Motion planning - Revision history

91.77.247.11 at 04:34, 19 November 2014

2014-11-19T04:34:50Z

← Older revision		Revision as of 06:34, 19 November 2014
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	~~Jayson~~ Berryhill is ~~how I'm called~~ and ~~my wife doesn't like it at all~~. ~~I've always cherished living~~ in ~~Mississippi~~. The preferred ~~hobby~~ for him and his children is ~~style~~ and he'll be ~~beginning something~~ else ~~along~~ with it~~. He is an order clerk and it's something he truly enjoy~~.<br><br>My ~~weblog ... real~~ psychic ([http://~~bigpolis~~.~~com~~/~~blogs~~/~~post/6503 resources~~])		Wilber Berryhill is what his wife loves to call him and he completely enjoys this name. Her family life in Alaska but her spouse desires them to transfer. He works as a bookkeeper. The preferred pastime for him and his children is fashion and he'll be starting some thing else alongside with it.<br><br>My web-site; free psychic readings - [http://test.jeka-nn.ru/node/129 test.jeka-nn.ru],

en>Waffleguy4: /* Configuration Space */

2014-02-06T06:28:17Z

Configuration Space

← Older revision		Revision as of 08:28, 6 February 2014
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	~~{{otheruses4\|the transfer matrix in wavelet theory\|the transfer matrix method in statistical physics\|Transfer-matrix method\|the transfer matrix method in optics\|Transfer-matrix method (optics)}}~~		Jayson Berryhill is how I'm called and my wife doesn't like it at all. I've always cherished living in Mississippi. The preferred hobby for him and his children is style and he'll be beginning something else along with it. He is an order clerk and it's something he truly enjoy.<br><br>My weblog ... real psychic ([http://bigpolis.com/blogs/post/6503 resources])

	~~In [[applied mathematics]], the~~ '''~~transfer matrix''' is a formulation~~ in terms of a [[block-Toeplitz matrix]] of the two-scale equation, which characterizes [[refinable function]]s. Refinable functions play an important role in [[wavelet]] theory and [[finite element]] theory.

	~~For the mask <math>h</math>, which is a vector with component indexes from <math>a</math> to <math>b</math>,~~
	~~the transfer matrix of <math>h</math>, we call it <math>T_h</math> here, is defined as~~
	~~:<math>~~
	~~(T_h)_{j,k} = h_{2\cdot j-k}.~~
	~~</math>~~
	~~More verbosely~~
	~~:<math>~~
	~~T_h =~~
	~~\begin{pmatrix}~~
	~~h_{a } & & & & & \\~~
	~~h_{a+2} & h_{a+1} & h_{a } & & & \\~~
	~~h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } & \\~~
	~~\ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\~~
	~~& h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\~~
	~~& & & h_{b } & h_{b-1} & h_{b-2} \\~~
	~~& & & & & h_{b }~~
	~~\end{pmatrix}~~.
	~~</math>~~
	The ~~effect of <math>T_h</math> can be expressed in terms of the [[downsampling]] operator "<math>\downarrow</math>":~~
	~~:<math>T_h\cdot x = (h*x)\downarrow 2.</math>~~

