Opencv之RANSAC算法用于直线拟合及特征点集匹配详解

1. 讲述Ransac拟合与最小二乘在曲线拟合上的优缺点
1. 讲述在进行特征点匹配时，最近邻匹配与Ransac匹配的不同之处
1. 另外，Ransac也被用于椭圆拟合、变换矩阵求解等

1. 直线拟合

1.1 原理

RANSAC(RANdom SAmple Consensus,随机采样一致)算法是从一组含有“外点”(outliers)的数据中正确估计数学模型参数的迭代算法。“外点”一般指的的数据中的噪声，比如说匹配中的误匹配和估计曲线中的离群点。故RANSAC也是一种“外点”检测算法。同时RANSAC是一个非确定性算法，在某种意义上说，它会产生一个在一定概率下合理的结果，其允许使用更多次的迭代来使其概率增加。
RANSAC算最早是由Fischler和Bolles在SRI上提出用来解决LDP(Location Determination Problem，位置确定问题)问题的。
对于RANSAC算法来说一个基本的假设就是数据是由“内点”和“外点”组成的。“内点”就是组成模型参数的数据，“外点”就是不适合模型的数据。同时RANSAC假设：在给定一组含有少部分“内点”的数据，存在一个程序可以估计出符合“内点”的模型
算法主要思想：

其过程如下图所示：
取点集中的两点确定一条直线，然后通过设定规则选取筛选内殿，拿最多的内点拟合出来的模型作为最终的可用模型

1.2 迭代次数推导

根据上面RANSAC基本原理的介绍，在这算法流程中存在两个重要的参数需要设置，迭代次数（采样次数）和距离阈值。
迭代的次数我们应该选择多大呢？这个值是否可以事先知道应该设为多少呢？还是只能凭经验决定呢？这个值其实是可以估算出来的。下面来推算一下。

1.3 与最小二乘区别

最小二乘法尽量去适应包括外点在内的所有点。因此，最小二乘法只适合与误差较小的情况。假使需要从一个噪音较大的数据集中提取模型（比方说只有20%的数据时符合模型的）时，最小二乘法就显得力不从心了。
RANSAC相当于一个概率模型，它通过计算内点出现的概率，找出噪点之外的点集拟合出的最优模型，通常更能表示系统属性。其相当于迭代使用最小二乘法+抽样测试。

1.4 代码实现

C++实现：

//====================================================================//
//Program:RANSAC直线拟合，并与最小二乘法结果进行对比
//====================================================================//
#include <iostream>
#include <opencv2/opencv.hpp>


//RANSAC 拟合2D 直线
//输入参数：points--输入点集
//        iterations--迭代次数
//        sigma--数据和模型之间可接受的差值,车道线像素宽带一般为10左右
//              （Parameter use to compute the fitting score）
//        k_min/k_max--拟合的直线斜率的取值范围.
//                     考虑到左右车道线在图像中的斜率位于一定范围内，
//                      添加此参数，同时可以避免检测垂线和水平线
//输出参数:line--拟合的直线参数,It is a vector of 4 floats
//              (vx, vy, x0, y0) where (vx, vy) is a normalized
//              vector collinear to the line and (x0, y0) is some
//              point on the line.
//返回值：无
void fitLineRansac(const std::vector<cv::Point2f>& points,
                   cv::Vec4f &line,
                   int iterations = 1000,
                   double sigma = 1.,
                   double k_min = -7.,
                   double k_max = 7.)
{
    unsigned int n = points.size();

    if(n<2)
    {
        return;
    }

    cv::RNG rng;
    double bestScore = -1.;
    for(int k=0; k<iterations; k++)
    {
        int i1=0, i2=0;
        while(i1==i2)
        {
            i1 = rng(n);
            i2 = rng(n);
        }
        const cv::Point2f& p1 = points[i1];
        const cv::Point2f& p2 = points[i2];

