Level Set Estimation via Trees

Tree-structured partitions provide a natural framework for rapid and accurate extraction of the level sets of a multivariate function f from noisy data. In general, a level set is the set S on which f exceeds some critical value (e.g., S = {x : f(x) ≥ γ}). Boundaries of level sets typically constitute manifolds embedded in the high-dimensional observation space. The identification of these boundaries is an important theoretical problem with applications for digital elevation maps, medical imaging, and pattern recognition. Because level set identification is intrinsically simpler than field denoising or estimation, explicit level set extraction methods can achieve higher accuracy than more indirect approaches (such as extracting a level set from an estimate of the function). The trees underlying our method are constructed by minimizing a complexity regularized data-fitting term over a family of dyadic partitions. Our method automatically adapts to spatially varying regularity of both the level set and the field underlying the data. Level set extraction using multiresolution trees can be implemented in near linear time and specifically aims to minimize an error metric sensitive to both the error in the location of the level set and the associated field estimation error.