柏林噪聲
隨機噪聲
如上圖所示隨機噪聲沒有任何規律可言，我們希望生成有一些意義的局部連續的隨機圖案
一維柏林噪聲
假設希望生成一段局部連續的隨機曲線，可以采用插值的方式：在固定點隨機分配y值（一般是整數點），其他的點使用插值算法
方法一：線性插值的方式
公式如下：
【數字圖像處理】二維（2D）線性插值的應用
y = a*y1 + (1-a)*y2
我們畫圖看看：
import math
import numpy as np
import matplotlib.pyplot as pltperm = [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]def perlin1D(x):# 整數x1和x2的坐標x1 = math.floor(x)x2 = x1 + 1# x1和x2的梯度值grad1 = perm[x1 % 255] * 2.0 - 255.0grad2 = perm[x2 % 255] * 2.0 - 255.0#x1和x2指向x的方向向量vec1 = x - x1vec2 = x - x2# x到x1的距離即vec1，利用公式3計算平滑參數t = 3 * pow(vec1, 2) - 2 * pow(vec1, 3)#梯度值與方向向量的乘積product1 = grad1 * vec1product2 = grad2 * vec2return product1 + t * (product2 - product1)def linear1D(x):# 整數x1和x2的坐標x1 = math.floor(x)x2 = x1 + 1# y值 隨機數grad1 = perm[x1 % 255] * 2.0 - 255.0grad2 = perm[x2 % 255] * 2.0 - 255.0t=x - x1return grad1 + t * (grad2 - grad1)def linear1D_plus(x):# 整數x1和x2的坐標x1 = math.floor(x)x2 = x1 + 1# x1和x2的梯度值grad1 = perm[x1 % 255] * 2.0 - 255.0grad2 = perm[x2 % 255] * 2.0 - 255.0#x1和x2指向x的方向向量vec1 = x - x1vec2 = x - x2t=x - x1#梯度值與方向向量的乘積product1 = grad1 * vec1product2 = grad2 * vec2return product1 + t * (product2 - product1)def draw1D():# 繪制散點圖x0=[]y0=[]for i in range(11):x0.append(i)y0.append( perm[i] * 2.0 - 255.0)plt.scatter(x0, y0,color='red')# 繪制1D的圖像x = np.linspace(0, 10, 100)y = np.zeros(100)y1 = np.zeros(100)y2 = np.zeros(100)for i in range(100):y[i] = perlin1D(x[i])y1[i] = linear1D(x[i])y2[i] =linear1D_plus(x[i])# 繪制圖像plt.plot(x, y,color='deepskyblue')plt.plot(x, y1,color='green')plt.plot(x, y2,color='orange')plt.show()draw1D()線性插值
x取[0,10]這個區間，y在整數點隨機取值，非整數點使用線性插值
ps 隨機值使用偽隨機數perm，柏林噪聲在圖像領域使用，顏色的取值范圍是[0,255]，所以perm的值也是[0,255]
上圖紅色的點是：整數點隨機取值的結果，綠色的線是線性插值。
y = t ? y 2 + ( 1 ? t ) ? y 1 = y 1 + t ( y 2 ? y 1 ) y = t*y2+ (1-t)*y1 = y1 + t(y2- y1 ) y=t?y2+(1?t)?y1=y1+t(y2?y1)
t = x ? x 1 t=x-x1 t=x?x1
線性插值plus
我們希望它更平滑一點，如果插值點x的值y與附近點x1,x2的位置相關
所以改進上述算法：
y = t ? y 2 ? w 2 + ( 1 ? t ) ? y 1 ? w 1 = y 1 ? w 1 + t ( y 2 ? w 2 ? y 1 ? w 1 ) y = t*y2*w2 + (1-t)*y1*w1 = y1*w1 + t(y2*w2 - y1*w1 ) y=t?y2?w2+(1?t)?y1?w1=y1?w1+t(y2?w2?y1?w1)
w是權重系數，也是柏林算法中的方向向量vec1 = x - x1
如圖中黃色的線
柏林噪聲
柏林噪聲在此基礎上再加強一步：
t = 3 t 2 ? 2 t 3 t=3t^2 -2t^3 t=3t2?2t3
算法步驟：
input: x
- 計算周圍的點：x1 , x2
- 計算x1 , x2 梯度 : grad1， grad2 隨機取[0,255]
- 方向向量: (vec1 =x-x1 ;vec2 = x-x2)
- 梯度值與方向向量的乘積 product=grad*vec
- 計算系數 t=3t^2 -2t^3
- 插值：y = product1 + t * (product2 - product1)
output :y
根據上述原理 可以畫一個不規則的圓形
def drawCircle():#畫圓形# 創建一個坐標系fig, ax = plt.subplots()# 定義圓心和半徑center = (0, 0)radius = 10# 生成圓的數據theta = np.linspace(0, 2*np.pi, 100)x = radius * np.cos(theta) + center[0]y = radius * np.sin(theta) + center[1]y1 = np.zeros(100)for i in range(100):y1[i] = y[i]+ perlin1D(theta[i]*5)/255*2# 畫出圓形ax.plot(x, y,color='orange')ax.plot(x, y1,color='deepskyblue')# 設置坐標軸范圍ax.set_xlim([-15, 15])ax.set_ylim([-15, 15])# 顯示圖像plt.show()
