Calculating the precision of homography on 2D plane

Question

I am trying to find a way to parametrize the precision of my homography calculation. I would like to obtain a value that describes the precision of the homography calculation for a measurement taken at a certain position.

I currently have succesfully calculated the homography (with cv::findHomography) and I can use it to map a point on my camera image onto a 2D map (using cv::perspectiveTransform). Now I want to track these objects on my 2D map and to do this I want to take in account that objects that are in the back of my camera image have a less precise position on my 2D map than the objects that are all the way in the front.

I have looked at the following example on this website that mentions plane fitting but I don't really understand how to fill the matrices correctly using this method. The visualisation of the result does seem to fit my needs. Is there any way to do this with standard OpenCV functions?

EDIT: Thanks Francesco for your recommendations. But, I think I am looking for something different than your answer. I am not looking to test the precision of the homography itself, but the relation between the density of measurements in one real camera view and the actual size on a map I create. I want to know that when I am 1 pixel off on my detection in the camera image, how many meters this will be on my map at this point.

I can of course calculate by taking some pixels around my measurement on my camera image and then use the homography to see how many meters on my map this represent every time I do a homography, but I don't want to calculate this every time. What I would like is to have a formula that tells me the relation between pixels in my image and pixels on my map so I can take this in account for my tracking on the map.

Answer 1

What you are looking for is called "predictive error bars" or "prediction uncertainty". You should definitely consult a good introductory book on estimation theory for details (e.g. this one). But briefly, the predictive uncertainty is the probability that...

• A certain pixel p in image 1 will is the mapping H(p') of a pixel p' in image 2 under the homography H...
• Given the uncertainty in H which is due to the errors in the matched pairs (q0, q0'), (q1, q1'), ..., that have been used to estimate H, ...
• But assuming the model is correct, that is, that the true map between images 1 and 2 is, in fact, a homography (although the estimated parameters of the homography itself may be affected by errors).

In order to estimate this probability distribution you'll need a model for the errors in the measurements, and a model for how they propagate through the (homography) model.