OpenCV>>相机相对姿态估计

环境

1. 使用3DS Max生成的3D场景
2. 摄像头视野为45度
3. 使用相同的相机以800x600分辨率渲染两个图像
4. 相机Z旋转==0度的图像A
5. 相机Z旋转==25度的图像B
6. 8个对应点（手动驱动），无异常值

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手头的任务

解析图像A&图B（上图）预期在Z轴上产生25度的诱导旋转

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实施

选项A:

1. 对应关系是手动生成的，以确保没有异常值（请参阅下面代码段中的"rotZ0"answers"rotZ25"）
2. 使用图像分辨率&FOV根据此链接
3. 相机固有矩阵由图像分辨率和相机FOV组成（根据此链接）
4. 使用'cc:：findFundamentalMat'导出基本矩阵
5. 基本矩阵（根据本链接）是相机内部矩阵'K'和基本矩阵'F'K.t（（）'`是内部矩阵转置
6. 对基本矩阵执行SVD
7. 解决4个可能的解决方案：[U*W*Vt]，[U*W.t（）*Vt][U*W.t（）*Vt.t（）]&[U*W*Vt.t（）]

选项B:

1. 对应关系是手动生成的，以确保没有异常值（请参见"rotZ0"answers"rotZ25"）
2. Essential矩阵使用'cv:：findEssentialMat'组成
3. 使用'cc:：recoverPose'估计相机姿势

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结果

以上两个选项都无法正确恢复相对相机姿势（预计在Z轴上旋转25度）

我做错了什么
如何正确解析相机的相对姿势

如有任何帮助，我们将不胜感激。

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完整代码

#define RAD2DEG(rad) (((rad) * 180)/M_PI)
#define DEG2RAD(deg) (((deg) * M_PI)/180)
#define FOV2FOCAL(pixelssensorsize, fov) ((pixelssensorsize) / (2 * tan((fov) / 2)))// http://books.google.co.il/books?id=bXzAlkODwa8C&pg=PA48&lpg=PA48&dq=expressing+focal+length+in+pixels&source=bl&ots=gY4972kxAC&sig=U1BUeNHhOHmYIrDrO0YDb1DrNng&hl=en&sa=X&ei=45dLU9u9DIyv7QbN2oGIDA&ved=0CGsQ6AEwCA#v=onepage&q=expressing%20focal%20length%20in%20pixels&f=false
// http://nghiaho.com/?page_id=846
void DecomposeRotation(IN const cv::Mat& R, OUT float& fX, OUT float& fY, OUT float& fZ) {// Taken from MatLab
    fX = (float)atan2(R.at<double>(2, 1), R.at<double>(2, 2));
    fY = (float)atan2(-R.at<double>(2, 0), sqrt(R.at<double>(2, 1)*R.at<double>(2, 1) + R.at<double>(2, 2)*R.at<double>(2, 2)));
    fZ = (float)atan2(R.at<double>(1, 0), R.at<double>(0, 0));
}
int _tmain(int argc, _TCHAR* argv[])
{
    // 25 deg rotation in the Z axis (800x600)
    const cv::Point2f rotZ0[] = { { 109, 250 }, { 175, 266 }, { 204, 279 }, { 221, 253 }, { 324, 281 }, { 312, 319 }, { 328, 352 }, { 322, 365 } };
    const cv::Point2f rotZ25[] = { { 510, 234 }, { 569, 622 }, { 593, 278 }, { 616, 257 }, { 716, 303 }, { 698, 340 }, { 707, 377 }, { 697, 390 } };
    const cv::Point2f rotZminus15[] = { { 37, 260 }, { 106, 275 }, { 135, 286 }, { 152, 260 }, { 258, 284 }, { 248, 324 }, { 266, 356 }, { 260, 370 } };

