MediaWiki-API-Ergebnis

{
    "batchcomplete": "",
    "continue": {
        "gapcontinue": "Rittal_Rack",
        "continue": "gapcontinue||"
    },
    "warnings": {
        "main": {
            "*": "Subscribe to the mediawiki-api-announce mailing list at <https://lists.wikimedia.org/mailman/listinfo/mediawiki-api-announce> for notice of API deprecations and breaking changes."
        },
        "revisions": {
            "*": "Because \"rvslots\" was not specified, a legacy format has been used for the output. This format is deprecated, and in the future the new format will always be used."
        }
    },
    "query": {
        "pages": {
            "237": {
                "pageid": 237,
                "ns": 0,
                "title": "Reproducing Polynomials with B-Splines",
                "revisions": [
                    {
                        "contentformat": "text/x-wiki",
                        "contentmodel": "wikitext",
                        "*": "<section begin=\"head\"/>\n[[Datei:bspline_family.png|right|150px|Family of B-splines up to N=4]]\nA B-Spline of order <math>N</math> is known to be able to reproduce any polynomial up to order <math>N</math>:\n\n<math>\n\\sum_{n \\in \\mathbb{Z}} c_{m,n} \\beta_N (t - n) = t^m\n</math>\n\nIn words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order <math>N</math>. This is needed for different applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework. In this case any kernel <math>\\varphi</math> reproducing polynomials (that is, satisfying the Strang-Fix conditions) can be used. However, among all possible kernels, the B-Splines have the smallest possible support.\n\nAn important question is how to obtain the coefficients <math>c_{m,n}</math> for the reproduction-formula. In this small article, I describe one way.\n<section end=\"head\"/>\n\nStarting from\n\n<math>\n\\sum_{n \\in \\mathbb{Z}} c_{m,n} \\varphi(t - n) = t^m\n</math>\n\nthe coefficients can be obtained using the dual of <math>\\varphi</math>, <math>\\tilde{\\varphi}</math><ref>P.L. Dragotti, M. Vetterli, T.Blu: \"Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang-Fix\", ''IEEE Transactions on Signal Processing'', vol. 55, No. 5, May 2007</ref> (I set <math>\\beta_N = \\varphi</math> for consistency with my notes):\n\n<math>\nc_{m,n} = \\int_{-\\infty}^{\\infty} t^m \\tilde{\\varphi}(t - n)\\,dt\n</math>\n\nHowever, even if the dual would be known, solving the infinite integral is only feasible when the dual has finite support. This is the case with the B-Spline itself but not with its dual!\n\nA closer look at the formula tells that this is nothing more than a convolution (under the assumption that <math>\\tilde{\\varphi}</math> is symmetric which is the case):\n\n<math>\nc_{m,n} = \\int t^m \\tilde{\\varphi}(-(n-t))\\,dt = \\int t^m \\tilde{\\varphi}(n-t)\\,dt = (t^m * \\tilde{\\varphi})(n)\n</math>\n\nNow, this can be transformed to fourier domain:\n\n<math>\n(t^m * \\tilde{\\varphi})(n) = \\mathcal{F}^{-1}\\left\\{ \\mathcal{F}\\left\\{t^m\\right\\} \\tilde{\\Phi}(\\omega)\\right\\} =  \\mathcal{F}^{-1}\\left\\{ j^m \\sqrt{2\\pi} \\delta^{(n)}(\\omega) \\tilde{\\Phi}(\\omega) \\right\\} = j^m \\sqrt{2\\pi} \\mathcal{F}^{-1}\\left\\{\\delta^{(n)}(\\omega) \\tilde{\\Phi}(\\omega) \\right\\}\n</math>\n\nWriting