OpenCV 之透视 n 点问题

阅读原文时间：2021年09月08日阅读：1

透视 n 点问题，源自相机标定，是计算机视觉的经典问题，广泛应用在机器人定位、SLAM、AR/VR、摄影测量等领域

1 PnP 问题

已知：相机的内参和畸变系数；世界坐标系中，n 个空间点坐标，以及投影在像平面上的像素坐标

求解：相机在世界坐标系下的位姿 R 和 t，即 {W} 到 {C} 的变换矩阵 $\;^w_c\bm{T} $，如下图：

世界坐标系中的 3d 空间点，与投影到像平面的 2d 像素点，两者之间的关系为：

$\quad s \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_1 \\ r_{21} & r_{22} & r_{23} & t_2 \\ r_{31} & r_{32} & r_{33} & t_3 \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w\\ 1 \end{bmatrix} $

根据给定空间点的数量，可将 PnP 问题分为两类：

第一类 3≤n≤5，选取的空间点较少，可通过联立方程组的方式求解，精度易受图像噪声影响，鲁棒性较差

第二类 n≥6，选取的空间点较多，可转化为求解超定方程的问题，一般侧重于鲁棒性和实时性的平衡

2 求解方法

2.1.1 转化为 Ax=0

令 $P = K\;[R\;\, t]$，$K$ 为相机内参矩阵，则 PnP 问题可简化为：已知 n 组 3d-2d 对应点，求解 $P_{3\times4}$

DLT (Direct Linear Transformation，直接线性变换)，便是直接利用这 n 组对应点，构建线性方程组来求解

$\quad s \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} p_{11} & p_{12} & p_{13} & p_{14} \\ p_{21} & p_{22} & p_{23} & p_{23} \\ p_{31} & p_{32} & p_{33} & p_{33} \end{bmatrix} \begin{bmatrix} X_w \\ Y_w \\ Z_w\\ 1 \end{bmatrix} $

简化符号 $X_w, Y_w, Z_w$ 为 $X, Y, Z$，展开得：

$\quad \begin{equation} \begin{cases} su= p_{11}X + p_{12}Y + p_{13}Z + p_{14}\\ \\sv=p_{21}X + p_{22}Y + p_{23}Z + p_{24} \\ \\s\;=p_{31}X + p_{32}Y + p_{33}Z + p_{34} \end{cases}\end{equation} \;\bm{=>} \; \begin{cases} Xp_{11} + Yp_{12} + Zp_{13} + p_{14} - uXp_{31} - uYp_{32} - uZp_{33} - up_{34} = 0 \\ \\ Xp_{21} + Yp_{22} + Zp_{23} + p_{24} - vXp_{31} - vYp_{32} - vZp_{33} - vp_{34} = 0 \end{cases}$

未知数有 11 个 ($p_{34}$可约掉)，则至少需要 6 组对应点，写成矩阵形式如下：

$\quad \begin{bmatrix} X_1&Y_1&Z_1&1 &0&0&0&0&-u_1X_1&-u_1Y_1&-u_1Z_1&-u_1 \\ 0&0&0&0& X_1&Y_1&Z_1&1&-v_1X_1&-v_1Y_1&-v_1Z_1&-v_1 \\ \vdots &\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots \\ \vdots &\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ X_n&Y_n&Z_n&1 &0&0&0&0&-u_nX_n&-u_nY_1&-u_nZ_n&-u_n \\ 0&0&0&0& X_n&Y_n&Z_n&1&-v_nX_n&-v_nY_n&-v_nZ_n&-v_n\end{bmatrix} \begin{bmatrix}p_{11}\\p_{12}\\p_{13}\\p_{14}\\ \vdots\\p_{32}\\p_{33}\\p_{34}\end{bmatrix}=\begin{bmatrix}0\\ \vdots\\ \vdots\\0\end{bmatrix}$

因此，求解 $P_{3\times4}$ 便转化成了 $Ax=0$ 的问题

2.1.2 SVD 求 R t

给定相机内参矩阵，则有 $K \begin{bmatrix} R & t \end{bmatrix} = \lambda \begin{bmatrix} p_1 & p_2 &p_3&p_4 \end{bmatrix}$

考虑 $\lambda$ 符号无关，得 $\lambda R = K^{-1}\begin{bmatrix} p_1 & p_2&p_3 \end{bmatrix}$

SVD 分解 $K^{-1}\begin{bmatrix} p_1&p_2&p_3\end{bmatrix}=\bm{U}\begin{bmatrix}d_{11} && \\ &d_{22}&\\&&&d_{33}\end{bmatrix} \bm{V^T}$

