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In the mathematical fields of [[numerical analysis]] and [[approximation theory]], '''box splines''' are [[piecewise]] [[polynomial]] [[Function (mathematics)|functions]] of several variables.<ref name="thebook">C. de Boor, K. Höllig, and S. Riemenschneider. Box Splines, volume 98 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.</ref> Box splines are considered as a multivariate generalization of [[B-spline|basis splines (B-splines)]] and are generally used for multivariate approximation/interpolation. Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower dimensional space.<ref>{{Cite doi| 10.1007/978-3-662-04919-8_17|noedit}}</ref> Box splines and simplex splines are well studied special cases of polyhedral splines which are defined as shadows of general [[polytopes]].
As head of Louis Vuitton, Yves Carcelle, who has died at the age of 66, built the French fashion house into one of the world's leading luxury brands.<br>Though it is only one of around 60 such brands owned by LVMH, Louis Vuitton is the conglomerate's driving force, and Carcelle expanded its appeal into areas such as eyewear and jewellery while travelling the world looking for new areas to colonise. One of his key decisions came in 1997 when he hired as creative director the then up-and-coming designer Marc Jacobs, whose introduction of the company's first ready-to-wear line helped seal its place at or near the top of fashion's premier league.<br><br><br>Born in 1948 in Paris, Carcelle graduated in mathematics from the city's �cole Polytechnique then gained a masters in business administration from the business school, Insead. His first job was as a salesman for Spontex a maker of household cleaning products, which he said was excellent training for a career in fashion: "You have to be a good salesmen, if nothing else," he said.<br><br>"And you have to know how to deal with people,"<br>The renowned fashion journalist Suzy Menkes wrote: "My favourite Yves Carcelle story - one he liked to tell with a glass of champagne in hand, perhaps at a private party at home in front of a bold Gilbert & George painting, with his then wife Rebecca at his side and with his boys listening in - was about his early days as a fledgling travelling salesman. Taking to the road with a girlfriend, the young Yves would send her into a hardware store asking with a flirtatious enthusiasm for a new product he was hoping to sell. Ten minutes later, he would be knocking on the same store door, offering to supply the goods. The success was instant."<br><br>Following his stint at Spontex, in 1974 he joined Blenda Pharm laboratories, then five years later was appointed director of the Absorba clothing brand at Poron. In 1985 he became president of the textiles firm Descamps, and his success in turning the company around led to his recruitment by Bernard Arnault as LVMH's director of strategy in 1989.<br><br>The following year he became [http://Www.Google.Co.uk/search?hl=en&gl=us&tbm=nws&q=chairman&gs_l=news chairman] and CEO, then in 1998 head of the LVMH fashion division, which as well as Louis Vuitton includes such brands as Givenchy, Donna Karan, Christian Dior, [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bags Outlet], Fendi, the jewellers Bulgari and cognac maker Hennessy.<br>Colleagues said Carcelle knew little about luxury when he joined Louis Vuitton. But he quickly won Arnault's trust and became one of his most respected lieutenants. "There was a lot of mutual respect between the two men even though they had very different personalities," an LVMH executive.<br><br>"Carcelle was very different from Arnault. Arnault is cold and not really somebody who easily gets excited about something, while Carcelle was very spontaneous and open."<br>Carcelle quadrupled Louis Vuitton's network of stores to just under 470, many of them in strategically important emerging markets such as China. He showed great skill in spreading the Louis Vuitton gospel: a rival told Suzy Menkes how when he and Carcelle would arrive in an often remote Chinese town, Carcelle would hand out beautifully wrapped small leather goods as gifts to the city's mayor and each member of his family.<br><br>In 2001 he went back to his old job following the departure of Marcello Bottoli after only 18 months. "It's not always easy to sum up what our brand means to people, but let me try," he said in 2004. "It's about reliability, quality, style, innovation and authenticity."<br>Carcelle became an expert in easing Louis Vuitton into new markets. When he wasn't globe-trotting, Carcelle was usually in his Paris office by 6am. Though he was softly-spoken, he could raise his voice when necessary - as he did when he let fly at a French government official who went to a Louis Vuitton store opening in Bangkok carrying a fake Vuitton bag.<br><br>Carcelle, who died of renal cancer, stepped down at the end of 2012 as Louis Vuitton's sales growth was starting to slow down after years of double-digit increases. There were calls for a change in management, particularly as Carcelle was in his early 60s. He stayed on as part of the LVMH executive committee and as vice president of the Fondation Louis Vuitton museum, designed by Frank Gehry on the Bois de Boulogne and due to open next month.<br><br>A keen sailor, he was an enthusiastic follower of the Louis Vuitton Cup, which until last year served as the eliminator series for the Americas Cup. Just before the onset of the global financial crisis he was asked if he was worried that his company could suffer.<br>"Not really," he replied. "The great thing about our business is that our customers are very rich - and then they are a bit less rich, but still rich, right?"<br><br>Yves Carcelle, businessman: born Paris 18 May 1948; Chevalier de la L�gion d'honneur 2014; twice married (five children); died 31 August 2014.
 