the inverse of this expression yields:\n\n<math>\nj^m \\sqrt{2\\pi} \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\pi}^{\\pi} \\delta^{(n)}(\\omega) \\tilde{\\Phi}(\\omega) e^{j\\omega n}\\,d\\omega = j^m \\int_{-\\infty}^{\\infty} \\delta^{(n)}(\\omega) \\underbrace{\\tilde{\\Phi}(\\omega) e^{j\\omega n}}_{f(\\omega)}\\,d\\omega\n</math>\n\nIt is known that<ref>http://en.wikipedia.org/wiki/Dirac_delta_function</ref>:\n\n<math>\n\\int \\delta^{(n)}(x) f(x)\\,dx = (-1)^n f^{(n)}(0) \n</math>\n\nso that\n\n<math>\nj^m \\int_{-\\infty}^{\\infty} \\delta^{(n)}(\\omega) f(\\omega)\\,d\\omega = j^m (-1)^m \\left. \\frac{\\partial^m}{\\partial \\omega^m} f(\\omega) \\right|_{\\omega = 0}\n</math>\n\nNow the whole procedure has been reduced to calculating the derivative of <math>f(\\omega)</math> and set the result to zero.\n\nAn open question is how to obtain the dual of <math>\\varphi</math>. As the reproduction formula spans a vector space, <math>\\varphi</math> must be at least bi-orthogonal to <math>\\tilde{\\varphi}</math>. This translates in fourier domain to<ref>S. Mallat: \"A Wavelet Tour of Signal Processing\", ''Academic Press'' 1999</ref>:\n\n<math>\n\\tilde{\\Phi}(\\omega) = \\frac{\\Phi(\\omega)}{\\sum_{k \\in \\mathbb{Z}} |\\Phi(\\omega + 2\\pi k)|^2}\n</math>\n\nThe fourier transform of a B-Spline of order <math>N</math> is (e.g. <ref>M.Unser: \"Splines - A Perfect Fit for Signal and Imaging Processing\", ''IEEE Signal Processing Magazine'' Nov. 1999</ref>):\n\n<math>\n\\Beta_N(\\omega) = \\Phi(\\omega) = \\left( \\frac{\\sin(\\omega/2)}{\\omega/2} \\right)^{N+1} =\n\\mathrm{sinc}^{N+1}(\\omega/2)\n</math>\n\nThe following derivation of the sum is borrowed from <ref>M.J.C.S. Reis, P.J.S.G. Ferreira, S.F.S.P. Soares: \"Linear combinations of B-splines as generating functions for signal approximation\", ''Elsevier Digital Signal Processing 15'', 2005</ref>. For this derivation to work, I set <math>L=N+1</math> temporarily:\n\n<math>\n\\sum_{k \\in \\mathbb{Z}} |\\Phi(\\omega + 2\\pi k)|^2 = \n\\sum_{k \\in \\mathbb{Z}} \\left|\\mathrm{sinc}\\left(\\frac{1}{2}(\\omega + 2\\pi k)\\right)^L \\right|^2 =\n\\sum_{k \\in \\mathbb{Z}} \\left|\\mathrm{sinc}\\left(\\frac{1}{2}(\\omega + 2\\pi k)\\right) \\right|^{2L}\n</math>\n\nand because <math>2L</math> is always even:\n\n<math>\n= \\sum_{k \\in \\mathbb{Z}}\\frac{\\sin^{2L}(\\frac{1}{2}(\\omega + 2\\pi k))}{\\left(\\frac{1}{2}(\\omega + 2\\pi k)\\right)^{2L}}\n= \\sum_{k \\in \\mathbb{Z}}\\frac{\\sin^{2L}(\\frac{\\omega}{2} + \\pi k))}{(\\frac{\\omega}{2} + \\pi k)^{2L}}\n</math>\n\nBecause of the periodicity it is known that\n\n<math>\n\\sin^{2L}(x + \\pi k) = \\sin^{2L}(x)\n</math>\n\nsuch that\n\n<math>\n= \\sin^{2L}\\left(\\frac{\\omega}{2}\\right) \\sum_{k \\in \\mathbb{Z}}\\frac{1}{(\\frac{\\omega}{2} + \\pi k)^{2L}}\n</math>\n\nAnd finally the following relation is used<ref>L.V. Ahlfors: \"Complex Analysis\", ''McGraw-Hill'', 1979</ref>:\n\n<math>\n\\sum_k \\frac{1}{(x + \\pi k)^{2L}} = -\\frac{1}{(2L-1)!} \\frac{d^{2L-1}}{dx^{2L-1}} \\cot{x}\n</math>\n\nin order to finally obtain:\n\n<math>\n\\sum_{k \\in \\mathbb{Z}} \\left|\\mathrm{sinc}\\left(\\frac{1}{2}(\\omega + 2\\pi k)\\right)^L \\right|^2 =\n-\\sin^{2L}\\left(\\frac{\\omega}{2}\\right) \\frac{1}{(2L-1)!