We consider the estimation of the Directions-Of-Arrival (DOA) of target signals in diffuse noise. The state-of-the-art MUltiple SIgnal Classification (MUSIC) algorithm necessitates accurate identification of the signal subspace. In diffuse noise, however, it is difficult to identify it directly from the observed spatial covariance matrix. In our approach, we estimate the target spatial covariance matrix, so that we can identify the orthogonal complement of the signal subspace as its null space. We present a unified framework for modeling noise covariance in a matrix space, which generalizes four state-of-the-art diffuse noise models. We propose two alternative algorithms for estimating the target spatial covariance matrix, namely Low-rank Matrix Completion (LMC) and Trace Norm Minimization (TNM). These rely on denoising of the observed spatial covariance matrix via orthogonal projection onto the orthogonal complement of the noise matrix subspace. The missing component lying in the noise matrix subspace is then completed by exploiting the low-rankness of the target spatial covariance matrix. Large-scale experiments with real-world noise show that TNM with a certain noise model outperforms conventional MUSIC based on Generalized EigenValue Decomposition (GEVD) by 5% in terms of the precision averaged over the dataset.