We introduce a novel basis for multivariate hierarchical tensor-product spline spaces. Our construction combines the truncation mechanism (Giannelli et al., 2012) with the idea of decoupling basis functions (Mokriš et al., 2014). While the first mechanism ensures the partition of unity property, which is essential for geometric modeling applications, the idea of decoupling allows us to obtain a richer set of basis functions than previous approaches. Consequently, we can guarantee the completeness property of the novel basis for large classes of multi-level spline spaces. In particular, completeness is obtained for the multi-level spline spaces defined on T-meshes for hierarchical splines of (multi-)degree p for example (i) with single knots and p-adic refinement and (ii) with knots of multiplicity m ≥ ( p + 1 ) / 3 and dyadic refinement (where each cell to be refined is subdivided into 2 d cells, with d being the number of variables) without any further restriction on the mesh configuration. Both classes (i), (ii) include multivariate quadratic hierarchical tensor-splines with dyadic refinement. 
TDHB-splines: The truncated decoupled basis of hierarchical tensor-product splines