design of regular wavelets using a three-step lifting scheme

Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING

(Draft) September 23, 2009

1

Design of Regular Wavelets Using a Three-Step

Lifting Scheme

Ramin Eslami*, Member, IEEE, and Hayder Radha, Fellow, IEEE

EDICS: DSP-BANK Filter bank design and theory, DSP-WAVL Wavelet theory and applications

Abstract- We propose structural multidimensional multi-channel filter banks with desirable numbers of

vanishing moments for the analysis and synthesis banks. For a two-channel filter bank, we use a three-

step lifting scheme as opposed to the conventional two-step lifting method in order to provide more

symmetry between the analysis and synthesis filters. We show that the resulting filters have more

regularity, lower frame bounds ratio, and better frequency selectivity. We also extend our design to a

general multi-channel framework. Three steps of lifting scheme provides us a degree of freedom which

we can benefit from toward a more flexible design.

Index terms- Lifting scheme, multidimensional filter banks, vanishing moments, regular wavelet

transform, frame bounds.

I. INTRODUCTION

It is well known that iterated filter banks lead to a wavelet scheme under certain conditions.

Meanwhile, regularity of wavelets is a critical characteristic that at least ensures providing a d -

dimensional multiresolution scheme in 2 ( )dL � [18], [22], [28]. Therefore, designing filters for a wavelet

scheme where they fulfill perfect reconstruction (which is equivalent to biorthogonality) and regularity

has been an attractive, yet challenging aspect of wavelets analysis especially for the multidimensional

Manuscript received September 23, 2009.

R. Eslami was with the Department of Electrical and Computer Engineering, Michigan State University, East

Lansing, Michigan 48824. He is now with the Department of Biomedical Engineering, University of Rochester,

Rochester, NY 14627 (phone: 585-276-4378; fax: 585-276-2127; e-mail: [email protected]).

H. Radha is with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing,

Michigan 48824 (phone: 517-432-9958; fax: 517-353-1980; e-mail: [email protected]).

ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME


2

framework.

A necessary condition for regularity is having vanishing moments at aliasing frequencies of wavelet

highpass filters. However, this is not a sufficient condition for the regularity of multidimensional wavelets

[5], [15].

Owing to the lack of spectral factorization for multidimensional signals, designing filters for such

signals is not trivial. However, there are a few methods developed for this case. One of the approaches is

the cascade algorithm proposed in [15], [16]. Although perfect reconstruction is achieved through this

design method, imposing vanishing moments is not always possible. One to multidimensional approach

[25], an extension of the McClellan transformation, is another method which is widely used in

multidimensional filter design. This approach can also be performed in the polyphase domain where one

can take advantage of a separable filtering for nonseparable (e.g. quincunx) filter banks [17], [19].

Lifting scheme is another representation of wavelets where it provides a fast algorithm for computing

the wavelet transform [6], [23], [24]. Lifting can be interpreted as a representation of a filter bank in the

polyphase domain using ladder structure [1], [19]. Since perfect reconstruction is guaranteed for any kind

of selection of lifting steps, the filter design for this approach can be accomplished by only imposing the

desired characteristics such as vanishing moments into the filters.

Kovačević and Sweldens [14] proposed a structural design of multidimensional wavelets of any order

and any number of analysis and synthesis vanishing moments. For two-channel schemes they used a two-

step lifting technique. This approach could be considered as a generalization of the design method

proposed by Phoong et al. [19], where they proposed a ladder structure for one-dimensional (1-D) filer

banks and also they designed quincunx filter bank (QFB) using transformation.

Ansari et al. [1] proposed a two-channel filter bank using a triplet of halfband filters (equivalent to a

three-step lifting), where they could address the restrictions in double-halfband filter bank (equivalent to a

two-step lifting) [19]. While the magnitudes of the normalized analysis filter pair for the double-halfband

filter bank in [19] have to be 1 and 0.5 at / 2ω π= , they can be set to an equal value of 2 2 in the



3

triple-halfband design. Extensions of triple-halfband design have also been proposed [4], [13], [26].

In this work, we generalize Ansari’s method [1] to a multidimensional filter bank design with any

number of analysis and synthesis vanishing moments using a structural approach based on Kovačević’s

method [14]. In our design we only consider FIR filters. We demonstrate that the designed filters achieve

better regularity, lower frame bounds ratio, and better frequency selectivity when compared to the filters

designed in [14].

