DCT快速变换

下载源代码

一、引言

DCT变换是数字图像处理中重要的变换,很多重要的图像算法、图像应用都是基于DCT变换的,如JPEG图像编码方式。对于大尺寸的二维数值矩阵,倘若采用普通的DCT变换来进行,其所花费的时间将是让人难以忍受甚至无法达到实用。而要克服这一难点,DCT变换的快速算法无非是非常吸引人的。

就目前而言,DCT变换的快速算法无非有以下两种方式:

1.由于FFT算法的普便采用,直接利用FFT来实现DCT变换的快速算法相比来说就相对容易。但是此种方法也有不足:计算过程会涉及到复数的运算。由于DCT变换前后的数据都是实数,计算过程中引入复数,而一对复数的加法相当于两对实数的加法,一对复数的乘法相当于四对实数的乘法和两对实数的加法,显然是增加了运算量,也给硬件存储提出了更高的要求。

2.直接在实数域进行DCT快速变换。显然,这种方法相比于前一种而言,计算量和硬件要求都要优于前者。

鉴于此,本文采用第二种方法来实现DCT变换的快速算法。

二、理论推导

  限于篇幅,在此不能罗列,具体推导过程可参见《DCT快速新算法及滤波器结构研究与子波变换域图像降噪研究》华南理工大学博士论文。

三、程序实现

DCT快速变换
考虑到DCT变换中的系数要重复计算,可使用查找表来提高运行的效率,只要开始DCT变换之前计算一次,DCT变换中就可以只查找而无需计算系数。
 

void initDCTParam(int deg)
{
      // deg 为DCT变换数据长度的幂

      if(bHasInit)
      {
             return; //不用再计算查找表
      }

      int total, halftotal, i, group, endstart, factor;

      total = 1 << deg;

      if(C != NULL) delete []C;

      C = (double *)new double[total];

      halftotal = total >> 1;

      for(i=0; i < halftotal; i++)
             C[total-i-1]=(double)(2*i+1);

      for(group=0; group < deg-1; group++)
      { 

             endstart=1 << (deg-1-group);

             int len = endstart >> 1;

             factor=1 << (group+1);

             for(int j = 0;j < len; j++)
                    C[endstart-j-1] = factor*C[total-j-1];
      }

      for(i=1; i < total; i++)
             C = 2.0*cos(C*PI/(total << 1)); ///C[0]空着,没使用

      bHasInit=true;
}

DCT变换过程可分为两部分:前向追底和后向回根

前向追底:

void dct_forward(double *f,int deg)
{
      // f中存储DCT数据

      int i_deg, i_halfwing, total, wing, wings, winglen, halfwing;

      double temp1,temp2;

      total = 1 << deg;

      for(i_deg = 0; i_deg < deg; i_deg++)
      {
             wings = 1 << i_deg;
             winglen = total >> i_deg;
             halfwing = winglen >> 1;
             for(wing = 0; wing < wings; wing++)
             {
                    for(i_halfwing = 0; i_halfwing < halfwing; i_halfwing++)
                    {
                           temp1 = f[wing*winglen+i_halfwing];
                           temp2 = f[(wing+1)*winglen-1-i_halfwing];
                           if(wing%2)
                                  swap(temp1,temp2); // 交换temp1与temp2的值

                           f[wing*winglen+i_halfwing] = temp1+temp2;
                           f[(wing+1)*winglen-1-i_halfwing] = 
                                (temp1-temp2)*C[winglen-1-i_halfwing];
                    }
             }
      }
}

后向回根:

void dct_backward(double *f,int deg)
{
      // f中存储DCT数据

      int total,i_deg,wing,wings,halfwing,winglen,i_halfwing,temp1,temp2;

      total = 1 << deg;

      for(i_deg = deg-1; i_deg >= 0; i_deg--)
      {
             wings = 1 << i_deg;
             winglen = 1 << (deg-i_deg);

             halfwing = winglen >> 1;

             for(wing = 0; wing < wings; wing++)
             {
                    for(i_halfwing = 0; i_halfwing < halfwing; i_halfwing++)
                    {  
                           //f[i_halfwing+wing*winglen] = f[i_halfwing+wing*winglen];
                           if(i_halfwing == 0)
                           {
                                    f[halfwing+wing*winglen+i_halfwing] = 
                                        0.5*f[halfwing+wing*winglen+i_halfwing];
                            }
                           else
                           {
                                  temp1=bitrev(i_halfwing,deg-i_deg-1);   // bitrev为位反序
                                  temp2=bitrev(i_halfwing-1,deg-i_deg-1); // 第一参数为要变换的数
                     // 第二参数为二进制长度
                                  f[halfwing+wing*winglen+temp1] =
                                       f[halfwing+wing*winglen+temp1]-f[halfwing+wing*winglen+temp2];
                           }     
                    }
             }
      }
}

