一.图像几何变换介绍
图像的几何空间变换是图像处理中的最基础的算法，是指对原始图像按需要改变其大小、形状和位置的变化，原始图像与目标函数之间的坐标变换函数为线性函数。二维图像的基本几何变换主要包括镜像、平移、缩放、旋转、错切(偏移)等操作，上述变换又称为仿射变换，在冈萨雷斯的数字图像处理第三版的第二章就做了相关介绍，数字图像处理第三版下载地址。
下面逐一介绍上述几种放射变换。
二.变换详解
常见几种变换公式如下表所示
在进行变换时，其中镜像、平移、缩放均以图像像素坐标系(图像左上顶点为坐标系原点)进行。
对于旋转和错切一般以图像中心为原点，此时在操作中需要加上坐标系转换公式。图像坐标系和旋转坐标系如下图所示，
旋转和错切变换分为3步，
将图像坐标转换到旋转坐标系；根据上述表格公式进行旋转或错切变换；将旋转坐标系转换到图像坐标系；
实际实现时可将三个矩阵合并进行计算，本文的实现即采用此种方法。
变换方式
前向映射
反向映射
由于旋转、缩放、变形变换中会出现漏点、不规则点问题，如果用前向映射，会导致得到的图像中有很多黑点，对于这种情况，通常采用反向映射。
反向映射需通过插值方法决定输出图像该位置的值，因此需要选择插值算法。通常有最近邻插值、双线性插值，双三次插值等，具体的差值方法、原理及优缺点自行百度吧，我们这里采用双线性插值。
三.C++实现
实现功能：镜像、旋转、平移、缩放、错切、组合变换。利用了OpenCV的Mat，指针实现的后期有时间再补充。编程环境：VS+OpenCV2.4.13
#include <opencv2/opencv.hpp>#include <iostream>using namespace std;using namespace cv;#define PI 3.1415927#define MAX(a,b) (((a)>(b))?(a):(b))// 单点双线性插值// [输入]ii--dst的行索引//jj--dst的列索引//u_src--jj反向映射到src中对应的列索引//v_src--ii反向映射到src中对应的行索引int Bilinear_interpolation_img(Mat src, Mat& dst, int ii, int jj, double u_src, double v_src){if (src.rows <= 0 || src.cols <= 0 || (src.channels() != 1 && src.channels() != 3) || src.depth() != CV_8U){printf("输入图像有误!\n");return 0;}if (u_src >= 0 && u_src <= src.cols - 1 && v_src >= 0 && v_src <= src.rows - 1){int x1 = int(u_src), x2 = (int)(u_src + 0.5), y1 = (int)v_src, y2 = (int)(v_src + 0.5);double pu = fabs(u_src - x1), pv = fabs(v_src - y2);if (src.channels() == 1){dst.at<uchar>(ii, jj) = (1 - pv)*(1 - pu)*src.at<uchar>(y2, x1) + (1 - pv)*pu*src.at<uchar>(y2, x2) +pv*(1 - pu)*src.at<uchar>(y1, x1) + pv*pu*src.at<uchar>(y1, x2);}else{dst.at<Vec3b>(ii, jj)[0] = (1 - pv)*(1 - pu)*src.at<Vec3b>(y2, x1)[0] + (1 - pv)*pu*src.at<Vec3b>(y2, x2)[0] +pv*(1 - pu)*src.at<Vec3b>(y1, x1)[0] + pv*pu*src.at<Vec3b>(y1, x2)[0];dst.at<Vec3b>(ii, jj)[1] = (1 - pv)*(1 - pu)*src.at<Vec3b>(y2, x1)[1] + (1 - pv)*pu*src.at<Vec3b>(y2, x2)[1] +pv*(1 - pu)*src.at<Vec3b>(y1, x1)[1] + pv*pu*src.at<Vec3b>(y1, x2)[1];dst.at<Vec3b>(ii, jj)[2] = (1 - pv)*(1 - pu)*src.at<Vec3b>(y2, x1)[2] + (1 - pv)*pu*src.at<Vec3b>(y2, x2)[2] +pv*(1 - pu)*src.at<Vec3b>(y1, x1)[2] + pv*pu*src.at<Vec3b>(y1, x2)[2];}}return 1;}//水平镜像、垂直镜像变换// [输入]way_mirror镜像方法：0水平镜像 1垂直镜像int affine_mirrorImg(Mat src, Mat& dst, int way_mirror){if (src.rows <= 0 || src.cols <= 0 || (src.channels() != 1 && src.channels() != 3) || src.depth() != CV_8U){printf("输入图像有误!\n");return 0;}if (way_mirror != 0 && way_mirror != 1){printf("输入镜像方法不为1或0,way_mirror: %d!\n",way_mirror);return 0;}int dst_h = src.rows, dst_w = src.cols;//目标图像宽高 初始化为原图宽高int ii = 0, jj = 0;double u_src = 0, v_src = 0;Mat M_mirr = (Mat_<double>(3, 3) << -1, 0, 0, 0, 1, 0, 0, 0, 1);if (way_mirror){M_mirr.at<double>(0,0) = 1;M_mirr.at<double>(1, 1) = -1;}Mat M_corrToSrc = (Mat_<double>(3, 3) << 1, 0, src.cols, 0, 1, 0, 0, 0, 1);if (way_mirror){M_corrToSrc.at<double>(0, 2) = 0;M_corrToSrc.at<double>(1, 2) = src.rows;}Mat