仿射變換與透視變換

仿射變換保證物體形狀的“平直性”和“平行性”。透視變換不能保證物體形狀的“平行性”。仿射變換是透視變換的特殊形式。

將透視變換寫成3*3矩陣形式，即為M;

以下面這張圖為例，實現仿射變換，包括旋轉，平移，縮放，剪切，以圖像中心為變換中心；

仿射變換

旋轉（逆時針旋轉30度）

Mat M=Mat::eye(3,3, CV_32FC1);
float alpha=PI/6;
float tx=0;
float ty=0;
float scale=1;

M.at<float>(0, 0) = cos(alpha);
M.at<float>(0, 1) = sin(alpha);
M.at<float>(0, 2) = (1-cos(alpha))*center.x - sin(alpha)* center.y;
M.at<float>(1, 0) = -sin(alpha);
M.at<float>(1, 1) = cos(alpha);
M.at<float>(1, 2) = sin(alpha)* center.x +(1-cos(alpha))*center.y;
M.at<float>(2, 0) = 0;
M.at<float>(2, 1) = 0;
M.at<float>(2, 2) = 1;

 

平移（水平和垂直方向各向右平移20個像素）

 Mat M=Mat::eye(3,3, CV_32FC1);
    float alpha=0;
    float tx=20;
    float ty=20;
    float scale=0;

    M.at<float>(0, 0) = cos(alpha);
    M.at<float>(0, 1) = sin(alpha);
    M.at<float>(0, 2) = (1-cos(alpha))*center.x - sin(alpha)* center.y + tx;
    M.at<float>(1, 0) = -sin(alpha);
    M.at<float>(1, 1) = cos(alpha);
    M.at<float>(1, 2) = sin(alpha)* center.x +(1-cos(alpha))*center.y + ty;
    M.at<float>(2, 0) = 0;
    M.at<float>(2, 1) = 0;
    M.at<float>(2, 2) = 1;

 

縮放（圖像縮小1/2）

Mat M=Mat::eye(3,3, CV_32FC1);
    float alpha=0;
    float tx=0;
    float ty=0;
    float scale=0.5;

    M.at<float>(0, 0) = cos(alpha) * scale;
    M.at<float>(0, 1) = sin(alpha) * scale;
    M.at<float>(0, 2) = (1-cos(alpha)* scale)*center.x - sin(alpha)* center.y* scale + tx;
    M.at<float>(1, 0) = -sin(alpha) * scale;
    M.at<float>(1, 1) = cos(alpha)* scale;
    M.at<float>(1, 2) = sin(alpha)* center.x* scale +(1-cos(alpha)* scale)*center.y + ty;
    M.at<float>(2, 0) = 0;
    M.at<float>(2, 1) = 0;
    M.at<float>(2, 2) = 1;

錯切變換（剪切變換）：

 

Mat M=Mat::eye(3,3, CV_32FC1);
    float alpha=PI/12;
    float tx=0;
    float ty=0;
    float scale=1;

    M.at<float>(0, 0) = 1;
    M.at<float>(0, 1) = sin(alpha);
    M.at<float>(0, 2) = (1-cos(alpha))*center.x - sin(alpha)* center.y;
    M.at<float>(1, 0) = 0;
    M.at<float>(1, 1) = 1;
    M.at<float>(1, 2) = sin(alpha)* center.x +(1-cos(alpha))*center.y;
    M.at<float>(2, 0) = 0;
    M.at<float>(2, 1) = 0;
    M.at<float>(2, 2) = 1;

 

 

透視變換（透視變換不保證平行性）

Mat M=Mat::eye(3,3, CV_32FC1);
    float alpha=0;
    float tx=0;
    float ty=0;
    float scale=1;

    M.at<float>(0, 0) = cos(alpha);
    M.at<float>(0, 1) = sin(alpha);
    M.at<float>(0, 2) =(1-cos(alpha))*center.x - sin(alpha)* center.y + tx;
    M.at<float>(1, 0) = -sin(alpha);
    M.at<float>(1, 1) = cos(alpha);
    M.at<float>(1, 2) =sin(alpha)* center.x +(1-cos(alpha))*center.y + ty;
    M.at<float>(2, 0) = 0.001;
    M.at<float>(2, 1) = 0.001;
    M.at<float>(2, 2) = 1;

上述操作使用同一個opencv接口函數：warpPerspective(srcImg, dstImg, M, Size(width, height));