柏林噪声（Perlin Noise）

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什么是柏林噪声？

1.固定一部分点的颜色。

2.“平滑”这些固定点之间的颜色。

3.用上面的方法生成几个不同频率的平滑噪音然后相加。

如何确定噪音的生成频率？只要改变固定点的个数就可以了，固定点多的频率就高，固定点少的频率就低。那么怎么平滑固定点之间的颜色？

perlin noise

步骤就变成了：

1.固定一部分点的gradient。

2″平滑”这些固定点中间的gradient。

3.用上面的方法生成几个不同频率的平滑噪音, 然后相加（准确来说这一步是属于分形噪声的部分）。

先上代码，之后一点点讲，这个代码是简化版的，只是为了帮助理解，真正用的时候还是要用官方的那篇代码，我把我写的代码的C++版放在了文章的末尾：

#include "pch.h" #include <iostream> #include <ctime> #include<fstream> using namespace std; struct vec3 { double x; double y; double z; vec3() {}; vec3(double x, double y,double z) :x(x), y(y),z(z) {} double operator*(vec3 t) { return x * t.x + y * t.y+z*t.z; } vec3 operator+(vec3 t) { return vec3(x + t.x, y + t.y,z+t.z); } }; class perlinNoise { public: vec3 g[12] = { vec3(1,1,0),vec3(-1,1,0),vec3(1,-1,0),vec3(-1,-1,0), vec3(1,0,1),vec3(-1,0,1),vec3(1,0,-1),vec3(-1,0,-1), vec3(0,1,1), vec3(0,-1,1),vec3(0,1,-1),vec3(0,-1,-1) }; vec3 vertex[25][25][25]; perlinNoise() { srand((int)time(0)); for (int i = 0; i < 25; i++) { for (int j = 0; j < 25; j++) { for (int k = 0; k < 25; k++) { int mrand = rand() % 12; vertex[i][j][k] = g[mrand]; } } } } double generateNoise(double x, double y,double z) { int X = (int)floor(x); int Y = (int)floor(y); int Z = (int)floor(z); double u = x - X; double v = y - Y; double w = z - Z; vec3 vec000 = vec3(u, v,w); vec3 vec010 = vec3(u, v, w) + vec3(0, -1, 0); vec3 vec100 = vec3(u, v, w) + vec3(-1, 0, 0); vec3 vec110 = vec3(u, v, w) + vec3(-1, -1, 0); vec3 vec001 = vec3(u, v, w) + vec3(0, 0, -1); vec3 vec011 = vec3(u, v, w) + vec3(0, -1, -1); vec3 vec101 = vec3(u, v, w) + vec3(-1, 0, -1); vec3 vec111 = vec3(u, v, w) + vec3(-1, -1, -1); double g000 = vec000 * vertex[X][Y][Z]; double g010 = vec010 * vertex[X][Y + 1][Z]; double g100 = vec100 * vertex[X + 1][Y][Z]; double g110 = vec110 * vertex[X + 1][Y + 1][Z]; double g001 = vec001 * vertex[X][Y][Z + 1]; double g011 = vec011 * vertex[X][Y + 1][Z + 1]; double g101 = vec101 * vertex[X + 1][Y][Z + 1]; double g111 = vec111 * vertex[X + 1][Y + 1][Z + 1]; u = fade(u); v = fade(v); w = fade(w); double lerpx1 = lerp(g000, g100, u); double lerpx2 = lerp(g010, g110, u); double lerpy1 = lerp(lerpx1, lerpx2, v); double lerpx3 = lerp(g001, g101, u); double lerpx4 = lerp(g011, g111, u); double lerpy2 = lerp(lerpx3, lerpx4, v); double lerpz = lerp(lerpy1, lerpy2, w); return lerpz; } double lerp(double a, double b, double t) { return a + t * (b - a); } double fade(double t) { return t * t * t * (t * (t * 6 - 15) + 10); } }; int main() { perlinNoise a; ofstream outfile("perlinNoise.ppm"); int X = 400, Y = 400; outfile << "P3" << endl << X << " " << Y << endl << "255" << endl; for (double i = 0; i < 20; i += 0.05) { for (double j = 0; j < 20; j += 0.05) { int temp = (a.generateNoise(i, j,7.) + 1.0) / 2.0*255.0; outfile << temp << " " << temp << " " << temp << " "; } } } 

