# How to convert coordinates between a base image and a projected/distorted version?

(Sorry, I meant to provide images, but can't due to low reputation. Please click the links.)

I have tried reading some papers on orthorectifiation (like A Comprehensive Study on the Rational Function Model for Photogrammetric Processing (2001)), but can't quite get to the right soltutions.

My problem is this:

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I have two images, represented in the graphic as the grey and orange squares (the real images however are not constrained to be squares). The contents of the images overlap on some points. These control points $(x_{i_{grey}}, y_{i_{grey}}) \rightarrow (x_{i_{orange}}, y_{i_{orange}})$ are known.

In reality, the second image must be projected in a certain perspective to have the control points match. This is seen in the next graphic:

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Now, I do not need to actually morph/modify the second image, I just want to convert some predefined coordinates $(x,y)$ (which are not the control points) from the base (grey) image to the target image (orange).

Example: Assume that the grey square has a side length of 100, thus the center would be at $(50,50)$. At what coordinates, given a mapping of control points, is this point found in the orange square in the first graphic?

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You have just unlocked 5 geek points!?!! –  user21436 Apr 7 '12 at 13:54
Do you mean the control points as "image features"? I guess your question can be reexpressed like this: given two images and a set of matching features, how to transform any image points from image 1 (gray) to image 2 (orange), right? If this is your question, you can compute the transformation matrix (homography or essential matrix) by using the known matching features. Once you have the transformation matrix, you can transform any image points from image 1 to image 2. You may refer to a book "An invitation to 3D vision" regarding how to compute the transformation matrix. –  Shiyu Apr 7 '12 at 14:20