M_trans = M_corrToSrc*M_mirr;Mat M_trans_inv = M_trans.inv();Mat dst_uv(3, 1, CV_64F);dst_uv.at<double>(2, 0) = 1;Mat src_uv(dst_uv);if (src.channels() == 3)dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC3); //RGB图初始elsedst = cv::Mat::zeros(dst_h, dst_w, CV_8UC1);//反向映射for (ii = 0; ii < dst_h; ++ii){for (jj = 0; jj < dst_w; ++jj){dst_uv.at<double>(0, 0) = jj;dst_uv.at<double>(1, 0) = ii;src_uv = M_trans_inv*dst_uv;u_src = src_uv.at<double>(0, 0);v_src = src_uv.at<double>(1, 0);// 边界问题if (u_src < 0) u_src = 0;if (v_src < 0) v_src = 0;if (u_src>src.cols - 1) u_src = src.cols - 1;if (v_src>src.rows - 1) v_src = src.rows - 1;//双线性插值Bilinear_interpolation_img(src, dst, ii, jj, u_src, v_src);}}return 1;}// 图像旋转(绕图像中心) 逆时针旋转为正// 可处理8位单通道或三通道图像int affine_rotateImg(Mat src, Mat& dst, double Angle){if (src.rows <= 0 || src.cols <= 0 || (src.channels() != 1 && src.channels() != 3) || src.depth() != CV_8U){printf("输入图像有误!\n");return 0;}double angle =0,cos_a=0,sin_a=0;//旋转角度int dst_h = src.rows, dst_w = src.cols;//目标图像宽高 初始化为原图宽高int ii = 0, jj = 0;double u_src = 0, v_src = 0;angle = Angle / 180 * CV_PI;cos_a = cos(angle);sin_a = sin(angle);dst_h = (int)(fabs(src.rows*cos_a) + fabs(src.cols*sin_a) + 0.5);dst_w = (int)(fabs(src.rows*sin_a) + fabs(src.cols*cos_a) + 0.5);if (src.channels()==3){dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC3); //RGB图初始}else{dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC1);}Mat M_toPhysics = (Mat_<double>(3, 3) << 1, 0, -0.5*src.cols, 0, -1, 0.5*src.rows, 0, 0, 1);Mat M_rotate = (Mat_<double>(3, 3) << cos_a, -sin_a, 0, sin_a, cos_a, 0, 0, 0, 1);Mat M_toPixel = (Mat_<double>(3, 3) << 1, 0, 0.5*dst.cols, 0, -1, 0.5*dst.rows, 0,0,1);Mat M_trans = M_toPixel*M_rotate*M_toPhysics;Mat M_trans_inv = M_trans.inv();Mat dst_uv(3, 1, CV_64F);dst_uv.at<double>(2, 0) = 1;Mat src_uv(dst_uv);//反向映射for (ii = 0; ii < dst_h; ++ii){for (jj = 0; jj < dst_w; ++jj){dst_uv.at<double>(0, 0) = jj;dst_uv.at<double>(1, 0) = ii;src_uv = M_trans_inv*dst_uv;u_src = src_uv.at<double>(0, 0);v_src = src_uv.at<double>(1, 0);//处理边界问题if (int(Angle) % 90 == 0){if (u_src < 0) u_src = 0;if (v_src < 0) v_src = 0;if (u_src>src.cols - 1) u_src = src.cols - 1;if (v_src>src.rows - 1) v_src = src.rows - 1;}//双线性插值Bilinear_interpolation_img(src, dst, ii, jj, u_src, v_src);}}return 1;}// 图像平移 在像素坐标系下进行 图像左顶点为原点，x轴为图像列，y轴为图像行// tx： x方向(图像列)平移量,向右平移为正// ty： y方向(图像行)平移量,向下平移为正int affine_moveImg(Mat src, Mat& dst, double tx, double ty){if (src.rows <= 0 || src.cols <= 0 || (src.channels() != 1 && src.channels() != 3) || src.depth() != CV_8U){printf("输入图像有误!