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Box spline
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<h2 id="definition">Definition</h2> <p>A box spline is a multivariate <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> ( R d → R ) {\displaystyle (\mathbb {R} ^{d}\to \mathbb {R} )} defined for a set of vectors, ξ ∈ R d , {\displaystyle \xi \in \mathbb {R} ^{d},} usually gathered in a matrix Ξ := [ ξ 1 … ξ N ] . {\displaystyle \mathbf {\Xi } :=\left[\xi _{1}\dots \xi _{N}\right].} </p><p>When the number of vectors is the same as the dimension of the domain (i.e., N = d {\displaystyle N=d} ) then the box spline is simply the (normalized) <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of the parallelepiped formed by the vectors in Ξ {\displaystyle \mathbf {\Xi } } : </p> M Ξ ( x ) := 1 ∣ det Ξ ∣ χ Ξ ( x ) = { 1 ∣ det Ξ ∣ if x = ∑ n = 1 d t n ξ n for some 0 ≤ t n < 1 0 otherwise . {\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}}} <p>Adding a new direction, ξ , {\displaystyle \xi ,} to Ξ , {\displaystyle \mathbf {\Xi } ,} or generally when N > d , {\displaystyle N>d,} the box spline is defined recursively:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> M Ξ ∪ ξ ( x ) = ∫ 0 1 M Ξ ( x − t ξ ) d t . {\displaystyle M_{\mathbf {\Xi } \cup \xi }(\mathbf {x} )=\int _{0}^{1}M_{\mathbf {\Xi } }(\mathbf {x} -t\xi )\,{\rm {d}}t.} <p>The box spline M Ξ {\displaystyle M_{\mathbf {\Xi } }} can be interpreted as the shadow of the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of the unit <a href="/facts/Hypercube/q2mRPbIz">hypercube</a> in R N {\displaystyle \mathbb {R} ^{N}} when projected down into R d . {\displaystyle \mathbb {R} ^{d}.} In this view, the vectors ξ ∈ Ξ {\displaystyle \xi \in \mathbf {\Xi } } are the geometric projection of the <a href="/facts/Standard_basis/sLc9L8xx">standard basis</a> in R N {\displaystyle \mathbb {R} ^{N}} (i.e., the edges of the hypercube) to R d . {\displaystyle \mathbb {R} ^{d}.} </p><p>Considering <a href="/facts/Tempered_distribution/yfaJ4mUp">tempered distributions</a> a box spline associated with a single direction vector is a <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac</a>-like <a href="/facts/Generalized_function/fCM4VsyT">generalized function</a> supported on t ξ {\displaystyle t\xi } for 0 ≤ t < 1 {\displaystyle 0\leq t<1} . Then the general box spline is defined as the convolution of distributions associated the single-vector box splines: </p> M Ξ = M ξ 1 ∗ M ξ 2 ∗ ⋯ ∗ M ξ N . {\displaystyle M_{\mathbf {\Xi } }=M_{\xi _{1}}\ast M_{\xi _{2}}\ast \dots \ast M_{\xi _{N}}.} <h2 id="properties">Properties</h2> <ul><li>Let κ {\displaystyle \kappa } be the minimum number of directions whose removal from Ξ {\displaystyle \Xi } makes the remaining directions <i>not</i> span R d {\displaystyle \mathbb {R} ^{d}} . Then the box spline has κ − 2 {\displaystyle \kappa -2} degrees of continuity: M Ξ ∈ C κ − 2 ( R d ) {\displaystyle M_{\mathbf {\Xi } }\in C^{\kappa -2}(\mathbb {R} ^{d})} .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li>When N ≥ d {\displaystyle N\geq d} (and vectors in Ξ {\displaystyle \Xi } span R d {\displaystyle \mathbb {R} ^{d}} ) the box spline is a compactly supported function whose support is a <a href="/facts/Zonohedron/w16kH9H3">zonotope</a> in R d {\displaystyle \mathbb {R} ^{d}} formed by the <a href="/facts/Minkowski_sum/dEQfoJ1A">Minkowski sum</a> of the direction vectors ξ ∈ Ξ {\displaystyle \xi \in \mathbf {\Xi } } .</li> <li>Since <a href="/facts/Zonohedron/w16kH9H3">zonotopes</a> are centrally symmetric, the support of the box spline is symmetric with respect to its center: c Ξ := 1 2 ∑ n = 1 N ξ n . {\displaystyle \mathbf {c} _{\Xi }:={\frac {1}{2}}\sum _{n=1}^{N}\xi _{n}.} </li> <li><a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of the box spline, in d {\displaystyle d} dimensions, is given by</li></ul> M ^ Ξ ( ω ) = exp ( − j c Ξ ⋅ ω ) ∏ n = 1 N sinc ( ξ n ⋅ ω ) . {\displaystyle {\hat {M}}_{\Xi }(\omega )=\exp(-j\mathbf {c} _{\Xi }\cdot \omega )\prod _{n=1}^{N}\operatorname {sinc} (\xi _{n}\cdot \omega ).} <h2 id="applications">Applications</h2> <p>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 <a href="/facts/Subdivision_surface/9AKT4pfM">subdivision surface</a> constructions. </p><p>Box splines have been useful in characterization of hyperplane arrangements.