图像旋转是非常常见的图像变换，通常应用于图像矫正，在OpenCV可以使用密集仿射变换函数cv::warpAffine()实现图像旋转。为了理解图像旋转的原理，本文实现了一个图像旋转算法。

图像旋转是指将图像绕某个中心点旋转一定角度后，得到一幅新的图像。图像旋转的示意图如图1所示。其中，四边形ABCD表示需要旋转的图像区域，它经过旋转角度后得到的图像区域为四边形 A'B'C'D'。点p(x,y)为图像内任意一点，它经过旋转角度后对应的点为p'(x',y')。

图1 图像旋转示意图

图像是如何进行旋转的？通常这个过程有三个步骤。

第一步，把图像内的坐标点绕旋转中心点旋转到对应的坐标上。由于图像是通过二维数组进行保存的，所以图像的坐标点一定要落在坐标系的第一象限内，并且要保证它们是整数坐标点。通常情况下，进行旋转后得到的坐标点不是整数点也不一定在第一象限内，因此需要对旋转后得到的点进行平移和取整，使得它们都是落在第一象限内的整数点。

图像内任意一点p(x,y)绕某个旋转点(X,Y)逆时针旋转角度后得到点p'(x',y')的计算公式如下：

点旋转的C++实现代码如下。（OpenCV3.45+VS2019）

//将点point2绕点point1逆时针旋转angle度后得到新的点newPointvoid rotatePoint(cv::Point& point1, cv::Point& point2, cv::Point& newPoint, double angle){	int dx, dy;	double dx1, dy1;	dy1 = -((double)point2.x - point1.x) * sin(angle) + ((double)point2.y - point1.y) * cos(angle);	dx1 = ((double)point2.x - point1.x) * cos(angle) + ((double)point2.y - point1.y) * sin(angle);	if (dx1 - (int)dx1 > 0.5)    //做一个四舍五入取整		dx = (int)dx1 + 1;	else	{		if (dx1 - (int)dx1 < -0.5)			dx = (int)dx1 - 1;		else			dx = (int)(dx1);	}	if (dy1 - (int)dy1 > 0.5)   //做一个四舍五入取整		dy = (int)dy1 + 1;	else	{		if (dy1 - (int)dy1 < -0.5)			dy = (int)dy1 - 1;		else			dy = (int)(dy1);	}	newPoint.x = point1.x + dx;	newPoint.y = point1.y + dy;}

用来平移坐标点的代码如下。

void translationPoint(cv::Point& point, int x, int y) //平移运算{	point.x = point.x + x;	point.y = point.y + y;}

注：平移量x与y的大小，可以根据旋转后图像的四个顶点A'、B'、C'、D'获得。

第二步，把图像对应的坐标像素大小赋给旋转后的坐标。即，图像内任意点p(x,y)对应的像素值为I(x,y)，那它旋转后得到的点p'(x',y')的像素值I(x',y')=I(x,y)。

如下图2所示。当我们需要旋转的图像区域在图片内时（这区域也可以是整张图片），如何确定旋转区域ABCD是很重要的，只有这样才能判断整张图片内的哪些点是四边形ABCD区域内的。

图2 图片中要旋转的区域

我们以图片的左上顶点为原点建立如图2所示的坐标系，其中四边形ABCD的四个顶点是已知的，分别为A(x0,y0)、B(x1,y1)、C(x2,y2)、D(x3,y3)。这时根据两点式可得到四条边的直线方程如下：

根据线性规划的知识，可以通过直线方程来表示四边形ABCD的区域。

注：因为四边形ABCD内的任意点p在直线AB上方，所以直线AB方程大于等于0；点p在直线BC左侧，所以直线BC方程小于等于0。同理可得，直线CD方程小于等于0、直线AD方程大于等于0。

