神奇的分形图像（一）-Fractal Image-Mandelbrot And Julia

article/2025/9/15 18:03:09

分析学原理不做过多阐述：这里以举例子的方式加以阐述，更能通俗的理解其概念。

分形几何学是一门以不规则几何形态为研究对象的几何学

从上图，你能很清晰的回答出它们是直线、矩形、长方体、维度分别是1、2、3维

但是如下图形呢我们通常引入豪斯多夫维度来衡量该图像的维度

这张图像来源是：https://www.zhihu.com/question/19931652

可能你不好理解为什么是n^3 = 4，其实你可以想象成Ⅰ图变成Ⅱ图是

一、首先分割成均等的三分

二、将中间（1/3长度的直线）变成尖角（总长度2/3长度的直线）

所以3 * length（Ⅱ） = 4 * length（Ⅰ）

则 $D = log_{3}(4) = 1.261$ 维度

今天要介绍的是Mandelbrot And Julia 曼德勃罗集和朱丽叶集的代码实现

曼德勃罗集和朱丽叶集都是利用公式 $Z_{n+1}= Z_{n}^2+c$ 迭代

相同的是迭代初值均为 $Z_{0}$ 而不同的迭代的参数

曼德勃罗集选用 $Z_{n+1}$ 继续迭代，而朱丽叶集采用固定参数迭代

其次需要理解一个知识点、迭代的条件是必须要收敛，否则结果无意义

这里参与迭代的参数是复数，如果对复数不了解请自行百度，复数表达方式为 $Z = a+b*i$

定义迭代结束的条件是 $\left | Z \right | <2$ 等价于 $sqrt(a^2 + b^2) < 2$

其中flag = 0实现曼德勃罗集 flag = 1实现朱丽叶集，参数c由滑条TrackBar调节。

#include <opencv2/opencv.hpp>
#include <iostream>
#include <math.h>
using namespace cv;
using namespace std;
const int trackbar_val_max = 200;class CMandelbrot
{
public:CMandelbrot(): x1(0), y1(), ScaleX(), ScaleY(),image(Mat()){}/*explicit CMandelbrot(int) : x1(0), y1(0), ScaleX(0), ScaleY(0){}*/CMandelbrot(float x1,float y1,float ScaleX,float ScaleY, Mat image){this->x1 = x1;this->y1 = y1;this->ScaleX = ScaleX;this->ScaleY = ScaleY;this->image = image;}
public:Mat   getImage(){return image;}float getX1(){return x1;}float getY1(){return y1;}float getScaleX(){return ScaleX;}float getScaleY(){return ScaleY;}
private:float x1;float y1;float ScaleX;float ScaleY;Mat image;
};
//逃逸时间算法实现
int mandelbrot(const complex<float> &z0, const int max,const complex<float> &z1)
{complex<float> z = z0;for(int t = 0; t < max; t++){if(z.real() * z.real() + z.imag() * z.imag()> 4.0f) return t;z = z * z + z1;}return max;
}
int mandelbrotFormula(const complex<float> &z0,const complex<float> &z1, const int maxIter=255) {int value = mandelbrot(z0, maxIter, z1);if(maxIter - value == 0)//if value == maxIter, set pixel = 0{return 0;}return cvRound(sqrt(value / (float) maxIter) * 255);// in order to overcome linear scaling is make no sense to change of gray
}
void sequentialManelbrot(Mat &img, const float x1, const float y1, const float scaleX, const float scaleY, complex<float> & c = complex<float>(0, 0) ,int flag = 0)
{if(flag == 0){if(img.channels() == 1){for(int i = 0; i < img.rows; i++){for(int j = 0; j < img.cols; j++){float x0 = j / scaleX + x1;//img x point convert to Mandelbrot setfloat y0 = i / scaleY + y1;//img y point convert to Mandelbrot setcomplex<float> z0(x0,y0);uchar value = (uchar)mandelbrotFormula(z0,z0);img.ptr<uchar>(i)[j] = value;}}}else{for(int i = 0; i < img.rows; i++){for(int j = 0; j < img.cols; j++){float x0 = j / scaleX + x1;//img x point convert to Mandelbrot setfloat y0 = i / scaleY + y1;//img y point convert to Mandelbrot setcomplex<float> z0(x0,y0);uchar value = (uchar)mandelbrotFormula(z0,z0);if(value > 150)img.at<Vec3b>(i, j) = Vec3b(0, 0, value);else if(value > 75)img.at<Vec3b>(i, j) = Vec3b(0, saturate_cast<uchar>(value + 75), saturate_cast<uchar>(value+150));else if(value > 50)img.at<Vec3b>(i, j) = Vec3b(0, value+50, 100);else if(value > 10)img.at<Vec3b>(i, j) = Vec3b(value+25, value+50, 100);else if(value > 0)img.at<Vec3b>(i, j) = Vec3b(value+25, 0, 100);elseimg.at<Vec3b>(i, j) = Vec3b(0, 0, 0);}}}}else{if(img.channels() == 1){for(int i = 0; i < img.rows; i++){for(int j = 0; j < img.cols; j++){float x0 = j / scaleX + x1;//img x point convert to Mandelbrot setfloat y0 = i / scaleY + y1;//img y point convert to Mandelbrot setcomplex<float> z0(x0,y0);uchar value = (uchar)mandelbrotFormula(z0,c);img.ptr<uchar>(i)[j] = value;}}}else{for(int i = 0; i < img.rows; i++){for(int j = 0; j < img.cols; j++){float x0 = j / scaleX + x1;//img x point convert to Mandelbrot setfloat y0 = i / scaleY + y1;//img y point convert to Mandelbrot setcomplex<float> z0(x0,y0);uchar value = (uchar)mandelbrotFormula(z0,c);if(value > 150)img.at<Vec3b>(i, j) = Vec3b(0, 0, value);else if(value > 75)img.at<Vec3b>(i, j) = Vec3b(0, value, saturate_cast<uchar>(value+150));else if(value > 50)img.at<Vec3b>(i, j) = Vec3b(0, value, 50);else if(value > 10)img.at<Vec3b>(i, j) = Vec3b(value+25, value, 50);else if(value > 0)img.at<Vec3b>(i, j) = Vec3b(value+25, 0, 50);elseimg.at<Vec3b>(i, j) = Vec3b(0, 0, 0);}}}}}
static float CalibrateRate(int pos)
{float rate = pos * 1.0f /  trackbar_val_max;if(rate < 0.5)rate = -2 * rate;elserate = 2 * rate;return rate;
}
static void on_trackbar_Real(int pos, void * userdata)
{int im_pos = getTrackbarPos("imaginary", "fractal");CMandelbrot *mm = (CMandelbrot *)userdata;float re_rate = CalibrateRate(pos);float im_rate = CalibrateRate(im_pos);
#ifdef _DEBUGcout << "re_rate" << re_rate << endl;cout << "im_rate" << im_rate << endl;
#endifsequentialManelbrot(mm->getImage(),mm->getX1(),mm->getY1(),mm->getScaleX(),mm->getScaleY(),complex<float>(re_rate,im_rate),1);imshow("fractal",mm->getImage());
}
static void on_trackbar_Imaginary(int pos , void * userdata)
{int re_pos = getTrackbarPos("real", "fractal");CMandelbrot *mm = (CMandelbrot *)userdata;float re_rate = CalibrateRate(re_pos);float im_rate = CalibrateRate(pos);
#ifdef _DEBUGcout << "re_rate" << re_rate << endl;cout << "im_rate" << im_rate << endl;
#endifsequentialManelbrot(mm->getImage(),mm->getX1(),mm->getY1(),mm->getScaleX(),mm->getScaleY(),complex<float>(re_rate,im_rate),1);imshow("fractal",mm->getImage());
}
int main()
{namedWindow("fractal",1);Mat mandelbrotImg2(800,600,CV_8UC3);float x1 = -1.4f, x2 = 1.16f;float y1 = -1.2f, y2 = 1.2f;float scaleX = mandelbrotImg2.cols / (x2 - x1);float scaleY = mandelbrotImg2.rows / (y2 - y1);CMandelbrot mm(x1,y1,scaleX,scaleY,mandelbrotImg2);int re = 40,im = 40;createTrackbar("real","fractal",&re,trackbar_val_max,on_trackbar_Real,&mm);createTrackbar("imaginary","fractal",&im,trackbar_val_max,on_trackbar_Imaginary,&mm);waitKey();
}

实现结果

Re = -0.4 Im = 0.65