分类: C/C++
2008-03-12 08:43:01
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[i] = 2.0*cos(C[i]*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[i] = param*f[i]; }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
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[i] = 1.0/(2.0*cos(C[i]*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
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[i] = param*f[i]; }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。
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; }