Reproducing Polynomials with B-Splines - Versionsgeschichte

Niki am 19. Juli 2010 um 15:45 Uhr

2010-07-19T15:45:23Z

Niki am 19. Juli 2010 um 15:39 Uhr

2010-07-19T15:39:57Z

Niki: /* References */

2010-07-19T15:37:55Z

References

Niki am 19. Juli 2010 um 15:37 Uhr

2010-07-19T15:37:13Z

Niki am 19. Juli 2010 um 15:36 Uhr

2010-07-19T15:36:01Z

Niki am 19. Juli 2010 um 15:31 Uhr

2010-07-19T15:31:22Z

Niki am 19. Juli 2010 um 15:30 Uhr

2010-07-19T15:30:45Z

Niki am 19. Juli 2010 um 15:27 Uhr

2010-07-19T15:27:23Z

Niki am 19. Juli 2010 um 15:26 Uhr

2010-07-19T15:26:27Z

Niki am 19. Juli 2010 um 15:24 Uhr

2010-07-19T15:24:55Z

@@ Zeile 56: / Zeile 56: @@
 </math>
-Now the whole procedure has been reduced to calculate the derivative of <math>f(\omega)</math> and set the result to zero.
+Now the whole procedure has been reduced to calculating the derivative of <math>f(\omega)</math> and set the result to zero.
-An open question is how to obtain the dual of <math>\varphi</math>. As the reproduction formula spans a vector space, the <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>:
+An 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>:
 <math>
@@ Zeile 71: / Zeile 71: @@
 </math>
-The 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> temprarily:
+The 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:
 <math>
@@ Zeile 98: / Zeile 98: @@
 </math>
-And finally the following relation is used:
+And finally the following relation is used<ref>L.V. Ahlfors: "Complex Analysis", ''McGraw-Hill'', 1979</ref>:
 <math>

@@ Zeile 18: / Zeile 18: @@
 </math>
-the coefficients can be obtained using the dual of <math>\varphi</math>, <math>\tilde{\varphi}</math> (I set <math>\beta_N = \varphi</math> for consistency with my notes):
+the 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):
 <math>
@@ Zeile 155: / Zeile 155: @@
 = References =
 <ref name="schoenberg">I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>

@@ Zeile 155: / Zeile 155: @@
 = References =
 <ref name="vetterli">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>
 <ref name="schoenberg">I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>
-</references>
 = Comments =

@@ Zeile 156: / Zeile 156: @@
 <references>
-<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>
+<ref name="vetterli">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>
-<ref>I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>
+<ref name="schoenberg">I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>
 </references>

@@ Zeile 1: / Zeile 1: @@
 <section begin="head"/>
 [[Datei:bspline_family.png|right|150px|Family of B-splines up to N=4]]
-A B-Spline of order <math>N</math> is known to be able to reproduce any polynomial up to order <math>N</math><ref>I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>:
+A B-Spline of order <math>N</math> is known to be able to reproduce any polynomial up to order <math>N</math>:
 <math>
@@ Zeile 7: / Zeile 7: @@
 </math>
-In 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<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>. 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.
+In 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.
 An 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.
@@ Zeile 155: / Zeile 155: @@
 = References =
-<references/>
+<references>
 = Comments =

← Nächstältere Version		Version vom 19. Juli 2010, 15:31 Uhr
Zeile 160:		Zeile 160:

	<comments />{{:{{TALKSPACE}}:{{PAGENAME}}}}		<comments />{{:{{TALKSPACE}}:{{PAGENAME}}}}
		+
		+	[[Kategorie:Weblog]]

@@ Zeile 1: / Zeile 1: @@
 <section begin="head"/>
-[[Datei:bspline_family.png|right|200px]]
+[[Datei:bspline_family.png|right|150px|Family of B-splines up to N=4]]
 A B-Spline of order <math>N</math> is known to be able to reproduce any polynomial up to order <math>N</math><ref>I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>:
@@ Zeile 149: / Zeile 149: @@
 </math>
-[[Datei:poly_repro_quad.png|center]]
+[[Datei:poly_repro_quad.png|center|cubic spline reproducing polynomial of order 2]]
-[[Datei:poly_repro_cubic.png|center]]
+[[Datei:poly_repro_cubic.png|center|cubic spline reproducing polynomial of order 3]]
 = References =

@@ Zeile 135: / Zeile 135: @@
 c_{m,n} = j^m \lim_{\omega \rightarrow 0} f(\omega)
 </math>
 = Examples for a cubic spline =
@@ Zeile 150: / Zeile 148: @@
 \end{array}
 </math>
 = References =

@@ Zeile 1: / Zeile 1: @@
 <section begin="head"/>
-[[Datei:poly_repro_lin.png|right]]
+[[Datei:bspline_family.png|right]]
 A B-Spline of order <math>N</math> is known to be able to reproduce any polynomial up to order <math>N</math><ref>I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>:
@@ Zeile 95: / Zeile 95: @@
 <math>
-= \sin^{2L}(\frac{\omega}{2}) \sum_{k \in \mathbb{Z}}\frac{1}{(\frac{\omega}{2} + \pi k)^{2L}}
+= \sin^{2L}\left(\frac{\omega}{2}\right) \sum_{k \in \mathbb{Z}}\frac{1}{(\frac{\omega}{2} + \pi k)^{2L}}
 </math>
@@ Zeile 136: / Zeile 136: @@
 </math>
-== Examples for a cubic spline ==
 For a cubic spline (N=3) the coefficients are:
-\begin{array}{lcr}
+<math>
 c_{0,n}   & = & 1 \\
 c_{1,n}   & = & n \\
@@ Zeile 146: / Zeile 149: @@
 c_{3,n}   & = & -n + n^3
 \end{array}
-== References ==
+= Comments =
-<references/>
+<comments />{{:{{TALKSPACE}}:{{PAGENAME}}}}