ill-posed inverse problem

Recovering a cause from an effect when the forward model destroys or barely preserves some information.

An inverse problem asks for the unknown cause $\mathbf x$ (a clean image) given the observed effect $\mathbf b$ and a forward model $H$ relating them ( $H\mathbf x = \mathbf b$ in the noise-free case). It is ill-posed when $H$ is singular or severely ill-conditioned -- some directions in $\mathbf x$ -space are barely (or not at all) constrained by $\mathbf b$ -- exactly the spatial-domain twin of a frequency-domain PSF with a near-zero or zero response. Plain least squares on an ill-posed $H$ reproduces inverse filtering's own blow-up; regularization is the standard fix.