位反序函数如下:

int bitrev(int bi,int deg)
{    
      int j = 1, temp = 0, degnum, halfnum;

      degnum = deg;

      //if(deg<0) return 0;

      if(deg == 0) return bi;

      halfnum = 1 << (deg-1);

      while(halfnum)
      {            
             if(halfnum&bi)
                    temp += j;

             j<<=1;

             halfnum >>= 1;
      }

      return temp;
}

完整实现一维DCT变换:

void fdct_1D_no_param(double *f,int deg)
{
      initDCTParam(deg);
      dct_forward(f,deg);
      dct_backward(f,deg);
      fbitrev(f,deg);     // 实现位反序排列
      f[0] = 1/(sqrt(2.0))*f[0];
}

void fdct_1D(double *f,int deg)
{
      fdct_1D_no_param(f,deg);
      int total = 1 << deg;
      double param = sqrt(2.0/total);
      for(int i = 0; i < total; i++)
             f = param*f;
}

利用一维DCT变换来实现二维DCT变换:

void fdct_2D(double *f,int deg_row,int deg_col)
{    
      int rows,cols,i_row,i_col;
      double two_div_sqrtcolrow;
      rows=1 << deg_row;
      cols=1 << deg_col;
      init2D_Param(rows,cols);
      two_div_sqrtcolrow = 2.0/(sqrt(double(rows*cols)));  

      for(i_row = 0; i_row < rows; i_row++)
      {
             fdct_1D_no_param(f+i_row*cols,deg_col);
      }

      for(i_col = 0; i_col < cols; i_col++)
      {
             for(i_row = 0; i_row < rows; i_row++)
             {
                    temp_2D[i_row] = f[i_row*cols+i_col];
             }

             fdct_1D_no_param(temp_2D, deg_row);

             for(i_row = 0; i_row < rows; i_row++)
             {
                    f[i_row*cols+i_col] = temp_2D[i_row]*two_div_sqrtcolrow;
             }          
      }

      bHasInit = false;
}

 

IDCT快速变换
初始化查找表:

void initIDCTParam(int deg)
{
      if(bHasInit)
             return;    //不用再计算查找表

      int total, halftotal, i, group, endstart, factor;
      total = 1 << deg;

      // if(C!=NULL) delete []C;
      // C=(double *)new double[total];

      // 由于正变换已经为C申请了空间,所以逆变换就需再申请空间了!

      halftotal = total >> 1;

      for(i = 0; i < halftotal; i++)
             C[total-i-1] = (double)(2*i+1);

      for(group = 0; group < deg-1; group++)
      { 
             endstart = 1 << (deg-1-group);
             int len = endstart>>1;
             factor = 1 << (group+1);
             for(int j = 0; j < len; j++)
                    C[endstart-j-1] = factor*C[total-j-1];
      }

      for(i = 1; i < total; i++)
             C = 1.0/(2.0*cos(C*PI/(total << 1)));       // C[0]空着没用

      bHasInit=true;
}

IDCT变换过程也可分为两部分:前向追底和后向回根
前向追底

void idct_forward(double *F,int deg)
{
      int total,i_deg,wing,wings,halfwing,winglen,i_halfwing,temp1,temp2;

      total = 1 << deg;
      for(i_deg = 0; i_deg < deg; i_deg++)
      {
             wings = 1 << i_deg;
             winglen = 1 << (deg-i_deg);
             halfwing = winglen >> 1;
             for(wing = 0; wing < wings; wing++)
             {
                    for(i_halfwing = halfwing-1; i_halfwing >= 0; i_halfwing--)
                    {
                           if(i_halfwing == 0)
                           {
                                  F[halfwing+wing*winglen+i_halfwing] = 
                                    2.0*F[halfwing+wing*winglen+i_halfwing];
                            }
                           else
                           { 
                                  temp1 = bitrev(i_halfwing,deg-i_deg-1);
                                  temp2 = bitrev(i_halfwing-1,deg-i_deg-1);
                                  F[halfwing+wing*winglen+temp1] = F[halfwing+wing*winglen+temp1]
                                          +F[halfwing+wing*winglen+temp2];
                           }
                    }
             }
      }
}

