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| {{other uses of|CDF}}
| | Hello, my title is Andrew and my wife doesn't like it at all. Distributing manufacturing is how he tends to make a residing. Mississippi is where her home is but her spouse desires them to transfer. The preferred hobby for him and his children is to perform lacross and he would never give it up.<br><br>Here is my page: are psychics real ([http://www.octionx.sinfauganda.co.ug/node/22469 check these guys out]) |
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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 | |
| 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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Hello, my title is Andrew and my wife doesn't like it at all. Distributing manufacturing is how he tends to make a residing. Mississippi is where her home is but her spouse desires them to transfer. The preferred hobby for him and his children is to perform lacross and he would never give it up.
Here is my page: are psychics real (check these guys out)