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Regularity, Moments, and Wavelet System Design

Module by: C. Sidney Burrus. E-mail the author

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We now look at a particular way to use the remaining N2-1N2-1 degrees of freedom to design the NN values of h(n)h(n) after satisfying (Reference) and (Reference), which insure the existence and orthogonality (or property of being a tight frame) of the scaling function and wavelets (Reference), (Reference), (Reference).

One of the interesting characteristics of the scaling functions and wavelets is that while satisfying (Reference) and (Reference) will guarantee the existence of an integrable scaling function, it may be extraordinarily irregular, even fractal in nature. This may be an advantage in analyzing rough or fractal signals but it is likely to be a disadvantage for most signals and images.

We will see in this section that the number of vanishing moments of h1(n)h1(n) and ψ(t)ψ(t) are related to the smoothness or differentiability of φ(t)φ(t) and ψ(t)ψ(t). Unfortunately, smoothness is difficult to determine directly because, unlike with differential equations, the defining recursion equation (Reference) does not involve derivatives.

We also see that the representation and approximation of polynomials are related to the number of vanishing or minimized wavelet moments. Since polynomials are often a good model for certain signals and images, this property is both interesting and important.

The number of zero scaling function moments is related to the “goodness" of the approximation of high-resolution scaling coefficients by samples of the signal. They also affect the symmetry and concentration of the scaling function and wavelets.

This section will consider the basic 2-band or multiplier-2 case defined in (Reference). The more general M-band or multiplier-M case is discussed in (Reference).

K-Regular Scaling Filters

Here we start by defining a unitary scaling filter to be an FIR filter with coefficients h(n)h(n) from the basic recursive equation (Reference) satisfying the admissibility conditions from (Reference) and orthogonality conditions from (Reference) as

∑ n h ( n ) = 2 and ∑ k h ( k ) h ( k + 2 m ) = δ ( m ) . ∑ n h ( n ) = 2 and ∑ k h ( k ) h ( k + 2 m ) = δ ( m ) .
(1)

The term “scaling filter" comes from Mallat's algorithm, and the relation to filter banks discussed in Chapter (Reference). The term “unitary" comes from the orthogonality conditions expressed in filter bank language, which is explained in Chapter (Reference).

A unitary scaling filter is said to be KK-regular if its z-transform has KK zeros at z=eiπz=eiπ. This looks like

H ( z ) = 1 + z - 1 2 K Q ( z ) H ( z ) = 1 + z - 1 2 K Q ( z )
(2)

where H(z)=∑nh(n)z-nH(z)=∑nh(n)z-n is the z-transform of the scaling coefficients h(n)h(n) and Q(z)Q(z) has no poles or zeros at z=eiπz=eiπ. Note that we are presenting a definition of regularity of h(n)h(n), not of the scaling function φ(t)φ(t) or wavelet ψ(t)ψ(t). They are related but not the same. Note also from (Reference) that any unitary scaling filter is at least K=1K=1 regular.

The length of the scaling filter is NN which means H(z)H(z) is an N-1N-1 degree polynomial. Since the multiple zero at z=-1z=-1 is order KK, the polynomial Q(z)Q(z) is degree N-1-KN-1-K. The existence of φ(t)φ(t) requires the zeroth moment be 22 which is the result of the linear condition in Equation 1. Satisfying the conditions for orthogonality requires N/2N/2 conditions which are the quadratic equations in Equation 1. This means the degree of regularity is limited by

1 ≤ K ≤ N 2 . 1 ≤ K ≤ N 2 .
(3)

Daubechies used the degrees of freedom to obtain maximum regularity for a given NN, or to obtain the minimum NN for a given regularity. Others have allowed a smaller regularity and used the resulting extra degrees of freedom for other design purposes.

Regularity is defined in terms of zeros of the transfer function or frequency response function of an FIR filter made up from the scaling coefficients. This is related to the fact that the differentiability of a function is tied to how fast its Fourier series coefficients drop off as the index goes to infinity or how fast the Fourier transform magnitude drops off as frequency goes to infinity. The relation of the Fourier transform of the scaling function to the frequency response of the FIR filter with coefficients h(n)h(n) is given by the infinite product (Reference). From these connections, we reason that since H(z)H(z) is lowpass and, if it has a high order zero at z=-1z=-1 (i.e., ω=πω=π), the Fourier transform of φ(t)φ(t) should drop off rapidly and, therefore, φ(t)φ(t) should be smooth. This turns out to be true.

We next define the kthkth moments of φ(t)φ(t) and ψ(t)ψ(t) as

m ( k ) = ∫ t k φ ( t ) d t m ( k ) = ∫ t k φ ( t ) d t
(4)

and

m 1 ( k ) = ∫ t k ψ ( t ) d t m 1 ( k ) = ∫ t k ψ ( t ) d t
(5)

and the discrete kthkth moments of h(n)h(n) and h1(n)h1(n) as

μ ( k ) = ∑ n n k h ( n ) μ ( k ) = ∑ n n k h ( n )
(6)

and

μ 1 ( k ) = ∑ n n k h 1 ( n ) . μ 1 ( k ) = ∑ n n k h 1 ( n ) .
(7)

The partial moments of h(n)h(n) (moments of samples) are defined as

ν ( k , ℓ ) = ∑ n ( 2 n + ℓ ) k h ( 2 n + ℓ ) . ν ( k , ℓ ) = ∑ n ( 2 n + ℓ ) k h ( 2 n + ℓ ) .
(8)

Note that μ(k)=ν(k,0)+ν(k,1)μ(k)=ν(k,0)+ν(k,1).

From these equations and the basic recursion equation (Reference) we obtain [2]

m ( k ) = 1 ( 2 k - 1 ) 2 ∑ ℓ = 1 k k ℓ μ ( ℓ ) m ( k - ℓ ) m ( k ) = 1 ( 2 k - 1 ) 2 ∑ ℓ = 1 k k ℓ μ ( ℓ ) m ( k - ℓ )
(9)

which can be derived by substituting (Reference) into Equation 4, changing variables, and using Equation 6. Similarly, we obtain

m 1 ( k ) = 1 2 k 2 ∑ ℓ = 0 k k ℓ μ 1 ( ℓ ) m ( k - ℓ ) . m 1 ( k ) = 1 2 k 2 ∑ ℓ = 0 k k ℓ μ 1 ( ℓ ) m ( k - ℓ ) .
(10)

These equations exactly calculate the moments defined by the integrals in Equation 4 and Equation 5 with simple finite convolutions of the discrete moments with the lower order continuous moments. A similar equation also holds for the multiplier-MM case described in (Reference)[2]. A Matlab program that calculates the continuous moments from the discrete moments using Equation 9 and Equation 10 is given in Appendix C.

Vanishing Wavelet Moments

Requiring the moments of ψ(t)ψ(t) to be zero has several interesting consequences. The following three theorems show a variety of equivalent characteristics for the KK-regular scaling filter, which relate both to our desire for smooth scaling functions and wavelets as well as polynomial representation.

Theorem 20 (Equivalent Characterizations of K-Regular Filters) A unitary scaling filter is K-regular if and only if the following equivalent statements are true:

  1. All moments of the wavelet filters are zero, μ1(k)=0μ1(k)=0, for k=0,1,⋯,(K-1)k=0,1,⋯,(K-1)
  2. All moments of the wavelets are zero, m1(k)=0m1(k)=0, for k=0,1,⋯,(K-1)k=0,1,⋯,(K-1)
  3. The partial moments of the scaling filter are equal for k=0,1,⋯,(K-1)k=0,1,⋯,(K-1)
  4. The frequency response of the scaling filter has a zero of order K at ω=πω=π, i.e. Equation 2.
  5. The kthkth derivative of the magnitude-squared frequency response of the scaling filter is zero at ω=0ω=0 for k=1,2,⋯,2K-1k=1,2,⋯,2K-1.
  6. All polynomial sequences up to degree (K-1)(K-1) can be expressed as a linear combination of shifted scaling filters.
  7. All polynomials of degree up to (K-1)(K-1) can be expressed as a linear combination of shifted scaling functions at any scale.

