霍夫变换可以用于直线检测，圆检测等。我需要解决的问题是二维点云线特征提取，相关文献上说霍夫变换可以提取，决定深入学习一下。

先上两张图：直线的表达式采用极坐标表示ρ=xcosθ+ysinθ，关于ρ和θ的几何意义如图。证明下面的图可以说明。

因此过某一点A的极坐标方程可以表示所有过A点的直线族。 同理过某一点B的极坐标方程可以表示过有过点B的直线族。两个直线族的集合就是同时过A点和B点的直线。

霍夫变换首先将极坐标系的纵横坐标轴ρ和θ离散化，构成一个网格m*n的数组。这样对栅格图像的每一个非0点进行霍夫变换，同时对相应的数组元素进行累加。

霍夫空间栅格图

OpenCV中的源码帮助理解一下：

1 static void icvHoughLinesStandard( const CvMat* img, float rho, float theta, 2                        int threshold, CvSeq *lines, int linesMax ) 3 { 4     cv::AutoBuffer
     
       _accum, _sort_buf; 5     cv::AutoBuffer
      
        _tabSin, _tabCos; 6  7     const uchar* image; 8     int step, width, height; 9     int numangle, numrho;10     int total = 0;11     int i, j;12     float irho = 1 / rho;13     double scale;14 15     CV_Assert( CV_IS_MAT(img) && CV_MAT_TYPE(img->type) == CV_8UC1 );16 17     image = img->data.ptr;18     step = img->step;19     width = img->cols;20     height = img->rows;21 22     numangle = cvRound(CV_PI / theta);//对角度进行等分23     numrho = cvRound(((width + height) * 2 + 1) / rho);//对半径进行等分？24 25     _accum.allocate((numangle+2) * (numrho+2));26     _sort_buf.allocate(numangle * numrho);27     _tabSin.allocate(numangle);28     _tabCos.allocate(numangle);29     int *accum = _accum, *sort_buf = _sort_buf;30     float *tabSin = _tabSin, *tabCos = _tabCos;31 32     memset( accum, 0, sizeof(accum[0]) * (numangle+2) * (numrho+2) );//定义了一个numangle*numrho数组33 34     float ang = 0;35     for(int n = 0; n < numangle; ang += theta, n++ )36     {37         tabSin[n] = (float)(sin((double)ang) * irho);38         tabCos[n] = (float)(cos((double)ang) * irho);39     }40 41     // stage 1. fill accumulator42     for( i = 0; i < height; i++ )43         for( j = 0; j < width; j++ )44         {45             if( image[i * step + j] != 0 )46                 for(int n = 0; n < numangle; n++ )47                 {48                     int r = cvRound( j * tabCos[n] + i * tabSin[n] );49                     r += (numrho - 1) / 2;50                     accum[(n+1) * (numrho+2) + r+1]++;51                 }52         }53 54     // stage 2. find local maximums55     for(int r = 0; r < numrho; r++ )56         for(int n = 0; n < numangle; n++ )57         {58             int base = (n+1) * (numrho+2) + r+1;59             if( accum[base] > threshold &&60                 accum[base] > accum[base - 1] && accum[base] >= accum[base + 1] &&61                 accum[base] > accum[base - numrho - 2] && accum[base] >= accum[base + numrho + 2] )62                 sort_buf[total++] = base;63         }64 65     // stage 3. sort the detected lines by accumulator value66     icvHoughSortDescent32s( sort_buf, total, accum );//快速排序
      
     

67 68     // stage 4. store the first min(total,linesMax) lines to the output buffer69     linesMax = MIN(linesMax, total);70     scale = 1./(numrho+2);71     for( i = 0; i < linesMax; i++ )//从数组中取前linesMax条线72     {73         CvLinePolar line;74         int idx = sort_buf[i];75         int n = cvFloor(idx*scale) - 1;76         int r = idx - (n+1)*(numrho+2) - 1;77         line.rho = (r - (numrho - 1)*0.5f) * rho;78         line.angle = n * theta;79         cvSeqPush( lines, &line );80     }81 }

 接下来需要思考如何用于霍夫变换进行二维激光点云的线特征提取。另外一个问题是三维点云的线特征如何提取？