How to find radial distortion coefficients between two sets of points? - matlab

Question

Title changed from: Specific transformation/distortion of an image using checkerboard coordinates
I have two checkerboard images. One of them is slightly more distorted than the other. I think it's a type of "barrel" distortion. I'm trying to compute the radial distortion parameters (or generate camera parameters) in order to distort one of the images to look like the other so that the checkerboard corners will line up.
Here is the binary undistorted image with its corners plotted in blue o's and the reference coordinates of the corners that we need to distort the image to plotted in red o's.
Most of the distortion is happening around the edges and the corners. I believe this is a type of radial distortion. How do I find the radial distortion coefficients between the two sets of coordinates representing the checkerboard corners?
Link to Image A (undistorted): http://imgur.com/rg4PNvp
Link to Image B (distorted): http://imgur.com/a/BIvid
I need to transform the checkerboard from Image B to have its corners line up with the corners from image A.
I have tried modifying the MATLAB Camera Calibration app generated script (link). I changed the world points that would be used to estimate camera parameters to equal my world points (corners) from Image A. However, this wasn't successful. The code I tried is can be seen in this pastebin: https://pastebin.com/D0StCb0p I used the same image in the imageFileNames because estimateCameraParameters requires at least 2 sets of coordinates.
Code:
% Define images to process
imageFileNames = {'C:\Users\asavelyev\Pictures\checkerB.tif',...
'C:\Users\asavelyev\Pictures\checkerB.tif',...
};
% Detect checkerboards in images
[imagePoints, boardSize, imagesUsed] = detectCheckerboardPoints(imageFileNames);
imageFileNames = imageFileNames(imagesUsed);
% Read the first image to obtain image size
originalImage = imread(imageFileNames{1});
[mrows, ncols, ~] = size(originalImage);
% AS: change these worldPoints to points of RGB image...
% Generate world coordinates of the corners of the squares
worldPoints = detectCheckerboardPoints('C:\Users\asavelyev\Pictures\checkerA.tif');
% Calibrate the camera
[cameraParams, imagesUsed, estimationErrors] = estimateCameraParameters(imagePoints, worldPoints, ...
'EstimateSkew', false, 'EstimateTangentialDistortion', false, ...
'NumRadialDistortionCoefficients', 2, 'WorldUnits', 'millimeters', ...
'InitialIntrinsicMatrix', [], 'InitialRadialDistortion', [], ...
'ImageSize', [mrows, ncols]);
% For example, you can use the calibration data to remove effects of lens distortion.
undistortedImage = undistortImage(originalImage, cameraParams);

Answer 1

This was solved by using this post from Mathematics StackOverflow: https://math.stackexchange.com/questions/302093/how-to-calculate-the-lens-distortion-coefficients-with-a-known-displacement-vect
Main issue I oversaw was the normalizion of the data points used in computing the K coefficients used in radial distortion.
Here is the MATLAB script I wrote for finding these coefficients:
% input images should be black and white checkerboards already thresholded into a binary image
% output image are the K coefficients used in radial distortion
function K = CalculateRadialDistortion(DistortedImg, UndistortedImg)
distortedCorners = detectCheckerboardPoints(DistortedImg);
undistortedCorners = detectCheckerboardPoints(UndistortedImg);
% normalize data
X1 = distortedCorners(:,1) - size(DistortedImg, 2)/2;
Y1 = distortedCorners(:,2) - size(DistortedImg, 1)/2;
X1p = undistortedCorners(:,1) - size(DistortedImg, 2)/2;
Y1p = undistortedCorners(:,2) - size(DistortedImg, 1)/2;
% X1p = (1+k1*r^2 + k2*r^4)X1 where r^2 = X1^2 + Y1^2
Rsq = X1.^2 + Y1.^2;
Rquad = Rsq.^2;
Rsqd = cat(1, Rsq, Rsq);
Rquadd = cat(1, Rquad, Rquad);
R = cat(2, Rsqd, Rquadd);
X1poX1 = X1p ./ X1;
X1poX1 = X1poX1 - 1;
Y1poY1 = Y1p ./ Y1;
Y1poY1 = Y1poY1 - 1;
X1Y1 = cat(1, X1poX1, Y1poY1);
K = linsolve(R, X1Y1);
end