The PhaseTree: Multiphase Signed Distance Fields

We introduce the PhaseTree, a novel hierarchical construction-tree representation for compactly modeling volumetric objects composed of multiple phases or materials across scales. An object is defined as a single construction tree that combines phase-aware primitives through composition and warping operators, yielding a unified multiphase signed distance representation that naturally supports complex topologies and interfaces between phases. The PhaseTree is compatible with standard signed distance field workflows: single-phase algorithms can be directly promoted to a PhaseTree, and conversely reduced without loss of information. As a result, our model integrates seamlessly with existing algorithms and rendering pipelines. We extend classical Sphere Tracing to robustly handle multiphase configurations and show that, despite the additional expressiveness, our implementation preserves the compactness and resolution independence of signed distance fields and incurs less than a 25% runtime overhead compared to single-phase Sphere Tracing.