柏林噪声C++

柏林噪声

随机噪声
在这里插入图片描述
如上图所示随机噪声没有任何规律可言，我们希望生成有一些意义的局部连续的随机图案

一维柏林噪声

在这里插入图片描述

假设希望生成一段局部连续的随机曲线，可以采用插值的方式：在固定点随机分配y值（一般是整数点），其他的点使用插值算法

方法一：线性插值的方式

公式如下：
在这里插入图片描述
【数字图像处理】二维（2D）线性插值的应用
y = a*y1 + (1-a)*y2

我们画图看看：
在这里插入图片描述

import math
import numpy as np
import matplotlib.pyplot as plt

perm = [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):
    # 整数x1和x2的坐标
    x1 = math.floor(x)
    x2 = x1 + 1

    # x1和x2的梯度值
    grad1 = perm[x1 % 255] * 2.0 - 255.0
    grad2 = perm[x2 % 255] * 2.0 - 255.0
    #x1和x2指向x的方向向量
    vec1 = x - x1
    vec2 = x - x2

    # x到x1的距离即vec1，利用公式3计算平滑参数
    t = 3 * pow(vec1, 2) - 2 * pow(vec1, 3)

    #梯度值与方向向量的乘积
    product1 = grad1 * vec1
    product2 = grad2 * vec2


    return product1 + t * (product2 - product1)

def linear1D(x):
    # 整数x1和x2的坐标
    x1 = math.floor(x)
    x2 = x1 + 1

    # y值 随机数
    grad1 = perm[x1 % 255] * 2.0 - 255.0
    grad2 = perm[x2 % 255] * 2.0 - 255.0

    t=x - x1

    return grad1 + t * (grad2 - grad1)

def linear1D_plus(x):
    # 整数x1和x2的坐标
    x1 = math.floor(x)
    x2 = x1 + 1

    # x1和x2的梯度值
    grad1 = perm[x1 % 255] * 2.0 - 255.0
    grad2 = perm[x2 % 255] * 2.0 - 255.0
    #x1和x2指向x的方向向量
    vec1 = x - x1
    vec2 = x - x2

    t=x - x1
    #梯度值与方向向量的乘积
    product1 = grad1 * vec1
    product2 = grad2 * vec2


    return product1 + t * (product2 - product1)

def draw1D():

    # 绘制散点图
    x0=[]
    y0=[]
    for i in range(11):
        x0.append(i)
        y0.append( perm[i] * 2.0 - 255.0)
    plt.scatter(x0, y0,color='red')
    
    # 绘制1D的图像
    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])
    
    # 绘制图像
    plt.plot(x, y,color='deepskyblue')
    plt.plot(x, y1,color='green')
    plt.plot(x, y2,color='orange')
    plt.show()

draw1D()

线性插值

x取[0,10]这个区间，y在整数点随机取值，非整数点使用线性插值
ps 随机值使用伪随机数perm，柏林噪声在图像领域使用，颜色的取值范围是[0,255]，所以perm的值也是[0,255]
上图红色的点是：整数点随机取值的结果，绿色的线是线性插值。
$y = t * y 2 + (1 - t) * y 1 = y 1 + t (y 2 - y 1)$
$t = x - x 1$

线性插值plus

我们希望它更平滑一点，如果插值点x的值y与附近点x1,x2的位置相关
所以改进上述算法：
$y = t * y 2 * w 2 + (1 - t) * y 1 * w 1 = y 1 * w 1 + t (y 2 * w 2 - y 1 * w 1)$
w是权重系数，也是柏林算法中的方向向量vec1 = x - x1
如图中黄色的线

柏林噪声

柏林噪声在此基础上再加强一步：
$t=3t^2 -2t^3$

算法步骤：
input: x

计算周围的点：x1 , x2
计算x1 , x2 梯度 : grad1， grad2 随机取[0,255]
方向向量: (vec1 =x-x1 ;vec2 = x-x2)
梯度值与方向向量的乘积 product=grad*vec
计算系数 t=3t^2 -2t^3
插值：y = product1 + t * (product2 - product1)
output :y

根据上述原理可以画一个不规则的圆形

def drawCircle():
    #画圆形
    # 创建一个坐标系
    fig, ax = plt.subplots()

    # 定义圆心和半径
    center = (0, 0)
    radius = 10

    # 生成圆的数据
    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
    # 画出圆形
    ax.plot(x, y,color='orange')
    ax.plot(x, y1,color='deepskyblue')
    # 设置坐标轴范围
    ax.set_xlim([-15, 15])
    ax.set_ylim([-15, 15])

    # 显示图像
    plt.show()

在这里插入图片描述

二维柏林噪声

头文件

#pragma once
#include<array>
class PerlinNoise2D
{
public:
	PerlinNoise2D();
	~PerlinNoise2D();
	
	float BasePerlinNoise2D(float x , float y); //输出数值的范围应该是[-1,1]
	float Octave2D_01(float x, float y, int octaves, float persistence = 0.5);//输出数值的范围限定在[0,1]
private:
	template<typename STLIterator>
	inline void shuffle(STLIterator begin, STLIterator end);
	/*
		生成最大值为max的随机数
	*/
	int random(int max);
	/*
	input: 方向向量(x,y) 哈希值 hash
	根据哈希值可以达到随机梯度
	输出：随机梯度与方向向量的乘积
	*/
	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)
{
	//得到周围四个点
	int _x = std::floor(x);
	int _y = std::floor(y);

	int ix = _x & 255;
	int iy = _y & 255;
	//hash函数得到随机索引值
	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];

	//根据索引值 得到方向向量和随机梯度的向量积

	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));
}

测试

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 )
{
    // 创建一个空白图像
    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); // 绘制像素点
		}
	}
	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);
    }

在这里插入图片描述

参考文献

Using Perlin Noise to Generate 2D Terrain and Water
FastNoiseSIMD github
libnoise
柏林噪声
 一篇文章搞懂柏林噪声算法，附代码讲解

游戏开发技术杂谈2：理解插值函数lerp

[Nature of Code] 柏林噪声
https://adrianb.io/2014/08/09/perlinnoise.html

柏林噪声C++

柏林噪声

一维柏林噪声

线性插值

线性插值plus

柏林噪声

二维柏林噪声

参考文献

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