圆检测（续）- RANSAC

继之前提到的两种方法之后，这里再列出基于RANSAC的圆检测，RANSAC(Random Sample Consensus)随机抽样一致性，略不同于霍夫圆变换那种基于投票的策略，这是一种对观测数据进行最大化模型检验的方法。下面来简单介绍一下它的原理：
1、原理
最小二乘法通常用在线性拟合参数中，但一旦最小二乘法输入的观测数据中包含有大量分散的干扰点时，它拟合出来的效果可能并不好，如可能会出现这样的情况：
least-squares-fit
可以看到，拟合出来的直线与期望有效点之间的重合率不高，也就代表着它的代价函数 $Cost(m,b) = \sum_{i=1}^{n}|y_i-(mx_i+b)|$ 虽然已经是最小的了，期望(不是概率论里面的期望)函数的值却不是最大的。
Ransac的思路是随机通过几个点用最小二乘法给出一个假设的直线，然后计算在直线内的inliers和在直线范围外的outliers。对所有可能的直线中找出inliers数目最多的那个，也就能找到最好的直线。
算法步骤：
（1）随机地抽取出所需要数目的点去拟合模型：
#2points
绿色的点代表采样的点
（2）用样本求出模型参数：
fitting
（3）在设定好直线的阈值范围中区分出inliers和outliers，并求内点占观测数据的比：
percentage
重复步骤1-3直到找出置信度最高的模型
RANSAC几点要注意的：
① 只有outliers%<50%时，得到的模型才是有保证的。
②inliers的阈值 $\delta$ 跟我们期望的模型抗噪能力相关，阈值越大，抗噪能力越弱，通常我们选取3个像素偏差的高斯模型作为噪声模型。
③ 重复1-3步骤的次数跟模型的outliers占比和我们所需要多高的置信度有关，可以用下面公式表示：

S = l o g ( 1 - P ) l o g ( 1 - p k )

$S = \frac{log(1-P)}{log(1-p^k)}$
其中，S是所需最小试验的次数，P是置信度，p是inliers占的百分比数，k是随机采样的数目。

2、实际例子
这里用网友Micka的代码来举例：

#include <opencv2/opencv.hpp>
#include <ctime>

float verifyCircle(cv::Mat dt, cv::Point2f center, float radius, std::vector<cv::Point2f> & inlierSet)
{
    unsigned int counter = 0;
    unsigned int inlier = 0;
    float minInlierDist = 2.0f;
    float maxInlierDistMax = 100.0f;
    float maxInlierDist = radius/25.0f;
    if(maxInlierDist<minInlierDist) maxInlierDist = minInlierDist;
    if(maxInlierDist>maxInlierDistMax) maxInlierDist = maxInlierDistMax;

    // choose samples along the circle and count inlier percentage
    for(float t =0; t<2*3.14159265359f; t+= 0.05f)
    {
        counter++;
        float cX = radius*cos(t) + center.x;
        float cY = radius*sin(t) + center.y;

        if(cX < dt.cols)
            if(cX >= 0)
                if(cY < dt.rows)
                    if(cY >= 0)
                        if(dt.at<float>(cY,cX) < maxInlierDist)
                        {
                            inlier++;
                            inlierSet.push_back(cv::Point2f(cX,cY));
                        }
    }

    return (float)inlier/float(counter);
}


inline void getCircle(cv::Point2f& p1,cv::Point2f& p2,cv::Point2f& p3, cv::Point2f& center, float& radius)
{
    float x1 = p1.x;
    float x2 = p2.x;
    float x3 = p3.x;

    float y1 = p1.y;
    float y2 = p2.y;
    float y3 = p3.y;

    // PLEASE CHECK FOR TYPOS IN THE FORMULA :)
    center.x = (x1*x1+y1*y1)*(y2-y3) + (x2*x2+y2*y2)*(y3-y1) + (x3*x3+y3*y3)*(y1-y2);
    center.x /= ( 2*(x1*(y2-y3) - y1*(x2-x3) + x2*y3 - x3*y2) );

    center.y = (x1*x1 + y1*y1)*(x3-x2) + (x2*x2+y2*y2)*(x1-x3) + (x3*x3 + y3*y3)*(x2-x1);
    center.y /= ( 2*(x1*(y2-y3) - y1*(x2-x3) + x2*y3 - x3*y2) );

    radius = sqrt((center.x-x1)*(center.x-x1) + (center.y-y1)*(center.y-y1));
}



std::vector<cv::Point2f> getPointPositions(cv::Mat binaryImage)
{
    std::vector<cv::Point2f> pointPositions;

    for(unsigned int y=0; y<binaryImage.rows; ++y)
    {
        //unsigned char* rowPtr = binaryImage.ptr<unsigned char>(y);
        for(unsigned int x=0; x<binaryImage.cols; ++x)
        {
            //if(rowPtr[x] > 0) pointPositions.push_back(cv::Point2i(x,y));
            if(binaryImage.at<unsigned char>(y,x) > 0) pointPositions.push_back(cv::Point2f(x,y));
        }
    }

    return pointPositions;
}



int main()
{
    clock_t starttime, endtime;
    starttime = clock();
    cv::Mat color = cv::imread("1.jpg");
    cv::Mat gray;

