Inverting a real-valued index grid

Well I just had to solve this remap inversion problem myself and I’ll outline my solution.

Given X, Y for the remap() function that does the following:

B[i, j] = A(X[i, j], Y[i, j])   

I computed Xinv, Yinv that can be used by the remap() function to invert the process:

A[x, y] = B(Xinv[x,y],Yinv[x,y])

First I build a KD-Tree for the 2D point set {(X[i,j],Y[i,j]} so I can efficiently find the N nearest neighbors to a given point (x,y). I use Euclidian distance for my distance metric. I found a great C++ header lib for KD-Trees on GitHub.

Then I loop thru all the (x,y) values in A‘s grid and find the N = 5 nearest neighbors {(X[i_k,j_k],Y[i_k,j_k]) | k = 0 .. N-1} in my point set.

• If distance d_k == 0 for some k then Xinv[x,y] = i_k and Yinv[x,y] = j_k, otherwise…

• Use Inverse Distance Weighting (IDW) to compute an interpolated value:

  • let weight w_k = 1 / pow(d_k, p) (I use p = 2)
  • Xinv[x,y] = (sum_k w_k * i_k)/(sum_k w_k)
  • Yinv[x,y] = (sum_k w_k * j_k)/(sum_k w_k)

Note that if B is a W x H image then X and Y are W x H arrays of floats. If A is a w x h image then Xinv and Yinv are w x h arrays for floats. It is important that you are consistent with image and map sizing.

Works like a charm! My first version I tried brute forcing the search and I never even waited for it to finish. I switched to a KD-Tree then I started to get reasonable run times. I f I ever get time I would like to add this to OpenCV.

The second image below is use remap() to remove the lens distortion from the first image. The third image is a result of inverting the process.