We introduce an algorithm for sketch vectorization with state-of-the-art accuracy and capable of handling complex sketches. We approach sketch vectorization as a surface extraction task from an unsigned distance field, which is implemented using a two-stage neural network and a dual contouring domain post processing algorithm. The first stage consists of extracting unsigned distance fields from an input raster image. The second stage consists of an improved neural dual contouring network more robust to noisy input and more sensitive to line geometry. To address the issue of under-sampling inherent in grid-based surface extraction approaches, we explicitly predict undersampling and keypoint maps. These are used in our post-processing algorithm to resolve sharp features and multi-way junctions. The keypoint and undersampling maps are naturally controllable, which we demonstrate in an interactive topology refinement interface. Our proposed approach produces far more accurate vectorizations on complex input than previous approaches with efficient running time.