二叉树递归分形，牛顿分形图案 分类： 视频图像处理 2015-07-24 10:16 50人阅读 评论(0) 收藏

1. 牛顿分形(Newton Fractal)
在复数域上使用牛顿迭代生成分形图像，函数公式F(z) = z^3 – 1在复数域上面有
三个根，一个是1，另外两个分别是复数-0.5+0.87i 与 -0.5 – 0.87i根据计算出来根
的值不同转换为RGB三种不同的颜色，根据迭代次数的多少设置颜色值的大小，
即颜色强度。



2. 曼德布罗特集合分形(Mandelbort Set Fractal) 使用复数函数公式F(z) = z^2 + c其中

c是一个复数



3. 递归分形树 (recursion tree)– 类似二叉树的递归生成树干，同时不断的缩小树干长
度，根据递归次数不同与角度不同可以得到不同的递归分形树，注意Java最大栈
深度是64，过度的归次数可能导致java栈溢出错误。递归次数建议不要超过32.



根据角度不同，可以生成不同的二叉递归树。

牛顿迭代与曼德尔波特分形算法需要复数范围内的加减乘除计算，请先google一下

然后就知道啦。本人实现的复数计算的类如下：

[java] view plaincopypackage com.gloomyfish.fractal;

public class Complex
{
private float real;
private float imaginary;

public Complex(float paramFloat1, float paramFloat2)
{
this.real = paramFloat1;
this.imaginary = paramFloat2;
}

public float real()
{
return this.real;
}

public float imaginary()
{
return this.imaginary;
}

public float modulus()
{
return (float)Math.sqrt(this.real * this.real + this.imaginary * this.imaginary);
}

public boolean equal(Complex paramComplex)
{
return ((this.real == paramComplex.real()) && (this.imaginary == paramComplex.imaginary()));
}

public Complex add(Complex paramComplex)
{
return new Complex(this.real + paramComplex.real(), this.imaginary + paramComplex.imaginary());
}

public Complex subtract(Complex paramComplex)
{
return new Complex(this.real - paramComplex.real(), this.imaginary - paramComplex.imaginary());
}

public Complex multiply(Complex paramComplex)
{
return new Complex(this.real * paramComplex.real() - (this.imaginary * paramComplex.imaginary()), this.real * paramComplex.imaginary() + this.imaginary * paramComplex.real());
}

public Complex divide(Complex paramComplex)
{
float f1 = paramComplex.real() * paramComplex.real() + paramComplex.imaginary() * paramComplex.imaginary();
float f2 = (this.real * paramComplex.real() + this.imaginary * paramComplex.imaginary()) / f1;
float f3 = (this.imaginary * paramComplex.real() - (this.real * paramComplex.imaginary())) / f1;

return new Complex(f2, f3);
}

public String toString()
{
String str = (this.imaginary >= 0.0F) ? "+" : "-";
return this.real + str + Math.abs(this.imaginary) + "i";
}
}
牛顿分形的算法代码如下：

[java] view plaincopypackage com.gloomyfish.fractal;

public class NewtonFractal extends Fractal {
private static final Complex ONE = new Complex(1.0F, 0.0F);
private static final Complex THREE = new Complex(3.0F, 0.0F);
public NewtonFractal(int widthImage, int heightImage) {
super(widthImage, heightImage);

// default start point and end point
// primary group,
this.x1 = -1.0f;
this.y1 = -1.0f;
this.x2 = 1.0f;
this.y2 = 1.0f;

// second group
// this.x1 = -3.0f;
// this.y1 = -1.76f;
// this.x2 = 3.0f;
// this.y2 = 1.76f;
// end comment

}

@Override
public void BuildFractal() {
int[] inPixels = new int[getWidth()*getHeight()];
getRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels );
int index = 0;
float xDelta = ((x2 - x1) / (float)width);
float yDelta = ((y2 - y1) / (float)height);
for(int row=0; row<height; row++) {
int ta = 0, tr = 0, tg = 0, tb = 0;
for(int col=0; col<width; col++) {
Complex localComplex2;
float f1 = this.x1 + col * xDelta;
float f2 = this.y2 - (row * yDelta);
Complex localComplex1 = new Complex(f1, f2);

int k = 0;
do {
Complex localComplex3 = localComplex1.multiply(localComplex1);
Complex localComplex4 = localComplex3.multiply(localComplex1);

localComplex2 = localComplex1;
localComplex1 = localComplex1.subtract(localComplex4.subtract(ONE).divide(THREE.multiply(localComplex3)));
}

while ((++k < MAX_ITERS) && (!(localComplex1.equal(localComplex2))));

int l = 20 * k % 10; // keep value scope between 0 and 255

// if root is 1 then
if (localComplex1.real() > 0.0F)
{
tr = tg = l;
tb = 255;
}

// if root is second complex = -0.5+0.87i
else if (localComplex1.imaginary() > 0.0F)
{
tr = tb = l;
tg = 255;
}
else
{
tr = 255;
tg = tb = l;
}

index = row * width + col;
ta = 255;
inPixels[index] = (ta << 24) | (tr << 16) | (tg << 8) | tb;
}
}
setRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels);
}

}
曼德尔波特分形算法如下：

[java] view plaincopypackage com.gloomyfish.fractal;

public class MandelbrotSetFractal extends Fractal {
private float delta = 0.01f;
public MandelbrotSetFractal(int widthImage, int heightImage) {
super(widthImage, heightImage);
this.delta = 0.01F;
this.x1 = (-(this.width / 2) * this.delta);
this.y1 = (-(this.height / 2) * this.delta);
this.x2 = (-this.x1);
this.y2 = (-this.y1);
}

