上一篇博客中讲到了goodFeatureToTrack()这个API函数能够获取图像中的强角点。但是获取的角点坐标是整数，但是通常情况下，角点的真实位置并不一定在整数像素位置，因此为了获取更为精确的角点位置坐标，需要角点坐标达到亚像素(subPixel)精度。
1. 求取亚像素精度的原理
找到一篇讲述原理非常清楚的文档
2. OpenCV源代码分析
OpenCV中有cornerSubPixel()这个API函数用来针对初始的整数角点坐标进行亚像素精度的优化，该函数原型如下：
voidcv::cornerSubPix( InputArray _image, InputOutputArray _corners,
Size win, Size zeroZone, TermCriteria criteria )
_image为输入的单通道图像；_corners为提取的初始整数角点(比如用goodFeatureToTrack提取的强角点)；win为求取亚像素角点的窗口大小，比如设置Size(11,11)，需要注意的是11为半径，则窗口大小为23x23；zeroZone是设置的“零区域”，在搜索窗口内，设置的“零区域”内的值不会被累加，权重值为0。如果设置为Size(-1,-1)，则表示没有这样的区域；critteria是条件阈值，包括迭代次数阈值和误差精度阈值，一旦其中一项条件满足设置的阈值，则停止迭代，获得亚像素角点。
这个API通过下面示例的语句进行调用：
cv::cornerSubPix(grayImg, pts, cv::Size(11, 11), cv::Size(-1, -1), cv::TermCriteria(CV_TERMCRIT_EPS + CV_TERMCRIT_ITER, 30, 0.1));
首先看criteria包含的两个条件阈值在代码中是怎么设置的。如下所示，最大迭代次数为100次，误差精度为eps*eps，也就是0.1*0.1。
const int MAX_ITERS = 100;int win_w = win.width * 2 + 1, win_h = win.height * 2 + 1;inti, j, k;int max_iters = (criteria.type & CV_TERMCRIT_ITER) ? MIN(MAX(criteria.maxCount, 1), MAX_ITERS) : MAX_ITERS;double eps = (criteria.type & CV_TERMCRIT_EPS) ? MAX(criteria.epsilon, 0.) : 0;
eps*= eps; //use square of error in comparsion operations
然后是高斯权重的计算，如下所示，窗口中心附近权重高，越往窗口边界权重越小。如果设置的有“零区域”，则权重值设置为0。计算出的权重分布如下图：
Mat maskm(win_h, win_w, CV_32F), subpix_buf(win_h+2, win_w+2, CV_32F);float* mask = maskm.ptr();for( i = 0; i < win_h; i++)
{float y = (float)(i - win.height)/win.height;float vy = std::exp(-y*y);for( j = 0; j < win_w; j++)
{float x = (float)(j - win.width)/win.width;
mask[i* win_w + j] = (float)(vy*std::exp(-x*x));
}
}//make zero_zone
if( zeroZone.width >= 0 && zeroZone.height >= 0 &&zeroZone.width* 2 + 1 < win_w && zeroZone.height * 2 + 1
{for( i = win.height - zeroZone.height; i <= win.height + zeroZone.height; i++)
{for( j = win.width - zeroZone.width; j <= win.width + zeroZone.width; j++)
{
mask[i* win_w + j] = 0;
}
}
}
接下来就是针对每个初始角点，按照上述公式，逐个进行迭代求取亚像素角点，代码如下。
① 代码中CI2为本次迭代获取的亚像素角点位置，CI为上次迭代获取的亚像素角点位置，CT是初始的整数角点位置。
② 每次迭代结束计算CI与CI2之间的欧式距离err，如果两者之间的欧式距离err小于设定的阈值，或者迭代次数达到设定的阈值，则停止迭代。
③停止迭代后，需要再次判断最终的亚像素角点位置和初始整数角点之间的差异，如果差值大于设定窗口尺寸的一半，则说明最小二乘计算中收敛性不好，丢弃计算得到的亚像素角点，仍然使用初始的整数角点。
//do optimization loop for all the points
for( int pt_i = 0; pt_i < count; pt_i++)
{
Point2f cT= corners[pt_i], cI =cT;int iter = 0;double err = 0;do{
Point2f cI2;double a = 0, b = 0, c = 0, bb1 = 0, bb2 = 0;
getRectSubPix(src, Size(win_w+2, win_h+2), cI, subpix_buf, subpix_buf.type());const float* subpix = &subpix_buf.at(1,1);//process gradient
for( i = 0, k = 0; i < win_h; i++, subpix += win_w + 2)
{double py = i -win.height;for( j = 0; j < win_w; j++, k++)
{double m =mask[k];double tgx = subpix[j+1] - subpix[j-1];double tgy = subpix[j+win_w+2] - subpix[j-win_w-2];double gxx = tgx * tgx *m;double gxy = tgx * tgy *m;double gyy = tgy * tgy *m;double px = j -win.width;
a+=gxx;
b+=gxy;
c+=gyy;
bb1+= gxx * px + gxy *py;
bb2+= gxy * px + gyy *py;
}
}double det=a*c-b*b;if( fabs( det ) <= DBL_EPSILON*DBL_EPSILON )break;//2x2 matrix inversion
double scale=1.0/det;
cI2.x= (float)(cI.x + c*scale*bb1 - b*scale*bb2);
cI2.y= (float)(cI.y - b*scale*bb1 + a*scale*bb2);
err= (cI2.x - cI.x) * (cI2.x - cI.x) + (cI2.y - cI.y) * (cI2.y -cI.y);
cI=cI2;if( cI.x < 0 || cI.x >= src.cols || cI.y < 0 || cI.y >=src.rows )break;