==Definition==
A box spline is a multivariate [[Function (mathematics)|function]] (<math>\mathbb{R}^d \to \mathbb{R} </math>) defined for a set of vectors, <math>\xi \in \mathbb{R}^d</math>, usually gathered in a matrix <math>\mathbf{\Xi} := \left[\xi_1 \dots \xi_N\right] </math>.
 
When the number of vectors is the same as the dimension of the domain (i.e., <math> N = d </math>) then the box spline is simply the (normalized) [[indicator function]] of the parallelepiped formed by the vectors in <math>\mathbf{\Xi}</math>:
:<math> M_{\mathbf{\Xi}}(\mathbf{x}) := \frac{1}{\mid{\det{\Xi}}\mid}\chi_{\mathbf{\Xi}}(\mathbf{x}) = \begin{cases} \frac{1}{\mid{\det{\Xi}}\mid} & \mathbf{x} = \sum_{n=1}^d{t_n \xi_n} \text{ for some } 0 \le t_n < 1 \\ 0 & \text{otherwise}\end{cases}.</math>
Adding a new direction, <math>\xi</math>, to <math>\mathbf{\Xi}</math>, or generally when <math>N > d</math>, the box spline is defined recursively:<ref name="thebook" />
:<math> M_{\mathbf{\Xi} \cup \xi}(\mathbf{x}) = \int_0^1{M_{\mathbf{\Xi}}(\mathbf{x}- t \xi) \, {\rm d}t}</math>.
 
[[File:Box Splines Square Grid Annotated Dark.png|thumb|right|Examples of bivariate box splines corresponding to 1, 2, 3 and 4 vectors in 2-D.]]
 
The box spline <math>M_{\mathbf{\Xi}}</math> can be interpreted as the shadow of the [[indicator function]] of the unit [[hypercube]] in <math>\mathbb{R}^N</math> when projected down into <math>\mathbb{R}^d</math>. In this view, the vectors <math>\xi \in \mathbf{\Xi}</math> are the geometric projection of the [[standard basis]] in <math>\mathbb{R}^N</math> (i.e., the edges of the hypercube) to <math>\mathbb{R}^d</math>.
 
Considering [[tempered distributions]] a box spline associated with a single direction vector is a [[Dirac delta function|Dirac]]-like [[generalized function]] supported on <math>t\xi</math> for <math>0 \le t < 1</math>. Then the general box spline is defined as the convolution of distributions associated the single-vector box splines:<ref name="boxTomo">{{Cite doi| 10.1109/TMI.2012.2191417|noedit}}</ref>
:<math>M_{\mathbf{\Xi}} = M_{\xi_1} \ast M_{\xi_2} \dots \ast M_{\xi_N}. </math>
 
==Properties==
* Let <math>\kappa</math> be the minimum number of directions whose removal from <math>\Xi</math> makes the remaining directions ''not'' span <math>\mathbb{R}^d</math>. Then the box spline has <math>\kappa-2</math> degrees of continuity: <math>M_{\mathbf{\Xi}} \in C^{\kappa-2}(\mathbb{R}^d)</math>.<ref name="thebook" />
 
* When <math>N\ge d</math> (and vectors in <math>\Xi</math> span <math>\mathbb{R}^d</math>) the box spline is a compactly supported function whose support is a [[Zonohedron|zonotope]] in <math>\mathbb{R}^d</math> formed by the [[Minkowski sum]] of the direction vectors <math>{\xi} \in \mathbf{\Xi}</math>.
 