This is a very powerful result [9], (Reference). It not only ties the number of zero moments to the regularity but also to the degree of polynomials that can be exactly represented by a sum of weighted and shifted scaling functions.

Theorem 21 If ψ(t)ψ(t) is KK-times differentiable and decays fast enough, then the first K-1K-1 wavelet moments vanish (Reference); i.e.,

d k d t k ψ ( t ) < ∞ , 0 ≤ k ≤ K d k d t k ψ ( t ) < ∞ , 0 ≤ k ≤ K
(11)

implies

m 1 ( k ) = 0 . 0 ≤ k ≤ K m 1 ( k ) = 0 . 0 ≤ k ≤ K
(12)

Unfortunately, the converse of this theorem is not true. However, we can relate the differentiability of ψ(t)ψ(t) to vanishing moments by

Theorem 22 There exists a finite positive integer LL such that if m1(k)=0m1(k)=0 for 0≤k≤K-10≤k≤K-1 then

d P d t P ψ ( t ) < ∞ d P d t P ψ ( t ) < ∞
(13)

for LP>KLP>K.

For example, a three-times differentiable ψ(t)ψ(t) must have three vanishing moments, but three vanishing moments results in only one-time differentiability.

These theorems show the close relationship among the moments of h1(n)h1(n), ψ(t)ψ(t), the smoothness of H(ω)H(ω) at ω=0ω=0 and ππ and to polynomial representation. It also states a loose relationship with the smoothness of φ(t)φ(t) and ψ(t)ψ(t) themselves.

Daubechies' Method for Zero Wavelet Moment Design

Daubechies used the above relationships to show the following important result which constructs orthonormal wavelets with compact support with the maximum number of vanishing moments.

Theorem 23 The discrete-time Fourier transform of h(n)h(n) having KK zeros at ω=πω=π of the form

H ( ω ) = 1 + e i ω 2 K L ( ω ) H ( ω ) = 1 + e i ω 2 K L ( ω )
(14)

satisfies

| H ( ω ) | 2 + | H ( ω + π ) | 2 = 2 | H ( ω ) | 2 + | H ( ω + π ) | 2 = 2
(15)

if and only if L(ω)=|L(ω)|2L(ω)=|L(ω)|2 can be written

L ( ω ) = P ( sin 2 ( ω / 2 ) ) L ( ω ) = P ( sin 2 ( ω / 2 ) )
(16)

with K≤N/2K≤N/2 where

P ( y ) = ∑ k = 0 K - 1 K - 1 + k k y k + y K R ( 1 2 - y ) P ( y ) = ∑ k = 0 K - 1 K - 1 + k k y k + y K R ( 1 2 - y )
(17)

and R(y)R(y) is an odd polynomial chosen so that P(y)≥0P(y)≥0 for 0≤y≤10≤y≤1.

If R=0R=0, the length NN is minimum for a given regularity K=N/2K=N/2. If N>2KN>2K, the second term containing RR has terms with higher powers of yy whose coefficients can be used for purposes other than regularity.

The proof and a discussion are found in Daubechies (Reference), (Reference). Recall from (Reference) that H(ω)H(ω) always has at least one zero at ω=πω=π as a result of h(n)h(n) satisfying the necessary conditions for φ(t)φ(t) to exist and have orthogonal integer translates. We are now placing restrictions on h(n)h(n) to have as high an order zero at ω=πω=π as possible. That accounts for the form of Equation 14. Requiring orthogonality in (Reference) gives Equation 15.

Because the frequency domain requirements in Equation 15 are in terms of the square of the magnitudes of the frequency response, spectral factorization is used to determine H(ω)H(ω) and therefore h(n)h(n) from |H(ω)|2|H(ω)|2. Equation Equation 14 becomes

| H ( ω ) | 2 = 1 + e i ω 2 2 K | L ( ω ) | 2 . | H ( ω ) | 2 = 1 + e i ω 2 2 K | L ( ω ) | 2 .
(18)

If we use the functional notation:

M ( ω ) = | H ( ω ) | 2 and L ( ω ) = | L ( ω ) | 2 M ( ω ) = | H ( ω ) | 2 and L ( ω ) = | L ( ω ) | 2
(19)

then Equation 18 becomes

M ( ω ) = | cos 2 ( ω / 2 ) | K L ( ω ) . M ( ω ) = | cos 2 ( ω / 2 ) | K L ( ω ) .
(20)

Since M(ω)M(ω) and L(ω)L(ω) are even functions of ωω they can be written as polynomials in cos(ω)cos(ω) and, using cos(ω)=1-2sin2(ω/2)cos(ω)=1-2sin2(ω/2), equation Equation 20 becomes

M ˜ ( sin 2 ( ω / 2 ) ) = | cos 2 ( ω / 2 ) | K P ( sin 2 ( ω / 2 ) ) M ˜ ( sin 2 ( ω / 2 ) ) = | cos 2 ( ω / 2 ) | K P ( sin 2 ( ω / 2 ) )
(21)

which, after a change of variables of y=sin2(ω/2)=1-cos2(ω/2)y=sin2(ω/2)=1-cos2(ω/2), becomes

M ˜ ( y ) = ( 1 - y ) K P ( y ) M ˜ ( y ) = ( 1 - y ) K P ( y )
(22)

where P(y)P(y) is an (N-K)(N-K) order polynomial which must be positive since it will have to be factored to find H(ω)H(ω) from Equation 19. This now gives Equation 14 in terms of new variables which are easier to use.

In order that this description supports an orthonormal wavelet basis, we now require that Equation 22 satisfies (Reference)

| H ( ω ) | 2 + | H ( ω + π ) | 2 = 2 | H ( ω ) | 2 + | H ( ω + π ) | 2 = 2
(23)

which using Equation 19 and Equation 22 becomes START HERE

M ( ω ) + M ( ω + π ) = ( 1 - y ) K P ( y ) + y K P ( 1 - y ) = 2 . M ( ω ) + M ( ω + π ) = ( 1 - y ) K P ( y ) + y K P ( 1 - y ) = 2 .
(24)

Equations of this form have an explicit solution found by using Bezout's theorem. The details are developed by Daubechies (Reference). If all the (N/2-1)(N/2-1) degrees of freedom are used to set wavelet moments to zero, we set K=N/2K=N/2 and the solution to Equation 24 is given by

P ( y ) = ∑ k = 0 K - 1 K - 1 + k k y k P ( y ) = ∑ k = 0 K - 1 K - 1 + k k y k
(25)

which gives a complete parameterization of Daubechies' maximum zero wavelet moment design. It also gives a very straightforward procedure for the calculation of the h(n)h(n) that satisfy these conditions. Herrmann derived this expression for the design of Butterworth or maximally flat FIR digital filters (Reference).

If the regularity is K<N/2K<N/2, P(y)P(y) must be of higher degree and the form of the solution is

P ( y ) = ∑ k = 0 K - 1 K - 1 + k k y k + y K R ( 1 2 - y ) P ( y ) = ∑ k = 0 K - 1 K - 1 + k k y k + y K R ( 1 2 - y )
(26)

where R(y)R(y) is chosen to give the desired filter length NN, to achieve some other desired property, and to give P(y)≥0P(y)≥0.

The steps in calculating the actual values of h(n)h(n) are to first choose the length NN (or the desired regularity) for h(n)h(n), then factor |H(ω)|2|H(ω)|2 where there will be freedom in choosing which roots to use for H(ω)H(ω). The calculations are more easily carried out using the z-transform form of the transfer function and using convolution in the time domain rather than multiplication (raising to a power) in the frequency domain. That is done in the Matlab program [hn,h1n] = daub(N) in Appendix C where the polynomial coefficients in Equation 25 are calculated from the binomial coefficient formula. This polynomial is factored with the roots command in Matlab and the roots are mapped from the polynomial variable yℓyℓ to the variable zℓzℓ in Equation 2 using first cos(ω)=1-2yℓcos(ω)=1-2yℓ, then with isin(ω)=cos2(ω)-1isin(ω)=cos2(ω)-1 and eiω=cos(ω)±isin(ω)eiω=cos(ω)±isin(ω) we use z=eiωz=eiω. These changes of variables are used by Herrmann (Reference) and Daubechies (Reference).

