1. 简介
计算机图形学中的应用非常广泛的变换是一种称为仿射变换的特殊变换，在仿射变换中的基本变换包括平移、旋转、缩放、剪切这几种。本文以及接下来的几篇文章重点介绍一下关于旋转的变换，包括二维旋转变换、三维旋转变换以及它的一些表达方式（旋转矩阵、四元数、欧拉角等）。
2. 绕原点二维旋转
首先要明确旋转在二维中是绕着某一个点进行旋转，三维中是绕着某一个轴进行旋转。二维旋转中最简单的场景是绕着坐标原点进行的旋转，如下图所示：
如图所示点v 绕 原点旋转θθ 角，得到点v’，假设 v点的坐标是(x, y) ，那么可以推导得到 v’点的坐标（x’, y’)(设原点到v的距离是r，原点到v点的向量与x轴的夹角是ϕϕ )
尽管图示中仅仅表示的是旋转一个锐角θθ的情形，但是我们推导中使用的是三角函数的基本定义来计算坐标的，因此当旋转的角度是任意角度（例如大于180度，导致v’点进入到第四象限）结论仍然是成立的。
旋转和平移 代码1
Rx 可以通过 getRotationMatrix2D 得到
Point center(face_img.cols/2, face_img.rows/2);
//cv::Mat rot_mat = cv::getRotationMatrix2D(center, -1 * arctan, 1.0);
cv::Mat Rx(2, 3, CV_32FC1);
double theta_r = roll * 3.1415926 / 180; /** 3.1415926 / 180*/
float cos_theta = cos(theta_r);
float sin_theta = sin(theta_r);
Rx.at<float>(0, 0) = cos_theta;
Rx.at<float>(0, 1) = -sin_theta;
Rx.at<float>(0, 2) = (1-cos_theta)*center.x + center.y * sin_theta;
Rx.at<float>(1, 0) = sin_theta;
Rx.at<float>(1, 1) = cos_theta;
Rx.at<float>(1, 2) = (1-cos_theta) * center.y - center.x* sin_theta;
//std::cout << rot_mat << std::endl;
cv::Mat rotated_ROI;
cv::warpAffine(face_img, rotated_ROI, Rx, face_img.size(), cv::INTER_LINEAR, cv::BORDER_CONSTANT, cv::Scalar::all(0));
3. 绕任意点的二维旋转
绕原点的旋转是二维旋转最基本的情况，当我们需要进行绕任意点旋转时，我们可以把这种情况转换到绕原点的旋转，思路如下：
1. 首先将旋转点移动到原点处
2. 执行如2所描述的绕原点的旋转
3. 再将旋转点移回到原来的位置
旋转和平移 代码2
Point center(face_img.cols/2, face_img.rows/2);
//cv::Mat rot_mat = cv::getRotationMatrix2D(center, -1 * arctan, 1.0);
cv::Mat Rx(3, 3, CV_32FC1);
double theta_r = roll * 3.1415926 / 180; /** 3.1415926 / 180*/
float cos_theta = cos(theta_r);
float sin_theta = sin(theta_r);
Rx.at<float>(0, 0) = cos_theta;
Rx.at<float>(0, 1) = -sin_theta;
Rx.at<float>(0, 2) = (1-cos_theta)*center.x + center.y * sin_theta;
Rx.at<float>(1, 0) = sin_theta;
Rx.at<float>(1, 1) = cos_theta;
Rx.at<float>(1, 2) = (1-cos_theta) * center.y - center.x* sin_theta;
Rx.at<float>(2, 0) = 0;
Rx.at<float>(2, 1) = 0;
Rx.at<float>(2, 2) = 1;
//std::cout << rot_mat << std::endl;
cv::Mat rotated_ROI;
//cv::warpAffine(face_img, rotated_ROI, rot_mat, face_img.size(), cv::INTER_LINEAR, cv::BORDER_CONSTANT, cv::Scalar::all(0));
warpPerspective(face_img, rotated_ROI, Rx, cv::Size(face_img.cols, face_img.rows));
cv::imshow("roll face", rotated_ROI);
绕X 轴 Y轴 Z轴旋转的结果
void warp_perspect_3_angle(cv::Mat face, float roll, float yaw, float pitch) {
cv::Mat face_img = face.clone();
int imgHeight = face_img.rows;
int imgWidth = face_img.cols;
float alpha, beta, gamma;
alpha = pitch * 3.1415926 / 180;
beta = yaw* 3.1415926 / 180;
gamma = roll * 3.1415926 / 180;