下面这些就是已经定义好的随机向量，它们是由立方体12条棱的中点决定的，至于为什么这样做，《GPU精粹》上说这样生成的图像比随机产生固定向量的“污点”更少：

 vec3 g[12] = { vec3(1,1,0),vec3(-1,1,0),vec3(1,-1,0),vec3(-1,-1,0), vec3(1,0,1),vec3(-1,0,1),vec3(1,0,-1),vec3(-1,0,-1), vec3(0,1,1), vec3(0,-1,1),vec3(0,1,-1),vec3(0,-1,-1) }; 

这个构造函数就是为每个点固定一个梯度，由生成的随机数来确定 使用g[12]中的哪一个梯度，由于空间原因只定义了（0，0，0）到（25,25,25）的点，所以最后输入的点的大小不能超过25，不过官方的代码可以输入更大的值。

 vec3 vertex[25][25][25]; perlinNoise() { srand((int)time(0)); for (int i = 0; i < 25; i++) { for (int j = 0; j < 25; j++) { for (int k = 0; k < 25; k++) { int mrand = rand() % 12; vertex[i][j][k] = g[mrand]; } } } } 

 int X = (int)floor(x); int Y = (int)floor(y); double u = x - X; double v = y - Y; vec2 vec00 = vec2(u, v); vec2 vec01 = vec2(u, v) + vec2(0, -1); vec2 vec10 = vec2(u, v) + vec2(-1, 0); vec2 vec11 = vec2(u, v) + vec2(-1, -1); 

在三维空间中求出这8个向量：

int X = (int)floor(x); int Y = (int)floor(y); int Z = (int)floor(z); double u = x - X; double v = y - Y; double w = z - Z; vec3 vec000 = vec3(u, v,w); vec3 vec010 = vec3(u, v, w) + vec3(0, -1, 0); vec3 vec100 = vec3(u, v, w) + vec3(-1, 0, 0); vec3 vec110 = vec3(u, v, w) + vec3(-1, -1, 0); vec3 vec001 = vec3(u, v, w) + vec3(0, 0, -1); vec3 vec011 = vec3(u, v, w) + vec3(0, -1, -1); vec3 vec101 = vec3(u, v, w) + vec3(-1, 0, -1); vec3 vec111 = vec3(u, v, w) + vec3(-1, -1, -1); 

接着，对每个顶点的梯度向量和距离向量做点积运算，我们就可以得出每个顶点的影响值：

double g000 = vec000 * vertex[X][Y][Z]; double g010 = vec010 * vertex[X][Y + 1][Z]; double g100 = vec100 * vertex[X + 1][Y][Z]; double g110 = vec110 * vertex[X + 1][Y + 1][Z]; double g001 = vec001 * vertex[X][Y][Z + 1]; double g011 = vec011 * vertex[X][Y + 1][Z + 1]; double g101 = vec101 * vertex[X + 1][Y][Z + 1]; double g111 = vec111 * vertex[X + 1][Y + 1][Z + 1]; 

下面是平滑曲线的函数，最初平滑曲线是这样的 w(t)=3t² – 2t³ ，它保证了w(0)=0, w(1)=1.和w’(0) = 0和w’(1)=0，而新版的这个函数还保证了w’’(0) = 0和w’’(1)=0，总之就是更平滑了

 double fade(double t) { return t * t * t * (t * (t * 6 - 15) + 10); } 

三次方插值和五次方插值对比：
我们需要用fade函数来变换u，v，w的值来让插值更加平滑：

u = fade(u); v = fade(v); w = fade(w); 

 double lerpx1 = lerp(g000, g100, u); double lerpx2 = lerp(g010, g110, u); double lerpy1 = lerp(lerpx1, lerpx2, v); double lerpx3 = lerp(g001, g101, u); double lerpx4 = lerp(g011, g111, u); double lerpy2 = lerp(lerpx3, lerpx4, v); double lerpz = lerp(lerpy1, lerpy2, w); return lerpz; 

还有一个内存问题
要想获得更大的频率，我们就必须要有更多的固定点，所以高频率的噪音绝对是有必要的. 而为了储存固定点的gradient, 要建立一个数组才行. 而当噪音的频率非常高的时候, 由于需要的固定点会很多, 25X25X25显然是不够的，这个数组也会非常的大. 为了降低内存的使用, perlin使用了1个256个元素的哈希表. 也就是说, 预先找出合理的, 足够随机的256个gradient, 存在一个表里. 然后每次需要某个固定点的gradient值的时候, 通过这个这个点的坐标, 伪随机的在表里选出一个值. 对于3d的情况, 如果我们想要坐标(i,j,k)的gradient g(i,j,k),而P里预存储了256个gradient, 那么:

g(i, j, k) = P[ ( i + P[ (j + P[k]) mod 256 ] ) mod 256 ] 

 int 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)))); 

