RANSAC（Random sample consensus）随机抽样一致性

1.算法介绍

RANSAC（Random Sample Consensus）是一种迭代的参数估计算法，用于从包含噪声和异常值的数据中拟合数学模型。它最初由Fischler和Bolles于1981年提出，被广泛应用于计算机视觉和计算机图形学等领域。

RANSAC的核心思想是随机选择数据中的一小部分样本，并根据这些样本拟合一个模型。然后，通过计算其他数据点到该模型的距离，并将距离小于一定阈值的数据点划分为内点，而距离大于阈值的数据点则划分为外点。重复此过程多次，并选择具有最多内点的模型作为最终的估计结果。

RANSAC算法的优点在于它对于大量异常值和噪声的数据有较好的鲁棒性。它可以估计出包含异常值的数据集中的准确模型，并且不要求事先知道异常值的数量，这种算法常被用于处理具有离群点或噪声的数据。

与最小二乘法相比，RANSAC的主要优点是对异常值的鲁棒性。最小二乘法试图最小化所有数据点的误差平方和，因此对异常值非常敏感。只要有一个数据点远离真实模型，就可能严重影响最小二乘法的结果。而RANSAC通过随机抽样和内点检测机制，能够有效地抵抗异常值的影响。

应用场景：

• 特征匹配：在图像配准和目标识别中，我们需要匹配两幅图像中的特征点。由于噪声和异常值的存在，直接匹配可能会得到错误的结果。RANSAC可以用于鲁棒地估计特征点之间的映射关系，从而提高匹配的准确性。

• 三维重建：在立体视觉和结构光扫描中，我们需要从多个视角的图像中重建三维结构。RANSAC可以用于估计相机的运动和场景的几何结构，从而实现鲁棒的三维重建。

• 运动估计：在视频分析和机器人导航中，我们需要估计物体或相机的运动。RANSAC可以用于从时间序列的图像数据中鲁棒地估计运动参数。

• 平面检测：在场景理解和物体检测中，我们需要从图像中检测平面。RANSAC可以用于从点云数据中鲁棒地检测平面。

• 校准：在相机校准和图像拼接中，我们需要估计图像之间的几何变换。RANSAC可以用于鲁棒地估计这些变换，从而实现准确的校准和拼接。

2.实现过程

下面的伪代码来源于维基百科，仔细读一下这份伪代码，它已经把算法的实现过程说的很明确了。

Given:
    data – A set of observations.
    model – A model to explain the observed data points.
    n – The minimum number of data points required to estimate the model parameters.
    k – The maximum number of iterations allowed in the algorithm.
    t – A threshold value to determine data points that are fit well by the model (inlier).
    d – The number of close data points (inliers) required to assert that the model fits well to the data.

Return:
    bestFit – The model parameters which may best fit the data (or null if no good model is found).


iterations = 0
bestFit = null
bestErr = something really large // This parameter is used to sharpen the model parameters to the best data fitting as iterations go on.

while iterations < k do
    maybeInliers := n randomly selected values from data
    maybeModel := model parameters fitted to maybeInliers
    confirmedInliers := empty set
    for every point in data do
        if point fits maybeModel with an error smaller than t then
             add point to confirmedInliers
        end if
    end for
    if the number of elements in confirmedInliers is > d then
        // This implies that we may have found a good model.
        // Now test how good it is.
        betterModel := model parameters fitted to all the points in confirmedInliers
        thisErr := a measure of how well betterModel fits these points
        if thisErr < bestErr then
            bestFit := betterModel
            bestErr := thisErr
        end if
    end if
    increment iterations
end while

return bestFit

3.以直线拟合为例

随机构造一些散点，添加一部分噪声，用RANSAC的算法来拟合这些散点，结果示例如下：

4.直线的描述

假设我们有一个点$P(x_0,y_0)$和一个法向量$N(a,b)$，那么直线的点法式方程可以表示为：

$$a(x-x_0)+b(y-y_0)=0$$

这个方程的含义是，对于直线上的任意一点$(x,y)$，它与点$P$的向量$<P,(x,y)>$与法向量$N$是垂直的。因为两个垂直的向量的点积为0，所以有这个等式。
给定两个点$P_1(x_1,y_1),P_2(x2,y_2)$,

• 直线的方向向量$D$：$D=(x_2-x_1,y_2-y_1)$

• 直线的法向量$N$：$N=(-(y_2-y_1),x_2-x_1)$
  将法向量$N$和已知$P1$代入上面的方程，得到：

$$-(y_2-y_1)(x-x_1)+(x_2-x_1)(y-y_1)=0$$

然后，我们可以将这个方程进一步整理为一般式方程ax + by + c = 0的形式：

$$(y_1-y_2)x+(x_2-x_1)y+(x_1{y}_2-x_2{y}_1)=0$$

所以，一般式方程的系数$a=y_1-y_2，b=x_2-x_1，c=x_1{y}_2-x_2{y}_1$。
点到直线的距离公式如下：

$$d=\frac{|a{x}_0+b{y}_0+c|}{\sqrt{(a^2+b^2)}}$$
有时候为了方便计算点到直线的距离，会对直线的一般式方程做归一化使得$a^2+b^2=1$这样上面的式子中分母项消失，变为$d=|a{x}_0+b{y}_0+c|$。

