Spline spaces with mixed orders of continuity over T-meshes

•A concept “Spline spaces with mixed orders of continuity over T-meshes” is presented.•Dimensional formulas are given for this type of bicubic spline spaces.•A set of bases are constructed with nice property from the geometric view.•Applications in image processing and FEM are presented.

In this paper, we introduce the concept of spline spaces with mixed orders of continuity over T-meshes. Then, the dimensions of the cubic spline spaces with continuity of order one and locally discontinuous over hierarchical T-meshes are presented by the B-net method. From the viewpoint of processing geometry data, a non-negative basis set with local support and partition of unity is constructed. Finally, the behavior of this type of spline is analyzed with the help of examples in image processing and finite element analysis.