        cv::Point2f dp = p2-p1;//直线的方向向量
        dp *= 1./norm(dp);
        double score = 0;

        if(dp.y/dp.x<=k_max && dp.y/dp.x>=k_min )
        {
            for(int i=0; i<n; i++)
            {
                cv::Point2f v = points[i]-p1;
                double d = v.y*dp.x - v.x*dp.y;//向量a与b叉乘/向量b的摸.||b||=1./norm(dp)
                //score += exp(-0.5*d*d/(sigma*sigma));//误差定义方式的一种
                if( fabs(d)<sigma )
                    score += 1;
            }
        }
        if(score > bestScore)
        {
            line = cv::Vec4f(dp.x, dp.y, p1.x, p1.y);
            bestScore = score;
        }
    }
}

int main()
{
    cv::Mat image(720,1280,CV_8UC3,cv::Scalar(125,125,125));

    //以车道线参数为(0.7657,-0.6432,534,548)生成一系列点
    double k = -0.6432/0.7657;
    double b = 548 - k*534;

    std::vector<cv::Point2f> points;

    for (int i = 360; i < 720; i+=10)
    {
        cv::Point2f point(int((i-b)/k),i);
        points.emplace_back(point);
    }

    //加入直线的随机噪声
    cv::RNG rng((unsigned)time(NULL));
    for (int i = 360; i < 720; i+=10)
    {
        int x = int((i-b)/k);
        x = rng.uniform(x-10,x+10);
        int y = i;
        y = rng.uniform(y-30,y+30);
        cv::Point2f point(x,y);
        points.emplace_back(point);
    }

    //加入噪声
    for (int i = 0; i < 720; i+=20)
    {
        int x = rng.uniform(1,640);
        int y = rng.uniform(1,360);

        cv::Point2f point(x,y);
        points.emplace_back(point);
    }





    int n = points.size();
    for (int j = 0; j < n; ++j)
    {
        cv::circle(image,points[j],5,cv::Scalar(0,0,0),-1);
    }


    //RANSAC 拟合
    if(1)
    {
        cv::Vec4f lineParam;
        fitLineRansac(points,lineParam,1000,10);
        double k = lineParam[1] / lineParam[0];
        double b = lineParam[3] - k*lineParam[2];

        cv::Point p1,p2;
        p1.y = 720;
        p1.x = ( p1.y - b) / k;

        p2.y = 360;
        p2.x = (p2.y-b) / k;

        cv::line(image,p1,p2,cv::Scalar(0,255,0),2);
    }


    //最小二乘法拟合
    if(1)
    {
        cv::Vec4f lineParam;
        cv::fitLine(points,lineParam,cv::DIST_L2,0,0.01,0.01);
        double k = lineParam[1] / lineParam[0];
        double b = lineParam[3] - k*lineParam[2];

        cv::Point p1,p2;
        p1.y = 720;
        p1.x = ( p1.y - b) / k;

        p2.y = 360;
        p2.x = (p2.y-b) / k;

        cv::line(image,p1,p2,cv::Scalar(0,0,255),2);
    }




    cv::imshow("image",image);
    cv::waitKey(0);

    return 0;
}

Opencv之RANSAC算法用于直线拟合及特征点集匹配详解-LMLPHP

Python 实现：

#!/usr/bin/env python3
#coding=utf-8

#============================#
#Program:RANSAC_Line.py
===========#

import numpy as np
import random
import math

import cv2

def fitLineRansac(points,iterations=1000,sigma=1.0,k_min=-7,k_max=7):
    """
    RANSAC 拟合2D 直线
    :param points:输入点集,numpy [points_num,1,2],np.float32
    :param iterations:迭代次数
    :param sigma:数据和模型之间可接受的差值,车道线像素宽带一般为10左右
                （Parameter use to compute the fitting score）
    :param k_min:
    :param k_max:k_min/k_max--拟合的直线斜率的取值范围.
                考虑到左右车道线在图像中的斜率位于一定范围内，
                添加此参数，同时可以避免检测垂线和水平线
    :return:拟合的直线参数,It is a vector of 4 floats
                (vx, vy, x0, y0) where (vx, vy) is a normalized
                vector collinear to the line and (x0, y0) is some
                point on the line.
    """
    line = [0,0,0,0]
    points_num = points.shape[0]

