What is the intrinsic dimension of patterns embedded in an image? We explore the application of separable shape tensors to a variety of classical and modern imaging modalities to answer this question, and others, when measuring the microstructure of materials, complex patterns in integrated circuits, and time-varying geospatial information. We demonstrate the effectiveness of using separable shape tensors as intuitive decompositions of segmented curves from images to dramatically reduce dimensionality in an unsupervised fashion. We subsequently explore subspace-based dimension reduction for regression tasks over these learned submanifolds to sketch a principled approach for unveiling nonlinear, reduced dimensionality.