5.源码

#include 
#include 
#include 

#include "opencv2/opencv.hpp"

// define point
struct Point
{
    float x;
    float y;
};

// define line
struct Line
{
    float a;
    float b;
    float c;
    Line(){};
    Line(Point p1, Point p2)
    {
        a = p2.y - p1.y;
        b = p1.x - p2.x;
        c = p2.x * p1.y - p1.x * p2.y;
        reciprocal_sqrt_asq_plus_bsq = 1.0 / std::sqrt(a * a + b * b);
    }

    Line(float a, float b, float c)
    {
        this->a = a;
        this->b = b;
        this->c = c;
        reciprocal_sqrt_asq_plus_bsq = 1.0 / std::sqrt(a * a + b * b);
    }

    // calculate the distance from a point to a line
    float Distance(const Point &p)
    {
        if (reciprocal_sqrt_asq_plus_bsq < 0.0)
            reciprocal_sqrt_asq_plus_bsq = 1.0 / std::sqrt(a * a + b * b);
        return std::abs(a * p.x + b * p.y + c) * reciprocal_sqrt_asq_plus_bsq;
    }

private:
    float reciprocal_sqrt_asq_plus_bsq = -1.0; // 1 / sqrt(a^2 + b^2)
    // calculate the point to line distance
    // d = |ax + by + c| / sqrt(a^2 + b^2)
};

class MyRansac
{
public:
    MyRansac(){};
    ~MyRansac(){};
    Line FitLine(std::vector<Point> &points, int max_iterations, float distance_threshold)
    {
        Line best_line;
        int best_line_num_inliers = 0;
        for (int i = 0; i < max_iterations; i++)
        {
            // randomly select two points
            int idx1 = rand() % points.size();
            int idx2 = rand() % points.size();
            while (idx1 == idx2)
                idx2 = rand() % points.size();
            Point p1 = points[idx1];
            Point p2 = points[idx2];

            // fit line
            Line line(p1, p2);

            // count inliers
            int num_inliers = 0;
            for (int j = 0; j < points.size(); j++)
            {
                Point p = points[j];
                float distance = line.Distance(p);
                if (distance < distance_threshold)
                    num_inliers++;
            }

            // update best line
            if (num_inliers > best_line_num_inliers)
            {
                best_line_num_inliers = num_inliers;
                best_line = line;
                std::cout << "best_line_num_inliers: " << best_line_num_inliers << std::endl;
            }
        }
        return best_line;
    }
};

class PointFactory
{
public:
    std::vector<Point> CreatePoints(float slope, float intercept, int num_points, float x_min, float x_max, float noise)
    {
        std::vector<Point> points;
        for (int i = 0; i < num_points; i++)
        {
            float x = x_min + (x_max - x_min) * rand() / RAND_MAX;
            // float y = y_min + (y_max - y_min) * rand() / RAND_MAX;
            float y_line = slope * x + intercept;
            float y_noise = y_line + noise * rand() / RAND_MAX;
            Point p;
            p.x = x;
            p.y = y_noise;
            points.push_back(p);
        }

        // add outlier
        int num_outliers = num_points / 10;
        for (int i = 0; i < num_outliers; i++)
        {
            float x = x_min + (x_max - x_min) * rand() / RAND_MAX;
            float y = x_min + (x_max - x_min) * rand() / RAND_MAX;
            Point p;
            p.x = x;
            p.y = y;
            points.push_back(p);
        }
        return points;
    }
};

// visualize points and line
void Visualize(std::vector<Point> &points, Line &line, float distance_threshold = 0.0)
{
    cv::Mat img(500, 500, CV_8UC3, cv::Scalar(255, 255, 255));
    for (int i = 0; i < points.size(); i++)
    {
        Point p = points[i];
        float distance = line.Distance(p);
        if (distance < distance_threshold)
            cv::circle(img, cv::Point(p.x, p.y), 2, cv::Scalar(255, 0, 0), -1);
        else
            cv::circle(img, cv::Point(p.x, p.y), 2, cv::Scalar(0, 0, 255), -1);
    }
    if (std::abs(line.b) > 1e-6)
        cv::line(img, cv::Point(0, -line.c / line.b), cv::Point(img.cols, -(line.a * img.cols + line.c) / line.b), cv::Scalar(0, 255, 0), 2);

    cv::imshow("RANSAC", img);
    cv::waitKey(0);
}

int main(int argc, char *argv[])
{
    std::srand(std::time(nullptr));
    // create points
    PointFactory point_factory;
    float slope = 1;
    float intercept = 10.0;
    int num_points = 100;
    float x_min = 0.0;
    float x_max = 500.0;
    float noise = 50.0;
    std::vector<Point> points = point_factory.CreatePoints(slope, intercept, num_points, x_min, x_max, noise);

    // fit line
    MyRansac ransac;
    float distance_threshold = 20.0;
    int max_iterations = 1000;
    Line line = ransac.FitLine(points, max_iterations, distance_threshold);

    // visualize
    Visualize(points, line, distance_threshold);

    return 0;
}

参考连接

[1] Random sample consensus