The nonparametric multiscale partition-based estimators presented in this thesis are powerful tools for signal reconstruction and set estimation from noisy observations. Unlike traditional wavelet-based multiscale methods, the spatially adaptive and computationally efficient methods presented in this thesis are (a) able to achieve near minimax optimal error convergence rates for broad classes of signals, including Besov spaces, and adapt to arbitrary levels of smoothness in one dimension; (b) capable of optimally reconstructing images consisting of smooth surfaces separated by smooth boundaries; (c) well-suited to a variety of observation models, including Poisson count statistics and unbinned observations of a point process, as well as the signal plus additive white Gaussian noise model; (d) amenable to energy-efficient decentralized estimation; and (e) flexible enough to facilitate the optimization of specialized criteria for tasks such as accurate set extraction.
Set estimation differs from signal reconstruction in that, while the goal of the latter is to estimate a function, the goal of the former is to determine where in its support the function meets some criterion. Because of their unique objectives, the two problems require distinct estimator evaluation metrics and analysis tools, yet both can be solved accurately and efficiently using the partition-based framework developed here. These methods are a key component of effective iterative inverse problem solvers for challenging tasks such as deblurring, tomographic reconstruction, and superresolution image reconstruction. Both signal reconstruction and set estimation arise routinely in a variety of scientific and engineering applications, and this thesis demonstrates the effectiveness of the proposed methods in the context of medical and astrophysical imaging, density estimation, distributed field estimation using wireless sensor networks, network traffic analysis, and digital elevation map processing.