=None
        h2[h>0]=None

        fig=plt.figure()
        if map=='ellipse':
            lon,lat=np.meshgrid(lon,lat)
            lon=sqrt(1-(lat/90)**2)*lon
            plt.contourf(lon, lat, h1, cmap='Greens_r')
            plt.contourf(lon, lat, h2, cmap='Blues_r')
        elif map=='cosine':
            lon,lat=np.meshgrid(lon,lat)
            lon=cos(np.radians(lat))*lon
            plt.contourf(lon, lat, h1, cmap='Greens_r')
            plt.contourf(lon, lat, h2, cmap='Blues_r')
        else:
            plt.contourf(lon, lat, h1, cmap='Greens_r')
            plt.contourf(lon, lat, h2, cmap='Blues_r')
        plt.show()

earth=Sphere(radius=2)
lon,lat,h=earth.create_sphere(octaves=10, persistence=0.5,multiplier=2,seed=57)
earth.draw_sphere(lon,lat,h,map='ellipse')