圖像旋轉是非常常見的圖像變換，通常應用于圖像矯正，在OpenCV可以使用密集仿射變換函數cv::warpAffine()實現圖像旋轉。為了理解圖像旋轉的原理，本文實現了一個圖像旋轉算法。

圖像旋轉是指將圖像繞某個中心點旋轉一定角度后，得到一幅新的圖像。圖像旋轉的示意圖如圖1所示。其中，四邊形ABCD表示需要旋轉的圖像區域，它經過旋轉角度后得到的圖像區域為四邊形 A'B'C'D'。點p(x,y)為圖像內任意一點，它經過旋轉角度后對應的點為p'(x',y')。

圖1 圖像旋轉示意圖

圖像是如何進行旋轉的？通常這個過程有三個步驟。

第一步，把圖像內的坐標點繞旋轉中心點旋轉到對應的坐標上。由于圖像是通過二維數組進行保存的，所以圖像的坐標點一定要落在坐標系的第一象限內，并且要保證它們是整數坐標點。通常情況下，進行旋轉后得到的坐標點不是整數點也不一定在第一象限內，因此需要對旋轉后得到的點進行平移和取整，使得它們都是落在第一象限內的整數點。

圖像內任意一點p(x,y)繞某個旋轉點(X,Y)逆時針旋轉角度后得到點p'(x',y')的計算公式如下：

?點旋轉的C++實現代碼如下。（OpenCV3.45+VS2019）

//將點point2繞點point1逆時針旋轉angle度后得到新的點newPoint void 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) //做一個四舍五入取整dx = (int)dx1 + 1;else{if (dx1 - (int)dx1 < -0.5)dx = (int)dx1 - 1;elsedx = (int)(dx1);}if (dy1 - (int)dy1 > 0.5) //做一個四舍五入取整dy = (int)dy1 + 1;else{if (dy1 - (int)dy1 < -0.5)dy = (int)dy1 - 1;elsedy = (int)(dy1);}newPoint.x = point1.x + dx;newPoint.y = point1.y + dy; }

用來平移坐標點的代碼如下。

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

注：平移量x與y的大小，可以根據旋轉后圖像的四個頂點A'、B'、C'、D'獲得。

第二步，把圖像對應的坐標像素大小賦給旋轉后的坐標。即，圖像內任意點p(x,y)對應的像素值為I(x,y)，那它旋轉后得到的點p'(x',y')的像素值I(x',y')=I(x,y)。

如下圖2所示。當我們需要旋轉的圖像區域在圖片內時（這區域也可以是整張圖片），如何確定旋轉區域ABCD是很重要的，只有這樣才能判斷整張圖片內的哪些點是四邊形ABCD區域內的。

?圖2 圖片中要旋轉的區域

我們以圖片的左上頂點為原點建立如圖2所示的坐標系，其中四邊形ABCD的四個頂點是已知的，分別為A(x0,y0)、B(x1,y1)、C(x2,y2)、D(x3,y3)。這時根據兩點式可得到四條邊的直線方程如下：

根據線性規劃的知識，可以通過直線方程來表示四邊形ABCD的區域。

注：因為四邊形ABCD內的任意點p在直線AB上方，所以直線AB方程大于等于0；點p在直線BC左側，所以直線BC方程小于等于0。同理可得，直線CD方程小于等于0、直線AD方程大于等于0。

實現代碼塊如下：

std::vector<cv::Point> newPoints;cv::Point newP;for (int i = 0; i < 4; ++i){if (points[i] != point) //判斷輸入的4個頂點是否與旋轉點point相同{rotatePoint(point, points[i], newP, angle); //頂點points[i]與旋轉點point不同，則進行旋轉計算newPoints.push_back(newP);}else{newPoints.push_back(points[i]);}}//獲取經旋轉后，新圖像的大小，其中w表示圖像寬長，h表示圖像高長。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);//獲取需要旋轉的四邊形區域的外接矩形表示區域范圍（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);//計算向x軸的平移量dx,向y軸的平移量dyint 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);//初始化輸出矩陣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);//實現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){//四邊形頂點A為points[0],頂點B為points[1],頂點C為points[2],頂點D為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); //將點point0繞點point旋轉angle度得到新的點point0translationPoint(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，對旋轉后的圖像進行插值。由于在第一步中對旋轉后的點進行了取整，這難免會使得新圖像存在間隙，所以需要對這些間隙進行填充。在OpenCV中常用的插值方法有以下5種：

圖3 常見的插值方法?