}while( ++iter < max_iters && err >eps );//if new point is too far from initial, it means poor convergence.//leave initial point as the result
if( fabs( cI.x - cT.x ) > win.width || fabs( cI.y - cT.y ) >win.height )
cI=cT;
corners[pt_i]=cI;
}
自己参照OpenCV源代码写了一个myCornerSubPix()接口函数以便加深理解，如下，仅供参考：
//获取窗口内子图像
bool getSubImg(cv::Mat srcImg, cv::Point2f currPoint, cv::Mat &subImg)
{int subH =subImg.rows;int subW =subImg.cols;int x = int(currPoint.x+0.5f);int y = int(currPoint.y+0.5f);int initx = x - subImg.cols / 2;int inity = y - subImg.rows / 2;if (initx < 0 || inity < 0 || (initx+subW)>=srcImg.cols || (inity+subH)>=srcImg.rows ) return false;
cv::Rect imgROI(initx, inity, subW, subH);
subImg=srcImg(imgROI).clone();return true;
}
//亚像素角点提取void myCornerSubPix(cv::Mat srcImg, vector<:point2f> &pts, cv::Size winSize, cv::Size zeroZone, cv::TermCriteria criteria)
{
//搜索窗口大小int winH = winSize.width * 2 + 1;int winW = winSize.height * 2 + 1;int winCnt = winH*winW;
//迭代阈值限制int MAX_ITERS = 100;int max_iters = (criteria.type & CV_TERMCRIT_ITER) ? MIN(MAX(criteria.maxCount, 1), MAX_ITERS) : MAX_ITERS;double eps = (criteria.type & CV_TERMCRIT_EPS) ? MAX(criteria.epsilon, 0.) : 0;
eps*= eps; //use square of error in comparsion operations//生成高斯权重
cv::Mat weightMask =cv::Mat(winH, winW, CV_32FC1);for (int i = 0; i < winH; i++)
{for (int j = 0; j < winW; j++)
{float wx = (float)(j - winSize.width) /winSize.width;float wy = (float)(i - winSize.height) /winSize.height;float vx = exp(-wx*wx);float vy = exp(-wy*wy);
weightMask.at(i, j) = (float)(vx*vy);
}
}
//遍历所有初始角点，依次迭代for (int k = 0; k < pts.size(); k++)
{doublea, b, c, bb1, bb2;
cv::Mat subImg= cv::Mat::zeros(winH+2, winW+2, CV_8UC1);
cv::Point2f currPoint=pts[k];
cv::Point2f iterPoint=currPoint;int iterCnt = 0;double err = 0;//迭代
do{
a= b = c = bb1 = bb2 = 0;//提取以当前点为中心的窗口子图像(为了方便求sobel微分，窗口各向四个方向扩展一行(列)像素)
if ( !getSubImg(srcImg, iterPoint, subImg)) break;
uchar*pSubData = (uchar*)subImg.data+winW+3;
//如下计算参考上述推导公式，窗口内累加for (int i = 0; i < winH; i ++)
{for (int j = 0; j < winW; j++)
{
//读取高斯权重值double m = weightMask.at(i, j);
//sobel算子求梯度double sobelx = double(pSubData[i*(winW+2) + j + 1] - pSubData[i*(winW+2) + j - 1]);double sobely = double(pSubData[(i+1)*(winW+2) + j] - pSubData[(i - 1)*(winW+2) +j]);double gxx = sobelx*sobelx*m;double gxy = sobelx*sobely*m;double gyy = sobely*sobely*m;
a+=gxx;
b+=gxy;
c+=gyy;
//邻域像素p的位置坐标double px = j -winSize.width;double py = i -winSize.height;
bb1+= gxx*px + gxy*py;
bb2+= gxy*px + gyy*py;
}
}double det = a*c - b*b;if (fabs(det) <= DBL_EPSILON*DBL_EPSILON)break;//求逆矩阵
double invA = c /det;double invC = a /det;double invB = -b /det;//角点新位置
cv::Point2f newPoint;
newPoint.x= (float)(iterPoint.x + invA*bb1 + invB*bb2);
newPoint.y= (float)(iterPoint.y + invB*bb1 + invC*bb2);//和上一次迭代之间的误差
err = (newPoint.x - iterPoint.x)*(newPoint.x - iterPoint.x) + (newPoint.y - iterPoint.y)*(newPoint.y -iterPoint.y);//更新角点位置
iterPoint =newPoint;
iterCnt++;if (iterPoint.x < 0 || iterPoint.x >= srcImg.cols || iterPoint.y < 0 || iterPoint.y >=srcImg.rows)break;
}while (err > eps && iterCnt
//判断求得的亚像素角点与初始角点之间的差异，即：最小二乘法的收敛性if (fabs(iterPoint.x - currPoint.x) > winSize.width || fabs(iterPoint.y - currPoint.y) >winSize.height)
iterPoint=currPoint;
//保存算出的亚像素角点
pts[k]=iterPoint;
}
}
夜已深，结束。