unrolled label of a Cap using chessboard pattern (OpenCV c++) - c++

Question

I´m trying to use a chessboard pattern, to get the information of the cylinder map and rectifie the "distortion" so that image shows the cap surface unrolled. I made a first test with a one shot calibration and cv::fisheye::undistortImage to get a un-distortion (attached two images).
*//runCalibrationFishEye
void runCalibrationFishEye(cv::Mat& image, cv::Matx33d&, cv::Vec4d&);
cv::Mat removeFisheyeLensDist(cv::Mat&, cv::Matx33d&, cv::Vec4d&);*
It is to remark that i am not interested in calibrate the image, to get metric values. I just want to use the chessboard information to unroll the image on the cylinder surface.
The final aim is to use the rectified images of 4 cameras and to stitch the rectified images to one unrolled image.
Do i need to make a full calibration of the camera? Or is there another way to get a remap of the cylinder surface?
I will try to implement this interesting unwarp method: https://dsp.stackexchange.com/questions/2406/how-to-flatten-the-image-of-a-label-on-a-food-jar/2409#2409
cap with chessboard
Rectification

Answer 1

I have found a similar approach, of another problem but with a similar Mathematics. And it was solved without a calibration pattern. Link here. Its a approximation, but the result is quite good enough.
the user Hammer gave an answer that helped me to get a solution. I have changed the way he do the mapping, using OpenCV remap. The formula to recalculate the coordinates is just as he gave it, using different values, and making a preprocessing to adjust the image (Rotation, zoom, and other adjustments).Unrolled image. I am now improving the distortion of the edges, so that it is not so pronounced at the edges. But the main question is solved.
cv::Point2f convert_pt(cv::Point2f point, int w, int h)
{
cv::Point2f pc(point.x - w / 2, point.y - h / 2);
float f = w;
float r = w;
float omega = w / 2;
float z0 = f - sqrt(r*r - omega*omega);
//Formula para remapear el cylindro
float zc = (2 * z0 + sqrt(4 * z0*z0 - 4 * (pc.x*pc.x / (f*f) + 1)*(z0*z0 - r*r))) / (2 * (pc.x*pc.x / (f*f) + 1));
cv::Point2f final_point(pc.x*zc / f, pc.y*zc / f);
final_point.x += w / 2;
final_point.y += h / 2;
return final_point;
}