  As interest in deep neural networks (DNNs) for image reconstruction tasks
grows, their reliability has been called into question (Antun et al., 2020;
Gottschling et al., 2020). However, recent work has shown that, compared to
total variation (TV) minimization, when appropriately regularized, DNNs show
similar robustness to adversarial noise in terms of $\ell^2$-reconstruction
error (Genzel et al., 2022). We consider a different notion of robustness,
using the $\ell^\infty$-norm, and argue that localized reconstruction artifacts
are a more relevant defect than the $\ell^2$-error. We create adversarial
perturbations to undersampled magnetic resonance imaging measurements (in the
frequency domain) which induce severe localized artifacts in the TV-regularized
reconstruction. Notably, the same attack method is not as effective against DNN
based reconstruction. Finally, we show that this phenomenon is inherent to
reconstruction methods for which exact recovery can be guaranteed, as with
compressed sensing reconstructions with $\ell^1$- or TV-minimization.