$\quad=> \lambda \approx d_{11}$ 和 $\begin{cases}\bm{R=UV^T} \\ \bm{t=\dfrac{K^{-1}p_4}{d_{11}}} \end{cases}$

当 n=3 时，PnP 即为 P3P，它有 4 个可能的解，求解方法是余弦定理 + 向量点积

2.2.1 余弦定理

根据投影几何的消隐点和消隐线，构建 3d-2d 之间的几何关系，如下：

根据余弦定理，则有

$\begin{cases} d_1^2 + d_2^2 - 2d_1d_2\cos\theta_{12} = p_{12}^2 \\ \\ d_2^2 + d_3^2 - 2d_2d_3\cos\theta_{23} = p_{23}^2 \\ \\ d_3^2 + d_1^2 + 2d_3d_2\cos\theta_23 = p_{31}^2 \end{cases}$

其中，只有 $d_1,\, d_2,\,d_3$ 是未知数，求解方程组即可

有个隐含的关键点：给定相机内参，以及 3d-2d 的投影关系，则消隐线之间的夹角 $\theta_{12}\; \theta_{23}\; \theta_{31}$ 是可计算得出的

2.2.2 向量点积

相机坐标系中，原点即为消隐点，原点到 3d-2d 的连线即为消隐线，如图所示：

如果知道 3d点投影到像平面的 2d点，在相机坐标系中的坐标 $U_1,\,U_2,\,U_3$，则 $\cos\theta_{23}= \dfrac {\overrightarrow{OU_2}\cdot \overrightarrow{OU_3}} {||\overrightarrow{OU_2}||\;||\overrightarrow{OU_3}||} $

具体到运算，可视为世界坐标系 {W} 和相机坐标系 {C} 重合，且 $Z = f$，则有

$\quad \begin{bmatrix} R & t \end{bmatrix} = \begin{bmatrix} 1 &0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \end{bmatrix} =>$ $\; s \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X_c \\ Y_c \\ Z_c \end{bmatrix} $

$K^{-1}$ 可用增广矩阵求得，且 $Z_c = f$，则有

$\quad \begin{bmatrix} X_c \\Y_c\\f \end{bmatrix} = s K^{-1}\begin{bmatrix} u\\v\\1 \end{bmatrix}$

记 $\vec u = \begin{bmatrix} X_c \\ Y_c \\ Z_c \end{bmatrix}$，则 $\cos\theta_{12}=\dfrac{(K^{-1}\vec{u_1})^T (K^{-1}\vec{u_2})}{||K^{-1}\vec{u_1}||\,||K^{-1}\vec{u_2}||}$，以此类推 $\cos\theta_{23}$ 和 $\cos\theta_{31}$

3 OpenCV 函数

OpenCV 中解 PnP 的方法有 9 种，目前实现了 7 种，还有 2 种未实现，对应论文如下：

- SOLVEPNP_P3P Complete Solution Classification for the Perspective-Three-Point Problem

- SOLVEPNP_AP3P An Efficient Algebraic Solution to the Perspective-Three-Point Problem

- SOLVEPNP_ITERATIVE 基于 L-M 最优化方法，求解重投影误差最小的位姿

- SOLVEPNP_EPNP EPnP: An Accurate O(n) Solution to the PnP Problem

- SOLVEPNP_SQPNP A Consistently Fast and Globally Optimal Solution to the Perspective-n-Point Problem

- SOLVEPNP_IPPE Infinitesimal Plane-based Pose Estimation 输入的 3D 点需要共面且 n ≥ 4

- SOLVEPNP_IPPE_SQUARE SOLVEPNP_IPPE 的一种特殊情况，要求输入 4 个共面点的坐标，并且按照特定的顺序排列

- SOLVEPNP_DLS (未实现) A Direct Least-Squares (DLS) Method for PnP 实际调用 SOLVEPNP_EPNP

- SOLVEPNP_UPLP (未实现) Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation 实际调用 SOLVEPNP_EPNP

solveP3P() 的输入是 3 组 3d-2d 对应点，定义如下：

// P3P has up to 4 solutions, and the solutions are sorted by reprojection errors(lowest to highest).  
int solveP3P (  
    InputArray     objectPoints,     // object points, 3x3 1-channel or 1x3/3x1 3-channel. vector<Point3f> can be also passed  
    InputArray      imagePoints,     // corresponding image points, 3x2 1-channel or 1x3/3x1 2-channel. vector<Point2f> can be also passed  
    InputArray     cameraMatrix,     // camera intrinsic matrix  
    InputArray       distCoeffs,     // distortion coefficients.If NULL/empty, the zero distortion coefficients are assumed.  
    OutputArrayOfArrays   rvecs,     // rotation vectors  
    OutputArrayOfArrays   tvecs,     // translation vectors  
    int                   flags      // solving method  
);