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Also, box splines can be used to compute the volume of polytopes.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>In the context of <a href="/facts/Multidimensional_sampling/10UJv920">multidimensional signal processing</a>, box splines can provide <a href="/facts/Reconstruction_filter/AB9SQoeR">multivariate interpolation kernels</a> (reconstruction filters) tailored to non-Cartesian <a href="/facts/Multidimensional_sampling/10UJv920">sampling lattices</a>,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> and <a href="/facts/Root_systems/TFaEdwMU">crystallographic lattices</a> (root lattices) that include many information-theoretically optimal sampling lattices.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Generally, optimal <a href="/facts/Sphere_packing/NY2FKyMf">sphere packing</a> and sphere covering lattices<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> are useful for sampling multivariate functions in 2-D, 3-D and higher dimensions.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> In the 2-D setting the three-direction box spline<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> is used for interpolation of hexagonally sampled images. In the 3-D setting, four-direction<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> and six-direction<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> box splines are used for interpolation of data sampled on the (optimal) <a href="/facts/Body-centered_cubic/92Co7BKM">body-centered cubic</a> and <a href="/facts/Face-centered_cubic/92Co7BKM">face-centered cubic</a> lattices respectively.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> The seven-direction box spline<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> has been used for modelling surfaces and can be used for interpolation of data on the Cartesian lattice<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> as well as the <a href="/facts/Body_centered_cubic/92Co7BKM">body centered cubic</a> lattice.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> Generalization of the four-<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> and six-direction<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> box splines to higher dimensions<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> can be used to build splines on <a href="/facts/Root_system/TFaEdwMU">root lattices</a>.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> Box splines are key ingredients of hex-splines<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> and Voronoi splines<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> that, however, are not refinable. </p><p>Box splines have found applications in high-dimensional filtering, specifically for fast bilateral filtering and non-local means algorithms.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> Moreover, box splines are used for designing efficient space-variant (i.e., non-convolutional) filters.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p><p>Box splines are useful basis functions for image representation in the context of <a href="/facts/Tomographic_reconstruction/rk84hL6s">tomographic reconstruction</a> problems as the spline spaces generated by box splines spaces are closed under <a href="/facts/X-ray_transform/jLvEpEU6">X-ray</a> and <a href="/facts/Radon_transform/JskFJRMS">Radon</a> transforms.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> 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.<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> </p><p>In the context of image processing, box spline frames have been shown to be effective in edge detection.<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Boor, C.; Höllig, K.; Riemenschneider, S. (1993). Box Splines. Applied Mathematical Sciences. Vol. 98. doi:10.1007/978-1-4757-2244-4. ISBN 978-1-4419-2834-4. <a href="978-1-4419-2834-4" target="_blank">978-1-4419-2834-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Prautzsch, H.; Boehm, W.; Paluszny, M. (2002). "Box splines". Bézier and B-Spline Techniques. Mathematics and Visualization. pp. 239–258. doi:10.1007/978-3-662-04919-8_17. ISBN 978-3-642-07842-2. <a href="978-3-642-07842-2" target="_blank">978-3-642-07842-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Boor, C.; Höllig, K.; Riemenschneider, S. (1993). Box Splines. Applied Mathematical