* Since [[Zonohedron|zonotopes]] are centrally symmetric, the support of the box spline is symmetric with respect to its center: <math>\mathbf{c}_\Xi := \frac{1}{2}\sum_{n=1}^N \xi_n .</math>
 
* [[Fourier transform]] of the box spline, in <math>d</math> dimensions, is given by
:: <math>\hat{M}_{\Xi}(\omega) = \exp{(-j\mathbf{c}_{\Xi}\cdot\omega)}\prod_{n=1}^N{{\rm sinc}(\xi_n\cdot\omega)}.</math>
 
==Applications==
Box splines have been useful in characterization of hyperplane arrangements.<ref name="boxHyperplane">De Concini, Corrado, and Claudio Procesi. Topics in hyperplane arrangements, polytopes and box-splines. Springer, 2011.</ref> Also, box splines can be
used to compute the volume of polytopes.<ref name="boxpolytope"> Zhiqiang Xu, Multivariate splines and polytopes,  Journal of Approximation Theory, Vol. 163, Issue 3, March 2011.</ref>
 
In the context of [[Multidimensional sampling|multidimensional signal processing]], box splines provide a flexible framework for designing (non-separable) basis functions acting as [[Reconstruction filter|multivariate interpolation kernels]] (reconstruction filters) geometrically tailored to non-Cartesian [[Multidimensional sampling|sampling lattices]]. This flexibility makes box splines suitable for designing (non-separble) interpolation filters for [[Cubic crystal system|crystallographic lattices]] which are optimal<ref name="optSamp">{{Cite doi|10.1109/TIT.2004.840864|noedit}}</ref> from the information-theoretic aspects for [[Multidimensional sampling|sampling]] multidimensional functions. Optimal sampling lattices have been studied in higher dimensions.<ref name="optSamp" /> Generally, optimal [[sphere packing]] and sphere covering lattices<ref>J. H. Conway, N. J. A. Sloane. Sphere packings, lattices and groups. Springer, 1999.</ref> are useful for sampling multivariate functions in 2-D, 3-D and higher dimensions.
 
For example, in the 2-D setting the three-direction box spline<ref>{{Cite doi| 10.1109/LSP.2006.871852|noedit}}</ref> is used for interpolation of hexagonally sampled images. In the 3-D setting, four-direction<ref name="fourDir">{{Cite doi| 10.1109/TVCG.2007.70429|noedit}}</ref> and six-direction<ref name="sixDir">{{Cite doi| 10.1109/TVCG.2008.115|noedit}}</ref> box splines are used for interpolation of data sampled on the (optimal) [[body centered cubic]] and [[face centered cubic]] lattices respectively.<ref>Entezari, Alireza. Optimal sampling lattices and trivariate box splines. [Vancouver, BC.]: Simon Fraser University, 2007. <http://summit.sfu.ca/item/8178>.</ref> The seven-direction box spline can be used for interpolation of data on the Cartesian lattice<ref>{{Cite doi| 10.1109/TVCG.2006.141|noedit}}</ref> as well as the [[body centered cubic]] lattice.<ref>{{Cite doi| 10.1109/TVCG.2012.130|noedit}}</ref> Generalization of the four-<ref name="fourDir" /> and six-direction<ref name="sixDir" /> box splines to higher dimensions<ref>Kim, Minho. Symmetric Box-Splines on Root Lattices. [Gainesville, Fla.]: University of Florida, 2008. <http://uf.catalog.fcla.edu/permalink.jsp?20UF021643670>.</ref> can be used to build splines on [[Root system|root lattices]]. Box splines are key ingredients of hex-splines<ref>{{Cite doi| 10.1109/TIP.2004.827231}}</ref> and Voronoi splines.<ref>{{Cite doi| 10.1109/TSP.2010.2051808|noedit}}</ref>
 
They have found applications in high-dimensional filtering, specifically for fast bilateral filtering and non-local means algorithms.<ref>{{Cite doi|10.1007/s10851-012-0379-2 |noedit}}</ref> Moreover, box splines are used for designing efficient space-variant (i.e., non-convolutional) filters.<ref>{{Cite doi| 10.1109/TIP.2010.2046953|noedit}}</ref>
 
Box splines are useful basis functions for image representation in the context of [[tomographic reconstruction]] problems as the box spline (function) spaces are closed under [[X-ray transform|X-ray]] and [[Radon transform|Radon]] transforms.<ref name="boxTomo" /><ref>{{Cite doi| 10.1109/ISBI.2010.5490105}}</ref>
 
