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混沌分形之迭代函数系统（IFS）

时间：2014-11-07 07:34:34      阅读：189      评论：0      收藏：0      [点我收藏+]

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      IFS是分形的重要分支。它是分形图像处理中最富生命力而且最具有广阔应用前景的领域之一。这一工作最早可以追溯到Hutchinson于1981年对自相似集的研究。美国科学家M.F.Barnsley于1985年发展了这一分形构型系统，并命名为迭代函数系统（Iterated Function System，IFS），后来又由Stephen Demko等人将其公式化，并引入到图像合成领域中。IFS将待生成的图像看做是由许多与整体相似的（自相似）或经过一定变换与整体相似的（自仿射）小块拼贴而成。
算法：
1.设定一个起始点（x0，y0）及总的迭代步数。
2.以概率P选取仿射变换W，形式为
     X1=a*x0 + b*y0 + e
     Y1=c*x0 + d*y0 + f
或
     X1=(a * x0*cosf(c/180)) - (b * y0*sinf(d/180)) + e
     Y1=(a * x0*sinf(c/180)) + (b * y0*cosf(d/180)) + f
3.以W作用点（x0，y0），得到新坐标（x1，y1）。
4.令x0=x1，y0=y1。
5.在屏幕上打出（x0，y0）。
6.重返第2步，进行下一次迭代，直到迭代次数大于总步数为止。

（1）三角形

class IFSTriangle : public FractalEquation
{
public:
    IFSTriangle()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        m_ParamA = 0.0f;
        m_ParamB = 0.5f;

        //‘IFS码赋值
        m[0][0] = 0.5f; m[0][1] = 0; m[0][2] = 0;  m[0][3] = 0.5f;  m[0][4] = 0;     m[0][5] = 0;    m[0][6] = 0.333f;
        m[1][0] = 0.5f; m[1][1] = 0; m[1][2] = 0;  m[1][3] = 0.5f;  m[1][4] = 0.5f;  m[1][5] = 0;    m[1][6] = 0.333f;
        m[2][0] = 0.5f; m[2][1] = 0; m[2][2] = 0;  m[2][3] = 0.5f;  m[2][4] = 0.25f; m[2][5] = 0.5f; m[2][6] = 0.334f;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        float a, b, c, d, e, f;    //‘仿射变幻中的系数

        float R = (float)rand()/RAND_MAX;

        if (R <= m[0][6]) 
        {
            a = m[0][0]; b = m[0][1]; c = m[0][2]; d = m[0][3]; e = m[0][4]; f = m[0][5];
        } 
        else if (R <= m[0][6] + m[1][6]) 
        {
            a = m[1][0]; b = m[1][1]; c = m[1][2]; d = m[1][3]; e = m[1][4]; f = m[1][5];
        } 
        else
        {
            a = m[2][0]; b = m[2][1]; c = m[2][2]; d = m[2][3]; e = m[2][4]; f = m[2][5];
        } 

        outX = (a * x) + (b * y) + e*FRACTAL_RADIUS;
        outY = (c * x) + (d * y) + f*FRACTAL_RADIUS;
        outZ = z;
    }

    bool IsValidParamA() const {return true;}
    bool IsValidParamB() const {return true;}

    void SetParamA(float v)
    {
        m_ParamA = v;
        m[2][1] = v; 
    }

    void SetParamB(float v)
    {
        m_ParamB = v;
        m[2][0] = v; 
        m[0][3] = v; 
    }

private:
    float m[3][7];      // ‘存放IFS码
};

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这里生成的是谢尔宾斯基三角形，但可以通过参数设置对其变形

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（2）皇冠

class IFSCrown : public FractalEquation
{
public:
    IFSCrown()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        m_ParamA = 2.0f;