Examine the Matlab program to see the details of just how this is carried out. The program uses the sort command to order the roots of H(z)H(1/z)H(z)H(1/z) after which it chooses the N-1N-1 smallest ones to give a minimum phase H(z)H(z) factorization. You could choose a different set of N-1N-1 roots in an effort to get a more linear phase or even maximum phase. This choice allows some variation in Daubechies wavelets of the same length. The MM-band generalization of this is developed by Heller in [9], (Reference). In (Reference), Daubechies also considers an alternation of zeros inside and outside the unit circle which gives a more symmetric h(n)h(n). A completely symmetric real h(n)h(n) that has compact support and supports orthogonal wavelets is not possible; however, symmetry is possible for complex h(n)h(n), biorthogonal systems, infinitely long h(n)h(n), and multiwavelets. Use of this zero moment design approach will also assure the resulting wavelets system is an orthonormal basis.

If all the degrees of freedom are used to set moments to zero, one uses K=N/2K=N/2 in Equation 14 and the above procedure is followed. It is possible to explicitly set a particular pair of zeros somewhere other than at ω=πω=π. In that case, one would use K=(N/2)-2K=(N/2)-2 in Equation 14. Other constraints are developed later in this chapter and in later chapters.

To illustrate some of the characteristics of a Daubechies wavelet system, Table 1 shows the scaling function and wavelet coefficients, h(n)h(n) and h1(n)h1(n), and the corresponding discrete scaling coefficient moments and wavelet coefficient moments for a length-8 Daubechies system. Note the N/2=4N/2=4 zero moments of the wavelet coefficients and the zeroth scaling coefficient moment of μ(0)=2μ(0)=2.

Table 1: Scaling Function and Wavelet Coefficients plus their Discrete Moments for Daubechies-8
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
0 0.23037781330890 0.01059740178507 1.414213 0 0
1 0.71484657055292 0.03288301166689 1.421840 0 1
2 0.63088076792986 -0.03084138183556 1.429509 0 2
3 -0.02798376941686 -0.18703481171909 0.359097 0 3
4 -0.18703481171909 0.02798376941686 -2.890773 12.549900 4
5 0.03084138183556 0.63088076792986 -3.453586 267.067254 5
6 0.03288301166689 -0.71484657055292 23.909120 3585.681937 6
7 -0.01059740178507 0.23037781330890      

Table 2 gives the same information for the length-6, 4, and 2 Daubechies scaling coefficients, wavelet coefficients, scaling coefficient moments, and wavelet coefficient moments. Again notice how many discrete wavelet moments are zero.

Table 3 shows the continuous moments of the scaling function φ(t)φ(t) and wavelet ψ(t)ψ(t) for the Daubechies systems with lengths six and four. The discrete moments are the moments of the coefficients defined by Equation 6 and Equation 7 with the continuous moments defined by Equation 4 and Equation 5 calculated using Equation 9 and Equation 10 with the programs listed in Appendix C.

Table 2: Daubechies Scaling Function and Wavelet Coefficients plus their Moments
  Daubechies N=6N=6        
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
0 0.33267055295008 -0.03522629188571 1.414213 0 0
1 0.80689150931109 -0.08544127388203 1.155979 0 1
2 0.45987750211849 0.13501102001025 0.944899 0 2
3 -0.13501102001025 0.45987750211849 -0.224341 3.354101 3
4 -0.08544127388203 -0.80689150931109 -2.627495 40.679682 4
5 0.03522629188571 0.33267055295008 5.305591 329.323717 5
  Daubechies N=4N=4        
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
0 0.48296291314453 0.12940952255126 1.414213 0 0
1 0.83651630373781 0.22414386804201 0.896575 0 1
2 0.22414386804201 -0.83651630373781 0.568406 1.224744 2
3 -0.12940952255126 0.48296291314453 -0.864390 6.572012 3
  Daubechies N=2N=2        
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
0 0.70710678118655 0.70710678118655 1.414213 0 0
1 0.70710678118655 -0.70710678118655 0.707107 0.707107 1
Table 3: Daubechies Scaling Function and Wavelet Continuous and Discrete Moments
  N = 6 N = 6      
k k μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) m ( k ) m ( k ) m 1 ( k ) m 1 ( k )
0 1.4142135 0 1.0000000 0
1 1.1559780 0 0.8174012 0
2 0.9448992 0 0.6681447 0
3 -0.2243420 3.3541019 0.4454669 0.2964635
4 -2.6274948 40.6796819 0.1172263 2.2824642
5 5.3055914 329.3237168 -0.0466511 11.4461157
  N = 4 N = 4      
k k μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) m ( k ) m ( k ) m 1 ( k ) m 1 ( k )
0 1.4142136 0 1.0000000 0
1 0.8965755 0 0.6343975 0
2 0.5684061 1.2247449 0.4019238 0.2165063
3 -0.8643899 6.5720121 0.1310915 0.7867785
4 -6.0593531 25.9598790 -0.3021933 2.0143421
5 -23.4373939 90.8156100 -1.0658728 4.4442798

These tables are very informative about the characteristics of wavelet systems in general as well as particularities of the Daubechies system. We see the μ(0)=2μ(0)=2 of Equation 1 and (Reference) that is necessary for the existence of a scaling function solution to (Reference) and the μ1(0)=m1(0)=0μ1(0)=m1(0)=0 of (Reference) and (Reference) that is necessary for the orthogonality of the basis functions. Orthonormality requires (Reference) which is seen in comparison of the h(n)h(n) and h1(n)h1(n), and it requires m(0)=1m(0)=1 from (Reference) and (Reference). After those conditions are satisfied, there are N/2-1N/2-1 degrees of freedom left which Daubechies uses to set wavelet moments m1(k)m1(k) equal zero. For length-6 we have two zero wavelet moments and for length-4, one. For all longer Daubechies systems we have exactly N/2-1N/2-1 zero wavelet moments in addition to the one m1(0)=0m1(0)=0 for a total of N/2N/2 zero wavelet moments. Note m(2)=m(1)2m(2)=m(1)2 as will be explained in Equation 32 and there exist relationships among some of the values of the even-ordered scaling function moments, which will be explained in Equation 51 through (Reference).

As stated earlier, these systems have a maximum number of zero moments of the wavelets which results in a high degree of smoothness for the scaling and wavelet functions. Figures (Reference) and (Reference) show the Daubechies scaling functions and wavelets for N=4,6,8,10,12,16,20,40N=4,6,8,10,12,16,20,40. The coefficients were generated by the techniques described in (Reference) and Chapter (Reference). The Matlab programs are listed in Appendix C and values of h(n)h(n) can be found in (Reference) or generated by the programs. Note the increasing smoothness as NN is increased. For N=2N=2, the scaling function is not continuous; for N=4N=4, it is continuous but not differentiable; for N=6N=6, it is barely differentiable once; for N=14N=14, it is twice differentiable, and similarly for longer h(n)h(n). One can obtain any degree of differentiability for sufficiently long h(n)h(n).

The Daubechies coefficients are obtained by maximizing the number of moments that are zero. This gives regular scaling functions and wavelets, but it is possible to use the degrees of

Figure 1: Daubechies Scaling Functions, N=4,6,8,...,40N=4,6,8,...,40
1: Φ D 4 Φ D 4
Daubechies Scaling Functions
2: Φ D 6 Φ D 6
Daubechies Scaling Functions
3: Φ D 8 Φ D 8
Daubechies Scaling Functions
4: Φ D 10 Φ D 10
Daubechies Scaling Functions
5: Φ D 12 Φ D 12
Daubechies Scaling Functions
6: Φ D 16 Φ D 16
Daubechies Scaling Functions
7: Φ D 20 Φ D 20
Daubechies Scaling Functions
8: Φ D 40 Φ D 40
Daubechies Scaling Functions

freedom to maximize the differentiability of φ(t)φ(t) rather than maximize the zero moments. This is not easily parameterized, and it gives only slightly greater smoothness than the Daubechies system (Reference).

