OpenCV 根据对应的三维点估计刚体变换的旋转平移矩阵（RT矩阵）

OpenCV通过estimateAffine3D() 提供了三维仿射变换模型的最小二乘估计方法，但是遗憾的是没有提供三维刚体变换模型（即旋转/平移矩阵，RT矩阵）的估计方法。

下面的代码提供了对该方法的一种实现。

struct TRigidTrans3D
{
double matR[9];

double X;
double Y;
double Z;
};

void GetRigidTrans3D(cv::Point3f* srcPoints, cv::Point3f* dstPoints, int pointsNum, TRigidTrans3D& transform)
{
double srcSumX = 0.0f;
double srcSumY = 0.0f;
double srcSumZ = 0.0f;

double dstSumX = 0.0f;
double dstSumY = 0.0f;
double dstSumZ = 0.0f;

for (int i = 0; i < pointsNum; ++ i)
{
srcSumX += srcPoints[i].x;
srcSumY += srcPoints[i].y;
srcSumZ += srcPoints[i].z;

dstSumX += dstPoints[i].x;
dstSumY += dstPoints[i].y;
dstSumZ += dstPoints[i].z;
}

cv::Point3f centerSrc, centerDst;

centerSrc.x = float(srcSumX / pointsNum);
centerSrc.y = float(srcSumY / pointsNum);
centerSrc.z = float(srcSumZ / pointsNum);

centerDst.x = float(dstSumX / pointsNum);
centerDst.y = float(dstSumY / pointsNum);
centerDst.z = float(dstSumZ / pointsNum);

cv::Mat srcMat(3, pointsNum, CV_32FC1);
cv::Mat dstMat(3, pointsNum, CV_32FC1);

float* srcDat = (float*)(srcMat.data);
float* dstDat = (float*)(dstMat.data);
for (int i = 0; i < pointsNum; ++ i)
{
srcDat[i] = srcPoints[i].x - centerSrc.x;
srcDat[pointsNum + i] = srcPoints[i].y - centerSrc.y;
srcDat[pointsNum * 2 + i] = srcPoints[i].z - centerSrc.z;

dstDat[i] = dstPoints[i].x - centerDst.x;
dstDat[pointsNum + i] = dstPoints[i].y - centerDst.y;
dstDat[pointsNum * 2 + i] = dstPoints[i].z - centerDst.z;
}

cv::Mat matS = srcMat * dstMat.t();

cv::Mat matU, matW, matV;
cv::SVDecomp(matS, matW, matU, matV);

cv::Mat matTemp = matU * matV;
double det = cv::determinant(matTemp);

double datM[] = {1, 0, 0, 0, 1, 0, 0, 0, det};
cv::Mat matM(3, 3, CV_64FC1, datM);

cv::Mat matR = matV.t() * matM * matU.t();

memcpy(transform.matR, matR.data, sizeof(double) * 9);

double* datR = (double*)(matR.data);
transform.X = centerDst.x - (centerSrc.x * datR[0] + centerSrc.y * datR[1] + centerSrc.z * datR[2]);
transform.Y = centerDst.y - (centerSrc.x * datR[3] + centerSrc.y * datR[4] + centerSrc.z * datR[5]);
transform.Z = centerDst.z - (centerSrc.x * datR[6] + centerSrc.y * datR[7] + centerSrc.z * datR[8]);
}