        //‘IFS码赋值
        m[0][0] = 0.5f;   m[0][1] = 0.5f;   m[0][2] = 0;  m[0][3] = 0;  m[0][4] = 0;    m[0][5] = 0;   m[0][6] = 0.2f;
        m[1][0] = 0.5f;   m[1][1] = 0.5f;   m[1][2] = 0;  m[1][3] = 0;  m[1][4] = 0.5f; m[1][5] = 0;   m[1][6] = 0.2f;
        m[2][0] = 0.25f;  m[2][1] = 0.25f;  m[2][2] = 0;  m[2][3] = 0;  m[2][4] = 2.0f; m[2][5] = 2.0f;m[2][6] = 0.3f;
        m[3][0] = 0.25f;  m[3][1] = 0.25f;  m[3][2] = 0;  m[3][3] = 0;  m[3][4] = -1.0f;m[3][5] = 2.0f;m[3][6] = 0.3f;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        float a, b, c, d, e, f;    //‘仿射变幻中的系数

        float R = (float)rand()/RAND_MAX;

        if (R <= m[0][6]) 
        {
            a = m[0][0]; b = m[0][1]; c = m[0][2]; d = m[0][3]; e = m[0][4]; f = m[0][5];
        } 
        else if (R <= m[0][6] + m[1][6]) 
        {
            a = m[1][0]; b = m[1][1]; c = m[1][2]; d = m[1][3]; e = m[1][4]; f = m[1][5];
        } 
        else if (R <= m[0][6] + m[1][6] + m[2][6]) 
        {
            a = m[2][0]; b = m[2][1]; c = m[2][2]; d = m[2][3]; e = m[2][4]; f = m[2][5];
        } 
        else 
        {
            a = m[3][0]; b = m[3][1]; c = m[3][2]; d = m[3][3]; e = m[3][4]; f = m[3][5];
        }

        outX = (a * x*cosf(c/180)) - (b * y*sinf(d/180)) + e*FRACTAL_RADIUS;
        outY = (a * x*sinf(c/180)) + (b * y*cosf(d/180)) + f*FRACTAL_RADIUS;
        outZ = z;
    }

    bool IsValidParamA() const {return true;}

    void SetParamA(float v)
    {
        m_ParamA = v;
        m[2][4] = v; 
        m[2][5] = v;
        m[3][4] = 1 - v;
        m[3][5] = v;
    }

private:
    float m[4][7];      // ‘存放IFS码
};

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（3）芦苇

// 芦苇
class IFSBulrush : public FractalEquation
{
public:
    IFSBulrush()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        m_ParamA = 10.0f;

        float k = m_ParamA*100.0f;

        //‘IFS码赋值
        m[0][0] = 0.5f;  m[0][1] = 0.5f;  m[0][2] = 0;  m[0][3] = 0;  m[0][4] = 0;    m[0][5] = 0;        m[0][6] = 0.3f;
        m[1][0] = 0.5f;  m[1][1] = 0.5f;  m[1][2] = k;  m[1][3] = k;  m[1][4] = 1;    m[1][5] = k/1600;   m[1][6] = 0.3f;
        m[2][0] = 0.5f;  m[2][1] = 0.5f;  m[2][2] = 0;  m[2][3] = 0;  m[2][4] = 0.5f; m[2][5] = 0;        m[2][6] = 0.4f;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        float a, b, c, d, e, f;    //‘仿射变幻中的系数

        float R = (float)rand()/RAND_MAX;

        if (R <= m[0][6]) 
        {
            a = m[0][0]; b = m[0][1]; c = m[0][2]; d = m[0][3]; e = m[0][4]; f = m[0][5];
        } 
        else if (R <= m[0][6] + m[1][6]) 
        {
            a = m[1][0]; b = m[1][1]; c = m[1][2]; d = m[1][3]; e = m[1][4]; f = m[1][5];
        } 
        else 
        {
            a = m[2][0]; b = m[2][1]; c = m[2][2]; d = m[2][3]; e = m[2][4]; f = m[2][5];
        } 

        outX = (a * x*cosf(c/180)) - (b * y*sinf(d/180)) + e*FRACTAL_RADIUS;
        outY = (a * x*sinf(c/180)) + (b * y*cosf(d/180)) + f*FRACTAL_RADIUS;
        outZ = z;
    }

    bool IsValidParamA() const {return true;}

    void SetParamA(float v)
    {
        m_ParamA = v;
        float k = m_ParamA*100.0f;
        m[1][2] = k;  
        m[1][3] = k;  
        m[1][5] = k/1600;
    }

private:
    float m[3][7];      // ‘存放IFS码
};

