# Why de-blurring a blurred image is an ill-conditioned problem? [closed]

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

• Condition number of a matrix is $cond(A) = \|A\| \|A^{-1}\|$ – dxdydz Mar 30 '19 at 20:14
• Can you show us the source or context of this comment? – usul Mar 30 '19 at 20:29
• @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
• 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 3:00
• @DanPiponi Seems like what you wrote in your comment would make a nice answer. – Federico Poloni Mar 31 '19 at 8:54