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Cohen–Daubechies–Feauveau wavelets are a family of biorthogonal wavelets that was made popular by
Ingrid Daubechies Baroness Ingrid Daubechies ( ; ; born 17 August 1954) is a Belgian-American physicist and mathematician. She is best known for her work with wavelets in image compression. Daubechies is recognized for her study of the mathematical methods that ...
. These are not the same as the orthogonal Daubechies wavelets, and also not very similar in shape and properties. However, their construction idea is the same. The
JPEG 2000 JPEG 2000 (JP2) is an image compression standard and coding system. It was developed from 1997 to 2000 by a Joint Photographic Experts Group committee chaired by Touradj Ebrahimi (later the JPEG president), with the intention of superseding their ...
compression standard uses the biorthogonal Le Gall–Tabatabai (LGT) 5/3 wavelet (developed by D. Le Gall and Ali J. Tabatabai) for
lossless compression Lossless compression is a class of data compression that allows the original data to be perfectly reconstructed from the compressed data with no loss of information. Lossless compression is possible because most real-world data exhibits statisti ...
and a CDF 9/7 wavelet for
lossy compression In information technology, lossy compression or irreversible compression is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to reduce data size ...
.


Properties

* The primal generator is a
B-spline In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of bre ...
if the simple factorization q_\text(X) = 1 (see below) is chosen. * The dual generator has the highest possible number of smoothness factors for its length. * All generators and wavelets in this family are symmetric.


Construction

For every positive integer ''A'' there exists a unique polynomial Q_A(X) of degree ''A'' − 1 satisfying the identity :\left(1 - \frac\right)^A\,Q_A(X) + \left(\frac\right)^A\,Q_A(2 - X) = 1. This is the same polynomial as used in the construction of the Daubechies wavelets. But, instead of a spectral factorization, here we try to factor :Q_A(X) = q_\text(X)\,q_\text(X), where the factors are polynomials with real coefficients and constant coefficient 1. Then :a_\text(Z) = 2 Z^d\,\left(\frac2\right)^A\,q_\text\left(1 - \frac\right) and :a_\text(Z) = 2 Z^d\,\left(\frac2\right)^A\,q_\text\left(1 - \frac\right) form a biorthogonal pair of scaling sequences. ''d'' is some integer used to center the symmetric sequences at zero or to make the corresponding discrete filters causal. Depending on the roots of Q_A(X), there may be up to 2^ different factorizations. A simple factorization is q_\text(X) = 1 and q_\text(X) = Q_A(X), then the primary scaling function is the
B-spline In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of bre ...
of order ''A'' − 1. For ''A'' = 1 one obtains the orthogonal
Haar wavelet In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be repr ...
.


Tables of coefficients

For ''A'' = 2 one obtains in this way the LeGall 5/3-wavelet: ---- For ''A'' = 4 one obtains the 9/7-CDF-wavelet. One gets Q_4(X) = 1 + 2X + \tfrac52 X^2 + \tfrac52 X^3, this polynomial has exactly one real root, thus it is the product of a linear factor 1 - cX and a quadratic factor. The coefficient ''c'', which is the inverse of the root, has an approximate value of −1.4603482098. For the coefficients of the centered scaling and wavelet sequences one gets numerical values in an implementation–friendly form


Numbering

There are two concurring numbering schemes for wavelets of the CDF family: * the number of smoothness factors of the lowpass filters, or equivalently the number of vanishing moments of the highpass filters, e.g. "2, 2"; * the sizes of the lowpass filters, or equivalently the sizes of the highpass filters, e.g. "5, 3". The first numbering was used in Daubechies' book ''Ten lectures on wavelets''. Neither of this numbering is unique. The number of vanishing moments does not tell about the chosen factorization. A filter bank 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). Similarly, the same wavelet may therefore be referred to as "CDF 5/3" (based on the filter sizes) or "biorthogonal 2, 2" (based on the vanishing moments).


Lifting decomposition

For the trivially factorized filterbanks a lifting decomposition can be explicitly given.


Even number of smoothness factors

Let n be the number of smoothness factors in the B-spline lowpass filter, which shall be even. Then define recursively : \begin a_0 &= \frac, \\ a_m &= \frac. \end The lifting filters are : s_(z) = a_m\cdot(2\cdot m + 1)\cdot(1 + z^). Conclusively, the interim results of the lifting are : \begin x_(z) &= z, \\ x_(z) &= 1, \\ x_(z) &= x_(z) + a_m\cdot(2\cdot m + 1)\cdot(z + z^) \cdot z^ \cdot x_(z), \end which leads to : x_(z) = 2^ \cdot (1 + z)^n \cdot z^. The filters x_ and x_ constitute the CDF-''n'',0 filterbank.


Odd number of smoothness factors

Now, let n be odd. Then define recursively : \begin a_0 &= \frac, \\ a_m &= \frac. \end The lifting filters are : s_(z) = a_m\cdot((2\cdot m + 1) + (2\cdot m - 1)\cdot z) / z^. Conclusively, the interim results of the lifting are : \begin x_(z) &= z, \\ x_(z) &= 1, \\ x_(z) &= x_(z) + a_0\cdot x_0(z), \\ x_(z) &= x_(z) + a_m\cdot((2\cdot m + 1)\cdot z + (2\cdot m - 1)\cdot z^) \cdot z^ \cdot x_(z), \end which leads to : x_(z) \sim (1 + z)^n, where we neglect the translation and the constant factor. The filters x_ and x_ constitute the CDF-''n'',1 filterbank.


Applications

The Cohen–Daubechies–Feauveau wavelet and other biorthogonal wavelets have been used to compress
fingerprint A fingerprint is an impression left by the friction ridges of a human finger. The recovery of partial fingerprints from a crime scene is an important method of forensic science. Moisture and grease on a finger result in fingerprints on surfa ...
scans for the
FBI The Federal Bureau of Investigation (FBI) is the domestic Intelligence agency, intelligence and Security agency, security service of the United States and Federal law enforcement in the United States, its principal federal law enforcement ag ...
. A standard for compressing fingerprints in this way was developed by Tom Hopper (FBI), Jonathan Bradley (
Los Alamos National Laboratory Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development Laboratory, laboratories of the United States Department of Energy National Laboratories, United States Department of Energy ...
) and Chris Brislawn (Los Alamos National Laboratory). By using wavelets, a compression ratio of around 20 to 1 can be achieved, meaning a 10 MB image could be reduced to as little as 500 kB while still passing recognition tests.


External links


JPEG 2000: How does it work?
*
CDF 9/7 Wavelet Transform for 2D Signals via Lifting: Source code in Python

Open Source 5/3-CDF-Wavelet implementation in C#, for arbitrary lengths


References

{{DEFAULTSORT:Cohen-Daubechies-Feauveau wavelet Biorthogonal wavelets