	~~==Properties==~~

	* <math>T_h\cdot x = T_x\cdot h</math>.
	* If you drop the first and ~~the last column and move the odd-indexed columns to the left and the even-indexed columns to the right, then you obtain a transposed [[Sylvester matrix]].~~
	* The determinant of a transfer matrix is ~~essentially a resultant.~~
	~~:More precisely:~~
	~~:Let <math>h_{\mathrm{e}}</math> be the even-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{e}})_k = h_{2k}</math>)~~ and ~~let <math>h_{\mathrm{o}}</math>~~ be ~~the odd-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{o}})_k = h_{2k+1}</math>).~~
	~~:Then <math>\det T_h = (-1)^{\lfloor\frac{b-a+1}{4}\rfloor}\cdot h_a\cdot h_b\cdot\mathrm{res}(h_{\mathrm{e}},h_{\mathrm{o}})</math>, where <math>\mathrm{res}</math> is the [[resultant]].~~
	~~:This connection allows for fast computation using the [[Euclidean algorithm]].~~
	* For the [[Trace (linear algebra)\|trace]] of the transfer matrix of [[convolution\|convolved]] masks holds
	~~:<math>\mathrm{tr}~T_{g*h} = \mathrm{tr}~T_g \cdot \mathrm{tr}~T_h</math>~~
	* For the [[determinant]] of the transfer matrix of convolved mask holds
	~~:<math>\det T_{g*h} = \det T_g \cdot \det T_h \cdot \mathrm{res}(g_-,h)</math>~~
	~~:where <math>g_-</math> denotes the mask~~ with ~~alternating signs, i.e. <math>(g_-)_k = (-1)^k \cdot g_k</math>.~~
	* If <math>T_{h}\cdot x = 0</math>, then <math>T_{gh}\cdot (g_-x) = 0</math>.
	: This is a concretion of the determinant property above. From the determinant property one knows that <math>T_{gh}</math> is [[Singular matrix\|singular]] whenever <math>T_{h}</math> is singular. This property also tells, how vectors from the [[null space]] of <math>T_{h}</math> can be converted to null space vectors of <math>T_{gh}</math>.
	* If <math>x</math> is an eigenvector of <math>T_{h}</math> with respect to the eigenvalue <math>\lambda</math>, i.e.
	~~: <math>T_{h}\cdot x = \lambda\cdot x</math>,~~
	~~:then <math>x(1,-1)</math>~~ is an ~~eigenvector of <math>T_{h(1,1)}</math> with respect to the same eigenvalue, i.e.~~
	~~: <math>T_{h(1,1)}\cdot (x(1,-1)) = \lambda\cdot (x*(1,-1))</math>.~~
	* Let <math>\lambda_a,\dots,\lambda_b</math> be the eigenvalues of <math>T_h</math>, which implies <math>\lambda_a+\dots+\lambda_b = \mathrm{tr}~T_h</math> and ~~more generally <math>\lambda_a^n+\dots+\lambda_b^n = \mathrm{tr}(T_h^n)</math>~~. ~~This sum is useful for estimating the [[spectral radius]] of~~ <~~math~~>~~T_h~~<~~/math~~>. ~~There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small <math>n</math>~~.
	~~:Let <math>C_k h</math> be the periodization of <math>h</math> with respect to period <math>2^k-1</math>~~. That is <math>C_k h</math> is a circular filter, which means that the component indexes are [[Modular arithmetic#Ring of congruence classes\|residue class]]es with respect to the modulus <math>2^k-1</math>. Then with the [[upsampling]] operator <math>\uparrow</math> it holds
	:<math>\mathrm{tr}(T_h^n) = \left(C_k h * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (~~C_k h\uparrow 2^{n-1})\right)_{~~[~~0]_{2^n-1}}</math>~~
	:~~Actually not <math>n-2<~~/~~math> convolutions are necessary, but only <math>2\cdot \log_2 n<~~/~~math> ones, when applying the strategy of efficient computation of powers~~. ~~Even more the approach can be further sped up using the [[Fast Fourier transform]].~~
	* From the previous statement we can derive an estimate of the [[spectral radius]] of <math>\varrho(T_h)</~~math>. It holds~~
	~~:<math>\varrho(T_h) \ge \frac{a}{\sqrt{\# h}} \ge \frac{1}{\sqrt{3\cdot \# h}}<~~/~~math>~~
	~~:where <math>\# h<~~/~~math> is the size of the filter and if all eigenvalues are real, it is also true that~~
	~~:<math>\varrho(T_h) \le a</math>,~~
	~~:where <math>a = \Vert C_2 h \Vert_2</math>.~~

	~~==See also==~~
	* [[Transfer matrix method]]
	* [[Hurwitz determinant]]

