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Why de-blurring a blurred image is an ill-conditioned problem? What's the intuitive explanation? How to show it using the condition number?

dxdydz
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  • What do you mean by the condition number? – მამუკა ჯიბლაძე Mar 30 '19 at 20:11
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    Condition number of a matrix is $cond(A) = |A| |A^{-1}|$ – dxdydz Mar 30 '19 at 20:14
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    Can you show us the source or context of this comment? – usul Mar 30 '19 at 20:29
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    @usul It is an observation that typically holds in numerical practice. If you want a reference, for instance there's Hansen, Nagy, O'Leary, Deblurring Images: matrices, spectra and filtering, p. 10: "the following properties generally hold for image deblurring problems: [...] The singular values decay to a value very close to zero. As a consequence, the condition number $cond(A) = \sigma_1/\sigma_N$ is very large, indicating that the solution is very sensitive to perturbation and rounding errors". – Federico Poloni Mar 30 '19 at 20:37
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    If you're talking about linear shift-invariant blurs then you mean convolution. The operation will be diagonal on a basis of exponential functions x->exp(ikx). A blurring operation acting on exp(ikx) will therefore multiply it by some number. If it's a 'blur', then by definition it multiplies high frequencies by small numbers which means that after adding a small perturbation, you can't undo this multiplication easily simply by dividing without adding enormous noise. Because this basis diagonalises the operation, nothing at any of the other frequencies can offer any help, they're independent. – Dan Piponi Mar 31 '19 at 03:00
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    @DanPiponi Seems like what you wrote in your comment would make a nice answer. – Federico Poloni Mar 31 '19 at 08:54
  • @FedericoPoloni I didn't want to put in too much effort as it's probably undergrad material and I expect it to get closed. – Dan Piponi Mar 31 '19 at 14:58
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    @DanPiponi Personally, I don't think this is undergraduate material, and I would hate to see it closed. I think that it's a nice question to ask for an "intuitive explanation". For instance, even if integrals and derivatives are first-year topics, I find this question a great one. And, while your answer is useful, it does not tell the whole story: what is the intuitive reason why the boundary conditions (which are not shift-invariant) don't matter? What is the intuitive reason why exp(ikx) are eigenvalues of shift-invariant blurs? – Federico Poloni Mar 31 '19 at 15:51

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