II. BACKGROUND

In this section we briefly outline some required background material in order to develop the proposed

filter design approach. Initially, we introduce notations that we use in this paper followed by a brief

background on two-channel filter banks. Then we explain about Neville filters and the two-step lifting

design proposed in [14].

A. Notations

A d -dimensional discrete signal [ ]x ∈n � with 1 2( , , , ) ddn n n= ∈n … Z has z -transform

( ) [ ]dX x−

∈=∑ n

nz n z

Z. Here 1 2( , , , ) d

dz z z= ∈z … � and we denote 1

id n

iiz

== ∏n

z . We also define Mz as

1 2( , , , )dM M MM z z z=z … , where [ ]1 2 dM M M M= � is a d d× integer matrix with iM as its i th

column. For two sequences [ ]x n and [ ]y n in 2 ( )d� Z , we denote their inner product by ,x y . For an

operator 2 2:A →� � , we use asterisk to denote the adjoint (or transpose conjugate) of the operator; hence

we have *, ,Ax y x A y= . We use asterisk to denote * 1( ) ( )X X −=z z , which is the conjugate of ( )X z on

the unit circle and is equivalent to the time reverse of the sequence [ ]x −n . We use x to represent the

greatest integer less than or equal to x .

We denote a multivariate polynomial in continuous time as ( ) dp a+∈

=∑ m

mmt t

Z where d∈t � ,

1 2( , , , )dm m m=m … , and { }0,1d dim i d+ = ∈ ≥ ≤ ≤mZ Z . Similarly, we denote a discrete-time polynomial as

[ ]p a=∑ m

mmn n . The degree of a monomial m

t or mn is denoted as

1

d

iim

==∑m and we use NP for the



4

set of polynomials with degree less than N .

We represent the d d× sampling matrix by M and define det( )MM � . We show downsampling and

upsampling operators with the sampling matrix M by M↓ and M↑ , respectively.

B. Two-Channel Filter Banks and Wavelets

We consider a multidimensional two-channel critically-sampled filter bank as depicted in Fig. 1. Here

M is the sampling matrix with size d d× , ( )H z and ( )H z� are lowpass analysis and synthesis filter pair

and likewise, ( )G z and ( )G z� are the highpass filter pair. We can express the filter bank in the polyphase

domain using the polyphase matrices [27], [28]

0 1

0 1

( ) ( )( )

( ) ( )p

H H

G G

=

z zA z

z z and 0 1

0 1

( ) ( )( )

( ) ( )p

H H

G G

=

z zS z

z z

� �

� �, (1)

where for perfect reconstruction we have ( ) ( )Tp p I=A z S z . Further, the analysis and synthesis lowpass

filters are expressed as 1

0( ) ( )i M

iiH H

==∑ c

z z z and 1

0( ) ( )i M

iiH H

−=

=∑ cz z z� � , and similarly the highpass

filters as 1

0( ) ( )i M

iiG G

==∑ c

z z z and 1

0( ) ( )i M

iiG G

−=

=∑ cz z z� � where 0 1{ , }c c are cosets of the sampling

matrix M assuming 0 =c 0 .

By iterating the filter bank, we obtain the analysis scaling and wavelet functions obeying [18]:

( ) [ ] ( )d h Mϕ ϕ∈

= − −∑nt n t n

Z, and ( ) [ ] ( )d g Mψ ϕ

∈= − −∑n

t n t nZ

. (2)

[Thus the wavelet analysis filters are [ ]h −n and [ ]g −n , equivalent to *( )H z and *( )G z .] And also we

have / 2, ( ) ( )j j

j Mϕ ϕ− −= −n

t t nM , where j ∈Z is the scale and we define det( )MM � . Similarly, we have

synthesis scaling and wavelet functions as

Fig. 1. A two-channel multidimensional filter bank.

M

x

M

MM

( )H z

H

( )H z�

rx

( )G z ( )G z�



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( ) [ ] ( )d h Mϕ ϕ∈

= −∑nt n t n��

Z, and ( ) [ ] ( )d g Mψ ϕ

∈= −∑n

t n t n� ��Z

. (3)

Vanishing moments in wavelets are critical regarding the regularity and also the approximation power

of wavelets [18], [22]. If the wavelet analysis and synthesis stages have N and N� vanishing moments,

respectively, we have

( ) 0dψ =∫m

t t t , for N<m , and ( ) 0dψ =∫m

t t t� , for N<m � .

