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：
void cv::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(, ), cv::Size(-, -), cv::TermCriteria(CV_TERMCRIT_EPS + CV_TERMCRIT_ITER, , 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 = ;
int win_w = win.width * + , win_h = win.height * + ;
int i, j, k;
int max_iters = (criteria.type & CV_TERMCRIT_ITER) ? MIN(MAX(criteria.maxCount, ), MAX_ITERS) : MAX_ITERS;
double eps = (criteria.type & CV_TERMCRIT_EPS) ? MAX(criteria.epsilon, .) : ;
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+, win_w+, CV_32F);
float* mask = maskm.ptr<float>(); for( i = ; i < win_h; i++ )
{
float y = (float)(i - win.height)/win.height;
float vy = std::exp(-y*y);
for( j = ; 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 >= && zeroZone.height >= &&
zeroZone.width * + < win_w && zeroZone.height * + < win_h )
{
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] = ;
}
}
}
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 = ; pt_i < count; pt_i++ )
{
Point2f cT = corners[pt_i], cI = cT;
int iter = ;
double err = ; do
{
Point2f cI2;
double a = , b = , c = , bb1 = , bb2 = ; getRectSubPix(src, Size(win_w+, win_h+), cI, subpix_buf, subpix_buf.type());
const float* subpix = &subpix_buf.at<float>(,); // process gradient
for( i = , k = ; i < win_h; i++, subpix += win_w + )
{
double py = i - win.height; for( j = ; j < win_w; j++, k++ )
{
double m = mask[k];
double tgx = subpix[j+] - subpix[j-];
double tgy = subpix[j+win_w+] - subpix[j-win_w-];
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 < || cI.x >= src.cols || cI.y < || 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 / ;
int inity = y - subImg.rows / ;
if (initx < || inity < || (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<cv::Point2f> &pts, cv::Size winSize, cv::Size zeroZone, cv::TermCriteria criteria)
{
//搜索窗口大小
int winH = winSize.width * + ;
int winW = winSize.height * + ;
int winCnt = winH*winW;
//迭代阈值限制
int MAX_ITERS = ;
int max_iters = (criteria.type & CV_TERMCRIT_ITER) ? MIN(MAX(criteria.maxCount, ), MAX_ITERS) : MAX_ITERS;
double eps = (criteria.type & CV_TERMCRIT_EPS) ? MAX(criteria.epsilon, .) : ;
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 = ; i < winH; i++)
{
for (int j = ; 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<float>(i, j) = (float)(vx*vy);
}
}
//遍历所有初始角点，依次迭代
for (int k = ; k < pts.size(); k++)
{
double a, b, c, bb1, bb2; cv::Mat subImg = cv::Mat::zeros(winH+, winW+, CV_8UC1);
cv::Point2f currPoint = pts[k];
cv::Point2f iterPoint = currPoint; int iterCnt = ;
double err = ;
//迭代
do
{
a = b = c = bb1 = bb2 = ;
//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+;
//如下计算参考上述推导公式，窗口内累加
for (int i = ; i < winH; i ++)
{
for (int j = ; j < winW; j++)
{
//读取高斯权重值
double m = weightMask.at<float>(i, j);
//sobel算子求梯度
double sobelx = double(pSubData[i*(winW+) + j + ] - pSubData[i*(winW+) + j - ]);
double sobely = double(pSubData[(i+)*(winW+) + j] - pSubData[(i - )*(winW+) + 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 < || iterPoint.x >= srcImg.cols || iterPoint.y < || iterPoint.y >= srcImg.rows)
break;
} while (err > eps && iterCnt < max_iters);
//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。