66.74.176.59: sp

2014-11-18T09:08:29Z

en>Bgwhite: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9866)

2014-01-20T08:28:01Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9866)

← Older revision		Revision as of 10:28, 20 January 2014
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	~~Maia Asay~~ is ~~her name but she never really liked that establish~~. ~~Alabama~~ is where ~~her home~~ is ~~but her husband wants them~~ to ~~maneuver~~. ~~To go~~ to ~~fitness~~ 1 ~~thing that I~~'~~ve done~~ for ~~countless~~. ~~Curing people exactly what he does~~ in ~~his day job and he will not put~~ it ~~back anytime ultimately~~. I'~~m not good~~ at ~~webdesign but make sure you check my website~~: ~~http~~://~~www~~.~~partybusphilly~~.~~com~~		{{Refimprove\|date=January 2010}}
			The '''Fast Wavelet Transform''' is a [[mathematics\|mathematical]] [[algorithm]] designed to turn a [[waveform]] or signal in the [[time domain]] into a [[sequence]] of coefficients based on an [[orthogonal basis]] of small finite waves, or [[wavelets]]. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain.

			It has as theoretical foundation the device of a finitely generated, orthogonal [[multiresolution analysis]] (MRA). In the terms given there, one selects a sampling scale ''J'' with [[sampling rate]] of 2<sup>J</sup> per unit interval, and projects the given signal ''f'' onto the space <math>V_J</math>; in theory by computing the [[dot product\|scalar product]]s

			:<math>s^{(J)}_n:=2^J \langle f(t),\phi(2^J t-n) \rangle,</math>

			where <math>\phi</math> is the [[Wavelet#Scaling_function\|scaling function]] of the chosen wavelet transform; in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so
			:<math> P_J[f](x):=\sum_{n\in\Z} s^{(J)}_n\,\phi(2^Jx-n)</math>
			is the [[orthogonal projection]] or at least some good approximation of the original signal in <math>V_J</math>.

			The MRA is characterised by its scaling sequence

			:<math>a=(a_{-N},\dots,a_0,\dots,a_N)</math> or, as [[Z-transform]], <math>a(z)=\sum_{n=-N}^Na_nz^{-n}</math>

			and its wavelet sequence

			:<math>b=(b_{-N},\dots,b_0,\dots,b_N)</math> or <math>b(z)=\sum_{n=-N}^Nb_nz^{-n}</math>

			(some coefficients might be zero). Those allow to compute the wavelet coefficients <math>d^{(k)}_n</math>, at least some range ''k=M,...,J-1'', without having to approximate the integrals in the corresponding scalar products. Instead, one can directly, with the help of convolution and decimation operators, compute those coefficients from the first approximation <math>s^{(J)}</math>.

			== Forward [[Discrete_wavelet_transform\|DWT]] ==
			One computes [[recursion\|recursively]], starting with the coefficient sequence <math>s^{(J)}</math> and counting down from ''k=J-1'' to some ''M<J'',

			[[Image:Wavelets_-_DWT.png\|thumb\|390px\|single application of a wavelet filter bank, with filters g=a<sup></sup>, h=b<sup></sup>]]
			:<math>
			s^{(k)}_n:=\frac12 \sum_{m=-N}^N a_m s^{(k+1)}_{2n+m}
			</math> or <math>
			s^{(k)}(z):=(\downarrow 2)(a^*(z)\cdot s^{(k+1)}(z))
			</math>
			and
			:<math>
			d^{(k)}_n:=\frac12 \sum_{m=-N}^N b_m s^{(k+1)}_{2n+m}
			</math> or <math>
			d^{(k)}(z):=(\downarrow 2)(b^*(z)\cdot s^{(k+1)}(z))
			</math>,

			for ''k=J-1,J-2,...,M'' and all <math>n\in\Z</math>. In the Z-transform notation:
			[[Image:Wavelets_-_Filter_Bank.png\|thumb\|400px\|recursive application of the filter bank]]
			:* The [[downsampling\|downsampling operator]] <math>(\downarrow 2)</math> reduces an infinite sequence, given by its [[Z-transform]], which is simply a [[Laurent series]], to the sequence of the coefficients with even indices, <math>(\downarrow 2)(c(z))=\sum_{k\in\Z}c_{2k}z^{-k}</math>.
			:* The starred Laurent-polynomial <math>a^(z)</math> denotes the adjoint filter, it has ''time-reversed'' adjoint coefficients, <math>a^(z)=\sum_{n=-N}^N a_{-n}^*z^{-n}</math>. (The adjoint of a real number being the number itself, of a complex number its conjugate, of a real matrix the transposed matrix, of a complex matrix its hermitian adjoint).
			:* Multiplication is polynomial multiplication, which is equivalent to the convolution of the coefficient sequences.

