calculate the transform matrix

I'm new to matlab. Can someone tell me how to calculate Projection Matrix. I have 2D-to-3D corresponding points based on that i want to calculate projection Matrix.

After that How can i calculate calibration matrix, rotation matrix and translation matrix using projection matrix.

Answer 1

[Arjun]

Hi @Soni Reddy,

I see that you need help in computing projection matrix given that you have corresponding 2D and 3D points.

To compute projection matrix, you can follow the given steps:

• First, the Direct Linear Transformation (DLT) method is used to compute the projection matrix (P).
• This involves setting up a system of linear equations derived from the correspondence between 3D world points and 2D image points, which is then solved using Singular Value Decomposition (SVD).
• Once the projection matrix is obtained, it is decomposed into the intrinsic matrix (K), rotation matrix (R), and translation vector (t).
• This decomposition is achieved using RQ decomposition, where the matrix (M), a 3x3 submatrix of (P), is decomposed into (K) and (R) by performing a QR decomposition on its transpose and then rearranging the results.
• The intrinsic matrix (K) is normalized to ensure that its bottom-right element is 1, and the translation vector (t) is extracted from the last column of (P).

Kindly refer to the code below for detailed implementation of the above steps:

function cameraCalibration(worldPoints, imagePoints) 

    % Ensure the points are in homogeneous coordinates 

    if size(worldPoints, 2) ~= 3 || size(imagePoints, 2) ~= 2 

        error('worldPoints should be Nx3 and imagePoints should be Nx2'); 

    end 

    % Number of points 

    n = size(worldPoints, 1);  

    % Construct the matrix A 

    A = []; 

    for i = 1:n 

        X = worldPoints(i, 1); 

        Y = worldPoints(i, 2); 

        Z = worldPoints(i, 3); 

        u = imagePoints(i, 1); 

        v = imagePoints(i, 2); 

        A = [A; 

             X, Y, Z, 1, 0, 0, 0, 0, -u*X, -u*Y, -u*Z, -u; 

             0, 0, 0, 0, X, Y, Z, 1, -v*X, -v*Y, -v*Z, -v]; 

    end 

    % Solve for p using SVD 

    [~, ~, V] = svd(A); 

    P = reshape(V(:, end), [4, 3])'; % Projection matrix 

    % Extract the 3x3 submatrix from P 

    M = P(:, 1:3); 
    
    % Perform RQ decomposition 

    [K, R] = rq(M); 

    % Ensure K has positive diagonal entries 

    T = diag(sign(diag(K))); 

    K = K * T; 

    R = T * R; 

    % Compute the translation vector 

    t = K \ P(:, 4); 

    % Normalize K 

    K = K / K(3, 3); 
    
    % Display the results 

    disp('Projection Matrix (P):'); 

    disp(P); 

    disp('Intrinsic Matrix (K):'); 

    disp(K); 

    disp('Rotation Matrix (R):'); 

    disp(R); 

    disp('Translation Vector (t):'); 

    disp(t); 

end 

function [R, Q] = rq(M) 

    % RQ decomposition using QR decomposition of the transposed matrix 

    [Q, R] = qr(flipud(M)'); 

    R = flipud(R'); 

    Q = flipud(Q'); 

    Q = Q(:, end:-1:1); 

    R = R(end:-1:1, :); 

end 

% Example usage 

worldPoints = [0, 0, 0; 1, 0, 0; 1, 1, 0; 0, 1, 0]; % Example 3D points 

imagePoints = [100, 100; 200, 100; 200, 200; 100, 200]; % Corresponding 2D points 

cameraCalibration(worldPoints, imagePoints); 

In the process above the intrinsic matrix is used as an alias for Calibration matrix.

I hope this helps!