    // convert to grayscale
    // you could load as grayscale if you want, but I used it for (colored) output too
    cv::cvtColor(color, gray, CV_BGR2GRAY);


    cv::Mat mask;

    float canny1 = 100;
    float canny2 = 20;

    cv::Mat canny;
    cv::Canny(gray, canny, canny1,canny2);
    //cv::imshow("canny",canny);

    mask = canny;



    std::vector<cv::Point2f> edgePositions;
    edgePositions = getPointPositions(mask);

    // create distance transform to efficiently evaluate distance to nearest edge
    cv::Mat dt;
    cv::distanceTransform(255-mask, dt,CV_DIST_L1, 3);

    //TODO: maybe seed random variable for real random numbers.

    unsigned int nIterations = 0;

    cv::Point2f bestCircleCenter;
    float bestCircleRadius;
    float bestCirclePercentage = 0;
    float minRadius = 10;   // TODO: ADJUST THIS PARAMETER TO YOUR NEEDS, otherwise smaller circles wont be detected or "small noise circles" will have a high percentage of completion

    //float minCirclePercentage = 0.2f;
    float minCirclePercentage = 0.05f;  // at least 5% of a circle must be present? maybe more...

    int maxNrOfIterations = edgePositions.size();   // TODO: adjust this parameter or include some real ransac criteria with inlier/outlier percentages to decide when to stop
    printf("%d\n", maxNrOfIterations);
    for(unsigned int its=0; its< maxNrOfIterations; ++its)
    {
        //RANSAC: randomly choose 3 point and create a circle:
        //TODO: choose randomly but more intelligent, 
        //so that it is more likely to choose three points of a circle. 
        //For example if there are many small circles, it is unlikely to randomly choose 3 points of the same circle.
        unsigned int idx1 = rand()%edgePositions.size();
        unsigned int idx2 = rand()%edgePositions.size();
        unsigned int idx3 = rand()%edgePositions.size();

        // we need 3 different samples:
        if(idx1 == idx2) continue;
        if(idx1 == idx3) continue;
        if(idx3 == idx2) continue;

        // create circle from 3 points:
        cv::Point2f center; float radius;
        getCircle(edgePositions[idx1],edgePositions[idx2],edgePositions[idx3],center,radius);

        // inlier set unused at the moment but could be used to approximate a (more robust) circle from alle inlier
        std::vector<cv::Point2f> inlierSet;

        //verify or falsify the circle by inlier counting:
        float cPerc = verifyCircle(dt,center,radius, inlierSet);

        // update best circle information if necessary
        if(cPerc >= bestCirclePercentage)
            if(radius >= minRadius)
            {
                bestCirclePercentage = cPerc;
                bestCircleRadius = radius;
                bestCircleCenter = center;
            }

    }

    std::cout << "bestCirclePerc: " << bestCirclePercentage << std::endl;
    std::cout << "bestCircleRadius: " << bestCircleRadius << std::endl;

    // draw if good circle was found
    if(bestCirclePercentage >= minCirclePercentage)
        if(bestCircleRadius >= minRadius);
    cv::circle(color, bestCircleCenter,bestCircleRadius, cv::Scalar(255,255,0),1);
    std::cout << "the used time is: "<<clock()-starttime <<std::endl;

    cv::imshow("output",color);
    cv::imshow("mask",mask);
    //cv::imwrite("../outputData/1_circle_normalized.png", normalized);
    cv::waitKey(0);

    return 0;
}

他的思路是：
1. 用Canny提取边缘点，用distanceTransform得到距离边缘点的距离图；
2. 随机抽取三个不同的点解方程，三个方程三个未知数，有解；
3. 将2得到的圆周上的点与1中对应位置的点进行比较，看是否属于inliers，随后输出百分比；
4. 找出最大百分比对应的圆就是RANSAC得到的圆。

3、比较霍夫变换跟RANSAC：
鲁棒性来说，霍夫变换要稳定一点；
速度来说，霍夫变换要快，而且其所需时间变化不大，100ms左右能够完成；
RANSAC跟HoughTranform的参数调节都很麻烦，相对来说，霍夫变换更加简单一点；
RANSAC拟合的程度可能会更高，但受到outliers%<50%这个条件限制。
所以综合来说，HoughTransform的应用更广，效率更高，某些情况下，它不能很好地找到合理的圆，这时可以将RANSAC加进去进行优化，可能精度会高很多。

参考资料： $\color{red}{RANSAC}$ Kavita Bala

原文链接：https://blog.csdn.net/purgle/article/details/51995250