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（4）万花筒

// 万花筒
class IFSPhantoscope : public FractalEquation
{
public:
    IFSPhantoscope()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        m_ParamA = 2.0f;

        float k = m_ParamA*100.0f;

        //‘IFS码赋值
        m[0][0] = 0.2f;  m[0][1] = 0.2f;  m[0][2] = 0;  m[0][3] = 0;  m[0][4] = 0.7f; m[0][5] = 0;     m[0][6] = 0.2f;
        m[1][0] = 0.2f;  m[1][1] = 0.2f;  m[1][2] = 0;  m[1][3] = 0;  m[1][4] =-0.7f; m[1][5] = 0;     m[1][6] = 0.2f;
        m[2][0] = 0.2f;  m[2][1] = 0.2f;  m[2][2] = 0;  m[2][3] = 0;  m[2][4] = 0;    m[2][5] = 0.7f;  m[2][6] = 0.2f;
        m[3][0] = 0.2f;  m[3][1] = 0.2f;  m[3][2] = 0;  m[3][3] = 0;  m[3][4] = 0;    m[3][5] = -0.7f; m[3][6] = 0.2f;
        m[4][0] = 0.85f; m[4][1] = 0.85f; m[4][2] = k;  m[4][3] = k;  m[4][4] = 0;    m[4][5] = 0;     m[4][6] = 0.2f;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        float a, b, c, d, e, f;    //‘仿射变幻中的系数

        float R = (float)rand()/RAND_MAX;

        if (R <= m[0][6]) 
        {
            a = m[0][0]; b = m[0][1]; c = m[0][2]; d = m[0][3]; e = m[0][4]; f = m[0][5];
        } 
        else if (R <= m[0][6] + m[1][6]) 
        {
            a = m[1][0]; b = m[1][1]; c = m[1][2]; d = m[1][3]; e = m[1][4]; f = m[1][5];
        } 
        else if (R <= m[0][6] + m[1][6] + m[2][6]) 
        {
            a = m[2][0]; b = m[2][1]; c = m[2][2]; d = m[2][3]; e = m[2][4]; f = m[2][5];
        } 
        else if (R <= m[0][6] + m[1][6] + m[2][6] + m[3][6]) 
        {
            a = m[3][0]; b = m[3][1]; c = m[3][2]; d = m[3][3]; e = m[3][4]; f = m[3][5];
        } 
        else 
        {
            a = m[4][0]; b = m[4][1]; c = m[4][2]; d = m[4][3]; e = m[4][4]; f = m[4][5];
        }

        outX = (a * x*cosf(c/180)) - (b * y*sinf(d/180)) + e*FRACTAL_RADIUS;
        outY = (a * x*sinf(c/180)) + (b * y*cosf(d/180)) + f*FRACTAL_RADIUS;
        outZ = z;
    }

    bool IsValidParamA() const {return true;}

    void SetParamA(float v)
    {
        m_ParamA = v;
        float k = m_ParamA*100.0f;
        m[4][2] = k;  
        m[4][3] = k;
    }

private:
    float m[5][7];      // ‘存放IFS码
};

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（5）Tree

class IFSTree : public FractalEquation
{
public:
    IFSTree()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        //‘IFS码赋值
        m[0][0] = 0.195f;  m[0][1] =-0.488f; m[0][2] = 0.344f; m[0][3] = 0.433f; m[0][4] = 0.4431f; m[0][5] = 0.2452f; m[0][6] = 0.25f;
        m[1][0] = 0.462f;  m[1][1] = 0.414f; m[1][2] =-0.252f; m[1][3] = 0.361f; m[1][4] = 0.2511f; m[1][5] = 0.5692f; m[1][6] = 0.25f;
        m[2][0] =-0.058f;  m[2][1] =-0.07f;  m[2][2] = 0.453f; m[2][3] =-0.111f; m[2][4] = 0.5976f; m[2][5] = 0.0969f; m[2][6] = 0.25f;
        m[3][0] =-0.035f;  m[3][1] = 0.07f;  m[3][2] =-0.469f; m[3][3] =-0.022f; m[3][4] = 0.4884f; m[3][5] = 0.5069f; m[3][6] = 0.2f;
        m[4][0] =-0.637f;  m[4][1] = 0.0f;   m[4][2] = 0.0f;   m[4][3] = 0.501f; m[4][4] = 0.8562f; m[4][5] = 0.2513f; m[4][6] = 0.05f;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        float a, b, c, d, e, f;    //‘仿射变幻中的系数

