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We have designed these “new" coiflet systems (e.g., N = 10 , 16 , 22 , 28 ) by using the Matlab optimization toolbox constrained optimization function. Wells and Tian [link] used Newton's method to design lengths N = 6 + 2 and N = 6 coiflets up to length 30 [link] . Selesnick [link] has used a filter design approach. Still another approach is given by Wei and Bovik [link] .

[link] also shows the result of designing a length-4 system, using the one degree of freedom to ask for one zero scaling functionmoment rather than one zero wavelet moment as we did for the Daubechies system. For length-4, we do not get any “extra" zero moments becausethere are not enough zero wavelet moments. Here we see a direct trade-off between zero scaling function moments and wavelet moments. Adding thesenew lengths to our traditional coiflets gives [link] .

Moments for Various Length- N and Degree- L Coiflets, where (*) is the number of zero wavelet moments, excluding the m 1 ( 0 ) = 0
N L m = 0 m 1 = 0 m = 0 m 1 = 0 Total zero Hölder
set set* actual actual* moments exponent
4 1 1 0 1 0 1 0.2075
6 2 1 1 2 1 3 1.0137
8 3 2 2 2 2 4 1.3887
10 3 3 2 4 2 6 1.0909
12 4 3 3 4 3 7 1.9294
14 5 4 4 4 4 8 1.7353
16 5 5 4 6 4 10 1.5558
18 6 5 5 6 5 11 2.1859
20 7 6 6 6 6 12 2.8531
22 7 7 6 8 6 14 2.5190
24 8 7 7 8 7 15 2.8300
26 9 8 8 8 8 16 3.4404
28 9 9 8 10 8 18 2.9734
30 10 9 9 10 9 19 3.4083

The fourth and sixth columns in [link] contain the number of zero wavelet moments, excluding the m 1 ( 0 ) = 0 which is zero because of orthogonality in all of these systems. The extra zero scaling function moments that occurafter a nonzero moment for N = 6 + 2 are also excluded from the count. This table shows coiflets for all even lengths. It shows the extra zero scalingfunction moments that are sometime achieved and how the total number of zero moments monotonically increases and how the “smoothness" as measuredby the Hölder exponent [link] , [link] , [link] increases with N and L .

When both scaling function and wavelet moments are set to zero, a larger number can be obtained than is expected from considering the degrees offreedom available. As stated earlier, of the N degrees of freedom available from the N coefficients, h ( n ) , one is used to insure existence of φ ( t ) through the linear constraint [link] , and N / 2 are used to insure orthonormality through the quadratic constraints [link] . This leaves N / 2 - 1 degrees of freedom to achieve other characteristics. Daubechies used these to set the first N / 2 - 1 wavelet moments to zero. If setting scaling function moments were independent ofsetting wavelet moments zero, one would think that the coiflet system would allow ( N / 2 - 1 ) / 2 wavelet moments to be set zero and the same number of scaling function moments. For the coifletsdescribed in [link] , one always obtains more than this. The structure of this problem allows more zero moments to be both set andachieved than the simple degrees of freedom would predict. In fact, the coiflets achieve approximately 2 N / 3 total zero moments as compared with the number of degrees of freedom which is approximately N / 2 , and which is achieved by the Daubechies wavelet system.

As noted earlier and illustrated in [link] , these coiflets fall into three classes. Those with scaling filter lengths of N = 6 + 2 (due to Tian) have equal number of zero scaling function and wavelet moments, but always has “extra" zero scaling function momentslocated after the first nonzero one. Lengths N = 6 (due to Daubechies) always have one more zero scaling function moment than zerowavelet moment and lengths N = 6 - 2 (new) always have two more zero scaling function moments than zero wavelet moments. These “extra" zeromoments are predicted by [link] to [link] , and there will be additional even-order zero moments for longer lengths. We have observedthat within each of these classes, the Hölder exponent increases monotonically.

Number of Zero Moments for The Three Classes of Coiflets ( = 1 , 2 , ), *excluding μ 1 ( 0 ) = 0 , †excluding Non-Contiguous zeros
N m = 0 m 1 = 0 * Total zero
Length achieved achieved moments
N = 6 + 2 ( N - 2 ) / 3 ( N - 2 ) / 3 ( 2 / 3 ) ( N - 2 )
N = 6 N / 3 ( N - 3 ) / 3 ( 2 / 3 ) ( N - 3 / 2 )
N = 6 - 2 ( N + 2 ) / 3 ( N - 4 ) / 3 ( 2 / 3 ) ( N - 1 )

The approach taken in some investigations of coiflets would specify the coiflet degree and then find the shortest filter that would achieve thatdegree. The lengths N = 6 - 2 were not found by this approach because they have the same coiflet degree as the system just two shorter.However, they achieve two more zero scaling function moments than the shorter length with the same degree. By specifying the number of zeromoments and/or the filter length, it is easier to see the complete picture.

[link] is just part of a large collection of zero moment wavelet system designs with a wide variety of trade-offs that wouldbe tailored to a particular application. In addition to the variety illustrated here, many (perhaps all) of these sets of specified zeromoments have multiple solutions. This is certainly true for length-6 as illustrated in [link] through [link] and for other lengths that we have found experimentally. The variety of solutions for each lengthcan have different shifts, different Hölder exponents, and different degrees of being approximately symmetric.

The results of this chapter and section show the importance of moments to the characteristics of scaling functions and wavelets. It may not,however, be necessary or important to use the exact criteria of Daubechies or Coifman, but understanding the effects of zero moments is very important. It may be that setting a few scaling function moments and afew wavelets moments may be sufficient with the remaining degrees of freedom used for some other optimization, either in the frequency domainor in the time domain. As is noted in the next section, an alternative might be to minimize a larger number of various moments rather than tozero a few [link] .

Examples of the Coiflet Systems are shown in [link] .

Minimization of moments rather than zero moments

Odegard has considered the case of minimization of a larger number of moments rather than setting N / 2 - 1 equal to zero [link] , [link] , [link] . This results in some improvement in representing or approximating a largerclass of signals at the expense of a better approximation of a smaller class. Indeed, Götze [link] has shown that even in the designed zero moments wavelet systems, the implementation of the system in finiteprecision arithmetic results in nonzero moments and, in some cases, non-orthogonal systems.

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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