    const double        dFOV = DEG2RAD(45);
    const cv::Point2d   res(800, 600);
    const cv::Point2d   pntPriciplePoint(res.x / 2, res.y / 2);
    const cv::Point2d   pntFocal(FOV2FOCAL(res.x, dFOV), FOV2FOCAL(res.y, dFOV));
    //transfer the vector of points to the appropriate opencv matrix structures
    const int                numPoints = sizeof(rotZ0) / sizeof(rotZ0[0]);
    std::vector<cv::Point2f> vecPnt1(numPoints);
    std::vector<cv::Point2f> vecPnt2(numPoints);
    for (int i = 0; i < numPoints; i++) {
        vecPnt2[i] = rotZ0[i];
        //vecPnt2[i] = rotZminus15[i];
        vecPnt1[i] = rotZ25[i];
    }
    //// Normalize points
    //for (int i = 0; i < numPoints; i++) {
    //  vecPnt1[i].x = (vecPnt1[i].x - pntPriciplePoint.x) / pntFocal.x;
    //  vecPnt1[i].y = (vecPnt1[i].y - pntPriciplePoint.y) / pntFocal.y;
    //  vecPnt2[i].x = (vecPnt2[i].x - pntPriciplePoint.x) / pntFocal.x;
    //  vecPnt2[i].y = (vecPnt2[i].y - pntPriciplePoint.y) / pntFocal.y;
    //}
    try {
        // http://docs.opencv.org/modules/calib3d/doc/camera_calibration_and_3d_reconstruction.html
        cv::Mat matK = cv::Mat::zeros(3, 3, CV_64F);
        matK.at<double>(0, 0) = pntFocal.x;
        matK.at<double>(1, 1) = pntFocal.y;
        matK.at<double>(0, 2) = pntPriciplePoint.x;
        matK.at<double>(1, 2) = pntPriciplePoint.y;
        matK.at<double>(2, 2) = 1;
        float x, y, z;
        cv::Mat R1, R2, R3, R4;
        cv::Mat t;
        cv::Mat matE;
#if 1 // Option [A]
        cv::Mat matF = cv::findFundamentalMat(vecPnt1, vecPnt2);
        matE = matK.t() * matF * matK; // http://en.wikipedia.org/wiki/Essential_matrix
        cv::Mat _tmp;
        cv::Mat U;
        cv::Mat Vt;
        cv::SVD::compute(matE, _tmp, U, Vt);
        cv::Matx33d W(0, -1, 0,
                      1,  0, 0,
                      0,  0, 1);
        R1 = U*cv::Mat(W)*Vt; // See http://stackoverflow.com/questions/14150152/extract-translation-and-rotation-from-fundamental-matrix for details
        R2 = U*cv::Mat(W)*Vt.t();
        R3 = U*cv::Mat(W).t()*Vt;
        R4 = U*cv::Mat(W).t()*Vt.t();
#else // Option [B] 
        matE = cv::findEssentialMat(vecPnt1, vecPnt2, pntFocal.x, pntPriciplePoint);// http://docs.opencv.org/trunk/modules/calib3d/doc/camera_calibration_and_3d_reconstruction.html
        cv::decomposeEssentialMat(matE, R1, R2, t);
        int iInliers = cv::recoverPose(matE, vecPnt1, vecPnt2, R4, t);// , pntFocal.x, pntPriciplePoint);
        R3 = cv::Mat::zeros(3, 3, CV_64F);
#endif
        DecomposeRotation(R1, x, y, z);
        std::cout << "Euler Angles R1 (X,Y,Z): " << RAD2DEG(x) << ", " << RAD2DEG(y) << ", " << RAD2DEG(z) << std::endl;
        DecomposeRotation(R2, x, y, z);
        std::cout << "             R2 (X,Y,Z): " << RAD2DEG(x) << ", " << RAD2DEG(y) << ", " << RAD2DEG(z) << std::endl;
        DecomposeRotation(R3, x, y, z);
        std::cout << "             R3 (X,Y,Z): " << RAD2DEG(x) << ", " << RAD2DEG(y) << ", " << RAD2DEG(z) << std::endl;
        DecomposeRotation(R4, x, y, z);
        std::cout << "             R4 (X,Y,Z): " << RAD2DEG(x) << ", " << RAD2DEG(y) << ", " << RAD2DEG(z) << std::endl;
        //cv::Mat res = matFrom.t() * matF * matTo;// Results in a null vector ( as it should ) http://en.wikipedia.org/wiki/Fundamental_matrix_(computer_vision)
        //res = matFrom.t() * matE * matTo;// Results in a null vector ( as it should )
    }
    catch (cv::Exception e) {
        _ASSERT(FALSE);
    }
    return 0;
}

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执行结果

选项A:

Euler Angles R1 (X,Y,Z): -26.2625, 8.70029, 163.643
             R2 (X,Y,Z): 16.6929, -29.9901, -3.81642
             R3 (X,Y,Z): 5.59033, -20.841, -19.9316
             R4 (X,Y,Z): -5.76906, 7.25413, -179.086

选项B:

Euler Angles R1 (X,Y,Z): -13.8355, 3.0098, 171.451
             R2 (X,Y,Z): 2.22802, -22.3479, -11.332
             R3 (X,Y,Z): 0, -0, 0
             R4 (X,Y,Z): 2.22802, -22.3479, -11.332