	~~==References==~~
	* {{cite article
	~~\|first=Gilbert\|last=Strang~~
	~~\|author-link=Gilbert Strang~~
	~~\|title=Eigenvalues of <math>(\downarrow 2){H}</math> and convergence of the cascade algorithm~~
	~~\|journal=IEEE Transactions on Signal Processing~~
	~~\|volume=44~~
	~~\|pages=233–238~~
	~~\|year=1996~~
	}}
	* {{cite thesis
	~~\|first=Henning~~
	~~\|last=Thielemann~~
	~~\|url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131~~
	~~\|title=Optimally matched wavelets~~
	~~\|type=PhD thesis~~
	~~\|year=2006~~
	~~}} (contains proofs of the above properties~~)

	~~[[Category:Wavelets]]~~
	~~[[Category:Numerical analysis]]~~

en>Mark viking: Fix typo from using the visual editor

2014-01-23T17:54:36Z

Fix typo from using the visual editor

← Older revision		Revision as of 19:54, 23 January 2014
Line 1:		Line 1:
	The ~~title~~ of the ~~writer~~ is ~~Numbers~~. ~~Body building~~ is one of the ~~things I adore most~~. ~~North Dakota~~ is ~~her birth place but she will have~~ to ~~move one day or another~~. ~~Hiring~~ is ~~her working day occupation now~~ but ~~she's always needed her personal business~~.<br><br>~~Look at my site~~ :: [http://~~www.january~~-~~yjm~~.~~com~~/~~xe/index.php?mid~~=~~video&document_srl~~=~~182582 at home std testing~~]		{{otheruses4\|the transfer matrix in wavelet theory\|the transfer matrix method in statistical physics\|Transfer-matrix method\|the transfer matrix method in optics\|Transfer-matrix method (optics)}}

			In [[applied mathematics]], the '''transfer matrix''' is a formulation in terms of a [[block-Toeplitz matrix]] of the two-scale equation, which characterizes [[refinable function]]s. Refinable functions play an important role in [[wavelet]] theory and [[finite element]] theory.

			For the mask <math>h</math>, which is a vector with component indexes from <math>a</math> to <math>b</math>,
			the transfer matrix of <math>h</math>, we call it <math>T_h</math> here, is defined as
			:<math>
			(T_h)_{j,k} = h_{2\cdot j-k}.
			</math>
			More verbosely
			:<math>
			T_h =
			\begin{pmatrix}
			h_{a } & & & & & \\
			h_{a+2} & h_{a+1} & h_{a } & & & \\
			h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } & \\
			\ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\
			& h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\
			& & & h_{b } & h_{b-1} & h_{b-2} \\
			& & & & & h_{b }
			\end{pmatrix}.
			</math>
			The effect of <math>T_h</math> can be expressed in terms of the [[downsampling]] operator "<math>\downarrow</math>":
			:<math>T_h\cdot x = (h*x)\downarrow 2.</math>