\n");return 0;}int dst_h = src.rows, dst_w = src.cols;int ii = 0, jj = 0;double u_src = 0, v_src = 0;if (src.channels() == 3){dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC3); //RGB图初始}else{dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC1);}Mat M_toPhysics = (Mat_<double>(3, 3) << 1, 0, tx, 0, 1, ty, 0, 0, 1);Mat M_trans_inv = M_toPhysics.inv();Mat dst_uv(3, 1, CV_64F);dst_uv.at<double>(2, 0) = 1;Mat src_uv(dst_uv);//反向映射for (ii = 0; ii < dst_h; ++ii){for (jj = 0; jj < dst_w; ++jj){dst_uv.at<double>(0, 0) = jj;dst_uv.at<double>(1, 0) = ii;src_uv = M_trans_inv*dst_uv;u_src = src_uv.at<double>(0, 0);v_src = src_uv.at<double>(1, 0);//双线性插值Bilinear_interpolation_img(src, dst, ii, jj, u_src, v_src);}}return 1;}// 缩放 以图像左顶点为原点// cx: 水平缩放尺度// cy: 垂直缩放尺度int affine_scalingImg(Mat src, Mat& dst, double cx, double cy){if (src.rows <= 0 || src.cols <= 0 || (src.channels() != 1 && src.channels() != 3) || src.depth() != CV_8U){printf("输入图像有误!\n");return 0;}int dst_h = (int)(cy*src.rows+0.5), dst_w =(int)(cx* src.cols+0.5);int ii = 0, jj = 0;double u_src = 0, v_src = 0;if (src.channels() == 3){dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC3); //RGB图初始}else{dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC1);}Mat M_scale = (Mat_<double>(3, 3) << cx, 0, 0, 0, cy, 0, 0, 0, 1);Mat M_trans_inv = M_scale.inv();Mat dst_uv(3, 1, CV_64F);dst_uv.at<double>(2, 0) = 1;Mat src_uv(dst_uv);//反向映射for (ii = 0; ii < dst_h; ++ii){for (jj = 0; jj < dst_w; ++jj){dst_uv.at<double>(0, 0) = jj;dst_uv.at<double>(1, 0) = ii;src_uv = M_trans_inv*dst_uv;u_src = src_uv.at<double>(0, 0);v_src = src_uv.at<double>(1, 0);// 边界问题if (u_src < 0) u_src = 0;if (v_src < 0) v_src = 0;if (u_src>src.cols - 1) u_src = src.cols - 1;if (v_src>src.rows - 1) v_src = src.rows - 1;//双线性插值Bilinear_interpolation_img(src, dst, ii, jj, u_src, v_src);}}return 1;}// 错切变换 以图像中心为偏移中心// [输入]sx--水平错切系数//sy--垂直错切系数int affine_miscut(Mat src, Mat& dst, double sx, double sy){if (src.rows <= 0 || src.cols <= 0 || (src.channels() != 1 && src.channels() != 3) || src.depth() != CV_8U){printf("输入图像有误!\n");return 0;}int dst_h = fabs(sy)*src.cols + src.rows, dst_w = fabs(sx)*src.rows + src.cols;int ii = 0, jj = 0;double u_src = 0, v_src = 0;if (src.channels() == 3){dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC3); //RGB图初始}else{dst = cv::Mat::zeros(dst_h, dst_w, CV_8UC1);}Mat M_toPhysics = (Mat_<double>(3, 3) << 1, 0, -0.5*src.cols, 0, -1, 0.5*src.rows, 0, 0, 1);Mat M_rotate = (Mat_<double>(3, 3) << 1, sx, 0, sy, 1, 0, 0, 0, 1);Mat M_toPixel = (Mat_<double>(3, 3) << 1, 0, 0.5*dst.cols, 0, -1, 0.5*dst.rows, 0, 0, 1);Mat M_trans = M_toPixel*M_rotate*M_toPhysics;Mat M_trans_inv = M_trans.inv();Mat dst_uv(3, 1, CV_64F);dst_uv.at<double>(2, 0) = 1;Mat src_uv(dst_uv);//反向映射for (ii = 0; ii < dst_h; ++ii){for (jj = 0; jj < dst_w; ++jj){dst_uv.at<double>(0, 0) = jj;dst_uv.at<double>(1, 0) = ii;src_uv = M_trans_inv*dst_uv;u_src = src_uv.at<double>(0, 0);v_src = src_uv.at<double>(1, 0);//双线性插值Bilinear_interpolation_img(src, dst, ii, jj, u_src, v_src);}}return 1;}// 组合变换示例// 缩放->旋转->错切(即偏移)// [输入]int affine_srm_combImg(Mat src, Mat& dst, double cx, double cy, double Angle, double sx, double sy){if (src.rows <= 0 || src.cols <= 0 || (src.channels() != 1 && src.channels() != 3) || src.depth() != CV_8U){printf("输入图像有误!