solvePnP() 实际上调用的是 solvePnPGeneric()，内部实现如下：

bool solvePnP(InputArray opoints, InputArray ipoints, InputArray cameraMatrix, InputArray distCoeffs, OutputArray rvec, OutputArray tvec, bool useExtrinsicGuess, int flags)
{
CV_INSTRUMENT_REGION();

vector<Mat> rvecs, tvecs;  
int solutions = solvePnPGeneric(opoints, ipoints, cameraMatrix, distCoeffs, rvecs, tvecs, useExtrinsicGuess, (SolvePnPMethod)flags, rvec, tvec);

if (solutions > 0)  
{  
    int rdepth = rvec.empty() ? CV\_64F : rvec.depth();  
    int tdepth = tvec.empty() ? CV\_64F : tvec.depth();  
    rvecs\[0\].convertTo(rvec, rdepth);  
    tvecs\[0\].convertTo(tvec, tdepth);  
}

return solutions > 0;

}　　

solvePnPGeneric() 除了求解相机位姿外，还可得到重投影误差，其定义如下：

bool solvePnPGeneric (  
    InputArray         objectPoints,    // object points, Nx3 1-channel or 1xN/Nx1 3-channel, N is the number of points. vector<Point3d> can be also passed  
    InputArray          imagePoints,    // corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel, N is the number of points. vector<Point2d> can be also passed  
    InputArray         cameraMatrix,    // camera intrinsic matrix  
    InputArray           distCoeffs,    // distortion coefficients  
    OutputArrayOfArrays        rvec,    // rotation vector  
    OutputArrayOfArrays        tvec,    // translation vector  
    bool  useExtrinsicGuess = false,    // used for SOLVEPNP\_ITERATIVE. If true, use the provided rvec and tvec as initial approximations, and further optimize them.  
    SolvePnPMethod  flags = SOLVEPNP\_ITERATIVE, // solving method  
    InputArray               rvec = noArray(),  // initial rotation vector when using SOLVEPNP\_ITERATIVE and useExtrinsicGuess is set to true  
    InputArray               tvec = noArray(),  // initial translation vector when using SOLVEPNP\_ITERATIVE and useExtrinsicGuess is set to true  
    OutputArray reprojectionError = noArray()   // optional vector of reprojection error, that is the RMS error  
);

solvePnP() 的一个缺点是鲁棒性不强，对异常点敏感，这在相机标定中问题不大，因为标定板的图案已知，并且特征提取较为稳定

然而，当相机拍摄实际物体时，因为特征难以稳定提取，会出现一些异常点，导致位姿估计的不准，因此，需要一种处理异常点的方法

RANSAC 便是一种高效剔除异常点的方法，对应 solvePnPRansac()，它是一个重载函数，共有 2 种参数形式，第 1 种形式如下：

bool solvePnPRansac (  
    InputArray         objectPoints,    // object points, Nx3 1-channel or 1xN/Nx1 3-channel, N is the number of points. vector<Point3d> can be also passed  
    InputArray          imagePoints,    // corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel, N is the number of points. vector<Point2d> can be also passed  
    InputArray         cameraMatrix,    // camera intrinsic matrix  
    InputArray           distCoeffs,    // distortion coefficients  
    OutputArray                rvec,    // rotation vector  
    OutputArray                tvec,    // translation vector  
    bool  useExtrinsicGuess = false,    // used for SOLVEPNP\_ITERATIVE. If true, use the provided rvec and tvec as initial approximations, and further optimize them.  
    int       iterationsCount = 100,    // number of iterations  
    float   reprojectionError = 8.0,    // inlier threshold value. It is the maximum allowed distance between the observed and computed point projections to consider it an inlier  
    double        confidence = 0.99,    // the probability that the algorithm produces a useful result  
    OutputArray inliers = noArray(),    // output vector that contains indices of inliers in objectPoints and imagePoints  
    int  flags = SOLVEPNP\_ITERATIVE     // solving method  
);

OpenCV 中还有 2 个位姿细化函数：通过迭代不断减小重投影误差，从而求得最佳位姿，solvePnPRefineLM() 使用 L-M 算法，solvePnPRefineVVS() 则用虚拟视觉伺服 (Virtual Visual Servoing)

solvePnPRefineLM() 的定义如下：

void solvePnPRefineLM (
InputArray objectPoints, // object points, Nx3 1-channel or 1xN/Nx1 3-channel, N is the number of points
InputArray imagePoints, // corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel
InputArray cameraMatrix, // camera intrinsic matrix
InputArray distCoeffs, // distortion coefficients
InputOutputArray rvec, // input/output rotation vector
InputOutputArray tvec, // input/output translation vector
TermCriteria criteria = TermCriteria(TermCriteria::EPS+TermCriteria::COUNT, 20, FLT_EPSILON) // Criteria when to stop the LM iterative algorithm
); 　　

