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| {{other uses of|CDF}}
| | Hi there. Let me start by introducing the author, her name is Sophia. My spouse and I reside in Kentucky. Invoicing is my profession. What I love performing is football but I don't have the time lately.<br><br>Feel free to surf to my weblog :: psychics online ([http://www.skullrocker.com/blogs/post/10991 click]) |
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| [[Image:Jpeg2000 2-level wavelet transform-lichtenstein.png|thumb|256px|An example of the 2D wavelet transform that is used in [[JPEG2000]]]]
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| '''Cohen-Daubechies-Feauveau wavelet''' are the historically first family of [[biorthogonal wavelet]]s, which was made popular by [[Ingrid Daubechies]].<ref>{{Cite doi|10.1002/cpa.3160450502}}</ref><ref>
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| {{cite book
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| |first=Ingrid|last=Daubechies
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| |title=Ten Lectures on wavelets
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| |journal=SIAM
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| |year=1992
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| }}</ref> These are not the same as the orthogonal [[Daubechies wavelet]]s, and also not very similar in shape and properties. However their construction idea is the same.
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| The [[JPEG 2000]] [[Image compression|compression]] standard uses the biorthogonal CDF 5/3 wavelet (also called the [[LeGall 5/3 wavelet]]) for lossless compression and a CDF 9/7 wavelet for lossy compression.
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| ==Properties==
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| * The [[primal generator]] is a [[B-spline]] if the simple factorization <math>q_{\mathrm{prim}}(X)=1</math> (see below) is chosen
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| * The [[dual generator]] has the maximum number of smoothness factors which is possible for its length.
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| * All generators and wavelets in this family are symmetric.
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| ==Construction==
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| For every positive integer ''A'' there exists a unique polynomial <math>Q_A(X)</math> of degree ''A−1'' satisfying the identity
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| :<math>(1-X/2)^A\,Q_A(X)+(X/2)^A\,Q_A(2-X)=1</math>.
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| This is the same polynomial as used in the construction of the
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| [[Daubechies wavelet]]s. But, instead of a spectral factorization,
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| here we try to factor
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| :<math>Q_A(X)=q_{\mathrm{prim}}(X)\,q_{\mathrm{dual}}(X)</math>,
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| where the factors are polynomials with real coefficients and
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| constant coefficient 1. Then,
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| :<math>a_{\mathrm{prim}}(Z)=2Z^d\,\left(\frac{1+Z}2\right)^A\,q_{\mathrm{prim}}(1-(Z+Z^{-1})/2)</math>
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| and
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| :<math>a_{\mathrm{dual}}(Z)=2Z^d\,\left(\frac{1+Z}2\right)^A\,q_{\mathrm{dual}}(1-(Z+Z^{-1})/2)</math>
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| form a biorthogonal pair of scaling sequences. ''d'' is some integer used to
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| center the symmetric sequences at zero or to make the corresponding discrete filters
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| causal.
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| Depending on the roots of <math>Q_A(X)</math>, there may be up to
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| <math>2^{A-1}</math> different factorizations. A simple factorization is
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| <math>q_{\mathrm{prim}}(X)=1</math> and <math>q_{\mathrm{dual}}(X)=Q_A(X)</math>, then
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| the <math>\mathrm{primary}</math> scaling function is the [[B-spline]] of order
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| ''A−1''. For ''A=1'' one obtains the orthogonal '''[[Haar wavelet]]'''.
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| ==Tables of coefficients==
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| [[File:Wavelet Bior2.2.svg|thumb|right|Cohen-Daubechies-Feauveau wavelet 5/3 used in JPEG 2000 standard.]]
