What will be a suitable option to circumvent a non-invertible matrix?

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Aboltabol 2023-9-27

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评论： Aboltabol 2023-9-28

Line 157 of this code-file (relevant part is reproduced below) used to calculate the optical flow between two images occassionally returns an error:

% .........
% Solve this system of linear equations, adding a small value along the
% diagonal to avoid potentially having a singular matrix
diag_small_value = sparse(1:2*N, 1:2*N, 1e-10);
A = A + diag_small_value;
xexact = A\b;
% .........
>>> Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate.

So, any downsides to using

pinv(A)*B

under a try-except block? Or, maybe we avoid the iterative computation since we already have the eigen vector though that will be weird .....

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Walter Roberson 2023-9-27

Note that "Matrix is singular" is a warning not an error, so try ; catch would not catch it.

You can consider testing rcond before using the \ operator.

I do not know anything about the mathematics of optical flow, so I do not know what equations are being solved or why they might give problems.

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Answer 1

Matt J 2023-9-27

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The approach is valid if A is symmetric, positive definite, but 1e-10 may not be large enough to add substantial conditioning.

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Aboltabol 2023-9-27

What is typically a good constant to add?

Matt J 2023-9-27

编辑：Matt J 2023-9-27

That depends on the maximum eigenvalue of A. Adding δ will give a new condition number of approximately,

Also, I don't know your application, so I don't know what accuracy you are prepared to sacrifice by modifying A.

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Answer 2

Bruno Luong 2023-9-27

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When you encount warning message like this, the first thing to ask is not "how do I change linear inversion algorithm?", but "what condition I forgot to take into account in order for my problem to be wellposed?".

In the optical flow, the answer could be that you forgot to impose some boundary condition so that the flow is uniquely determine.

If you won't fix that, any algorithm will fail to give you the correct answer.