Training-free Spatially Grounded Geometric Shape Encoding

In this work, we introduce a novel training-free encoding framework, dubbed XShapeEnc, that encodes an arbitrary spatially grounded 2D geometric shape into a compact representation exhibiting five favorable properties, including invertibility, adaptivity, generality and controllability. Specifically, a 2D spatially grounded geometric shape is decomposed into its normalized geometry within the unit disk and its pose vector, where the pose is further transformed into a harmonic pose field that also lies within the unit disk. A set of orthogonal Zernike basis is constructed to encode shape geometry and pose either independently or jointly, with controllable relative emphasis on shape geometry or shape pose. We demonstrate the theoretical validity, efficiency, discriminability, and wide applicability of XShapeEnc via extensive analysis and experiments across a wide range of shape-aware tasks and our self-curated XShapeCorpus dataset. We envision XShapeEnc as a foundational tool for research beyond one-dimensional sequential data and data-driven, learning-based encoding paradigms, paving the way for a unified spatial encoding framework for frontier 2D spatial intelligence.