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| For ''A=2'' one obtains in this way the '''LeGall 5/3-wavelet''':
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| {|class="wikitable"
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| !A
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| !Q<sub>A</sub>(X)
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| !q<sub>prim</sub>(X)
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| !q<sub>dual</sub>(X)
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| !a<sub>prim</sub>(Z)
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| !a<sub>dual</sub>(Z)
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| |-
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| |2
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| |<math>1 + X</math>
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| |1
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| |<math>1 + X</math>
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| |<math>\frac12(1+Z)^2\,Z</math>
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| |<math>\frac12(1+Z)^2\,\left(-\frac12 + 2\,Z - \frac12\,Z^2\right)</math>
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| |}
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| ----
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| For ''A=4'' one obtains the '''9/7-CDF-wavelet'''. One gets <math>Q_4(X)=1 + 2\,X + 5/2\,X^2 + 5/2\,X^3</math>, this polynomial has exactly one real root, thus it is the product of a linear factor <math>1-c\,X</math> and a quadratic factor. The coefficient ''c'', which is the inverse of the root, has an approximate value of −1.4603482098.
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| {|class="wikitable"
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| !A
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| !Q<sub>A</sub>(X)
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| !q<sub>prim</sub>(X)
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| !q<sub>dual</sub>(X)
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| |-
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| |4
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| |<math>1 + 2\,X + 5/2\,X^2 + 5/2\,X^3</math>
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| |<math>1-c\,X</math>
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| |<math>1 + (c + 2)*\,X + (c^2 + 2*c + 5/2)\,X^2</math>
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| |}
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| For the coefficients of the centered scaling and wavelet sequences one gets numerical values in an implementation–friendly form
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| {|class="wikitable"
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| !''k''
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| !Analysis lowpass filter
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| ''(1/2 a<sub>dual</sub>)''
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| !Analysis highpass filter
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| ''(b<sub>dual</sub>)''
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| !'''Synthesis lowpass filter'''
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| ''(a<sub>prim</sub>)''
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| !'''Synthesis highpass filter'''
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| ''(1/2 b<sub>prim</sub>)''
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| |----
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| | -4
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| | 0.026748757411
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| | 0
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| | 0
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| | 0.026748757411
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| |----
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| | -3
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| | -0.016864118443
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| | 0.091271763114
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| | -0.091271763114
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| | 0.016864118443
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| |----
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| | -2
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| | -0.078223266529
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| | -0.057543526229
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| | -0.057543526229
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| | -0.078223266529
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| |----
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| | -1
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| | 0.266864118443
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| | -0.591271763114
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| | 0.591271763114
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| | -0.266864118443
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| |----
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| | 0
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| | 0.602949018236
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| | 1.11508705
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| | 1.11508705
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| | 0.602949018236
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| |----
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| | 1
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| | 0.266864118443
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| | -0.591271763114
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| | 0.591271763114
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| | -0.266864118443
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| |----
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| | 2
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| | -0.078223266529
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| | -0.057543526229
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| | -0.057543526229
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| | -0.078223266529
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| |----
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| | 3
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| | -0.016864118443
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| | 0.091271763114
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| | -0.091271763114
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| | 0.016864118443
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| |----
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| | 4
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| | 0.026748757411
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| | 0
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| | 0
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| | 0.026748757411
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| |----
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| |}
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| ==Numbering==
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| There are two concurring numbering schemes for wavelets of the CDF family.
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| * the number of smoothness factors of the lowpass filters, or equivalently the number of [[Moment (mathematics)|vanishing moments]] of the highpass filters, e.g. 2,2
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| * the sizes of the lowpass filters, or equivalently the sizes of the highpass filters, e.g. 5,3
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| The first numbering was used in Daubechies' book ''Ten lectures on wavelets''.
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| Neither of this numbering is unique. The number of vanishing moments does not tell about the chosen factorization. A filterbank with filter sizes 7 and 9 can have 6 and 2 vanishing moments when using the trivial factorization, or 4 and 4 vanishing moments as it is the case for the JPEG 2000 wavelet. The same wavelet may therefore be referred to as "CDF 9/7" (based on the filter sizes) or "biorthogonal 4.4" (based on the vanishing moments).
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| ==Lifting decomposition==
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| For the trivially factorized filterbanks a [[Lifting scheme|lifting decomposition]] can be explicitly given.<ref>
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| {{cite thesis
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| |first=Henning
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| |last=Thielemann
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| |url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131
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| |title=Optimally matched wavelets
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| |type=PhD thesis
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| |year=2006
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| |chapter=section 3.2.4
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| }}</ref>
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| ===Even number of smoothness factors===
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| Let <math>n</math> be the number of smoothness factors in the B-spline lowpass filter,
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| which shall be even.