Sciences. Vol. 98. doi:10.1007/978-1-4757-2244-4. ISBN 978-1-4419-2834-4. <a href="978-1-4419-2834-4" target="_blank">978-1-4419-2834-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Boor, C.; Höllig, K.; Riemenschneider, S. (1993). Box Splines. Applied Mathematical Sciences. Vol. 98. doi:10.1007/978-1-4757-2244-4. ISBN 978-1-4419-2834-4. <a href="978-1-4419-2834-4" target="_blank">978-1-4419-2834-4</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>De Concini, C.; Procesi, C. (2010). Topics in Hyperplane Arrangements, Polytopes and Box-Splines. doi:10.1007/978-0-387-78963-7. ISBN 978-0-387-78962-0. <a href="978-0-387-78962-0" target="_blank">978-0-387-78962-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Xu, Z. (2011). "Multivariate splines and polytopes". Journal of Approximation Theory. 163 (3): 377–387. arXiv:0806.1127. doi:10.1016/j.jat.2010.10.005. S2CID 10063913. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Entezari, Alireza. Optimal sampling lattices and trivariate box splines. 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[Vancouver, BC.]: Simon Fraser University, 2007. <http://summit.sfu.ca/item/8178>. <a href="http://summit.sfu.ca/item/8178" target="_blank">http://summit.sfu.ca/item/8178</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Peters, Jorg; Wittman, M. (1997). "Box-spline based CSG blends". Proceedings of the fourth ACM symposium on Solid modeling and applications - SMA '97. pp. 195. doi:10.1145/267734.267783. ISBN 0897919467. S2CID 10064302. <a href="0897919467" target="_blank">0897919467</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Entezari, A.; Moller, T. (2006). "Extensions of the Zwart-Powell Box Spline for Volumetric Data Reconstruction on the Cartesian Lattice". IEEE Transactions on Visualization and Computer Graphics. 12 (5): 1337–1344. doi:10.1109/TVCG.2006.141. PMID 17080870. S2CID 232110. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Minho Kim (2013). "Quartic Box-Spline Reconstruction on the BCC Lattice". IEEE Transactions on Visualization and Computer Graphics. 19 (2): 319–330. doi:10.1109/TVCG.2012.130. PMID 22614329. S2CID 7338997. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Entezari, A.; Van De Ville, D.; Moller, T. (2008). "Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice" (PDF). IEEE Transactions on Visualization and Computer Graphics. 14 (2): 313–328. doi:10.1109/TVCG.2007.70429. PMID 18192712. 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N.; MuñOz-Barrutia, A.; Unser, M. (2010). "Fast Space-Variant Elliptical Filtering Using Box Splines". IEEE Transactions on Image Processing. 19 (9): 2290–2306. arXiv:1003.2022. Bibcode:2010ITIP...19.2290C. doi:10.1109/TIP.2010.2046953. PMID 20350851. S2CID 16383503. <a href="/wiki/Arrate_Mu%C3%B1oz_Barrutia" target="_blank">/wiki/Arrate_Mu%C3%B1oz_Barrutia</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Entezari, A.; Nilchian, M.; Unser, M. (2012). "A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems" (PDF). IEEE Transactions on Medical Imaging. 31 (8): 1532–1541. doi:10.1109/TMI.2012.2191417. PMID 22453611. S2CID 3787118. <a href="http://bigwww.epfl.ch/publications/entezari1201.pdf" target="_blank">http://bigwww.epfl.ch/publications/entezari1201.pdf</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Entezari, A.; Unser, M. (2010). "A box spline calculus for computed tomography". 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro. p. 600. doi:10.1109/ISBI.2010.5490105. ISBN 978-1-4244-4125-9. S2CID 17368057. <a href="978-1-4244-4125-9" target="_blank">978-1-4244-4125-9</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Entezari, A.; Nilchian, M.; Unser, M. (2012). "A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems" (PDF). IEEE Transactions on Medical Imaging. 31 (8): 1532–1541. doi:10.1109/TMI.2012.2191417. PMID 22453611. S2CID 3787118. <a href="http://bigwww.epfl.ch/publications/entezari1201.pdf" target="_blank">http://bigwww.epfl.ch/publications/entezari1201.pdf</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Guo, W.; Lai, M. J. (2013). "Box Spline Wavelet Frames for Image Edge Analysis". SIAM Journal on Imaging Sciences. 6 (3): 1553. doi:10.1137/120881348. S2CID 2708993. <a href="https://zenodo.org/record/894468" target="_blank">https://zenodo.org/record/894468</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> </ol>