后向回根

void idct_backward(double *F, int deg)
{
      int i_deg,i_halfwing,total,wing,wings,winglen,halfwing;

      double temp1, temp2;
      total = 1 << deg;
      for(i_deg = deg-1; i_deg >= 0; i_deg--)
      {
             wings = 1 << i_deg;
             winglen = total >> i_deg;
             halfwing = winglen >> 1;
             for(wing = 0; wing < wings; wing++)
             {
                    for(i_halfwing = 0; i_halfwing < halfwing; i_halfwing++)
                    {
                           temp1 = F[wing*winglen+i_halfwing];
                           temp2 = F[(wing+1)*winglen-1-i_halfwing]*C[winglen-1-i_halfwing];
                           if(wing % 2)
                           {
                                  F[wing*winglen+i_halfwing] = (temp1-temp2)*0.5;
                                  F[(wing+1)*winglen-1-i_halfwing] = (temp1+temp2)*0.5;
                           }
                           else
                           {
                                  F[wing*winglen+i_halfwing] = (temp1+temp2)*0.5;
                                  F[(wing+1)*winglen-1-i_halfwing] = (temp1-temp2)*0.5;
                           }
                    }
             }
      }
}

完整实现一维IDCT变换:

void fidct_1D_no_param(double *F, int deg)
{
      initIDCTParam(deg);
      F[0] = F[0]*sqrt(2.0);
      fbitrev(F, deg);
      idct_forward(F, deg);
      idct_backward(F, deg);
}

void fdct_1D(double *f, int deg)
{
      fdct_1D_no_param(f, deg);
      int total = 1 << deg;

      double param = sqrt(2.0/total);
      for(int i = 0; i < total; i++)
             f = param*f;
}

利用一维IDCT变换来实现二维IDCT变换:

void fidct_2D(double *F, int deg_row, int deg_col)
{
      int rows,cols,i_row,i_col;

      double     sqrtcolrow_div_two;
      rows = 1 << deg_row;
      cols = 1 << deg_col;
      init2D_Param(rows,cols);
      sqrtcolrow_div_two = (sqrt(double(rows*cols)))/2.0;

      for(i_row = 0; i_row < rows; i_row++)
      {
             fidct_1D_no_param(F+i_row*cols,deg_col);
      }

      for(i_col = 0; i_col < cols; i_col++)
      {
             for(i_row = 0; i_row < rows; i_row++)
             {
                    temp_2D[i_row] = F[i_row*cols+i_col];
             }

             fidct_1D_no_param(temp_2D, deg_row);
             for(i_row = 0; i_row < rows; i_row++)
             {
                    F[i_row*cols+i_col] = temp_2D[i_row]*sqrtcolrow_div_two;
             }
      }

      bHasInit=false;
}

多线程的考量由于DCT变换要花费一定的时间,特别是在数据矩阵尺寸比较大的时候。此时,如果没有增加一个线程来执行DCT变换,操作界面可能因程序忙于DCT变换的计算而失去响应,所以,增加一个用来进行DCT变换的线程是十分必要的。
首先定义一个结构

typedef struct
{    
      int row;
      int col;
      double *data;
      //double *data2;
      //double *data3; // 在计算彩色图象的数据矩阵时,彩色图象用RGB三个分量

      bool m_bfinished;

      DWORD m_start,m_end; //以毫秒计,用来计算DCT变换所用的时间;
}RUNINFO;

DCT变换进程函数:

UINT ThreadProcfastDct(LPVOID pParam)
{
      RUNINFO *pinfo = (RUNINFO*)pParam;
      pinfo->m_start = ::GetTickCount();
      fdct_2D((double *)pinfo->data, GetTwoIndex(pinfo->row), GetTwoIndex(pinfo->col));
      pinfo->m_end = ::GetTickCount();
      pinfo->m_bfinished = true;

      return 1;
}

IDCT变换进程函数:

UINT ThreadProcfastIDct(LPVOID pParam)
{
      RUNINFO *pinfo = (RUNINFO*)pParam;     
      pinfo->m_start = ::GetTickCount();
      fidct_2D((double *)pinfo->data, GetTwoIndex(pinfo->row), GetTwoIndex(pinfo->col));
      pinfo->m_end = ::GetTickCount();
      pinfo->m_bfinished = true;

      return 1;
}

四、程序运行

图1 普通DCT变换


图2 快速DCT变换


图3 快速IDCT变换

从以上可以看出,采用上述快速DCT变换对一幅256灰度的256*256的图像进行DCT正变换只需94ms,IDCT逆变换也只需94ms,而如果采用普通DCT变换,所需时间要575172ms。由此可见,DCT快速变换的巨大的优势,计算速度快,效率高。

点赞 (0)

发表评论

电子邮件地址不会被公开。 必填项已用*标注