4 应用实例

当手持标定板旋转不同角度时，利用相机内参 + solvePnP()，便可求出相机相对标定板的位姿

#include "opencv2/imgproc.hpp"
#include "opencv2/highgui.hpp"
#include "opencv2/calib3d.hpp"

using namespace std;
using namespace cv;

Size kPatternSize = Size(9, 6);
float kSquareSize = 0.025;
// camera intrinsic parameters and distortion coefficient
const Mat cameraMatrix = (Mat_(3, 3) << 5.3591573396163199e+02, 0.0, 3.4228315473308373e+02, 0.0, 5.3591573396163199e+02, 2.3557082909788173e+02, 0.0, 0.0, 1.0); const Mat distCoeffs = (Mat_(5, 1) << -2.6637260909660682e-01, -3.8588898922304653e-02, 1.7831947042852964e-03,
-2.8122100441115472e-04, 2.3839153080878486e-01);

int main()
{
// 1) read image
Mat src = imread("left07.jpg");
if (src.empty())
return -1;
// prepare for subpixel corner
Mat src_gray;
cvtColor(src, src_gray, COLOR_BGR2GRAY);

// 2) find chessboard corners and subpixel refining  
vector<Point2f> corners;  
bool patternfound = findChessboardCorners(src, kPatternSize, corners);  
if (patternfound) {  
    cornerSubPix(src\_gray, corners, Size(11, 11), Size(-1, -1), TermCriteria(TermCriteria::EPS + TermCriteria::MAX\_ITER, 30, 0.1));  
}  
else {  
    return -1;  
}

// 3) object coordinates  
vector<Point3f> objectPoints;  
for (int i = 0; i < kPatternSize.height; i++)  
{  
    for (int j = 0; j < kPatternSize.width; j++)  
    {  
        objectPoints.push\_back(Point3f(float(j \* kSquareSize), float(i \* kSquareSize), 0));  
    }  
}

// 4) Rotation and Translation vectors  
Mat rvec, tvec;  
solvePnP(objectPoints, corners, cameraMatrix, distCoeffs, rvec, tvec);

// 5) project estimated pose on the image  
drawFrameAxes(src, cameraMatrix, distCoeffs, rvec, tvec, 2\*kSquareSize);  
imshow("Pose estimation", src);  
waitKey();

}　　

当标定板旋转不同角度时，相机所对应的各个位姿如下：

OpenCV 中有一个实时目标跟踪例程，位于 "opencv\samples\cpp\tutorial_code\calib3d\real_time_pose_estimation" 中，实现步骤如下：

1) 读取目标的三维模型和网格 -> 2) 获取视频流 -> 3) ORB 特征检测 -> 4) 3d-2d 特征匹配 -> 5) 相机位姿估计 -> 6) 卡尔曼滤波

例程中设计了一个 PnPProblem 类来实现位姿估计，其中 2 个重要的函数 estimatePoseRANSAC() 和 backproject3DPoint() 定义如下：

class PnPProblem  
{  
public:  
    explicit PnPProblem(const double param\[\]);  // custom constructor  
    virtual ~PnPProblem();

    cv::Point2f backproject3DPoint(const cv::Point3f& point3d);

    void estimatePoseRANSAC(const std::vector<cv::Point3f>& list\_points3d, const std::vector<cv::Point2f>& list\_points2d,  
                            int flags, cv::Mat& inliers, int iterationsCount, float reprojectionError, double confidence);  
    // ...  
}

// Custom constructor given the intrinsic camera parameters  
PnPProblem::PnPProblem(const double params\[\])  
{  
    // intrinsic camera parameters  
    \_A\_matrix = cv::Mat::zeros(3, 3, CV\_64FC1);  
    \_A\_matrix.at<double>(0, 0) = params\[0\];       //  \[ fx   0  cx \]  
    \_A\_matrix.at<double>(1, 1) = params\[1\];       //  \[  0  fy  cy \]  
    \_A\_matrix.at<double>(0, 2) = params\[2\];       //  \[  0   0   1 \]  
    \_A\_matrix.at<double>(1, 2) = params\[3\];  
    \_A\_matrix.at<double>(2, 2) = 1;  
    // rotation matrix, translation matrix, rotation-translation matrix  
    \_R\_matrix = cv::Mat::zeros(3, 3, CV\_64FC1);  
    \_t\_matrix = cv::Mat::zeros(3, 1, CV\_64FC1);  
    \_P\_matrix = cv::Mat::zeros(3, 4, CV\_64FC1);  
}