二維柏林噪聲
頭文件
#pragma once
#include<array>
class PerlinNoise2D
{
public:PerlinNoise2D();~PerlinNoise2D();float BasePerlinNoise2D(float x , float y); //輸出數值的范圍應該是[-1,1]float Octave2D_01(float x, float y, int octaves, float persistence = 0.5);//輸出數值的范圍限定在[0,1]
private:template<typename STLIterator>inline void shuffle(STLIterator begin, STLIterator end);/*生成最大值為max的隨機數*/int random(int max);/*input: 方向向量(x,y) 哈希值 hash根據哈希值可以達到隨機梯度輸出：隨機梯度與方向向量的乘積*/inline float Grad(int hash, float x, float y);inline float Grad2(int hash, float x, float y);inline float Fade(const float t);inline float Lerp(const float a, const float b, const float t);inline float RemapClamp_01(float x);float Octave2D(float x, float y, int octaves, float persistence = 0.5);
private:std::array<int, 256> m_permutation;};cpp
#include "PerlinNoise2D.h"
#include <random>
#include <numeric>
PerlinNoise2D::PerlinNoise2D()
{std::iota(m_permutation.begin(), m_permutation.end(), 0);shuffle(m_permutation.begin(), m_permutation.end());
}PerlinNoise2D::~PerlinNoise2D()
{}int PerlinNoise2D::random(int max)
{return (std::random_device{}() % (max)+1);
}/*
洗牌算法
*/
template<typename STLIterator>
inline void PerlinNoise2D::shuffle(STLIterator begin, STLIterator end)
{if (begin == end){return;}using difference_type = typename std::iterator_traits<STLIterator>::difference_type;for (STLIterator it = begin + 1; it < end; ++it){int n = random(static_cast<int>(it - begin));std::iter_swap(it, begin + static_cast<difference_type>(n));}}inline float PerlinNoise2D::Grad(int hash, float x, float y)
{float z = 0.34567;switch (hash & 0xF){case 0x0: return x + y;case 0x1: return -x + y;case 0x2: return x - y;case 0x3: return -x - y;case 0x4: return x + z;case 0x5: return -x + z;case 0x6: return x - z;case 0x7: return -x - z;case 0x8: return y + z;case 0x9: return -y + z;case 0xA: return y - z;case 0xB: return -y - z;case 0xC: return y + x;case 0xD: return -y + z;case 0xE: return y - x;case 0xF: return -y - z;default: return 0; // never happens}
}inline float PerlinNoise2D::Grad2(int hash, float x, float y)
{const double PI = 3.14159265359;const int numPoints = 36;hash = hash % numPoints;double angle = 2 * PI * hash / numPoints;double gradx = cos(angle);double grady = sin(angle);return gradx * x + grady*y;
}inline float PerlinNoise2D::Fade(const float t)
{return t * t * t * (t * (t * 6 - 15) + 10);
}inline float PerlinNoise2D::Lerp(const float a, const float b, const float t)