Figure 2: Daubechies Wavelets, N=4,6,8,...,40N=4,6,8,...,40
1: φ D 4 φ D 4
Daubechies Wavelets
2: φ D 6 φ D 6
Daubechies Wavelets
3: φ D 8 φ D 8
Daubechies Wavelets
4: φ D 10 φ D 10
Daubechies Wavelets
5: φ D 12 φ D 12
Daubechies Wavelets
6: φ D 16 φ D 16
Daubechies Wavelets
7: φ D 20 φ D 20
Daubechies Wavelets
8:
Daubechies Wavelets

Examples of Daubechies scaling functions resulting from choosing different factors in the spectral factorization of |H(ω)|2|H(ω)|2 in Equation 18 can be found in (Reference).

Non-Maximal Regularity Wavelet Design

If the length of the scaling coefficient filter is longer than twice the desired regularity, i.e., N>2KN>2K, then the parameterization of Equation 26 should be used and the coefficients in the polynomial R(y)R(y) must be determined. One interesting possibility is that of designing a system that has one or more zeros of H(ω)H(ω) between ω=π/2ω=π/2 and ω=πω=π with the remaining zeros at ππ to contribute to the regularity. This will give better frequency separation in the filter bank in exchange for reduced regularity and lower degree polynomial representation.

If a zero of H(ω)H(ω) is set at ω=ω0ω=ω0, then the conditions of

M ( ω 0 ) = 0 and d M ( ω ) d ω ω = ω 0 = 0 M ( ω 0 ) = 0 and d M ( ω ) d ω ω = ω 0 = 0
(27)

are imposed with those in Equation 26, giving a set of linear simultaneous equations that can be solved to find the scaling coefficients h(n)h(n).

A powerful design system based on a Remez exchange algorithm allows the design of an orthogonal wavelet system that has a specified regularity and an optimal Chebyshev behavior in the stopband of H(ω)H(ω). This and a variation that uses a constrained least square criterion [10] is described in [5] and another Chebyshev design in (Reference).

An alternative approach is to design the wavelet system by setting an optimization criterion and using a general constrained optimization routine, such as that in the Matlab optimization toolbox, to design the h(n)h(n) with the existence and orthogonality conditions as constraints. This approach was used to generate many of the filters described in Table 7. Jun Tian used a Newton's method (Reference), (Reference) to design wavelet systems with zero moments.

Relation of Zero Wavelet Moments to Smoothness

We see from Theorems (Reference) and (Reference) that there is a relationship between zero wavelet moments and smoothness, but it is not tight. Although in practical application the degree of differentiability may not be the most important measure, it is an important theoretical question that should be addressed before application questions can be properly posed.

First we must define smoothness. From the mathematical point of view, smoothness is essentially the same as differentiability and there are at least two ways to pose that measure. The first is local (the Hölder measure) and the second is global (the Sobolev measure). Numerical algorithms for estimating the measures in the wavelet setting have been developed by Rioul (Reference) and Heller and Wells (Reference), (Reference) for the Hölder and Sobolev measures respectively.

Definition 1 (Hölder continuity) Let φ : I R → C φ : I R → C and let 0 < α < 1. Then the function φ is Hölder continuous of order α if there exists a constant c such that

φ(x) − φ(y) ≤ c x − y α for all x , y ∈ I R φ(x) − φ(y) ≤ c x − y α for all x , y ∈ I R
(28)

Based on the above definition, one observes φφ has to be a constant if α>1α>1. Hence, this is not very useful for determining regularity of order α>1α>1. However, using the above definition, Hölder regularity of any order r>0r>0 is defined as follows:

Definition 2 (Hölder regularity) A function φ : ℝ → ℂ φ : ℝ → ℂ is regular of order r = P + α (0<α≤1) r = P + α (0<α≤1) if φ ∈ C P φ ∈ C P and its Pth derivative is Hölder continuous of order α α

Definition 3 (Sobolev regularity) Letφ:ℝ→ℂ, thenφis said to belong to the Sobolev space of order s ( φ ∈ H s ) if Letφ:ℝ→ℂ, thenφis said to belong to the Sobolev space of order s ( φ ∈ H s ) if

∫ IR ( 1 + ω 2 ) s Φ ˆ ( ω ) 2 d ω < ∞ ∫ IR ( 1 + ω 2 ) s Φ ˆ ( ω ) 2 d ω < ∞
(29)

Notice that, although Sobolev regularity does not give the explicit order of differentiability, it does yield lower and upper bounds on rr, the Hölder regularity, and hence the differentiability of φφ if φ∈L2φ∈L2. This can be seen from the following inclusions:

H s + 1 / 2 ⊂ C r ⊂ H s H s + 1 / 2 ⊂ C r ⊂ H s
(30)

A very interesting and important result by Volkmer (Reference) and by Eirola (Reference) gives an exact asymptotic formula for the Hölder regularity index (exponent) of the Daubechies scaling function.

Theorem 24 The limit of the Hölder regularity index of a Daubechies scaling function as the length of the scaling filter goes to infinity is (Reference)

lim N → ∞ α N N = 1 - log ( 3 ) 2 log ( 2 ) = ( 0 . 2075 ⋯ ) lim N → ∞ α N N = 1 - log ( 3 ) 2 log ( 2 ) = ( 0 . 2075 ⋯ )
(31)

This result, which was also proven by A. Cohen and J. P. Conze, together with empirical calculations for shorter lengths, gives a good picture of the smoothness of Daubechies scaling functions. This is illustrated in Figure 3 where the Hölder index is plotted versus scaling filter length for both the maximally smooth case and the Daubechies case.

The question of the behavior of maximally smooth scaling functions was empirically addressed by Lang and Heller in (Reference). They use an algorithm by Rioul to calculate the Hölder smoothness of scaling functions that have been designed to have maximum Hölder smoothness and the results are shown in Figure 3 together with the smoothness of the Daubechies scaling functions as functions of the length of the scaling filter. For the longer lengths, it is possible to design systems that give a scaling function over twice as smooth as with a Daubechies design. In most applications, however, the greater Hölder smoothness is probably not important.

Figure 3: Hölder Smoothness versus Coefficient Length for Daubechies' (++) and Maximally Smooth (o) Wavelets.
Hölder Smoothness versus Coefficient Length for Daubechies'
Figure 4: Number of Zeros at ω=πω=π versus Coefficient Length for Daubechies' (++) and Maximally Smooth (o) Wavelets.
Number of Zeros at w=pi versus Coefficient Length for Daubechies'

Figure 4 shows the number of zero moments (zeros at ω=πω=π) as a function of the number of scaling function coefficients for both the maximally smooth and Daubechies designs.

One case from this figure is for N=26N=26 where the Daubechies smoothness is SH=4.005SH=4.005 and the maximum smoothness is SH=5.06SH=5.06. The maximally smooth scaling function has one more continuous derivative than the Daubechies scaling function.

Recent work by Heller and Wells (Reference), (Reference) gives a better connection of properties of the scaling coefficients and the smoothness of the scaling function and wavelets. This is done both for the scale factor or multiplicity of M=2M=2 and for general integer MM.

The usual definition of smoothness in terms of differentiability may not be the best measure for certain signal processing applications. If the signal is given as a sequence of numbers, not as a function of a continuous variable, what does smoothness mean? Perhaps the use of the variation of a signal may be a useful alternative [3], [7], (Reference).

Vanishing Scaling Function Moments

While the moments of the wavelets give information about flatness of H(ω)H(ω) and smoothness of ψ(t)ψ(t), the moments of φ(t)φ(t) and h(n)h(n) are measures of the “localization" and symmetry characteristics of the scaling function and, therefore, the wavelet transform. We know from (Reference) that ∑nh(n)=2∑nh(n)=2 and, after normalization, that ∫φ(t)dt=1∫φ(t)dt=1. Using Equation 9, one can show [2] that for K≥2K≥2, we have

m ( 2 ) = m 2 ( 1 ) . m ( 2 ) = m 2 ( 1 ) .
(32)

This can be seen in Table 3. A generalization of this result has been developed by Johnson (Reference) and is given in Equation 51 through (Reference).