        float R = (float)rand()/RAND_MAX;

        if (R <= m[0][6]) 
        {
            a = m[0][0]; b = m[0][1]; c = m[0][2]; d = m[0][3]; e = m[0][4]; f = m[0][5];
        } 
        else if (R <= m[0][6] + m[1][6]) 
        {
            a = m[1][0]; b = m[1][1]; c = m[1][2]; d = m[1][3]; e = m[1][4]; f = m[1][5];
        } 
        else if (R <= m[0][6] + m[1][6] + m[2][6]) 
        {
            a = m[2][0]; b = m[2][1]; c = m[2][2]; d = m[2][3]; e = m[2][4]; f = m[2][5];
        } 
        else if (R <= m[0][6] + m[1][6] + m[2][6] + m[3][6]) 
        {
            a = m[3][0]; b = m[3][1]; c = m[3][2]; d = m[3][3]; e = m[3][4]; f = m[3][5];
        } 
        else 
        {
            a = m[4][0]; b = m[4][1]; c = m[4][2]; d = m[4][3]; e = m[4][4]; f = m[4][5];
        }

        outX = a*x + b*y + e;
        outY = c*x + d*y + f;
        outZ = z;
    }

private:
    float m[5][7];      // ‘存放IFS码
};

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这个图形我很喜欢，所以还专门将其生成图像：

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（6）大脑

class IFSBrain : public FractalEquation
{
public:
    IFSBrain()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        //‘IFS码赋值
        //0.03  0       0       0.45    0   0     0.05;    
        //-0.03 0       0       -0.45   0   0.4   0.15;    
        //0.56  -0.56   0.56    0.56    0   0.4   0.4;      
        //0.56  0.56    -0.56   0.56    0   0.4   0.4;
        m[0][0] = 0.03f;  m[0][1] = 0.0f;   m[0][2] = 0.0f;  m[0][3] = 0.45f; m[0][4] = 0.0f; m[0][5] = 0.0f; m[0][6] = 0.05f;
        m[1][0] =-0.03f;  m[1][1] = 0.0f;   m[1][2] = 0.0f;  m[1][3] =-0.45f; m[1][4] = 0.0f; m[1][5] = 0.4f; m[1][6] = 0.15f;
        m[2][0] = 0.56f;  m[2][1] =-0.56f;  m[2][2] = 0.56f; m[2][3] = 0.56f; m[2][4] = 0.0f; m[2][5] = 0.4f; m[2][6] = 0.4f;
        m[3][0] = 0.56f;  m[3][1] = 0.56f;  m[3][2] =-0.56f; m[3][3] = 0.56f; m[3][4] = 0.0f; m[3][5] = 0.4f; m[3][6] = 0.4f;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        float a, b, c, d, e, f;    //‘仿射变幻中的系数

        float R = (float)rand()/RAND_MAX;

        if (R <= m[0][6]) 
        {
            a = m[0][0]; b = m[0][1]; c = m[0][2]; d = m[0][3]; e = m[0][4]; f = m[0][5];
        } 
        else if (R <= m[0][6] + m[1][6]) 
        {
            a = m[1][0]; b = m[1][1]; c = m[1][2]; d = m[1][3]; e = m[1][4]; f = m[1][5];
        } 
        else if (R <= m[0][6] + m[1][6] + m[2][6]) 
        {
            a = m[2][0]; b = m[2][1]; c = m[2][2]; d = m[2][3]; e = m[2][4]; f = m[2][5];
        } 
        else
        {
            a = m[3][0]; b = m[3][1]; c = m[3][2]; d = m[3][3]; e = m[3][4]; f = m[3][5];
        } 

        outX = a*x + b*y + e;
        outY = c*x + d*y + f;
        outZ = z;
    }

private:
    float m[4][7];      // ‘存放IFS码
};

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－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－

关于基类FractalEquation的定义及相关软件见：混沌与分形

混沌分形之迭代函数系统（IFS）

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原文地址：http://www.cnblogs.com/WhyEngine/p/4069166.html
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