实现代码块如下：

std::vector<cv::Point> newPoints;	cv::Point newP;	for (int i = 0; i < 4; ++i)	{		if (points[i] != point)  //判断输入的4个顶点是否与旋转点point相同		{			rotatePoint(point, points[i], newP, angle);  //顶点points[i]与旋转点point不同，则进行旋转计算			newPoints.push_back(newP);		}		else		{			newPoints.push_back(points[i]);		}	}	//获取经旋转后，新图像的大小，其中w表示图像宽长，h表示图像高长。	int w = 0, h = 0;	int suw[4] = { newPoints[1].x - newPoints[0].x,newPoints[1].x - newPoints[3].x,		newPoints[2].x - newPoints[0].x,newPoints[2].x - newPoints[3].x };	int suh[4] = { newPoints[2].y - newPoints[0].y ,newPoints[2].y - newPoints[1].y,		newPoints[3].y - newPoints[0].y,newPoints[3].y - newPoints[1].y };	w = absMax4(suw);	h = absMax4(suh);	//获取需要旋转的四边形区域的外接矩形表示区域范围（x_min,y_min)、(x_max,y_max)	int y_max, y_min, x_max, x_min;	int points_x[4] = { points[0].x,points[1].x,points[2].x,points[3].x };	int points_y[4] = { points[0].y,points[1].y,points[2].y,points[3].y };	y_max = Max4(points_y);	y_min = Min4(points_y);	x_max = Max4(points_x);	x_min = Min4(points_x);	//计算向x轴的平移量dx,向y轴的平移量dy	int dx, dy;	int a[4] = { newPoints[0].x,newPoints[1].x,newPoints[2].x,newPoints[3].x };	int b[4] = { newPoints[0].y,newPoints[1].y,newPoints[2].y,newPoints[3].y };	dx = Min4(a);	dy = Min4(b);	//初始化输出矩阵	if(inputMat.type() == CV_8UC1)		cv::Mat(h, w, CV_8UC1, cv::Scalar::all(255)).copyTo(outputMat);	if(inputMat.type() == CV_8UC3)		cv::Mat(h, w, CV_8UC3, cv::Scalar(255, 255, 255)).copyTo(outputMat);  //实现I(x',y')=I(x,y)	double z1, z2, z3, z4;	for (int i = y_min; i < y_max; ++i)	{		for (int j = x_min; j < x_max; ++j)		{				//四边形顶点A为points[0],顶点B为points[1],顶点C为points[2],顶点D为points[3].			z1 = i - (double)points[0].y -				(j - (double)points[0].x) * ((double)points[0].y - points[1].y) / ((double)points[0].x - points[1].x);			z2 = j - (double)points[1].x -				(i - (double)points[1].y) * ((double)points[1].x - points[2].x) / ((double)points[1].y - points[2].y);			z3 = i - (double)points[2].y -				(j - (double)points[2].x) * ((double)points[2].y - points[3].y) / ((double)points[2].x - points[3].x);			z4 = j - (double)points[0].x -				(i - (double)points[0].y) * ((double)points[0].x - points[3].x) / ((double)points[0].y - points[3].y);			if (z1 >= 0 && z2 <= 0 && z3 <= 0 && z4 >= 0)			{				cv::Point point0(j, i);				rotatePoint(point, point0, point0, angle);  //将点point0绕点point旋转angle度得到新的点point0				translationPoint(point0, -dx, -dy);     //平移				if (point0.x >= 0 && point0.x < w && point0.y >= 0 && point0.y < h)				{					if (inputMat.type() == CV_8UC1)					{						uchar* str = inputMat.ptr<uchar>(i);						outputMat.at<uchar>(point0.y, point0.x) = str[j];					}					if (inputMat.type() == CV_8UC3)					{						cv::Vec3b* str = inputMat.ptr<cv::Vec3b>(i);						outputMat.at<cv::Vec3b>(point0.y, point0.x) = str[j];					}				}			}		}	}

第三步，对旋转后的图像进行插值。由于在第一步中对旋转后的点进行了取整，这难免会使得新图像存在间隙，所以需要对这些间隙进行填充。在OpenCV中常用的插值方法有以下5种：

