Image undistortion in a certain area

I have a distorted image and I have calibrate the camera. I can do image undistortion on the whole image. Now, I want to undistortion on part of that, which mean as long as I know the position of the area and scale factor, then I can do image undistortion. From the principle, it is feasible. I need someone to help me realize it. You should be really familiar the principle of camera calibration including all of the equations so that we can make some change. Also, you need to know linear algebra and coordinate transformation. If you are not so familiar with it, it is fine, as long as you are familiar with opencv and visual studio, I can advice you to add the change. Specifically, we need to change the cv::initUndistortRectifyMap function by adding bias and scaling terms into it. The deliverable would be the api to undistort the image.