上一篇部落格中講到了goodFeatureToTrack（）這個API函式能夠獲取影象中的強角點。但是獲取的角點座標是整數，但是通常情況下，角點的真實位置並不一定在整數畫素位置，因此為了獲取更為精確的角點位置座標，需要角點座標達到亞畫素（subPixel）精度。

1. 求取亞畫素精度的原理

　　找到一篇講述原理非常清楚的文件

 2. OpenCV原始碼分析

　　OpenCV中有cornerSubPixel（）這個API函式用來針對初始的整數角點座標進行亞畫素精度的優化，該函式原型如下：

void cv::cornerSubPix( InputArray _image, InputOutputArray _corners,
                       Size win, Size zeroZone, TermCriteria criteria )
 

　　_image為輸入的單通道影象；_corners為提取的初始整數角點（比如用goodFeatureToTrack提取的強角點）；win為求取亞畫素角點的視窗大小，比如設定Size（11,11），需要注意的是11為半徑，則視窗大小為23x23；zeroZone是設定的“零區域”，在搜尋視窗內，設定的“零區域”內的值不會被累加，權重值為0。如果設定為Size(-1,-1)，則表示沒有這樣的區域；critteria是條件閾值，包括迭代次數閾值和誤差精度閾值，一旦其中一項條件滿足設定的閾值，則停止迭代，獲得亞畫素角點。

　　這個API通過下面示例的語句進行呼叫：

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包含的兩個條件閾值在程式碼中是怎麼設定的。如下所示，最大迭代次數為100次，誤差精度為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;
    int i, 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

　　然後是高斯權重的計算，如下所示，視窗中心附近權重高，越往視窗邊界權重越小。如果設定的有“零區域”，則權重值設定為0。計算出的權重分佈如下圖：

Mat maskm(win_h, win_w, CV_32F), subpix_buf(win_h+2, win_w+2, CV_32F);
    float* mask = maskm.ptr<float>();

    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 < 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] = 0;
            }
        }
    }

　　接下來就是針對每個初始角點，按照上述公式，逐個進行迭代求取亞畫素角點，程式碼如下。

　　① 程式碼中CI2為本次迭代獲取的亞畫素角點位置，CI為上次迭代獲取的亞畫素角點位置，CT是初始的整數角點位置。

　　② 每次迭代結束計算CI與CI2之間的歐式距離err，如果兩者之間的歐式距離err小於設定的閾值，或者迭代次數達到設定的閾值，則停止迭代。

　　③停止迭代後，需要再次判斷最終的亞畫素角點位置和初始整數角點之間的差異，如果差值大於設定視窗尺寸的一半，則說明最小二乘計算中收斂性不好，丟棄計算得到的亞畫素角點，仍然使用初始的整數角點。

// 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<float>(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;
    }

　　自己參照OpenCV原始碼寫了一個myCornerSubPix（）介面函式以便加深理解，如下，僅供參考：

//獲取視窗內子影象
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;
}

//亞畫素角點提取
void myCornerSubPix(cv::Mat srcImg, vector<cv::Point2f> &pts, cv::Size winSize, cv::Size zeroZone, cv::TermCriteria criteria)
{
　　//搜尋視窗大小
    int winH = winSize.width * 2 + 1;
    int winW = winSize.height * 2 + 1;
    int winCnt = winH*winW;

　　//迭代閾值限制
    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

    //生成高斯權重
    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<float>(i, j) = (float)(vx*vy);
        }
    }
　　//遍歷所有初始角點，依次迭代
    for (int k = 0; k < pts.size(); k++)
    {
        double a, 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;
            //提取以當前點為中心的視窗子影象（為了方便求sobel微分，視窗各向四個方向擴充套件一行（列）畫素）
            if ( !getSubImg(srcImg, iterPoint, subImg)) break;
            uchar *pSubData = (uchar*)subImg.data+winW+3;
            //如下計算參考上述推導公式，視窗內累加
            for (int i = 0; i < winH; i ++)
            {
                for (int j = 0; j < winW; j++)
                {
　　　　　　　　　　　　//讀取高斯權重值
                    double m = weightMask.at<float>(i, j);
                    //sobel運算元求梯度
                    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;
                    //鄰域畫素p的位置座標
                    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;
            //求逆矩陣
            double invA = c / det;
            double invC = a / det;
            double invB = -b / det;
            //角點新位置
            cv::Point2f newPoint;
            newPoint.x = (float)(iterPoint.x + invA*bb1 + invB*bb2);
            newPoint.y = (float)(iterPoint.y + invB*bb1 + invC*bb2);
            //和上一次迭代之間的誤差
            err = (newPoint.x - iterPoint.x)*(newPoint.x - iterPoint.x) + (newPoint.y - iterPoint.y)*(newPoint.y - iterPoint.y);
            //更新角點位置
            iterPoint = newPoint;
            iterCnt++;
            if (iterPoint.x < 0 || iterPoint.x >= srcImg.cols || iterPoint.y < 0 || iterPoint.y >= srcImg.rows)
                break;
        } while (err > eps && iterCnt < max_iters);
        //判斷求得的亞畫素角點與初始角點之間的差異，即：最小二乘法的收斂性
        if (fabs(iterPoint.x - currPoint.x) > winSize.width || fabs(iterPoint.y - currPoint.y) > winSize.height)
            iterPoint = currPoint;
　　　　//儲存算出的亞畫素角點
        pts[k] = iterPoint;
    }
}

　　夜已深，結束。