Curvilinear targets are common in many imaging modalities. Detection of such targets can be challenging because of
their multiscale structure, their frequent obscuration in natural imagery, their turns, intersections, and merges, and the
prevalence of false positive detections based on local information. Using a spatial spectroscopy approach, we introduce
image analysis methods that use the concept of gauge frames to simplify the identification of curvilinear targets. Fast
computational approximation methods are described for gauge fields, and an experiment is described illustrating the
power of higher-order derivatives for understanding even relatively simple geometric structures. Methods for extracting
coherent curvilinear objects that exploit the larger-scale commonalities of points in the object are described.