==References==
{{reflist}}
 
[[Category:Splines]]

Latest revision as of 12:15, 7 January 2015

As head of Louis Vuitton, Yves Carcelle, who has died at the age of 66, built the French fashion house into one of the world's leading luxury brands.
Though it is only one of around 60 such brands owned by LVMH, Louis Vuitton is the conglomerate's driving force, and Carcelle expanded its appeal into areas such as eyewear and jewellery while travelling the world looking for new areas to colonise. One of his key decisions came in 1997 when he hired as creative director the then up-and-coming designer Marc Jacobs, whose introduction of the company's first ready-to-wear line helped seal its place at or near the top of fashion's premier league.


Born in 1948 in Paris, Carcelle graduated in mathematics from the city's �cole Polytechnique then gained a masters in business administration from the business school, Insead. His first job was as a salesman for Spontex a maker of household cleaning products, which he said was excellent training for a career in fashion: "You have to be a good salesmen, if nothing else," he said.

"And you have to know how to deal with people,"
The renowned fashion journalist Suzy Menkes wrote: "My favourite Yves Carcelle story - one he liked to tell with a glass of champagne in hand, perhaps at a private party at home in front of a bold Gilbert & George painting, with his then wife Rebecca at his side and with his boys listening in - was about his early days as a fledgling travelling salesman. Taking to the road with a girlfriend, the young Yves would send her into a hardware store asking with a flirtatious enthusiasm for a new product he was hoping to sell. Ten minutes later, he would be knocking on the same store door, offering to supply the goods. The success was instant."

Following his stint at Spontex, in 1974 he joined Blenda Pharm laboratories, then five years later was appointed director of the Absorba clothing brand at Poron. In 1985 he became president of the textiles firm Descamps, and his success in turning the company around led to his recruitment by Bernard Arnault as LVMH's director of strategy in 1989.

The following year he became chairman and CEO, then in 1998 head of the LVMH fashion division, which as well as Louis Vuitton includes such brands as Givenchy, Donna Karan, Christian Dior, Celine Bags Outlet, Fendi, the jewellers Bulgari and cognac maker Hennessy.
Colleagues said Carcelle knew little about luxury when he joined Louis Vuitton. But he quickly won Arnault's trust and became one of his most respected lieutenants. "There was a lot of mutual respect between the two men even though they had very different personalities," an LVMH executive.

"Carcelle was very different from Arnault. Arnault is cold and not really somebody who easily gets excited about something, while Carcelle was very spontaneous and open."
Carcelle quadrupled Louis Vuitton's network of stores to just under 470, many of them in strategically important emerging markets such as China. He showed great skill in spreading the Louis Vuitton gospel: a rival told Suzy Menkes how when he and Carcelle would arrive in an often remote Chinese town, Carcelle would hand out beautifully wrapped small leather goods as gifts to the city's mayor and each member of his family.

In 2001 he went back to his old job following the departure of Marcello Bottoli after only 18 months. "It's not always easy to sum up what our brand means to people, but let me try," he said in 2004. "It's about reliability, quality, style, innovation and authenticity."
Carcelle became an expert in easing Louis Vuitton into new markets. When he wasn't globe-trotting, Carcelle was usually in his Paris office by 6am. Though he was softly-spoken, he could raise his voice when necessary - as he did when he let fly at a French government official who went to a Louis Vuitton store opening in Bangkok carrying a fake Vuitton bag.

Carcelle, who died of renal cancer, stepped down at the end of 2012 as Louis Vuitton's sales growth was starting to slow down after years of double-digit increases. There were calls for a change in management, particularly as Carcelle was in his early 60s. He stayed on as part of the LVMH executive committee and as vice president of the Fondation Louis Vuitton museum, designed by Frank Gehry on the Bois de Boulogne and due to open next month.

A keen sailor, he was an enthusiastic follower of the Louis Vuitton Cup, which until last year served as the eliminator series for the Americas Cup. Just before the onset of the global financial crisis he was asked if he was worried that his company could suffer.
"Not really," he replied. "The great thing about our business is that our customers are very rich - and then they are a bit less rich, but still rich, right?"

Yves Carcelle, businessman: born Paris 18 May 1948; Chevalier de la L�gion d'honneur 2014; twice married (five children); died 31 August 2014.

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