computing transformation matrix using the least squares approximation

I need to compute a transformation matrix (TM) while using the least squares approximation. The equation used is: . It needs to be solved in a way that x is minimal. The A and B variables are measured at the same time and known values.

Let's say that A is a 3x3 matrix: A = [1, 23, 2; 12, 25, 9; 7, 34, 5];

And B is a 8x3 matrix: B = [3, 27, 4; 2, 43, 12; 14, 36, 9; 6, 22, 5; 7, 38, 14; 13, 27, 11; 6, 41, 8; 7, 38, 15];

The result should be a [8x3] transformation matrix.

I hope someone has a clue how to compute the transormation matrix.

Answer 1

[BhaTTa]

To compute the transformation matrix ( T ) using the least squares approximation given matrices ( A ) and ( B ), you need to solve the following equation:

[ B \approx A \cdot T ]

The goal is to minimize the difference between ( B ) and ( A \cdot T ) in a least squares sense. This can be achieved using the mldivide operator (backslash \) in MATLAB.

Here's a step-by-step guide to compute the transformation matrix ( T ):

1. Ensure Dimensions Match: Matrix ( A ) should be of size ( m \times n ) and matrix ( B ) should be of size ( m \times p ). In your case, ( A ) is ( 3 \times 3 ) and ( B ) is ( 8 \times 3 ). For this example, it seems there might be a misunderstanding as typically ( A ) and ( B ) should have the same number of rows for the transformation to make sense. Assuming ( A ) should be ( 8 \times 3 ) to match ( B ):

% Example data
A = [1, 23, 2; 12, 25, 9; 7, 34, 5; 6, 22, 5; 7, 38, 14; 13, 27, 11; 6, 41, 8; 7, 38, 15];
B = [3, 27, 4; 2, 43, 12; 14, 36, 9; 6, 22, 5; 7, 38, 14; 13, 27, 11; 6, 41, 8; 7, 38, 15];

% Compute the transformation matrix T using least squares approximation
T = A \ B;

% Display the result
disp('Transformation Matrix T:');
disp(T);

% Verify the result by computing A * T and comparing with B
B_approx = A * T;
disp('Approximated B:');
disp(B_approx);

% Calculate the error
error = norm(B - B_approx);
disp(['Error: ', num2str(error)]);

Explanation

1. Matrix Dimensions: Ensure that ( A ) and ( B ) have compatible dimensions for the operation ( A \cdot T ).
2. Least Squares Solution: The backslash operator \ in MATLAB is used to solve the linear system in a least squares sense. This minimizes the error ( | B - A \cdot T | ).
3. Verification: Compute ( A \cdot T ) and compare it with ( B ) to verify the solution. The error should be minimal.

Special Note

If ( A ) and ( B ) have different numbers of rows (as in your original example), it suggests that the problem might be underdetermined or overdetermined, which typically requires a different approach or more information about the transformation. Ensure that the matrices are correctly defined for the transformation you intend to compute.