Box spline

In the mathematical fields of

Definition

A box spline is a multivariate function ${\displaystyle (\mathbb {R} ^{d}\to \mathbb {R} )}$ defined for a set of vectors, ${\displaystyle \xi \in \mathbb {R} ^{d},}$ usually gathered in a matrix ${\displaystyle \mathbf {\Xi } :=\left[\xi _{1}\dots \xi _{N}\right].}$

When the number of vectors is the same as the dimension of the domain (i.e., ${\displaystyle N=d}$) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in ${\displaystyle \mathbf {\Xi } }$:

${\displaystyle M_{\mathbf {\Xi } }(\mathbf {x} ):={\frac {1}{\mid {\det {\Xi }}\mid }}\chi _{\mathbf {\Xi } }(\mathbf {x} )={\begin{cases}{\frac {1}{\mid {\det {\Xi }}\mid }}&{\text{if }}\mathbf {x} =\sum _{n=1}^{d}{t_{n}\xi _{n}}{\text{ for some }}0\leq t_{n}<1\\0&{\text{otherwise}}.\end{cases}}}$

Adding a new direction, ${\displaystyle \xi ,}$ to ${\displaystyle \mathbf {\Xi } ,}$ or generally when ${\displaystyle N>d,}$ the box spline is defined recursively:^[1]

${\displaystyle M_{\mathbf {\Xi } \cup \xi }(\mathbf {x} )=\int _{0}^{1}M_{\mathbf {\Xi } }(\mathbf {x} -t\xi )\,{\rm {d}}t.}$

The box spline ${\displaystyle M_{\mathbf {\Xi } }}$ can be interpreted as the shadow of the indicator function of the unit hypercube in ${\displaystyle \mathbb {R} ^{N}}$ when projected down into ${\displaystyle \mathbb {R} ^{d}.}$ In this view, the vectors ${\displaystyle \xi \in \mathbf {\Xi } }$ are the geometric projection of the standard basis in ${\displaystyle \mathbb {R} ^{N}}$ (i.e., the edges of the hypercube) to ${\displaystyle \mathbb {R} ^{d}.}$

Considering

supported on ${\displaystyle t\xi }$ for ${\displaystyle 0\leq t<1}$. Then the general box spline is defined as the convolution of distributions associated the single-vector box splines:

${\displaystyle M_{\mathbf {\Xi } }=M_{\xi _{1}}\ast M_{\xi _{2}}\ast \dots \ast M_{\xi _{N}}.}$

Properties

• Let ${\displaystyle \kappa }$ be the minimum number of directions whose removal from ${\displaystyle \Xi }$ makes the remaining directions not span ${\displaystyle \mathbb {R} ^{d}}$. Then the box spline has ${\displaystyle \kappa -2}$ degrees of continuity: ${\displaystyle M_{\mathbf {\Xi } }\in C^{\kappa -2}(\mathbb {R} ^{d})}$.[1]
• When ${\displaystyle N\geq d}$ (and vectors in ${\displaystyle \Xi }$ span ${\displaystyle \mathbb {R} ^{d}}$) the box spline is a compactly supported function whose support is a zonotope in ${\displaystyle \mathbb {R} ^{d}}$ formed by the
  Minkowski sum
  of the direction vectors ${\displaystyle \xi \in \mathbf {\Xi } }$.
• Since zonotopes are centrally symmetric, the support of the box spline is symmetric with respect to its center: ${\displaystyle \mathbf {c} _{\Xi }:={\frac {1}{2}}\sum _{n=1}^{N}\xi _{n}.}$
• Fourier transform of the box spline, in ${\displaystyle d}$ dimensions, is given by

${\displaystyle {\hat {M}}_{\Xi }(\omega )=\exp(-j\mathbf {c} _{\Xi }\cdot \omega )\prod _{n=1}^{N}\operatorname {sinc} (\xi _{n}\cdot \omega ).}$

Applications

For applications, linear combinations of shifts of one or more box splines on a lattice are used. Such splines are efficient, more so than linear combinations of simplex splines, because they are refinable and, by definition, shift invariant. 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.^[4]

In the context of

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.^[23]

References