三维重建笔记_多视几何_求基本矩阵F_六/七/八点法_误差评估

旧城等待， 2023-07-02 06:28 12阅读 0赞

**目录**

1 “8-点法”求基本矩阵F(Fundamental Matrix)

2 “7-点法”求F

2.1 参考1：计算机视觉中的数学方法

2.2 参考2：SFU 浙江大学 计算机视觉课程

2.3.1 误差评估的方法

2.3.2 相机矩阵求解的几点建议

2.3.3 退化情况

3 “6-点法”求F

--------------------

# 1 “8-点法”求基本矩阵F(Fundamental Matrix) #

参考：计算机视觉中的数学方法，吴福朝 4.3.1 基本矩阵估计

奇异值分解（Singular Value Decomposition, SVD）

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70][]

## 参考代码实现 ##

(ORB-SLAM3 TwoViewReconstruction.cc)

cv::Mat TwoViewReconstruction::ComputeF21(const vector<cv::Point2f> &vP1,const vector<cv::Point2f> &vP2)
    {
        const int N = vP1.size();
    
        cv::Mat A(N,9,CV_32F);
    
        for(int i=0; i<N; i++)
        {
            const float u1 = vP1[i].x;
            const float v1 = vP1[i].y;
            const float u2 = vP2[i].x;
            const float v2 = vP2[i].y;
    
            A.at<float>(i,0) = u2*u1;
            A.at<float>(i,1) = u2*v1;
            A.at<float>(i,2) = u2;
            A.at<float>(i,3) = v2*u1;
            A.at<float>(i,4) = v2*v1;
            A.at<float>(i,5) = v2;
            A.at<float>(i,6) = u1;
            A.at<float>(i,7) = v1;
            A.at<float>(i,8) = 1;
        }
    
        cv::Mat u,w,vt;
    
        cv::SVDecomp(A,w,u,vt,cv::SVD::MODIFY_A | cv::SVD::FULL_UV);
    
        cv::Mat Fpre = vt.row(8).reshape(0, 3);
    
        cv::SVDecomp(Fpre,w,u,vt,cv::SVD::MODIFY_A | cv::SVD::FULL_UV);
    
        w.at<float>(2)=0;
    
        return  u*cv::Mat::diag(w)*vt;
    }
    
    
    void TwoViewReconstruction::FindFundamental(vector<bool> &vbMatchesInliers, float &score, cv::Mat &F21)
    {
        // Number of putative matches
        const int N = vbMatchesInliers.size();
    
        // Normalize coordinates
        vector<cv::Point2f> vPn1, vPn2;
        cv::Mat T1, T2;
        Normalize(mvKeys1,vPn1, T1);
        Normalize(mvKeys2,vPn2, T2);
        cv::Mat T2t = T2.t();
    
        // Best Results variables
        score = 0.0;
        vbMatchesInliers = vector<bool>(N,false);
    
        // Iteration variables
        vector<cv::Point2f> vPn1i(8);
        vector<cv::Point2f> vPn2i(8);
        cv::Mat F21i;
        vector<bool> vbCurrentInliers(N,false);
        float currentScore;
    
        // Perform all RANSAC iterations and save the solution with highest score
        for(int it=0; it<mMaxIterations; it++)
        {
            // Select a minimum set
            for(int j=0; j<8; j++)
            {
                int idx = mvSets[it][j];
    
                vPn1i[j] = vPn1[mvMatches12[idx].first];
                vPn2i[j] = vPn2[mvMatches12[idx].second];
            }
    
            cv::Mat Fn = ComputeF21(vPn1i,vPn2i);
    
            F21i = T2t*Fn*T1;
    
            currentScore = CheckFundamental(F21i, vbCurrentInliers, mSigma);
    
            if(currentScore>score)
            {
                F21 = F21i.clone();
                vbMatchesInliers = vbCurrentInliers;
                score = currentScore;
            }
        }
    }

# 2 “7-点法”求F #

## 2.1 参考1：计算机视觉中的数学方法 ##

*  吴福朝 4.3.2 基本矩阵估计
 *  ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 1][]
 *  ![20200403081235536.png][]
 *  ![20200403081257899.png][]

## **2.2 参考2：**SFU 浙江大学 计算机视觉课程 ##

*  谭平教授 [https://www.jiqizhixin.com/articles/19052902][https_www.jiqizhixin.com_articles_19052902]
 *  ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 2][]
 *  ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 3][]
 *  当得到1个实数解、2个虚数解时，直接舍弃虚数解，实数解就是F的唯一解；
 *  当得到3个实数解时，称之为critical condition,这意味着，F的7个点和两个相机中心点，这9个点，通过三维空间中某个特定曲面。在这种特定情况下，7个对应点不能唯一确定一个F。

### 2.3.1 误差评估的方法 ###

有3种方式

代数误差 algebraic error

*  最原始的方式，一般在初始化时使用。

几何误差 geometric error

*  退而求其次的方式

重投影误差 re-projection error

*  golden rule 目前最好的方式
 *  ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 4][]

### 2.3.2 相机矩阵求解的几点建议 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 5][]

### 2.3.3 退化情况 ###

![202001280912450.png][]

Planar scene: 纯平面的场景

# 3 “6-点法”求F #

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 6][]

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[20200403081235536.png]: https://img-blog.csdnimg.cn/20200403081235536.png
[20200403081257899.png]: https://img-blog.csdnimg.cn/20200403081257899.png
[https_www.jiqizhixin.com_articles_19052902]: https://www.jiqizhixin.com/articles/19052902
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[202001280912450.png]: https://img-blog.csdnimg.cn/202001280912450.png
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