A more general picture of the effects of zero moments can be seen by next considering two approximations. Indeed, this analysis gives a very important insight into the effects of zero moments. The mixture of zero scaling function moments with other specifications is addressed later in "Coiflets and Related Wavelet Systems".

Approximation of Signals by Scaling Function Projection

The orthogonal projection of a signal f(t)f(t) on the scaling function subspace VjVj is given and denoted by

P j { f ( t ) } = ∑ k 〈 f ( t ) , φ j , k ( t ) 〉 φ j , k ( t ) P j { f ( t ) } = ∑ k 〈 f ( t ) , φ j , k ( t ) 〉 φ j , k ( t )
(33)

which gives the component of f(t)f(t) which is in VjVj and which is the best least squares approximation to f(t)f(t) in VjVj.

As given in Equation 5, the ℓthℓth moment of ψ(t)ψ(t) is defined as

m 1 ( ℓ ) = ∫ t ℓ ψ ( t ) d t . m 1 ( ℓ ) = ∫ t ℓ ψ ( t ) d t .
(34)

We can now state an important relation of the projection Equation 33 as an approximation to f(t)f(t) in terms of the number of zero wavelet moments and the scale.

Theorem 25 If m1(ℓ)=0m1(ℓ)=0 for ℓ=0,1,⋯,Lℓ=0,1,⋯,L then the L2L2 error is

ϵ 1 = ∥ f ( t ) - P j { f ( t ) } ∥ 2 ≤ C 2 - j ( L + 1 ) , ϵ 1 = ∥ f ( t ) - P j { f ( t ) } ∥ 2 ≤ C 2 - j ( L + 1 ) ,
(35)

where CC is a constant independent of jj and LL but dependent on f(t)f(t) and the wavelet system (Reference), (Reference).

This states that at any given scale, the projection of the signal on the subspace at that scale approaches the function itself as the number of zero wavelet moments (and the length of the scaling filter) goes to infinity. It also states that for any given length, the projection goes to the function as the scale goes to infinity. These approximations converge exponentially fast. This projection is illustrated in (Reference).

Approximation of Scaling Coefficients by Samples of the Signal

A second approximation involves using the samples of f(t)f(t) as the inner product coefficients in the wavelet expansion of f(t)f(t) in Equation 33. We denote this sampling approximation by

S j { f ( t ) } = ∑ k 2 - j / 2 f ( k / 2 j ) φ j , k ( t ) S j { f ( t ) } = ∑ k 2 - j / 2 f ( k / 2 j ) φ j , k ( t )
(36)

and the scaling function moment by

m ( ℓ ) = ∫ t ℓ φ ( t ) d t m ( ℓ ) = ∫ t ℓ φ ( t ) d t
(37)

and can state (Reference) the following

Theorem 26 If m(ℓ)=0m(ℓ)=0 for ℓ=1,2,⋯,Lℓ=1,2,⋯,L then the L2L2 error is

ϵ 2 = ∥ S j { f ( t ) } - P j { f ( t ) } ∥ 2 ≤ C 2 2 - j ( L + 1 ) , ϵ 2 = ∥ S j { f ( t ) } - P j { f ( t ) } ∥ 2 ≤ C 2 2 - j ( L + 1 ) ,
(38)

where C2C2 is a constant independent of jj and LL but dependent on f(t)f(t) and the wavelet system.

This is a similar approximation or convergence result to the previous theorem but relates the projection of f(t)f(t) on a jj-scale subspace to the sampling approximation in that same subspace. These approximations are illustrated in (Reference).

Figure 5: Approximation and Projection of f(t)f(t) at a Finite Scale
Approximation and Projection of f(t) at a Finite Scale

This “vector space" illustration shows the nature and relationships of the two types of approximations. The use of samples as inner products is an approximation within the expansion subspace VjVj. The use of a finite expansion to represent a signal f(t)f(t) is an approximation from L2L2 onto the subspace VjVj. Theorems "Approximation of Signals by Scaling Function Projection" and (Reference) show the nature of those approximations, which, for wavelets, is very good.

An illustration of the effects of these approximations on a signal is shown in Figure 6 where a signal with a very smooth component (a sinusoid) and a discontinuous component (a square wave) is expanded in a wavelet series using samples as the high resolution scaling function coefficients. Notice the effects of projecting onto lower and lower resolution scales.

If we consider a wavelet system where the same number of scaling function and wavelet moments are set zero and this number is as large as possible, then the following is true (Reference), (Reference):

Theorem 27 If m(ℓ)=m1(ℓ)=0m(ℓ)=m1(ℓ)=0 for ℓ=1,2,⋯,Lℓ=1,2,⋯,L and m1(0)=0m1(0)=0, then the L2L2 error is

ϵ 3 = ∥ f ( t ) - S j { f ( t ) } ∥ 2 ≤ C 3 2 - j ( L + 1 ) , ϵ 3 = ∥ f ( t ) - S j { f ( t ) } ∥ 2 ≤ C 3 2 - j ( L + 1 ) ,
(39)

where C3C3 is a constant independent of jj and LL, but dependent on f(t)f(t) and the wavelet system.

Here we see that for this wavelet system called a Coifman wavelet system, that using samples as the inner product expansion coefficients is an excellent approximation. This justifies that using samples of a signal as input to a filter bank gives a proper wavelet analysis. This approximation is also illustrated in (Reference) and in (Reference).

Coiflets and Related Wavelet Systems

From the previous approximation theorems, we see that a combination of zero wavelet and zero scaling function moments used with samples of the signal may give superior results to wavelets with only zero wavelet moments. Not only does forcing zero scaling function moments give a better approximation of the expansion coefficients by samples, it often causes the scaling function to be more symmetric. Indeed, that characteristic may be more important than the sample approximation in certain applications.

Daubechies considered the design of these wavelets which were suggested by Coifman (Reference), (Reference), (Reference). Gopinath (Reference), [2] and Wells (Reference), (Reference) show how zero scaling function moments give a better approximation of high-resolution scaling coefficients by samples. Tian and Wells (Reference), (Reference) have also designed biorthogonal systems with mixed zero moments with very interesting properties.

Figure 6: Approximations to f(t)f(t) at a Different Finite Scales
Approximations to f(t) at a Different Finite Scales

The Coifman wavelet system (Daubechies named the basis functions “coiflets") is an orthonormal multiresolution wavelet system with

∫ t k φ ( t ) d t = m ( k ) = 0 , for k = 1 , 2 , ⋯ , L - 1 ∫ t k φ ( t ) d t = m ( k ) = 0 , for k = 1 , 2 , ⋯ , L - 1
(40)
∫ t k ψ ( t ) d t = m 1 ( k ) = 0 , for k = 1 , 2 , ⋯ , L - 1 . ∫ t k ψ ( t ) d t = m 1 ( k ) = 0 , for k = 1 , 2 , ⋯ , L - 1 .
(41)

This definition imposes the requirement that there be at least L-1L-1 zero scaling function moments and at least L-1L-1 wavelet moments in addition to the one zero moment of m1(0)m1(0) required by orthogonality. This system is said to be of order or degree LL and sometime has the additional requirement that the length of the scaling function filter h(n)h(n), which is denoted NN, is minimum (Reference), (Reference). In the design of these coiflets, one obtains more total zero moments than N/2-1N/2-1. This was first noted by Beylkin, et al (Reference). The length-4 wavelet system has only one degree of freedom, so it cannot have both a scaling function moment and wavelet moment of zero (see Table 6). Tian (Reference), (Reference) has derived formulas for four length-6 coiflets. These are:

h = - 3 + 7 16 2 , 1 - 7 16 2 , 7 - 7 8 2 , 7 + 7 8 2 , 5 + 7 16 2 , 1 - 7 16 2 , h = - 3 + 7 16 2 , 1 - 7 16 2 , 7 - 7 8 2 , 7 + 7 8 2 , 5 + 7 16 2 , 1 - 7 16 2 ,
(42)

or

h = - 3 - 7 16 2 , 1 + 7 16 2 , 7 + 7 8 2 , 7 - 7 8 2 , 5 - 7 16 2 , 1 + 7 16 2 , h = - 3 - 7 16 2 , 1 + 7 16 2 , 7 + 7 8 2 , 7 - 7 8 2 , 5 - 7 16 2 , 1 + 7 16 2 ,
(43)