} \\frac{d^{2L-1}}{d\\left(\\frac{\\omega}{2}\\right)^{2L-1}} \\cot{\\left(\\frac{\\omega}{2}\\right)}\n</math>\n\nand with <math>L = N+1</math>:\n\n<math>\n\\sum_{k \\in \\mathbb{Z}} |\\Phi(\\omega + 2\\pi k)|^2 =\n-\\sin^{2(N+1)}\\left(\\frac{\\omega}{2}\\right) \\frac{1}{(2N+1)!} \\frac{d^{2N+1}}{d\\left(\\frac{\\omega}{2}\\right)^{2N+1}} \\cot{\\left(\\frac{\\omega}{2}\\right)}\n</math>\n\nTherefore, together with <math>\\Phi(\\omega)</math> this yields:\n\n<math>\n\\tilde{\\Phi}(\\omega) = \\frac{(2N+1)!}{\\left(\\frac{\\omega}{2}\\right) \\sin\\left(\\frac{\\omega}{2}\\right)^{N+1} \\frac{d^{2N+1}}{d\\left(\\frac{\\omega}{2}\\right)^{2N+1}} \\cot{\\left(\\frac{\\omega}{2}\\right)}}\n</math>\n\nand finally substituting for <math>t(\\omega)</math>:\n\n<math>\nf(\\omega) = \\frac{(2N+1)!}{\\left(\\frac{\\omega}{2}\\right) \\sin\\left(\\frac{\\omega}{2}\\right)^{N+1} \\frac{d^{2N+1}}{d\\left(\\frac{\\omega}{2}\\right)^{2N+1}} \\cot{\\left(\\frac{\\omega}{2}\\right)}} e^{j \\omega n}\n</math>\n\nAs this function is not well defined it is better to use the limit:\n\n<math>\nc_{m,n} = j^m \\lim_{\\omega \\rightarrow 0} f(\\omega)\n</math>\n\n= Examples for a cubic spline =\n\nFor a cubic spline (N=3) the coefficients are:\n\n<math>\n\\begin{array}{lcl}\nc_{0,n}   & = & 1 \\\\\nc_{1,n}   & = & n \\\\\nc_{2,n}   & = & \\frac{1}{3}\\left( -1 + 3n^2 \\right) \\\\\nc_{3,n}   & = & -n + n^3\n\\end{array}\n</math>\n\n[[Datei:poly_repro_quad.png|center|cubic spline reproducing polynomial of order 2]]\n\n[[Datei:poly_repro_cubic.png|center|cubic spline reproducing polynomial of order 3]]\n\n= References =\n\n<ref name=\"schoenberg\">I.J. Schoenberg: \"Cardinal interpolation and spline functions\", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>\n\n<references/>\n\n= Comments =\n\n<comments />{{:{{TALKSPACE}}:{{PAGENAME}}}}\n\n[[Kategorie:Weblog]]"
                    }
                ]
            },
            "323": {
                "pageid": 323,
                "ns": 0,
                "title": "Retis",
                "revisions": [
                    {
                        "contentformat": "text/x-wiki",
                        "contentmodel": "wikitext",
                        "*": "__NOTOC__\n\n== Willkommen auf retis.nobaq.net.! ==\n\n=== Wieso bin ich hier? ===\n\nVermutlich bist du hier weil du auf eine Seite willst die auf diesem Server gehostet ist oder gehostet war. F\u00fcr Fragen: niki at nobaq punkt net. Bitte folge den Links (ein paar gehostete Seiten):\n\n* http://www.nobaq.net\n* http://www.hyperlex.at\n* http://Niki.Hammler.net\n* http://www.nobaq.at\n* http://www.nobaq.eu\n* http://www.hammler.info\n* http://niki.priv.at\n* https://secure.nobaq.net\n\n=== Wer ist das? ===\n\nDieser Webserver ist retis.nobaq.net (92.82.102.173/29 bzw. 2001:7b8:3cd:3::173). Es ist der Webserver von Nikolaus Hammler. Der Name retis ist lateinisch und bedeutet Netz. Der virtuelle Server befindet sich auf [[Server|radix.intra.nobaq.net]]. Fuer weitere Nodes & Server, siehe [[Netzwerk]].\n\n=== Dienste ===\n\n* Webserver (apache), horde, zarafa-webaccess, ...\n* XMPP/Jabber (prosody)\n* FTPS (pureftp) zum Zugriff auf Webspace"
                    }
                ]
            }
        }
    }
}