图3 常见的插值方法

在本文中采用的插值方法与最近邻插值类似，即把最近四个方向（上下左右）的平均值作为插值。

//灰度图(CV_8UC1)的插值代码for (int i = 1; i < outputMat.rows - 1; ++i)		{			for (int j = 1; j < outputMat.cols - 1; ++j)			{				if (outputMat.at<uchar>(i, j) == 255)				{					int sum = 0;					uchar* str1 = outputMat.ptr<uchar>(i - 1);					sum = str1[j - 1] + str1[j] + str1[j + 1];					uchar* str2 = outputMat.ptr<uchar>(i);					sum = sum + str2[j - 1] + str2[j + 1];					uchar* str3 = outputMat.ptr<uchar>(i + 1);					sum = sum + str3[j - 1] + str3[j] + str3[j + 1];					sum = sum / 8;					outputMat.at<uchar>(i, j) = (uchar)sum;				}			}		}///彩色图(CV_8UC3)的插值代码for (int i = 1; i < outputMat.rows - 1; ++i)		{			for (int j = 1; j < outputMat.cols - 1; ++j)			{				if (outputMat.at<cv::Vec3b>(i, j) == cv::Vec3b(255, 255, 255))				{					int sum[3] = { 0,0,0 };					uchar r, g, b;					for (int k = 0; k < 3; k++)					{						cv::Vec3b* str1 = outputMat.ptr<cv::Vec3b>(i - 1);						sum[k] = str1[j - 1][k] + str1[j][k] + str1[j + 1][k];						cv::Vec3b* str2 = outputMat.ptr<cv::Vec3b>(i);						sum[k] = sum[k] + str2[j - 1][k] + str2[j + 1][k];						cv::Vec3b* str3 = outputMat.ptr<cv::Vec3b>(i + 1);						sum[k] = sum[k] + str3[j - 1][k] + str3[j][k] + str3[j + 1][k];						sum[k] = sum[k] / 8;					}					r = (uchar)sum[0];					g = (uchar)sum[1];					b = (uchar)sum[2];					outputMat.at<cv::Vec3b>(i, j) = cv::Vec3b(r, g, b);				}			}		}