    if points_num<2:
        return line

    bestScore = -1
    for k in range(iterations):
        i1,i2 = random.sample(range(points_num), 2)
        p1 = points[i1][0]
        p2 = points[i2][0]

        dp = p1 - p2 #直线的方向向量
        dp *= 1./np.linalg.norm(dp) # 除以模长，进行归一化

        score = 0
        a = dp[1]/dp[0]
        if a <= k_max and a>=k_min:
            for i in range(points_num):
                v = points[i][0] - p1
                dis = v[1]*dp[0] - v[0]*dp[1]#向量a与b叉乘/向量b的摸.||b||=1./norm(dp)
                # score += math.exp(-0.5*dis*dis/(sigma*sigma))误差定义方式的一种
                if math.fabs(dis)<sigma:
                    score += 1
        if score > bestScore:
            line = [dp[0],dp[1],p1[0],p1[1]]
            bestScore = score

    return line



if __name__ == '__main__':
    image = np.ones([720,1280,3],dtype=np.ubyte)*125

    # 以车道线参数为(0.7657, -0.6432, 534, 548)生成一系列点
    k = -0.6432 / 0.7657
    b = 548 - k * 534

    points = []
    for i in range(360,720,10):
        point = (int((i-b)/k),i)
        points.append(point)

    # 加入直线的随机噪声
    for i in range(360,720,10):
        x = int((i-b)/k)
        x = random.sample(range(x-10,x+10),1)
        y = i
        y = random.sample(range(y - 30, y + 30),1)

        point = (x[0],y[0])
        points.append(point)

    # 加入噪声
    for i in range(0,720,20):
        x = random.sample(range(1, 640), 1)
        y = random.sample(range(1, 360), 1)
        point = (x[0], y[0])
        points.append(point)

    for point in points:
        cv2.circle(image,point,5,(0,0,0),-1)


    points = np.array(points).astype(np.float32)
    points = points[:,np.newaxis,:]

    # RANSAC 拟合
    if 1:
        [vx, vy, x, y] = fitLineRansac(points,1000,10)
        k = float(vy) / float(vx)  # 直线斜率
        b = -k * x + y

        p1_y = 720
        p1_x = (p1_y-b) / k
        p2_y = 360
        p2_x = (p2_y-b) / k

        p1 = (int(p1_x),int(p1_y))
        p2 = (int(p2_x), int(p2_y))

        cv2.line(image,p1,p2,(0,255,0),2)

    # 最小二乘法拟合
    if 1:
        [vx, vy, x, y] = cv2.fitLine(points, cv2.DIST_L2, 0, 0.1, 0.01)
        k = float(vy) / float(vx)  # 直线斜率
        b = -k * x + y

        p1_y = 720
        p1_x = (p1_y - b) / k
        p2_y = 360
        p2_x = (p2_y - b) / k

        p1 = (int(p1_x), int(p1_y))
        p2 = (int(p2_x), int(p2_y))

        cv2.line(image, p1, p2, (0, 0, 255), 2)


    cv2.imshow('image',image)
    cv2.waitKey(0)

2. 特征匹配

基于特征的图像匹配中会存在误匹配对，因此为提高匹配率，在粗匹配的基础上实现精匹配，可采用下面两种方法：
用RANSAC算法来寻找最佳单应性矩阵H，在此先提取SIFT特征点进行最近邻粗匹配，然后采取Ransac进行细匹配，最后再进行变换矩阵求解
代码实现如下：