接下来是fbm的介绍（后面所有代码均以文末代码为基础写的）

这样做的结果是一个包含不同大小特性的函数。低频率函数的大幅度振动构成基本的形状，而高频率函数的小幅度振动构成小范围的细节信息。Perlin将不同倍程的噪声相加后的函数（每个噪声都比前一个倍程的噪声振幅减半）叫做1/f噪声，不过如今通常会使用分形布朗运动（fraction Brownian motion）或者fbm这个名词来描述它。

下面是perlin noise+fbm的代码

ImprovedNoise pNoise; int octaves = 4; ofstream outfile("分形噪声.ppm"); int xx=400, yy=400; outfile << "P3" << endl << xx << " " << yy << " " << endl << "255" << endl; for (double i = 0; i < 20; i+=0.05) { for (double j = 0; j < 20; j+=0.05) { //* double sum = 0,maxValue=0,frequency = 1, amplitude = 1; for (int k = 0; k < octaves; k++, frequency *= 2.0, amplitude *= 0.5) { sum += pNoise.noise(i*frequency, j*frequency, 7.415)*amplitude; maxValue += amplitude; } sum /= maxValue; int b = (sum+1)*255.0/2.0; outfile << b << " " << b << " " << b << endl; //* } } 

sum += abs(pNoise.noise(i*frequency, j*frequency, 7.415))*amplitude; 

double sum = 0frequency = 1, amplitude = 1; for (int k = 0; k < octaves; k++, frequency *= 2.0, amplitude *= 0.5) { sum += abs(pNoise.noise(i*frequency, j*frequency, 7.415))*amplitude; } sum = sin(sum + i * frequency / 16.0); int b = (sum+1)*255.0/2.0; outfile << b << " " << b << " " << b << endl; 

if (b < 50) { outfile << 255 << " " << b << " " << b << endl; } else if(b<100) { outfile << b << " " << 255 << " " << b << endl; } else if (b < 150) { outfile << 175 << " " << 175 << " " << b << endl; } else if (b < 200) { outfile << b << " " << 125 << " " << 125 << endl; } else { outfile << b << " " << b << " " << 255 << endl; } 

一些Perlin噪声产生美图，这些作品基本上都来自这位大神的文章
1.nimitz发明了一种对每层噪音添加旋转的方法，得到的图形看起来像翻滚的岩浆

2.除了操作噪音本身之外，还可以操作噪音所在的空间（坐标系）。

3.1/z变换，最简单的一种共形变换，1/z再进展到把模长也除掉就会得到星光状的图像。

Perlin Noise代码的C++版

#include "pch.h" #include <iostream> #include <cmath> #include<fstream> using namespace std; class ImprovedNoise { public : ImprovedNoise() { int permutation[512] = { 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 }; for (int i = 0; i < 256; i++) p[256 + i] = p[i] = permutation[i]; } double noise(double x, double y, double z) { int X = (int)floor(x) & 255, Y = (int)floor(y) & 255, Z = (int)floor(z) & 255; x -=floor(x); y -=floor(y); z -=floor(z); double u = fade(x), v = fade(y), w = fade(z); int 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)))); } double fade(double t) { return t * t * t * (t * (t * 6 - 15) + 10); } double lerp(double t, double a, double b) { return a + t * (b - a); } double grad(int hash, double x, double y, double z) { int h = hash & 15; double u = h < 8 ? x : y, v = h < 4 ? y : h == 12 || h == 14 ? x : z; return ((h & 1) == 0 ? u : -u) + ((h & 2) == 0 ? v : -v); } int p[512]; }; int main() { ImprovedNoise pNoise; int octaves = 4; ofstream outfile("noise.ppm"); int xx=400, yy=400; outfile << "P3" << endl << xx << " " << yy << " " << endl << "255" << endl; for (double i = 0; i < 20; i+=0.05) { for (double j = 0; j < 20; j+=0.05) { double sum = pNoise.noise(i, j, 7.415); int b = (sum + 1)*255.0 / 2.0; outfile << b << " " << b << " " << b << endl; } } //cout << count; return 0; }