			==Properties==

			* <math>T_h\cdot x = T_x\cdot h</math>.
			* If you drop the first and the last column and move the odd-indexed columns to the left and the even-indexed columns to the right, then you obtain a transposed [[Sylvester matrix]].
			* The determinant of a transfer matrix is essentially a resultant.
			:More precisely:
			:Let <math>h_{\mathrm{e}}</math> be the even-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{e}})_k = h_{2k}</math>) and let <math>h_{\mathrm{o}}</math> be the odd-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{o}})_k = h_{2k+1}</math>).
			:Then <math>\det T_h = (-1)^{\lfloor\frac{b-a+1}{4}\rfloor}\cdot h_a\cdot h_b\cdot\mathrm{res}(h_{\mathrm{e}},h_{\mathrm{o}})</math>, where <math>\mathrm{res}</math> is the [[resultant]].
			:This connection allows for fast computation using the [[Euclidean algorithm]].
			* For the [[Trace (linear algebra)\|trace]] of the transfer matrix of [[convolution\|convolved]] masks holds
			:<math>\mathrm{tr}~T_{g*h} = \mathrm{tr}~T_g \cdot \mathrm{tr}~T_h</math>
			* For the [[determinant]] of the transfer matrix of convolved mask holds
			:<math>\det T_{g*h} = \det T_g \cdot \det T_h \cdot \mathrm{res}(g_-,h)</math>
			:where <math>g_-</math> denotes the mask with alternating signs, i.e. <math>(g_-)_k = (-1)^k \cdot g_k</math>.
			* If <math>T_{h}\cdot x = 0</math>, then <math>T_{gh}\cdot (g_-x) = 0</math>.
			: This is a concretion of the determinant property above. From the determinant property one knows that <math>T_{gh}</math> is [[Singular matrix\|singular]] whenever <math>T_{h}</math> is singular. This property also tells, how vectors from the [[null space]] of <math>T_{h}</math> can be converted to null space vectors of <math>T_{gh}</math>.
			* If <math>x</math> is an eigenvector of <math>T_{h}</math> with respect to the eigenvalue <math>\lambda</math>, i.e.
			: <math>T_{h}\cdot x = \lambda\cdot x</math>,
			:then <math>x(1,-1)</math> is an eigenvector of <math>T_{h(1,1)}</math> with respect to the same eigenvalue, i.e.
			: <math>T_{h(1,1)}\cdot (x(1,-1)) = \lambda\cdot (x*(1,-1))</math>.
			* Let <math>\lambda_a,\dots,\lambda_b</math> be the eigenvalues of <math>T_h</math>, which implies <math>\lambda_a+\dots+\lambda_b = \mathrm{tr}~T_h</math> and more generally <math>\lambda_a^n+\dots+\lambda_b^n = \mathrm{tr}(T_h^n)</math>. This sum is useful for estimating the [[spectral radius]] of <math>T_h</math>. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small <math>n</math>.
			:Let <math>C_k h</math> be the periodization of <math>h</math> with respect to period <math>2^k-1</math>. That is <math>C_k h</math> is a circular filter, which means that the component indexes are [[Modular arithmetic#Ring of congruence classes\|residue class]]es with respect to the modulus <math>2^k-1</math>. Then with the [[upsampling]] operator <math>\uparrow</math> it holds
			:<math>\mathrm{tr}(T_h^n) = \left(C_k h * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (C_k h\uparrow 2^{n-1})\right)_{[0]_{2^n-1}}</math>
			:Actually not <math>n-2</math> convolutions are necessary, but only <math>2\cdot \log_2 n</math> ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the [[Fast Fourier transform]].
			* From the previous statement we can derive an estimate of the [[spectral radius]] of <math>\varrho(T_h)</math>. It holds
			:<math>\varrho(T_h) \ge \frac{a}{\sqrt{\# h}} \ge \frac{1}{\sqrt{3\cdot \# h}}</math>
			:where <math>\# h</math> is the size of the filter and if all eigenvalues are real, it is also true that
			:<math>\varrho(T_h) \le a</math>,
			:where <math>a = \Vert C_2 h \Vert_2</math>.

			==See also==
			* [[Transfer matrix method]]
			* [[Hurwitz determinant]]

			==References==
			* {{cite article
			\|first=Gilbert\|last=Strang
			\|author-link=Gilbert Strang
			\|title=Eigenvalues of <math>(\downarrow 2){H}</math> and convergence of the cascade algorithm
			\|journal=IEEE Transactions on Signal Processing
			\|volume=44
			\|pages=233–238
			\|year=1996
			}}
			* {{cite thesis
			\|first=Henning
			\|last=Thielemann
			\|url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131
			\|title=Optimally matched wavelets
			\|type=PhD thesis
			\|year=2006
			}} (contains proofs of the above properties)

			[[Category:Wavelets]]
			[[Category:Numerical analysis]]

74.14.218.53 at 23:34, 14 August 2012

2012-08-14T23:34:49Z

New page

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