//RANSAC算法
int main()
{
    Mat img_object = imread("./data/101.png", IMREAD_GRAYSCALE);
    Mat img_scene = imread("./data/100.png", IMREAD_GRAYSCALE);

    if (img_object.empty() || img_scene.empty())
    {
        cout << "Could not open or find the image!\n" << endl;
        return -1;
    }
    //-- Step 1: Detect the keypoints using SURF Detector, compute the descriptors
    int minHessian = 800; // default: 400
    Ptr<SURF> surf = SURF::create(800);
    std::vector<KeyPoint> keypoints_object, keypoints_scene;
    Mat descriptors_object, descriptors_scene;
    surf->detectAndCompute(img_object, noArray(), keypoints_object, descriptors_object);
    surf->detectAndCompute(img_scene, noArray(), keypoints_scene, descriptors_scene);

    //-- Step 2: Matching descriptor vectors with a FLANN based matcher
    // Since SURF is a floating-point descriptor NORM_L2 is used
    Ptr<DescriptorMatcher> matcher = DescriptorMatcher::create(DescriptorMatcher::FLANNBASED);
    std::vector< std::vector<DMatch> > knn_matches;
    matcher->knnMatch(descriptors_object, descriptors_scene, knn_matches, 2);

    //-- Filter matches using the Lowe's ratio test
    const float ratio_thresh = 0.75f;
    std::vector<DMatch> good_matches;
    for (size_t i = 0; i < knn_matches.size(); i++)
    {
        if (knn_matches[i][0].distance < ratio_thresh * knn_matches[i][1].distance)
        {
            good_matches.push_back(knn_matches[i][0]);
        }
    }

    //-- Draw matches
    Mat img_matches;
    drawMatches(img_object, keypoints_object, img_scene, keypoints_scene, good_matches, img_matches, Scalar::all(-1),
        Scalar::all(-1), std::vector<char>(), DrawMatchesFlags::NOT_DRAW_SINGLE_POINTS);

    //-- Localize the object
    std::vector<Point2f> obj;
    std::vector<Point2f> scene;

    for (size_t i = 0; i < good_matches.size(); i++)
    {
        //-- Get the keypoints from the good matches
        obj.push_back(keypoints_object[good_matches[i].queryIdx].pt);
        scene.push_back(keypoints_scene[good_matches[i].trainIdx].pt);
    }
    vector<uchar>inliers;
    Mat H = findHomography(obj, scene, inliers, RANSAC);


    //-- Draw matches with RANSAC
    Mat img_matches_ransac;
    std::vector<DMatch> good_matches_ransac;
    for (size_t i = 0; i < inliers.size(); i++)
    {
        if (inliers[i])
        {
            good_matches_ransac.push_back(good_matches[i]);
        }
    }
    drawMatches(img_object, keypoints_object, img_scene, keypoints_scene, good_matches_ransac, img_matches_ransac, Scalar::all(-1),
        Scalar::all(-1), std::vector<char>(), DrawMatchesFlags::NOT_DRAW_SINGLE_POINTS);
    namedWindow("img_matches", WINDOW_NORMAL);
    imshow("img_matches", img_matches);
    imwrite("img_matches.jpg", img_matches);

    namedWindow("img_matches_ransac", WINDOW_NORMAL);
    imshow("img_matches_ransac", img_matches_ransac);
    imwrite("img_matches_ransac.jpg", img_matches_ransac);
	waitKey();

	return 0;
}

只进行knn匹配与加上Ransac匹配的效果对比图如下：

参考：

1.https://blog.csdn.net/leonardohaig/article/details/104570965?spm=1001.2014.3001.5506
2.https://blog.csdn.net/H19981118/article/details/122014318?spm=1001.2014.3001.5506

明月醉窗台

Opencv之RANSAC算法用于直线拟合及特征点集匹配详解

Opencv之RANSAC算法用于直线拟合及特征点集匹配详解

1. 直线拟合

1.1 原理

1.2 迭代次数推导

1.3 与最小二乘区别

1.4 代码实现

2. 特征匹配

参考：