\n");return 0;}double angle, cos_a, sin_a;int dst_s_h, dst_s_w, dst_sr_h, dst_sr_w, dst_srm_h, dst_srm_w;angle = Angle / 180 * CV_PI;cos_a = cos(angle);sin_a = sin(angle);dst_s_h = (int)(cy*src.rows + 0.5);dst_s_w = (int)(cx* src.cols + 0.5);dst_sr_h = (int)(fabs(dst_s_h*cos_a) + fabs(dst_s_w*sin_a) + 0.5);dst_sr_w = (int)(fabs(dst_s_h*sin_a) + fabs(dst_s_w*cos_a) + 0.5);dst_srm_h = fabs(sy)*dst_sr_w + dst_sr_h;dst_srm_w = fabs(sx)*dst_sr_h + dst_sr_w;int ii = 0, jj = 0;double u_src = 0, v_src = 0;if (src.channels() == 3){dst = cv::Mat::zeros(dst_srm_h, dst_srm_w, CV_8UC3); //RGB图初始}else{dst = cv::Mat::zeros(dst_srm_h, dst_srm_w, CV_8UC1);}Mat M_scale = (Mat_<double>(3, 3) << cx, 0, 0, 0, cy, 0, 0, 0, 1);Mat M_toPhysics = (Mat_<double>(3, 3) << 1, 0, -0.5*dst_s_w, 0, -1, 0.5*dst_s_h, 0, 0, 1);Mat M_rotate = (Mat_<double>(3, 3) << cos_a, -sin_a, 0, sin_a, cos_a, 0, 0, 0, 1);Mat M2 = M_rotate*M_toPhysics;Mat M_mis = (Mat_<double>(3, 3) << 1, sx, 0, sy, 1, 0, 0, 0, 1);Mat M_toPixel = (Mat_<double>(3, 3) << 1, 0, 0.5*dst.cols, 0, -1, 0.5*dst.rows, 0, 0, 1);Mat M3 = M_toPixel*M_mis;Mat M_trans = M3*M2*M_scale;Mat M_trans_inv = M_trans.inv();Mat dst_uv(3, 1, CV_64F);dst_uv.at<double>(2, 0) = 1;Mat src_uv(dst_uv);//反向映射for (ii = 0; ii < dst_srm_h; ++ii){for (jj = 0; jj < dst_srm_w; ++jj){dst_uv.at<double>(0, 0) = jj;dst_uv.at<double>(1, 0) = ii;src_uv = M_trans_inv*dst_uv;u_src = src_uv.at<double>(0, 0);v_src = src_uv.at<double>(1, 0);//处理边界问题if (int(Angle) % 90 == 0){if (u_src < 0) u_src = 0;if (v_src < 0) v_src = 0;if (u_src>src.cols - 1) u_src = src.cols - 1;if (v_src>src.rows - 1) v_src = src.rows - 1;}//双线性插值Bilinear_interpolation_img(src, dst, ii, jj, u_src, v_src);}}return 1;}void main(){Mat src = imread("autumn.jpg", 1),dst;//水平、垂直镜像int way_mirror = 1;//affine_mirrorImg(src, dst, way_mirror);//旋转double angle_r = 250;//int flag = affine_rotateImg(src, dst, angle_r);//if (flag == 0)//{//return;//}//平移double tx = 50, ty = -50;//affine_moveImg(src, dst, tx, ty);//尺度变换(缩放)double cx = 1.5, cy = 1.5;affine_scalingImg(src, dst, cx, cy);//错切(偏移)double sx = 0.2, sy = 0.2;//affine_trans_deviation(src, dst, sx, sy);//affine_miscut(src, dst, sx, sy);//组合变换 缩放->旋转->错切(即偏移)//affine_srm_combImg(src, dst, cx, cy, angle_r, sx, sy);// 显示 Mat src_resize, dst_resize;//affine_scalingImg(src, src_resize, 0.4, 0.3);//affine_scalingImg(dst, dst_resize, 0.4, 0.3);namedWindow("src", 0);namedWindow("dst", 0);imshow("src", src);imshow("dst", dst);waitKey(0);system("pause");}
程序运行结果如下：
注：图片显示部分是截图效果，为了可能看起来不够标准，实际程序运行是没有问题的~
水平镜像
垂直镜像
平移
缩放
错切
旋转
组合变换
参考链接
博客及程序写作过程中难免出错，欢迎指正~
