Locally stationary covariance and signal estimation with macrotiles

A macrotile estimation algorithm is introduced to estimate the covariance of locally stationary processes. A macrotile algorithm uses a penalized method to optimize the partition of the space in orthogonal subspaces, and the estimation is computed with a projection operator. It is implemented by searching for a best basis among a dictionary of orthogonal bases and by constructing an adaptive segmentation of this basis to estimate the covariance coefficients. The macrotile algorithm provides a consistent estimation of the covariance of locally stationary processes, using a dictionary of local cosine bases. This estimation is computed with a fast algorithm. Macrotile algorithms apply to other estimation problems such as the removal of additive noise in signals. This simpler problem is used as an intuitive, guide to better understand the case of covariance estimation. Examples of removal of white noise from sounds illustrate the results.