Most change-detection methods carry a hidden passenger: the assumption that the domain is flat. On a line or a plane, a change is a boundary you can draw with a ruler. On a sphere, even defining “region” requires different mathematics. CRISP — Change Region Identification and SeParation — confronts this directly, locating change regions in signal-plus-noise models on d-dimensional spheres, including the case of multiple spatially complex regions with irregular boundaries.
The problem is not merely technical. You cannot rescue Euclidean change-detection by projecting a sphere onto a plane and running the old algorithms. Every projection distorts something — area, angle, distance — and the distortion is not uniform. What looks like a compact region on the sphere becomes a smeared artifact in the projection. The bias is geometric, not statistical, and no amount of data corrects it.
CRISP works because it respects the curvature of the domain from the start. Its estimators are built for the sphere's intrinsic metric, not adapted from flat-space ancestors. This matters beyond spheres. Climate data lives on the Earth. Neural signals live on the cortex. Directional data lives on circles and tori. In every case, the geometry of the space constrains what counts as a change, what counts as a boundary, and what counts as a region. Flatland statistics break the moment the domain curves — and most interesting domains curve.