Downscaling of remotely sensed precipitation images and outputs of general circulation models has been a subject of intense interest in hydrometeorology. The problem of downscaling is basically one of resolution enhancement, that is, appropriately adding details or high frequency features onto a low-resolution observation or simulated rainfall field. Invoking the property of rainfall self similarity, this mathematically ill-posed problem has been approached in the past within a stochastic framework resulting in ensemble of possible high-resolution realizations. In this work, we recast the rainfall downscaling into an ill-posed inverse problem and introduce a class of nonlinear estimators to properly regularize it and obtain the best high-resolution estimate in an optimal sense. This regularization capitalizes on two main observations: (1) precipitation fields are sparse when transformed into an appropriately chosen domain (e.g., wavelet), and (2) small-scale organized precipitation features tend to recur within and across different storm environments. We demonstrate the promise of the proposed methodology through downscaling and error analysis of level III precipitation reflectivity snapshots provided by the ground-based next generation Doppler weather radars in a ground validation sites of the Tropical Rainfall Measuring Mission.