@Override
public void BuildFractal() {
int[] inPixels = new int[getWidth()*getHeight()];
getRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels );
int index = 0;
for(int row=0; row<height; row++) {
int ta = 0, tr = 0, tg = 0, tb = 0;
float f1 = y2 - (row * delta);
for(int col=0; col<width; col++) {
float f5;
int i1;
float f2 = x1 + col * delta;
Complex localComplex1 = new Complex(f2, f1);
Complex localComplex2 = new Complex(0.0F, 0.0F);

int k = 0;
int l = 0;
do
{
localComplex2 = localComplex2.multiply(localComplex2).add(localComplex1);
f5 = localComplex2.modulus();
k = (f5 > 2.0F) ? 1 : 0; }
while ((++l < 32) && (k == 0));

index = row * width + col;
if (k != 0) {
i1 = 255 - (255 * l / 32);
i1 = Math.min(i1, 240);
tr = tg = tb = i1;
}
else
{
i1 = (int)(100.0F * f5) / 2 + 1;

int i2 = 101 * i1 & 0xFF;
int i3 = 149 * i1 & 0xFF;
int i4 = 199 * i1 & 0xFF;
tr = i2;
tg = i3;
tb = i4;
}

ta = 255;
inPixels[index] = (ta << 24) | (tr << 16) | (tg << 8) | tb;
}
}
setRGB(fractalImage, 0, 0, getWidth(), getHeight(), inPixels);
}

}
递归分形树代码如下：

[java] view plaincopypackage com.gloomyfish.fractal;

import java.awt.BorderLayout;
import java.awt.Color;
import java.awt.Dimension;
import java.awt.Font;
import java.awt.FontFormatException;
import java.awt.Graphics;
import java.awt.Graphics2D;
import java.awt.RenderingHints;
import java.io.IOException;
import java.io.InputStream;
import java.util.Date;

import javax.swing.JComponent;
import javax.swing.JFrame;

public class FractalTree extends JComponent {

/**
*
*/
private static final long serialVersionUID = 8812325148970066491L;

private int maxRecursions = 8; //never make this too big or it'll take forever
private double angle = 0.2 * Math.PI; //angle in radians
private double shrink = 1.8; //relative size of new branches
public FractalTree() {
super();
}
protected void paintComponent(Graphics g) {
Graphics2D g2 = (Graphics2D) g;
g2.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);
g2.setPaint(Color.WHITE);
g2.fillRect(0, 0, 400, 400);
renderTree(g2);
g2.setPaint(Color.RED);
try {
g2.setFont(loadFont());
} catch (FontFormatException e) {
// TODO Auto-generated catch block
e.printStackTrace();
} catch (IOException e) {
// TODO Auto-generated catch block
e.printStackTrace();
}
g2.drawString("Created by Gloomyfish " + new Date(System.currentTimeMillis()), 10, 320);
}

/**
* create fractal tree using recursion
* @param Graphics2D g2
*/
private void renderTree(Graphics2D g2) {
g2.setPaint(new Color(128, 96, 64));
recursion(400.0d / 2.0d, 400.0d -1.0d, 0.0d, -1.0d, 400.0d / 2.3d, 0, g2);
}

// http://www.cg.info.hiroshima-cu.ac.jp/~miyazaki/knowledge/teche31.html void recursion(double posX, double posY, double dirX, double dirY, double size, int n, Graphics2D g2)
{
int x1, x2, y1, y2;
x1 = (int)posX;
y1 = (int)posY;
x2 = (int)(posX + size * dirX);
y2 = (int)(posY + size * dirY);
g2.drawLine(x1, y1, x2, y2);

if(n >= maxRecursions) return;
double posX2, posY2, dirX2, dirY2, size2;
int n2;

// calculate the new start point coordinate
posX2 = posX + size * dirX;
posY2 = posY + size * dirY;
size2 = size / shrink; // make different length of line.
n2 = n + 1;

// rotate angle and get the new directX, directY
// http://www.jimloy.com/geometry/trigz.htm // sin(theta + angle) = sin(theta) * cos(angle) + cos(theta) * sin(angle)
// cos(theta + angle) = -sin(angle) * sin(theta) + cos(theta) * cos(angle)
dirX2 = Math.cos(angle) * dirX + Math.sin(angle) * dirY;
dirY2 = -Math.sin(angle) * dirX + Math.cos(angle) * dirY;
recursion(posX2, posY2, dirX2, dirY2, size2, n2, g2);

dirX2 = Math.cos(-angle) * dirX + Math.sin(-angle) * dirY;
dirY2 = -Math.sin(-angle) * dirX + Math.cos(-angle) * dirY;
recursion(posX2, posY2, dirX2, dirY2, size2, n2, g2);
}

/**
* http://en.wikipedia.org/wiki/Mandelbrot_set * http://www.urbanfonts.com/fonts/sans-serif-fonts.htm * @return
* @throws FontFormatException
* @throws IOException
*/
public Font loadFont() throws FontFormatException, IOException{
String fontFileName = "AMERSN.ttf";
InputStream is = this.getClass().getResourceAsStream(fontFileName);
Font actionJson = Font.createFont(Font.TRUETYPE_FONT, is);
Font actionJsonBase = actionJson.deriveFont(Font.BOLD, 12);
return actionJsonBase;
}

public static void main(String[] args) {
JFrame frame = new JFrame("Fractal Tree UI - GloomyFish");
frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
frame.getContentPane().setLayout(new BorderLayout());

// Display the window.
frame.getContentPane().add(new FractalTree(), BorderLayout.CENTER);
frame.setPreferredSize(new Dimension(450,400));
frame.pack();
frame.setVisible(true);
}
}