or

h = - 3 + 15 16 2 , 1 - 15 16 2 , 3 - 15 8 2 , 3 + 15 8 2 , 13 + 15 16 2 , 9 - 15 16 2 , h = - 3 + 15 16 2 , 1 - 15 16 2 , 3 - 15 8 2 , 3 + 15 8 2 , 13 + 15 16 2 , 9 - 15 16 2 ,
(44)

or

h = - 3 - 15 16 2 , 1 + 15 16 2 , 3 + 15 8 2 , 3 - 15 8 2 , 13 - 15 16 2 , 9 + 15 16 2 , h = - 3 - 15 16 2 , 1 + 15 16 2 , 3 + 15 8 2 , 3 - 15 8 2 , 13 - 15 16 2 , 9 + 15 16 2 ,
(45)

with the first formula Equation 42 giving the same result as Daubechies (Reference), (Reference) (corrected) and that of Odegard [1] and the third giving the same result as Wickerhauser (Reference). The results from Equation 42 are included in Table 4 along with the discrete moments of the scaling function and wavelet, μ(k)μ(k) and μ1(k)μ1(k) for k=0,1,2,3k=0,1,2,3. The design of a length-6 Coifman system specifies one zero scaling function moment and one zero wavelet moment (in addition to μ1(0)=0μ1(0)=0), but we, in fact, obtain one extra zero scaling function moment. That is the result of m(2)=m(1)2m(2)=m(1)2 from [2]. In other words, we get one more zero scaling function moment than the two degrees of freedom would seem to indicate. This is true for all lengths N=6ℓN=6ℓ for ℓ=1,2,3,⋯ℓ=1,2,3,⋯ and is a result of the interaction between the scaling function moments and the wavelet moments described later.

The property of zero wavelet moments is shift invariant, but the zero scaling function moments are shift dependent (Reference). Therefore, a particular shift for the scaling function must be used. This shift is two for the length-6 example in Table 4, but is different for the solutions in Equation 44 and Equation 45. Compare this table to the corresponding one for Daubechies length-6 scaling functions and wavelets given in Table 2 where there are two zero discrete wavelet moments – just as many as the degrees of freedom in that design.

The scaling function from Equation 42 is fairly symmetric, but not around its center and the other three designs in Equation 43, Equation 44, and Equation 45 are not symmetric at all. The scaling function from Equation 42 is also fairly smooth, and from Equation 44 only slightly less so but the scaling function from Equation 43 is very rough and from Equation 45 seems to be fractal. Examination of the frequency response H(ω)H(ω) and the zero location of the FIR filters h(n)h(n) shows very similar frequency responses for Equation 42 and Equation 44 with Equation 43 having a somewhat irregular but monotonic frequency response and Equation 45 having a zero on the unit circle at ω=π/3ω=π/3, i.e., not satisfying Cohen's condition (Reference) for an orthognal basis. It is also worth noticing that the design in Equation 42 has the largest Hölder smoothness. These four designs, all satisfying the same necessary conditions, have very different characteristics. This tells us to be very careful in using zero moment methods to design wavelet systems. The designs are not unique and some are much better than others.

Table 4 contains the scaling function and wavelet coefficients for the length-6 and 12 designed by Daubechies and length-8 designed by Tian together with their discrete moments. We see the extra zero scaling function moments for lengths 6 and 12 and also the extra zero for lengths 8 and 12 that occurs after a nonzero one.

The continuous moments can be calculated from the discrete moments and lower order continuous moments (Reference), [2], (Reference) using Equation 9 and Equation 10. An important relationship of the discrete moments for a system with K-1K-1 zero wavelet moments is found by calculating the derivatives of the magnitude squared of the discrete time Fourier transform of h(n)h(n) which is H(ω)=∑nh(n)e-iωnH(ω)=∑nh(n)e-iωn and has 2K-12K-1 zero derivatives of the magnitude squared at ω=0ω=0. This gives [2] the kthkth derivative for kk even and 1<k<2K-11<k<2K-1

∑ ℓ = 0 k k ℓ ( - 1 ) ℓ μ ( ℓ ) μ ( k - ℓ ) = 0 . ∑ ℓ = 0 k k ℓ ( - 1 ) ℓ μ ( ℓ ) μ ( k - ℓ ) = 0 .
(46)

Solving for μ(k)μ(k) in terms of lower order discrete moments and using μ(0)=2μ(0)=2 gives for kk even

μ ( k ) = 1 2 2 ∑ ℓ = 1 k - 1 k ℓ ( - 1 ) ℓ μ ( ℓ ) μ ( k - ℓ ) μ ( k ) = 1 2 2 ∑ ℓ = 1 k - 1 k ℓ ( - 1 ) ℓ μ ( ℓ ) μ ( k - ℓ )
(47)

which allows calculating the even-order discrete scaling function moments in terms of the lower odd-order discrete scaling function moments for k=2,4,⋯,2K-2k=2,4,⋯,2K-2. For example:

μ ( 2 ) = 1 2 μ 2 ( 1 ) μ ( 4 ) = - 1 2 2 [ 8 μ ( 1 ) μ ( 3 ) - 3 μ 4 ( 1 ) ] ⋯ ⋯ μ ( 2 ) = 1 2 μ 2 ( 1 ) μ ( 4 ) = - 1 2 2 [ 8 μ ( 1 ) μ ( 3 ) - 3 μ 4 ( 1 ) ] ⋯ ⋯
(48)

which can be seen from values in Table 2.

Johnson (Reference) noted from Beylkin (Reference) and Unser (Reference) that by using the moments of the autocorrelation function of the scaling function, a relationship of the continuous scaling function moments can be derived in the form

∑ ℓ = 0 k k ℓ ( - 1 ) k - ℓ m ( ℓ ) m ( k - ℓ ) = 0 ∑ ℓ = 0 k k ℓ ( - 1 ) k - ℓ m ( ℓ ) m ( k - ℓ ) = 0
(49)

where 0<k<2K0<k<2K if K-1K-1 wavelet moments are zero. Solving for m(k)m(k) in terms of lower order moments gives for kk even

m ( k ) = - 1 2 ∑ ℓ = 1 k - 1 k ℓ ( - 1 ) ℓ m ( ℓ ) m ( k - ℓ ) m ( k ) = - 1 2 ∑ ℓ = 1 k - 1 k ℓ ( - 1 ) ℓ m ( ℓ ) m ( k - ℓ )
(50)

which allows calculating the even-order scaling function moments in terms of the lower odd-order scaling function moments for k=2,4,⋯,2K-2k=2,4,⋯,2K-2. For example (Reference):

