Box spline

In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables.

Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower-dimensional space.

) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in

can be interpreted as the shadow of the indicator function of the unit hypercube in

Considering tempered distributions a box spline associated with a single direction vector is a Dirac-like generalized function supported on

They therefore form the starting point for many subdivision surface constructions.

Box splines have been useful in characterization of hyperplane arrangements.

[3] Also, box splines can be used to compute the volume of polytopes.

[8] In the 2-D setting the three-direction box spline[9] is used for interpolation of hexagonally sampled images.

In the 3-D setting, four-direction[10] and six-direction[11] box splines are used for interpolation of data sampled on the (optimal) body-centered cubic and face-centered cubic lattices respectively.

[5] The seven-direction box spline[12] has been used for modelling surfaces and can be used for interpolation of data on the Cartesian lattice[13] as well as the body centered cubic lattice.

Box splines have found applications in high-dimensional filtering, specifically for fast bilateral filtering and non-local means algorithms.

[19] Moreover, box splines are used for designing efficient space-variant (i.e., non-convolutional) filters.

[20] Box splines are useful basis functions for image representation in the context of tomographic reconstruction problems as the spline spaces generated by box splines spaces are closed under X-ray and Radon transforms.

[21][22] In this application while the signal is represented in shift-invariant spaces, the projections are obtained, in closed-form, by non-uniform translates of box splines.

[21] In the context of image processing, box spline frames have been shown to be effective in edge detection.