In this letter, we propose a robust algorithm for determining a unitary transform of a discrete-time signal, whose samples are sensed in impulsive noise environments. The proposed algorithm estimates the transformed signal coefficients by solving an $\ellb _{\bf 1}$-regularized least absolute deviation ( $\ellb_{\bf 1}$-LAD) regression problem, leading to the weighted median (WM) as the optimal operator for computing each transform coefficient. Numerical simulations on synthetic data are presented in order to compare the performance of the proposed algorithm to those yielded by previously reported methods. Furthermore, an example of audio denoising using the short time Fourier transform of a digital audio record, in the presence of impulsive noise, is also shown.