在本文中采用的插值方法與最近鄰插值類似，即把最近四個方向（上下左右）的平均值作為插值。

//灰度圖(CV_8UC1)的插值代碼 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;}}} ///彩色圖(CV_8UC3)的插值代碼 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);}}}

整個算法的完整代碼如下：

#include<iostream> #include<opencv2/opencv.hpp>//計算點point2繞點point1逆時針旋轉angle度后得到新的點newPoint void 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) //做一個四舍五入dx = (int)dx1 + 1;else{if (dx1 - (int)dx1 < -0.5)dx = (int)dx1 - 1;elsedx = (int)(dx1);}if (dy1 - (int)dy1 > 0.5) //做一個四舍五入dy = (int)dy1 + 1;else{if (dy1 - (int)dy1 < -0.5)dy = (int)dy1 - 1;elsedy = (int)(dy1);}newPoint.x = point1.x + dx;newPoint.y = point1.y + dy; }void translationPoint(cv::Point& point, int x, int y) //平移運算 {point.x = point.x + x;point.y = point.y + y; }int Max4(int a[4]) //獲取四個數中的最大值 {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]) //獲取四個數中的最小值 {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) //判斷輸入的4個頂點是否與旋轉點point相同{rotatePoint(point, points[i], newP, angle); //頂點points[i]與旋轉點point不同，則進行旋轉計算newPoints.push_back(newP);}else{newPoints.push_back(points[i]);}}//獲取經旋轉后，新圖像的大小，其中w表示圖像寬長，h表示圖像高長。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);//獲取需要旋轉的四邊形區域的外接矩形表示區域范圍（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);//計算向x軸的平移量dx,向y軸的平移量dyint 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);//初始化輸出矩陣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);//實現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){//四邊形頂點A為points[0],頂點B為points[1],頂點C為points[2],頂點D為points[3].//直線ABz1 = i - (double)points[0].y -(j - (double)points[0].x) * ((double)points[0].y - points[1].y) / ((double)points[0].x - points[1].x);//直線BCz2 = j - (double)points[1].x -(i - (double)points[1].y) * ((double)points[1].x - points[2].x) / ((double)points[1].y - points[2].y);//直線CDz3 = i - (double)points[2].y -(j - (double)points[2].x) * ((double)points[2].y - points[3].y) / ((double)points[2].x - points[3].x);//直線ADz4 = 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); //將點point0繞點point旋轉angle度得到新的點point0translationPoint(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:/圖像處理/圖片一/thee2.jpg");//cv::Mat srcImage = cv::imread("F:/圖像處理/圖片一/thee2.jpg", 0);if (srcImage.empty()){printf("圖片讀取失敗！\n");return -1;}//需要旋轉的圖像區域四個頂點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("原圖", srcImage);cv::imshow("旋轉得到的圖像", outputImage);cv::waitKey(0);return 0; }

算法測試：

輸入的圖像(大小460X613)：

?

//旋轉區域1，其中srcImage為輸入圖像 std::vector<cv::Point> points;points.push_back(cv::Point(0, 0)); //頂點Apoints.push_back(cv::Point(srcImage.cols, 0)); //頂點Bpoints.push_back(cv::Point(srcImage.cols, srcImage.rows)); //頂點Cpoints.push_back(cv::Point(0, srcImage.rows)); //旋轉點為Point(40, 70)，旋轉角度0.4

旋轉得到的結果如下：

?

//旋轉區域2，其中srcImage為輸入圖像 std::vector<cv::Point> points;points.push_back(cv::Point(70, 0)); //頂點Apoints.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)); //旋轉點為Point(30, 50)，旋轉角度0.2

旋轉得到的結果如下：

?

總結

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