{return (a + (b - a) * t);
}
float PerlinNoise2D::BasePerlinNoise2D(float x, float y)
{//得到周圍四個點int _x = std::floor(x);int _y = std::floor(y);int ix = _x & 255;int iy = _y & 255;//hash函數得到隨機索引值int AA = m_permutation[(m_permutation[ix & 255] + iy) & 255];int BA = m_permutation[(m_permutation[(ix + 1) & 255] + iy) & 255];int AB = m_permutation[(m_permutation[ix & 255] + iy+1) & 255];int BB = m_permutation[(m_permutation[ix+1 & 255] + iy+1) & 255];//根據索引值 得到方向向量和隨機梯度的向量積float fx = x - _x;float fy = y - _y;float g1 = Grad2(AA,fx,fy);float g2 = Grad2(BA, fx - 1, fy);float g3 = Grad2(AB, fx, fy - 1);float g4 = Grad2(BB, fx - 1, fy - 1);//插值float u = Fade(fx);float v = Fade(fy);float x1 = Lerp(g1, g2, u);float x2 = Lerp(g3, g4, u);return Lerp(x1, x2, v);
}float PerlinNoise2D::Octave2D(float x, float y, int octaves, float persistence)
{float result = 0.0;float amplitude = 1.0;for (int i = 0; i < octaves; i++){result += (BasePerlinNoise2D(x, y) * amplitude);x *= 2;y *= 2;amplitude *= persistence;}return result;
}inline float PerlinNoise2D::RemapClamp_01(float x)
{if (x <= -1.0) {return 0.0;}else if (1.0 <= x){return 1.0;}return (x * 0.5 + 0.5);
}float PerlinNoise2D::Octave2D_01(float x, float y, int octaves, float persistence )
{return RemapClamp_01(Octave2D(x,y, octaves));
}
測試
class PerlinNoiseTest
{
public:PerlinNoiseTest() {};~PerlinNoiseTest() {};void drawImage(float frequency=4.0, int octaves=2,int width = 400, int height = 400);
};#include "PerlinNoiseTest.h"
#include "PerlinNoise2D.h"
#include <opencv2/opencv.hpp>
# include <algorithm>
#include <iostream>
using namespace cv;
void PerlinNoiseTest::drawImage(float frequency, int octaves, int width , int height )
{// 創建一個空白圖像cv::Mat image(height, width, CV_8UC3, cv::Scalar(255, 255, 255));frequency = std::clamp((double)frequency, 0.1, 64.0);const double fx = (frequency / width);const double fy = (frequency / height);int maxcolor = 0;PerlinNoise2D perlin;for (std::int32_t y = 0; y < height; ++y){for (std::int32_t x = 0; x < width; ++x){int color = 255*perlin.Octave2D_01((x * fx), (y * fy), octaves);maxcolor = max(maxcolor , color);image.at<cv::Vec3b>(y, x) = cv::Vec3b(color, color, color); // 繪制像素點}}std::cout << "maxcolor: "<< maxcolor;imshow("Generated Texture", image);imwrite("D:\\code\\noise\\image\\PerlinNoiseTest.jpg", image);waitKey(0);
}int main() {PerlinNoiseTest perlinTest;perlinTest.drawImage(20,1,400,400);}
參考文獻
Using Perlin Noise to Generate 2D Terrain and Water
FastNoiseSIMD github
libnoise
柏林噪聲
一篇文章搞懂柏林噪聲算法，附代碼講解
游戲開發技術雜談2：理解插值函數lerp
[Nature of Code] 柏林噪聲
https://adrianb.io/2014/08/09/perlinnoise.html