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| Then define recursively
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| : <math>a_0 = \frac{1}{n}</math>
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| : <math>a_m = \frac{1}{(n^2-4\cdot m^2)\cdot a_{m-1}}</math>
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| The lifting filters are
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| : <math>s_{m}(z) = a_m\cdot(2\cdot m + 1)\cdot(1 + z^{(-1)^m})</math>
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| Conclusively the interim results of the lifting are
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| : <math>x_{-1}(z) = z</math>
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| : <math>x_{0}(z) = 1</math>
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| : <math>x_{m+1}(z) = x_{m-1}(z) + a_m\cdot(2\cdot m+1)\cdot(z+z^{-1}) \cdot z^{(-1)^m} \cdot x_{m}(z)</math>
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| which leads to
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| : <math>x_{n/2}(z) = 2^{-n/2} \cdot (1+z)^n \cdot z^{n/2 \bmod 2 - n/2}</math>
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| The filters <math>x_{n/2}</math> and <math>x_{n/2-1}</math> constitute the CDF-n,0 filterbank.
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| ===Odd number of smoothness factors===
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| Now, let <math>n</math> be odd.
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| Then define recursively
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| : <math>a_0 = \frac{1}{n}</math>
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| : <math>a_m = \frac{1}{(n^2-(2\cdot m-1)^2)\cdot a_{m-1}}</math>
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| The lifting filters are
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| : <math>s_{m}(z) = a_m\cdot((2\cdot m + 1) + (2\cdot m - 1)\cdot z) / z^{m \bmod 2}</math>
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| Conclusively the interim results of the lifting are
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| : <math>x_{-1}(z) = z</math>
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| : <math>x_{0}(z) = 1</math>
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| : <math>x_{1}(z) = x_{-1}(z)+a_0\cdot x_0(z)</math>
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| : <math>x_{m+1}(z) = x_{m-1}(z) + a_m\cdot((2\cdot m+1)\cdot z + (2\cdot m-1)\cdot z^{-1}) \cdot z^{(-1)^m} \cdot x_{m}(z)</math>
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| which leads to
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| : <math>x_{(n+1)/2}(z) \sim (1+z)^n</math>
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| where we neglect the translation and the constant factor.
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| The filters <math>x_{(n+1)/2}</math> and <math>x_{(n-1)/2}</math> constitute the CDF-n,1 filterbank.
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| ==Applications==
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| The Cohen-Daubechies-Feauveau wavelet and other biorthogonal wavelets have been used to compress fingerprint scans for the FBI.<ref name="cipra94">
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| {{cite book
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| |first=Barry|last=Cipra
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| |title= What's Happening in the Mathematical Sciences (Vol.2) Parlez-vous Wavelets?
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| |journal=American Mathematical Society
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| |year=1994
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| }}</ref> A standard for compressing fingerprints in this way was developed by Tom Hopper (FBI), Jonathan Bradley (Los Alamos National Laboratory) and Chris Brislawn (Los Alamos National Laboratory).<ref name="cipra94"/> By using wavelets, a compression ratio of around 20 to 1 can be achieved, meaning a 10MB image could be reduced to as little as 500KB while still passing recognition tests.<ref name="cipra94"/>
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| ==External links==
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| * [http://faculty.gvsu.edu/aboufade/web/wavelets/student_work/EF/how-works.html JPEG 2000: How does it work?]
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| * [http://www.embl.de/~gpau/misc/dwt97.c Fast discrete CDF 9/7 wavelet transform source code in C language (lifting implementation)]
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| * [http://www.olhovsky.com/content/wavelet/2dwavelet97lift.py CDF 9/7 Wavelet Transform for 2D Signals via Lifting: Source code in Python]
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| ==References==
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| <references/>
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| [[Category:Biorthogonal wavelets]]
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Hi there. Let me start by introducing the author, her name is Sophia. My spouse and I reside in Kentucky. Invoicing is my profession. What I love performing is football but I don't have the time lately.
Feel free to surf to my weblog :: psychics online (click)