  One of the most prominent methods for uncertainty quantification in
high-dimen-sional statistics is the desparsified LASSO that relies on
unconstrained $\ell_1$-minimization. The majority of initial works focused on
real (sub-)Gaussian designs. However, in many applications, such as magnetic
resonance imaging (MRI), the measurement process possesses a certain structure
due to the nature of the problem. The measurement operator in MRI can be
described by a subsampled Fourier matrix. The purpose of this work is to extend
the uncertainty quantification process using the desparsified LASSO to design
matrices originating from a bounded orthonormal system, which naturally
generalizes the subsampled Fourier case and also allows for the treatment of
the case where the sparsity basis is not the standard basis. In particular we
construct honest confidence intervals for every pixel of an MR image that is
sparse in the standard basis provided the number of measurements satisfies $n
\gtrsim\max\{ s\log^2 s\log p, s \log^2 p \}$ or that is sparse with respect to
the Haar Wavelet basis provided a slightly larger number of measurements.