Mat Rot = Mat::eye(3, 3, CV_32FC1);
Rot.at<float>(0, 0) = cos(beta) * cos(gamma);
Rot.at<float>(0, 1) = cos(beta) * sin(gamma);
Rot.at<float>(0, 2) = -sin(beta);
Rot.at<float>(1, 0) = sin(alpha) * sin(beta) * cos(gamma) - cos(alpha) * sin(gamma);
Rot.at<float>(1, 1) = sin(alpha) * sin(beta) * sin(gamma) + cos(alpha) * cos(gamma);
Rot.at<float>(1, 2) = sin(alpha) * cos(beta);
Rot.at<float>(2, 0) = cos(alpha) * sin(beta) * cos(gamma) + sin(alpha) * sin(gamma);
Rot.at<float>(2, 1) = cos(alpha) * sin(beta) * sin(gamma) - sin(alpha) * cos(gamma);
Rot.at<float>(2, 2) = cos(alpha) * cos(beta);
Mat invRot;
invert(Rot, invRot, DECOMP_SVD);
float fx = imgWidth/2;
float fy = imgHeight/2;
float cx = imgWidth / 2;
float cy = imgHeight / 2;
Mat point3D = Mat::zeros(3, 1, CV_32FC1);
Mat oldPoint3D = Mat::zeros(3, 1, CV_32FC1);
Mat dstImg = face_img.clone();
dstImg.setTo(0);
uchar* pImgData = (uchar*)face_img.data;
uchar* pDstData = (uchar*)dstImg.data;
for (int j = 0; j < imgHeight; j++)
{
for (int i = 0; i < imgWidth; i++)
{
float X = (i - cx) / fx;
float Y = (j - cy) / fy;
float Z = 1;
point3D.at<float>(0, 0) = X;
point3D.at<float>(1, 0) = Y;
point3D.at<float>(2, 0) = Z;
//求旋转前坐标点
oldPoint3D = invRot*point3D;
float oldX = oldPoint3D.at<float>(0, 0);
float oldY = oldPoint3D.at<float>(1, 0);
float oldZ = oldPoint3D.at<float>(2, 0);
//重投影到二维平面
if (oldZ > 1e-3)
{
float u = ((fx*oldX + cx*oldZ) / oldZ);
float v = ((fy*oldY + cy*oldZ) / oldZ);
int u0 = floor(u);
int v0 = floor(v);
int u1 = u0 + 1;
int v1 = v0 + 1;
if (u0 >= 0 && v0 >= 0 && u1 < imgWidth && v1 < imgHeight)
{
float dx = u - u0;
float dy = v - v0;
float weight1 = (1 - dx)*(1 - dy);
float weight2 = dx*(1 - dy);
float weight3 = (1 - dx)*dy;
float weight4 = dx*dy;
pDstData[j*imgWidth * 3 + i * 3 + 0] = weight1*pImgData[v0*imgWidth * 3 + u0 * 3 + 0] +
weight2*pImgData[v0*imgWidth * 3 + u1 * 3 + 0] +
weight3*pImgData[v1*imgWidth * 3 + u0 * 3 + 0] +
weight4*pImgData[v1*imgWidth * 3 + u1 * 3 + 0];
pDstData[j*imgWidth * 3 + i * 3 + 1] = weight1*pImgData[v0*imgWidth * 3 + u0 * 3 + 1] +
weight2*pImgData[v0*imgWidth * 3 + u1 * 3 + 1] +
weight3*pImgData[v1*imgWidth * 3 + u0 * 3 + 1] +
weight4*pImgData[v1*imgWidth * 3 + u1 * 3 + 1];
pDstData[j*imgWidth * 3 + i * 3 + 2] = weight1*pImgData[v0*imgWidth * 3 + u0 * 3 + 2] +
weight2*pImgData[v0*imgWidth * 3 + u1 * 3 + 2] +
weight3*pImgData[v1*imgWidth * 3 + u0 * 3 + 2] +
weight4*pImgData[v1*imgWidth * 3 + u1 * 3 + 2];
}
}
}
}
imshow("show", dstImg);
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