// Estimate the pose given a list of 2D/3D correspondences with RANSAC and the method to use  
void PnPProblem::estimatePoseRANSAC (  
    const std::vector<Point3f>& list\_points3d,        // list with model 3D coordinates  
    const std::vector<Point2f>& list\_points2d,        // list with scene 2D coordinates  
    int flags, Mat& inliers, int iterationsCount,     // PnP method; inliers container  
    float reprojectionError, float confidence)        // RANSAC parameters  
{  
    // distortion coefficients, rotation vector and translation vector  
    Mat distCoeffs = Mat::zeros(4, 1, CV\_64FC1);  
    Mat rvec = Mat::zeros(3, 1, CV\_64FC1);  
    Mat tvec = Mat::zeros(3, 1, CV\_64FC1);  
    // no initial approximations  
    bool useExtrinsicGuess = false;           

    // PnP + RANSAC  
    solvePnPRansac(list\_points3d, list\_points2d, \_A\_matrix, distCoeffs, rvec, tvec, useExtrinsicGuess, iterationsCount, reprojectionError, confidence, inliers, flags);

    // converts Rotation Vector to Matrix  
    Rodrigues(rvec, \_R\_matrix);  
    \_t\_matrix = tvec;                            // set translation matrix  
    this->set\_P\_matrix(\_R\_matrix, \_t\_matrix);    // set rotation-translation matrix  
}

// Backproject a 3D point to 2D using the estimated pose parameters  
cv::Point2f PnPProblem::backproject3DPoint(const cv::Point3f& point3d)  
{  
    // 3D point vector \[x y z 1\]'  
    cv::Mat point3d\_vec = cv::Mat(4, 1, CV\_64FC1);  
    point3d\_vec.at<double>(0) = point3d.x;  
    point3d\_vec.at<double>(1) = point3d.y;  
    point3d\_vec.at<double>(2) = point3d.z;  
    point3d\_vec.at<double>(3) = 1;

    // 2D point vector \[u v 1\]'  
    cv::Mat point2d\_vec = cv::Mat(4, 1, CV\_64FC1);  
    point2d\_vec = \_A\_matrix \* \_P\_matrix \* point3d\_vec;

    // Normalization of \[u v\]'  
    cv::Point2f point2d;  
    point2d.x = (float)(point2d\_vec.at<double>(0) / point2d\_vec.at<double>(2));  
    point2d.y = (float)(point2d\_vec.at<double>(1) / point2d\_vec.at<double>(2));

    return point2d;  
}

PnPProblem 类的调用如下：实例化 -> estimatePoseRansac() 估计位姿 -> backproject3DPoint() 画出位姿

// Intrinsic camera parameters: UVC WEBCAM
double f = 55; // focal length in mm
double sx = 22.3, sy = 14.9; // sensor size
double width = 640, height = 480; // image size
double params_WEBCAM[] = { width * f / sx, // fx
height * f / sy, // fy
width / 2, // cx
height / 2 }; // cy
// instantiate PnPProblem class
PnPProblem pnp_detection(params_WEBCAM);

// RANSAC parameters
int iterCount = 500; // number of Ransac iterations.
float reprojectionError = 2.0; // maximum allowed distance to consider it an inlier.
float confidence = 0.95; // RANSAC successful confidence.

// OpenCV requires solvePnPRANSAC to minimally have 4 set of points
if (good_matches.size() >= 4)
{
// -- Step 3: Estimate the pose using RANSAC approach
pnp_detection.estimatePoseRANSAC(list_points3d_model_match, list_points2d_scene_match,
pnpMethod, inliers_idx, iterCount, reprojectionError, confidence);

// ... ..

}

// … …　

float fp = 5;
vector pose2d;
pose2d.push_back(pnp_detect_est.backproject3DPoint(Point3f(0, 0, 0))); // axis center
pose2d.push_back(pnp_detect_est.backproject3DPoint(Point3f(fp, 0, 0))); // axis x
pose2d.push_back(pnp_detect_est.backproject3DPoint(Point3f(0, fp, 0))); // axis y
pose2d.push_back(pnp_detect_est.backproject3DPoint(Point3f(0, 0, fp))); // axis z

draw3DCoordinateAxes(frame_vis, pose2d); // draw axes

// … …

　　实时目标跟踪的效果如下：