Cohen-Daubechies-Feauveau wavelet

Cohen-Daubechies-Feauveau wavelets (CDF wavelets) are historically the first family of biorthogonal wavelets . They were designed by Albert Cohen , Ingrid Daubechies and Jean-Christophe Feauveau and presented in 1990. CDF wavelets are to be distinguished from the orthogonal Daubechies wavelets , which have different shapes and properties. Both Wavelettypen go back to the same design idea, CDF wavelet renounce in favor of symmetry on orthogonality of wavelets (Daubechies wavelets in the reverse is true).

The JPEG-2000 - compression standard uses the biorthogonal CDF 5/3 wavelet (also LeGall-5/3 wavelet called) lossless compression and the CDF 9/7 wavelet for lossy compression.

Example of a 2D wavelet transform used in the JPEG2000 standard

properties

• The prime generator is a B-spline if simple factorization (see below) is chosen ${\displaystyle q_{\mathrm {prim} }(X)=1}$
• The dual generator has the maximum number of smoothness factors that is possible for the length
• All generators and wavelets in this family are symmetrical.

construction

For every positive integer there is a unique polynomial of degree that satisfies the following identity : ${\displaystyle A}$ ${\displaystyle Q_{A}(X)}$ ${\displaystyle A-1}$

${\displaystyle (1-X/2)^{A}\,Q_{A}(X)+(X/2)^{A}\,Q_{A}(2-X)=1}$.

It is the same polynomial used in the construction of the Daubechies wavelets. Instead of a spectral factorization, an attempt is made here

${\displaystyle Q_{A}(X)=q_{\mathrm {prim} }(X)\,q_{\mathrm {dual} }(X)}$

to factorize. The factors are polynomials with real coefficients and an absolute term . ${\displaystyle q_{\mathrm {prim} }(X),q_{\mathrm {dual} }(X)}$ ${\displaystyle q_{\mathrm {prim} }(0)=q_{\mathrm {dual} }(0)=1}$

In this case, shape

${\displaystyle a_{\mathrm {prim} }(Z)=2Z^{d}\,\left({\frac {1+Z}{2}}\right)^{A}\,q_{\mathrm {prim} }(1-(Z+Z^{-1})/2)}$

and

${\displaystyle a_{\mathrm {dual} }(Z)=2Z^{d}\,\left({\frac {1+Z}{2}}\right)^{A}\,q_{\mathrm {dual} }(1-(Z+Z^{-1})/2)}$

a biorthogonal pair of scaling sequences. is an integer used to center the symmetric sequence on zero or to make the corresponding discrete filters causal. ${\displaystyle d}$

There are up to different factorizations depending on the roots of . A simple factorization is and . In this case the primary scaling function is the B-spline of the order . For we get the orthogonal Haar wavelet . ${\displaystyle Q_{A}(X)}$ ${\displaystyle 2^{A-1}}$ ${\displaystyle q_{\mathrm {prim} }(X)=1}$ ${\displaystyle q_{\mathrm {dual} }(X)=Q_{A}(X)}$ ${\displaystyle A-1}$ ${\displaystyle A=1}$

Coefficient table

Cohen-Daubechies-Feauveau wavelet 5/3, as used in the JPEG 2000 standard.

For we get the LeGall 5/3 wavelet : ${\displaystyle A=2}$

A.	Q A (X)	q prime (X)	q dual (X)	a prime (Z)	a dual (Z)
2	${\displaystyle 1+X}$	1	${\displaystyle 1+X}$	${\displaystyle {\frac {1}{2}}(1+Z)^{2}\,Z}$	${\displaystyle {\frac {1}{2}}(1+Z)^{2}\,\left(-{\tfrac {1}{2}}+2\,Z-{\tfrac {1}{2}}\,Z^{2}\right)}$
				${\displaystyle ={\frac {1}{2}}\,\left(Z+2Z^{2}+Z^{3}\right)}$	${\displaystyle ={\frac {1}{4}}\,\left(-1+2Z+6Z^{2}+2Z^{3}-Z^{4}\right)}$

For we get the 9/7 CDF wavelet . One receives . This polynomial has exactly one real root and is therefore the product of the linear factor and a quadratic factor. The coefficient , which is the inverse of the root, has a value of about −1.4603482098. ${\displaystyle A=4}$ ${\displaystyle Q_{4}(X)=1+2\,X+5/2\,X^{2}+5/2\,X^{3}}$ ${\displaystyle 1-c\,X}$ ${\displaystyle c}$

A.	Q A (X)	q prime (X)	q dual (X)
4th	${\displaystyle 1+2\,X+5/2\,X^{2}+5/2\,X^{3}}$	${\displaystyle 1-c\,X}$	${\displaystyle 1+(c+2)*\,X+(c^{2}+2*c+5/2)\,X^{2}}$

For the coefficients of the centered scaling and wavelet sequences, numerical values ​​are obtained in an implementation-friendly form:

k	Analysis low pass filter (1/2 a dual )	Analysis high pass filter (b dual )	Synthesis low pass filter (a prim )	Synthesis high pass filter (1/2 b prim )
−4	0.026748757411	0	0	0.026748757411
−3	−0.016864118443	0.091271763114	−0.091271763114	0.016864118443
−2	−0.078223266529	−0.057543526229	−0.057543526229	−0.078223266529
−1	0.266864118443	−0.591271763114	0.591271763114	−0.266864118443
0	0.602949018236	1.11508705	1.11508705	0.602949018236
1	0.266864118443	−0.591271763114	0.591271763114	−0.266864118443
2	−0.078223266529	−0.057543526229	−0.057543526229	−0.078223266529
3	−0.016864118443	0.091271763114	−0.091271763114	0.016864118443
4th	0.026748757411	0	0	0.026748757411

Number designation

There are two parallel numbering schemes for CDF family wavelets.

• The number of smoothness factors of the low pass filters, or (equivalently) the number of vanishing moments of the high pass filters, e.g. B. 2.2
• The lengths of the low pass filters, or (equivalently) the lengths of the high pass filters, e.g. B. 5.3

The first scheme was used in Daubechies' book "Ten lectures on wavelets". None of the names is clear. The number of vanishing moments says nothing about the chosen factorization. A filter bank with filter lengths of 7 and 9 has 6 and 2 vanishing moments, if one uses a trivial factorization, or 4 and 4 vanishing moments, as in the case of the JPEG-2000 wavelet. The same wavelet can therefore be called “CDF 9/7” (based on the filter lengths) or “biorthogonal 4/4” (based on the vanishing moments).

Lifting disassembly

A lifting decomposition can be given explicitly for the trivially factored filter banks.

literature

• A. Cohen, I. Daubechies and JC Feauveau: Biorthogonal bases of compactly supported wavelets . Comm. Pure & Appl. Math 45, 1992, p. 485 to 560 .
• I. Daubechies: Ten Lectures on wavelets . SIAM, 1992.

Web links

• JPEG 2000: How does it work?
• Fast discrete CDF 9/7 wavelet transform source code in C language (lifting implementation) archived on March 5th, 2012 ( Memento of March 5th, 2012 in the Internet Archive )
• CDF 9/7 Wavelet Transform for 2D Signals via Lifting: Source code in Python
• Open source 5/3 CDF wavelet implementation in C #, for any size

Individual evidence

1. ^ Albert Cohen, Ingrid Daubechies and Jean-Christophe Feauveau: Biorthogonal Bases of Compactly Supported Wavelets , in Communications on Pure and Applied Mathematics , Volume 45, Issue 5, Wiley 1992
2. ↑ See section 3.2.4 of the elaboration under [1]