			It follows that

			:<math>P_k[f](x):=\sum_{n\in\Z} s^{(k)}_n\,\phi(2^kx-n)</math>

			is the orthogonal projection of the original signal ''f'' or at least of the first approximation <math>P_J[f](x)</math> onto the [[linear subspace\|subspace]] <math>V_k</math>, that is, with sampling rate of 2<sup>k</sup> per unit interval. The difference to the first approximation is given by

			:<math>P_J[f](x)=P_k[f](x)+D_k[f](x)+\dots+D_{J-1}[f](x)</math>,

			where the difference or detail signals are computed from the detail coefficients as

			:<math>D_k[f](x):=\sum_{n\in\Z} d^{(k)}_n\,\psi(2^kx-n)</math>,

			with <math>\psi</math> denoting the ''mother wavelet'' of the wavelet transform.

			==Inverse DWT==
			Given the coefficient sequence <math>s^{(M)}</math> for some ''M<J'' and all the difference sequences <math>d^{(k)}</math>, ''k=M,...,J-1'', one computes recursively
			:<math>
			s^{(k+1)}_n:=\sum_{k=-N}^N a_k s^{(k)}_{2n-k}+\sum_{k=-N}^N b_k d^{(k)}_{2n-k}
			</math> or <math>
			s^{(k+1)}(z)=a(z)\cdot(\uparrow 2)(s^{(k)}(z))+b(z)\cdot(\uparrow 2)(d^{(k)}(z))
			</math>
			for ''k=J-1,J-2,...,M'' and all <math>n\in\Z</math>. In the Z-transform notation:
			:* The [[upsampling\|upsampling operator]] <math>(\uparrow 2)</math> creates zero-filled holes inside a given sequence. That is, every second element of the resulting sequence is an element of the given sequence, every other second element is zero or <math>(\uparrow 2)(c(z)):=\sum_{n\in\Z}c_nz^{-2n}</math>. This linear operator is, in the [[Hilbert space]] <math>\ell^2(\Z,\R)</math>, the adjoint to the downsampling operator <math>(\downarrow 2)</math>.

			==See also==
			* [[Lifting scheme]]

			==References==
			{{reflist}}

			* A.N. Akansu ''Multiplierless Suboptimal PR-QMF Design'' Proc. SPIE 1818, Visual Communications and Image Processing, p. 723, November, 1992
			* A.N. Akansu ''Multiplierless 2-band Perfect Reconstruction Quadrature Mirror Filter (PR-QMF) Banks'' US Patent 5,420,891, 1995
			* A.N. Akansu ''Multiplierless PR Quadrature Mirror Filters for Subband Image Coding'' IEEE Trans. Image Processing, p. 1359, September 1996
			* M.J. Mohlenkamp, M.C. Pereyra ''Wavelets, Their Friends, and What They Can Do for You'' (2008 EMS) p. 38
			* B.B. Hubbard ''The World According to Wavelets: The Story of a Mathematical Technique in the Making (1998 Peters) p. 184
			* S.G. Mallat ''A Wavelet Tour of Signal Processing'' (1999 Academic Press) p. 255
			* A. Teolis ''Computational Signal Processing with Wavelets'' (1998 Birkhäuser) p. 116
			* Y. Nievergelt ''Wavelets Made Easy'' (1999 Springer) p. 95

			==Further reading==
			G. Beylkin, R. Coifman, V. Rokhlin, "Fast wavelet transforms and numerical algorithms" ''Comm. Pure Appl. Math.'', 44 (1991) pp. 141–183

			[[Category:Wavelets]]
			[[Category:Discrete transforms]]

en>Wvbailey: /* Propositional formula with "feedback" */ clarify by using a black-box analogy

2012-07-31T20:39:34Z

Propositional formula with "feedback": clarify by using a black-box analogy

New page

Maia Asay is her name but she never really liked that establish. Alabama is where her home is but her husband wants them to maneuver. To go to fitness 1 thing that I've done for countless. Curing people exactly what he does in his day job and he will not put it back anytime ultimately. I'm not good at webdesign but make sure you check my website: http://www.partybusphilly.com

Propositional formula - Revision history

66.74.176.59: sp

en>Bgwhite: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9866)

en>Wvbailey: /* Propositional formula with "feedback" */ clarify by using a black-box analogy