Pixel domain multi-resolution minimum variance painting of CMB maps

This work introduces an unbiased minimum variance painting of the pixel domain CMB maps for both the missing and available sky regions. In the missing region, it generates CMB realizations that have identical statistical properties to the expected CMB signal; and in the available region, it significantly alleviate the unwanted residuals, e.g., the residual of EB-leakage. The time cost of this method follows $\sim$$O(N_{side}^3)$ ($N_{side}$ is the map resolution parameter), which is similar to the fast spherical harmonic transforms and is much better than the traditional minimum variance method that follows $\sim$$O(N_{side}^6)$. This method is the basis of a series of minimum variance estimations for analyzing CMB datasets, and more works based on it will follow in the future.