m ( 2 ) = m 2 ( 1 ) m ( 4 ) = 4 m ( 3 ) m ( 1 ) - 3 m 4 ( 1 ) m ( 6 ) = 6 m ( 5 ) m ( 1 ) + 10 m 2 ( 3 ) + 60 m ( 3 ) m 3 ( 1 ) + 45 m 6 ( 1 ) m ( 8 ) = 8 m ( 7 ) m ( 1 ) + 56 m ( 5 ) m ( 3 ) - 168 m ( 5 ) m 3 ( 1 ) + 2520 m ( 3 ) m 5 ( 1 ) - 840 m ( 3 ) m 2 ( 1 ) - 1575 m 8 ( 1 ) ⋯ ⋯ m ( 2 ) = m 2 ( 1 ) m ( 4 ) = 4 m ( 3 ) m ( 1 ) - 3 m 4 ( 1 ) m ( 6 ) = 6 m ( 5 ) m ( 1 ) + 10 m 2 ( 3 ) + 60 m ( 3 ) m 3 ( 1 ) + 45 m 6 ( 1 ) m ( 8 ) = 8 m ( 7 ) m ( 1 ) + 56 m ( 5 ) m ( 3 ) - 168 m ( 5 ) m 3 ( 1 ) + 2520 m ( 3 ) m 5 ( 1 ) - 840 m ( 3 ) m 2 ( 1 ) - 1575 m 8 ( 1 ) ⋯ ⋯
(51)
Table 4: Coiflet Scaling Function and Wavelet Coefficients plus their Discrete Moments
  Length-N=6N=6, Degree L=2L=2      
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
-2 -0.07273261951285 0.01565572813546 1.414213 0 0
-1 0.33789766245781 -0.07273261951285 0 0 1
0 0.85257202021226 -0.38486484686420 0 -1.163722 2
1 0.38486484686420 0.85257202021226 -0.375737 -3.866903 3
2 -0.07273261951285 -0.33789766245781 -2.872795 -10.267374 4
3 -0.01565572813546 -0.07273261951285      
  Length-N=8N=8, Degree L=3L=3      
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
-4 0.04687500000000 0.01565572813546 1.414213 0 0
-3 -0.02116013576461 -0.07273261951285 0 0 1
-2 -0.14062500000000 -0.38486484686420 0 0 2
-1 0.43848040729385 1.38486484686420 -2.994111 0.187868 3
0 1.38486484686420 -0.43848040729385 0 11.976447 4
1 0.38486484686420 -0.14062500000000 -45.851020 -43.972332 5
2 -0.07273261951285 0.02116013576461 63.639610 271.348747 6
3 -0.01565572813546 0.04687500000000      
  Length-N=12N=12, Degree L=4L=4      
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
-4 0.016387336463 0.000720549446 1.414213 0 0
-3 -0.041464936781 0.001823208870 0 0 1
-2 -0.067372554722 -0.005611434819 0 0 2
-1 0.386110066823 -0.023680171946 0 0 3
0 0.812723635449 0.059434418646 0 11.18525 4
1 0.417005184423 0.076488599078 -5.911352 175.86964 5
2 -0.076488599078 -0.417005184423 0 1795.33634 6
3 -0.059434418646 -0.812723635449 -586.341304 15230.54650 7
4 0.023680171946 -0.386110066823 3096.310009 117752.68833 8
5 0.005611434819 0.067372554722      
6 -0.001823208870 0.041464936781      
7 -0.000720549446 -0.016387336463      

if the wavelet moments are zero up to k=K-1k=K-1. Notice that setting m(1)=m(3)=0m(1)=m(3)=0 causes m(2)=m(4)=m(6)=m(8)=0m(2)=m(4)=m(6)=m(8)=0 if sufficient wavelet moments are zero. This explains the extra zero moments in Table 4. It also shows that the traditional specification of zero scaling function moments is redundant. In Table 4m(8)m(8) would be zero if more wavelet moments were zero.

Table 5: Discrete and Continuous Moments for the Coiflet Systems
  N=6N=6, L = 2 L = 2    
k k μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) m ( k ) m ( k ) m 1 ( k ) m 1 ( k )
0 1.4142135623 0 1.0000000000 0
1 0 0 0 0
2 0 -1.1637219122 0 -0.2057189138
3 -0.3757374752 -3.8669032118 -0.0379552166 -0.3417891854
4 -2.8727952940 -10.2673737288 -0.1354248688 -0.4537580992
5 -3.7573747525 -28.0624304008 -0.0857053279 -0.6103378310
  N=8N=8, L = 3 L = 3    
k k μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) m ( k ) m ( k ) m 1 ( k ) m 1 ( k )
0 1.4142135623 0 1.0000000000 0
1 0 0 0 0
2 0 0 0 0
3 -2.9941117777 0.1878687376 -0.3024509630 0.0166054072
4 0 11.9764471108 0 0.5292891854
5 -45.8510203537 -43.9723329775 -1.0458570134 -0.9716604635

To see the continuous scaling function and wavelet moments for these systems, Table 5 shows both the continuous and discrete moments for the length-6 and 8 coiflet systems. Notice the zero moment m(4)=μ(4)=0m(4)=μ(4)=0 for length-8. The length-14, 20, and 26 systems also have the “extra" zero scaling moment just after the first nonzero moment. This always occurs for length-N=6ℓ+2N=6ℓ+2 coiflets.

(Reference) shows the length-6, 8, 10, and 12 coiflet scaling functions φ(t)φ(t) and wavelets ψ(t)ψ(t). Notice their approximate symmetry and compare this to Daubechies' classical wavelet systems and her more symmetric ones achieved by using the different factorization mentioned in "Daubechies' Method for Zero Wavelet Moment Design" and shown in (Reference). The difference between these systems and truly symmetric ones (which requires giving up orthogonality, realness, or finite support) is probably negligible in many applications.

Figure 7: Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
1: Φ C 6 Φ C 6
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
2: ψ C 6 ψ C 6
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
3: Φ C 8 Φ C 8
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
4: ψ C 8 ψ C 8
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
5: Φ C 10 Φ C 10
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
6: ψ C 10 ψ C 10
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
7: Φ C 12 Φ C 12
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets
8: ψ C 12 ψ C 12
Length-6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets

Coifman Wavelet Systems from a Specified Filter Length

The preceding section shows that Coifman systems do not necessarily have an equal number of scaling function and wavelet moments equal to zero. Lengths N=6ℓ+2N=6ℓ+2 have equal numbers of zero scaling function and wavelet moments, but always have even-order “extra" zero scaling function moments located after the first nonzero one. Lengths N=6ℓN=6ℓ always have an “extra" zero scaling function moment. Indeed, both will have several even-order “extra" zero moments for longer NN as a result of the relationships illustrated in Equation 51 through (Reference). Lengths N=6ℓ-2N=6ℓ-2 do not occur for the original definition of a Coifman system if one looks only at the degree with minimum length. If we specify the length of the coefficient vector, all even lengths become possible, some with the same coiflet degree.

We examine the general Coifman wavelet system defined in Equation 40 and Equation 41 and allow the number of specified zero scaling function and wavelet moments to differ by at most one. That will include all the reported coiflets plus length-10, 16, 22, and N=6ℓ-2N=6ℓ-2. The length-10 was designed by Odegard [1] by setting the number of zero scaling functions to 3 and the number of zero wavelet moment to 2 rather than 2 and 2 for the length-8 or 3 and 3 for the length-12 coiflets. The result in Table 6 shows that the length-10 design again gives one extra zero scaling function moment which is two more than the number of zero wavelet moments. This is an even-order moment predicted by (Reference) and results in a total number of zero moments between that for length-8 and length-12, as one would expect. A similar approach was used to design length-16, 22, and 28.

Table 6: Coiflet Scaling Function and Wavelet Coefficients plus their Discrete Moments
  Length-N=4N=4, Degree L=1L=1      
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
-1 0.224143868042 0.129409522551 1.414213 0 0
0 0.836516303737 0.482962913144 0 -0.517638 1
1 0.482962913144 -0.836516303737 0.189468 0.189468 2
2 -0.129409522551 0.224143868042 -0.776457 0.827225 3
  Length-N=10N=10, Degree L=3L=3      
n n h ( n ) h ( n ) h 1 ( n ) h 1 ( n ) μ ( k ) μ ( k ) μ 1 ( k ) μ 1 ( k ) k k
-2 0.032128481856 0.000233764788 1.414213 0 0
-1 -0.075539271956 -0.000549618934 0 0 1
0 -0.096935064502 -0.013550370057 0 0 2
1 0.491549094027 0.033777338659 0 3.031570 3
2 0.805141083557 0.304413564385 0 24.674674 4
3 0.304413564385 -0.805141083557 -14.709025 138.980052 5
4 -0.033777338659 0.491549094027 64.986095 710.373341 6
5 -0.013550370057 0.096935064502      
6 0.000549618934 -0.075539271956      
7 0.000233764788 0.032128481856      

We have designed these “new" coiflet systems (e.g., N=10,16,22,28N=10,16,22,28) by using the Matlab optimization toolbox constrained optimization function. Wells and Tian (Reference) used Newton's method to design lengths N=6ℓ+2N=6ℓ+2 and N=6ℓN=6ℓ coiflets up to length 30 [1]. Selesnick [11] has used a filter design approach. Still another approach is given by Wei and Bovik (Reference).