整个算法的完整代码如下：

#include<iostream>#include<opencv2/opencv.hpp>    //计算点point2绕点point1逆时针旋转angle度后得到新的点newPointvoid rotatePoint(cv::Point& point1, cv::Point& point2, cv::Point& newPoint, double angle){	int dx, dy;	double dx1, dy1;	dy1 = -((double)point2.x - point1.x) * sin(angle) + ((double)point2.y - point1.y) * cos(angle);	dx1 = ((double)point2.x - point1.x) * cos(angle) + ((double)point2.y - point1.y) * sin(angle);	if (dx1 - (int)dx1 > 0.5)    //做一个四舍五入		dx = (int)dx1 + 1;	else	{		if (dx1 - (int)dx1 < -0.5)			dx = (int)dx1 - 1;		else			dx = (int)(dx1);	}	if (dy1 - (int)dy1 > 0.5)   //做一个四舍五入		dy = (int)dy1 + 1;	else	{		if (dy1 - (int)dy1 < -0.5)			dy = (int)dy1 - 1;		else			dy = (int)(dy1);	}	newPoint.x = point1.x + dx;	newPoint.y = point1.y + dy;}void translationPoint(cv::Point& point, int x, int y) //平移运算{	point.x = point.x + x;	point.y = point.y + y;}int Max4(int a[4])    //获取四个数中的最大值{	int max = a[0];	for (int i = 1; i < 4; i++)	{		if (max < a[i])			max = a[i];	}	return max;}int Min4(int a[4])    //获取四个数中的最小值{	int min = a[0];	for (int i = 1; i < 4; i++)	{		if (min > a[i])			min = a[i];	}	return min;}int absMax4(int a[4]){	int max = 0, m;	for (int i = 0; i < 4; i++)	{		if (a[i] < 0)			m = -a[i];		else m = a[i];		if (max < m)			max = m;	}	return max;}void rotateImage(cv::Mat inputMat, cv::Mat& outputMat, std::vector<cv::Point> points, cv::Point point, double angle){	std::vector<cv::Point> newPoints;	cv::Point newP;	for (int i = 0; i < 4; ++i)	{		if (points[i] != point)  //判断输入的4个顶点是否与旋转点point相同		{			rotatePoint(point, points[i], newP, angle);  //顶点points[i]与旋转点point不同，则进行旋转计算			newPoints.push_back(newP);		}		else		{			newPoints.push_back(points[i]);		}	}	//获取经旋转后，新图像的大小，其中w表示图像宽长，h表示图像高长。	int w = 0, h = 0;	int suw[4] = { newPoints[1].x - newPoints[0].x,newPoints[1].x - newPoints[3].x,		newPoints[2].x - newPoints[0].x,newPoints[2].x - newPoints[3].x };	int suh[4] = { newPoints[2].y - newPoints[0].y ,newPoints[2].y - newPoints[1].y,		newPoints[3].y - newPoints[0].y,newPoints[3].y - newPoints[1].y };	w = absMax4(suw);	h = absMax4(suh);	//获取需要旋转的四边形区域的外接矩形表示区域范围（x_min,y_min)、(x_max,y_max)	int y_max, y_min, x_max, x_min;	int points_x[4] = { points[0].x,points[1].x,points[2].x,points[3].x };	int points_y[4] = { points[0].y,points[1].y,points[2].y,points[3].y };	y_max = Max4(points_y);	y_min = Min4(points_y);	x_max = Max4(points_x);	x_min = Min4(points_x);	//计算向x轴的平移量dx,向y轴的平移量dy	int dx, dy;	int a[4] = { newPoints[0].x,newPoints[1].x,newPoints[2].x,newPoints[3].x };	int b[4] = { newPoints[0].y,newPoints[1].y,newPoints[2].y,newPoints[3].y };	dx = Min4(a);	dy = Min4(b);	//初始化输出矩阵	if(inputMat.type() == CV_8UC1)		cv::Mat(h, w, CV_8UC1, cv::Scalar::all(255)).copyTo(outputMat);	if(inputMat.type() == CV_8UC3)		cv::Mat(h, w, CV_8UC3, cv::Scalar(255, 255, 255)).copyTo(outputMat); //实现I(x',y')=I(x,y)	double z1, z2, z3, z4;	for (int i = y_min; i < y_max; ++i)	{		for (int j = x_min; j < x_max; ++j)		{				//四边形顶点A为points[0],顶点B为points[1],顶点C为points[2],顶点D为points[3].      //直线AB			z1 = i - (double)points[0].y -				(j - (double)points[0].x) * ((double)points[0].y - points[1].y) / ((double)points[0].x - points[1].x);			//直线BC			z2 = j - (double)points[1].x -				(i - (double)points[1].y) * ((double)points[1].x - points[2].x) / ((double)points[1].y - points[2].y);		  //直线CD			z3 = i - (double)points[2].y -				(j - (double)points[2].x) * ((double)points[2].y - points[3].y) / ((double)points[2].x - points[3].x);			//直线AD			z4 = j - (double)points[0].x -				(i - (double)points[0].y) * ((double)points[0].x - points[3].x) / ((double)points[0].y - points[3].y);			if (z1 >= 0 && z2 <= 0 && z3 <= 0 && z4 >= 0)			{				cv::Point point0(j, i);				rotatePoint(point, point0, point0, angle);  //将点point0绕点point旋转angle度得到新的点point0				translationPoint(point0, -dx, -dy);				if (point0.x >= 0 && point0.x < w && point0.y >= 0 && point0.y < h)				{					if (inputMat.type() == CV_8UC1)					{						uchar* str = inputMat.ptr<uchar>(i);						outputMat.at<uchar>(point0.y, point0.x) = str[j];					}					if (inputMat.type() == CV_8UC3)					{						cv::Vec3b* str = inputMat.ptr<cv::Vec3b>(i);						outputMat.at<cv::Vec3b>(point0.y, point0.x) = str[j];					}				}			}		}	}	if (inputMat.type() == CV_8UC1)	{   //插值		for (int i = 1; i < outputMat.rows - 1; ++i)		{			for (int j = 1; j < outputMat.cols - 1; ++j)			{				if (outputMat.at<uchar>(i, j) == 255)				{					int sum = 0;					uchar* str1 = outputMat.ptr<uchar>(i - 1);					sum = str1[j - 1] + str1[j] + str1[j + 1];					uchar* str2 = outputMat.ptr<uchar>(i);					sum = sum + str2[j - 1] + str2[j + 1];					uchar* str3 = outputMat.ptr<uchar>(i + 1);					sum = sum + str3[j - 1] + str3[j] + str3[j + 1];					sum = sum / 8;					outputMat.at<uchar>(i, j) = (uchar)sum;				}			}		}	}	if (inputMat.type() == CV_8UC3)	{   //插值		for (int i = 1; i < outputMat.rows - 1; ++i)		{			for (int j = 1; j < outputMat.cols - 1; ++j)			{				if (outputMat.at<cv::Vec3b>(i, j) == cv::Vec3b(255, 255, 255))				{					int sum[3] = { 0,0,0 };					uchar r, g, b;					for (int k = 0; k < 3; k++)					{						cv::Vec3b* str1 = outputMat.ptr<cv::Vec3b>(i - 1);						sum[k] = str1[j - 1][k] + str1[j][k] + str1[j + 1][k];						cv::Vec3b* str2 = outputMat.ptr<cv::Vec3b>(i);						sum[k] = sum[k] + str2[j - 1][k] + str2[j + 1][k];						cv::Vec3b* str3 = outputMat.ptr<cv::Vec3b>(i + 1);						sum[k] = sum[k] + str3[j - 1][k] + str3[j][k] + str3[j + 1][k];						sum[k] = sum[k] / 8;					}					r = (uchar)sum[0];					g = (uchar)sum[1];					b = (uchar)sum[2];					outputMat.at<cv::Vec3b>(i, j) = cv::Vec3b(r, g, b);				}			}		}	}}int main(){	cv::Mat srcImage = cv::imread("F:/图像处理/图片一/thee2.jpg");	//cv::Mat srcImage = cv::imread("F:/图像处理/图片一/thee2.jpg", 0);	if (srcImage.empty())	{		printf("图片读取失败！\n");		return -1;	}  //需要旋转的图像区域四个顶点	std::vector<cv::Point> points;  	points.push_back(cv::Point(0, 0));	points.push_back(cv::Point(srcImage.cols, 0));	points.push_back(cv::Point(srcImage.cols, srcImage.rows));	points.push_back(cv::Point(0, srcImage.rows));  	cv::Mat outputImage;	rotateImage(srcImage, outputImage, points, cv::Point(40, 70), 0.4);	cv::imshow("原图", srcImage);	cv::imshow("旋转得到的图像", outputImage);	cv::waitKey(0);	return 0;}

算法测试：

输入图像(大小460X613)：

输入图像

//旋转区域1，其中srcImage为输入图像std::vector<cv::Point> points;	points.push_back(cv::Point(0, 0));    //顶点A	points.push_back(cv::Point(srcImage.cols, 0));  //顶点B	points.push_back(cv::Point(srcImage.cols, srcImage.rows));  //顶点C	points.push_back(cv::Point(0, srcImage.rows));//旋转点为Point(40, 70)，旋转角度0.4

旋转得到的结果如下：

//旋转区域2，其中srcImage为输入图像std::vector<cv::Point> points;	points.push_back(cv::Point(70, 0));    //顶点A	points.push_back(cv::Point(srcImage.cols-50, 0));	points.push_back(cv::Point(srcImage.cols-100, srcImage.rows-100));	points.push_back(cv::Point(0, srcImage.rows-50));//旋转点为Point(30, 0)，旋转角度0.4

旋转得到的结果如下：