A couple of days ago, I read somewhere (unfortunately can't find it anymore), that one cavet of inverse problems is that "similar looking" data $y$, given by $$ \mathcal{A}x = y, $$ where $\mathcal{A}$ is the Radon transform for example, can have very different sources $x$. I'm wondering if there is any mathematical construction of even a numerical example where functions/images $x_1, x_2$ look quite different, however the produced (noise free) data $y_1, y_2$ looks very similar.
Thanks in advance!