Table 6 also shows the result of designing a length-4 system, using the one degree of freedom to ask for one zero scaling function moment rather than one zero wavelet moment as we did for the Daubechies system. For length-4, we do not get any “extra" zero moments because there are not enough zero wavelet moments. Here we see a direct trade-off between zero scaling function moments and wavelet moments. Adding these new lengths to our traditional coiflets gives Table 7.

Table 7: Moments for Various Length-NN and Degree-LL Coiflets, where (*) is the number of zero wavelet moments, excluding the m1(0)=0m1(0)=0
N N L L m = 0 m = 0 m 1 = 0 m 1 = 0 m = 0 m = 0 m 1 = 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 Table 7 contain the number of zero wavelet moments, excluding the m1(0)=0m1(0)=0 which is zero because of orthogonality in all of these systems. The extra zero scaling function moments that occur after a nonzero moment for N=6ℓ+2N=6ℓ+2 are also excluded from the count. This table shows coiflets for all even lengths. It shows the extra zero scaling function moments that are sometime achieved and how the total number of zero moments monotonically increases and how the “smoothness" as measured by the Hölder exponent (Reference), (Reference), (Reference) increases with NN and LL.

When both scaling function and wavelet moments are set to zero, a larger number can be obtained than is expected from considering the degrees of freedom available. As stated earlier, of the NN degrees of freedom available from the NN coefficients, h(n)h(n), one is used to insure existence of φ(t)φ(t) through the linear constraint Equation 1, and N/2N/2 are used to insure orthonormality through the quadratic constraints Equation 1. This leaves N/2-1N/2-1 degrees of freedom to achieve other characteristics. Daubechies used these to set the first N/2-1N/2-1 wavelet moments to zero. If setting scaling function moments were independent of setting wavelet moments zero, one would think that the coiflet system would allow (N/2-1)/2(N/2-1)/2 wavelet moments to be set zero and the same number of scaling function moments. For the coiflets described in Table 7, one always obtains more than this. The structure of this problem allows more zero moments to be both set and achieved than the simple degrees of freedom would predict. In fact, the coiflets achieve approximately 2N/32N/3 total zero moments as compared with the number of degrees of freedom which is approximately N/2N/2, and which is achieved by the Daubechies wavelet system.

As noted earlier and illustrated in Table 8, these coiflets fall into three classes. Those with scaling filter lengths of N=6ℓ+2N=6ℓ+2 (due to Tian) have equal number of zero scaling function and wavelet moments, but always has “extra" zero scaling function moments located after the first nonzero one. Lengths N=6ℓN=6ℓ (due to Daubechies) always have one more zero scaling function moment than zero wavelet moment and lengths N=6ℓ-2N=6ℓ-2 (new) always have two more zero scaling function moments than zero wavelet moments. These “extra" zero moments are predicted by Equation 51 to (Reference), and there will be additional even-order zero moments for longer lengths. We have observed that within each of these classes, the Hölder exponent increases monotonically.

Table 8: Number of Zero Moments for The Three Classes of Coiflets (ℓ=1,2,⋯ℓ=1,2,⋯), *excluding μ1(0)=0μ1(0)=0, †excluding Non-Contiguous zeros
N N m=0m=0† m1=0m1=0* Total zero
Length achieved achieved moments
N = 6 ℓ + 2 N = 6 ℓ + 2 ( N - 2 ) / 3 ( N - 2 ) / 3 ( N - 2 ) / 3 ( N - 2 ) / 3 ( 2 / 3 ) ( N - 2 ) ( 2 / 3 ) ( N - 2 )
N = 6 ℓ N = 6 ℓ N / 3 N / 3 ( N - 3 ) / 3 ( N - 3 ) / 3 ( 2 / 3 ) ( N - 3 / 2 ) ( 2 / 3 ) ( N - 3 / 2 )
N = 6 ℓ - 2 N = 6 ℓ - 2 ( N + 2 ) / 3 ( N + 2 ) / 3 ( N - 4 ) / 3 ( N - 4 ) / 3 ( 2 / 3 ) ( N - 1 ) ( 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 that degree. The lengths N=6ℓ-2N=6ℓ-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 zero moments and/or the filter length, it is easier to see the complete picture.

Table 7 is just part of a large collection of zero moment wavelet system designs with a wide variety of trade-offs that would be tailored to a particular application. In addition to the variety illustrated here, many (perhaps all) of these sets of specified zero moments have multiple solutions. This is certainly true for length-6 as illustrated in Equation 42 through Equation 45 and for other lengths that we have found experimentally. The variety of solutions for each length can 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 a few wavelets moments may be sufficient with the remaining degrees of freedom used for some other optimization, either in the frequency domain or 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 to zero a few [6].

Examples of the Coiflet Systems are shown in (Reference).

Minimization of Moments Rather than Zero Moments

Odegard has considered the case of minimization of a larger number of moments rather than setting N/2-1N/2-1 equal to zero [8], [7], [6]. This results in some improvement in representing or approximating a larger class of signals at the expense of a better approximation of a smaller class. Indeed, Götze [4] has shown that even in the designed zero moments wavelet systems, the implementation of the system in finite precision arithmetic results in nonzero moments and, in some cases, non-orthogonal systems.

References

  1. Burrus, C. S. and Odegard, J. E. (1998, March). Coiflet Systems and Zero Moments. [Also CML Technical Report, Oct. 1996]. IEEE Transactions on Signal Processing, 46(3), 761–766.
  2. Gopinath, R. A. and Burrus, C. S. (1992, May). On the Moments of the Scaling Function. In Proceedings of the IEEE International Symposium on Circuits and Systems. (Vol. 2, p. 963–966). ISCAS-92, San Diego, CA
  3. Gopinath, Ramesh A. and Burrus, C. Sidney. (1995, February). On Cosine–Modulated Wavelet Orthonormal Bases. [Also a Tech. Report No. CML TR-91-27, March 1992.]. IEEE Transactions on Image Processing, 4(2), 162–176.
  4. Götze, J. and Odegard, J. E. and Rieder, P. and Burrus, C. S. (1996, May 12-14). Approximate Moments and Regularity of Efficiently Implemented Orthogonal Wavelet Transforms. In Proceedings of the IEEE International Symposium on Circuits and Systems. (Vol. 2, pp. II-405–408). IEEE ISCAS-96, Atlanta
  5. Lang, M. and Selesnick, I. and Odegard, J. E. and Burrus, C. S. (1994, October). Constrained FIR Filter Design for 2-Band Filter Banks and Orthonormal Wavelets. In Proceedings of the IEEE Digital Signal Processing Workshop. (p. 211–214). Yosemite
  6. Odegard, Jan E. and Burrus, C. Sidney. (1996, May 12-15). New Class of Wavelets for Signal Approximation. In Proceedings of the IEEE International Symposium on Circuits and Systems. (Vol. 2, pp. II-189–192). IEEE ISCAS-96, Atlanta
  7. Odegard, Jan E. and Burrus, C. Sidney. (1996, May 7–10). Toward a New Measure of Smoothness for the Design of Wavelet Basis. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing. (Vol. 3, pp. III-1467–1470). IEEE ICASSP-96, Atlanta
  8. Odegard, J. E. and Burrus, C. S. (1997, to be submitted). Wavelets with New Moment Approximation Properties. IEEE Transactions on Signal Processing.
  9. Steffen, P. and Heller, P. N. and Gopinath, R. A. and Burrus, C. S. (1993, December). Theory of Regular -Band Wavelet Bases. [Special Transaction issue on wavelets; Rice contribution also in Tech. Report No. CML TR-91-22, Nov. 1991.]. IEEE Transactions on Signal Processing, 41(12), 3497–3511.
  10. Selesnick, Ivan W. and Lang, Markus and Burrus, C. Sidney. (1997, to appear). Magnitude Squared Design of Recursive Filters with the Chebyshev Norm Using a Constrained Rational Remez Algorithm. IEEE Transactions on Signal Processing.
  11. Selesnick, Ivan W. and Odegard, Jan E. and Burrus, C. Sidney. (1996, September 2–4). Nearly Symmetric Orthogonal Wavelets with Non-Integer DC Group Delay. In Proceedings of the IEEE Digital Signal Processing Workshop. (p. 431–434). Loen, Norway

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