These conditions are expressed for the highpass filters and polynomial ( )P z as

( ) *( ) ( ) 0M G P↓ =z z , for ( ) NP ∈z P , (4)

and

( ) ( ) ( ) 0M G P↓ =z z� , for ( )N

P ∈z P� . (5)

[Note that wavelet analysis highpass filter is *( )G z .] As a result, ( )G z has N zeros and ( )G z� has N�

zeros at =ω 0 , but ( )H z and ( )H z� have N� and N zeros at =ω π , respectively. Therefore, if N N> � , the

wavelet transform has more approximation power in the decomposition section and is more regular in

reconstruction. This setting is more desirable in compression since more regular filters are required to

compensate for the quantization in signal reconstruction [18].

C. Neville Filters

Neville filters is a notion introduced in honor of Neville who introduced 1-D polynomial interpolation

algorithm [21] and is referred to a class of interpolating filters, as described below.

Definition 1 [14]: A filter ( )L z is a Neville filter of order N and shift d∈τ � if the following

condition is satisfied:

[ ] [ ] [ ]l p p∗ = +n n n τ , for [ ] Np ∈n P . (6)

This condition is equivalent to having

[ ]( )l − =∑ m m

kk k τ , for N<m . (7)

Remark 1: Here we point out some of the important properties of the Neville filters we need later.



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1) The filter [ ]l −n or *( )L z will be a Neville filter of order N and shift −τ if [ ]l n is a Neville filter of

order N and shift τ . 2) The upsampled version of a Neville filter of order N with shift τ [i.e.

( ) ( )MQ L=z z ] is also a Neville filter of order N but with shift M τ . 3) If 1( )L z and 2 ( )L z are Neville

filters of order 1N and 2N , and shift 1τ and 2τ , then 1 2( ) ( )L Lz z is a Neville filter of order 1 2min( , )N N

and shift 1 2+τ τ . 4) The monomial filter mz ( d∈m � ) is a Neville filter of shift m and order infinity.

Hence, one only needs to build Neville filters with shifts within a hypercube. Neville filters with different

shifts can be obtained using this property.

Remark 2: To find a Neville filter of order N and shift τ , one needs to solve a system of equations

given by (7). For a d -dimensional filter, there are 1N d

qd

+ − =

equations and hence the filter length

will be q .

As a result, a 1-D Neville filter of order N has N nonzero coefficients. In this case, the system of

equations has a Vandermonde matrix which is always invertible. We choose the support of the filter in the

vicinity of τ to avoid extrapolation [14]. Consequently, we select the support of 1-D Neville filters as

2 , ( 1) 2N N− − , where x denotes the greatest integer less than or equal to x . Below we show

that Neville filters with different supports can be obtained from the filters with support

2 , ( 1) 2N N− − .

Proposition 1: If ( )L z is a 1-D Neville filter of order N and shift τ having support on

2 , ( 1) 2N k N k− + − + ( k ∈� ), then 1( ) ( )kL z z L z= where 1( )L z is a Neville filters of order N and

shift kτ − which has the support on 2 , ( 1) 2N N− − .

Proof: This proposition is proved using the property 4 of Remark 1. □

An important class of filters are Deslauriers-Dubuc [10] filters. These filters are Neville filters with

shift 1/ 2 [14]; thus, they can interpolate polynomials at half integers. The coefficients of these filters

when the order N is even are obtained through 2 1

( ) 1( ) ( )

N k kN kk

R z a z z− +

== +∑ , where



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2 1

1( 1) ( 2 0.5 )

( 2 )!( 2 1 )!( 0.5)

Nk N

ik

N ia

N k N k k

+ −=

− + −=

− − + −∏

.

Hence, for even N the filter ( ) ( )NR z is linear phase. Table I shows a few Deslauriers-Dubuc filters. In the

next section we show how we can benefit from Neville filters in order to design two-channel filter banks.

D. Kovačević’s Design for Two-Channel Filter Banks

A two-channel filter bank using a lifting framework provides perfect reconstruction for any kind of

lifting steps. Therefore, one can design filters by simply imposing vanishing moments on the filters and

derive the appropriate lifting steps. This has been done by Kovačević and Sweldens in [14] for filter

banks with two lifting steps (see Fig. 2). In this case if we express the filter bank in the polyphase domain

we have (note that the normalization factors 0K and 1K are not included)

1 ( ) ( ) ( )

( ) 1p

Q L Q

L

− = −

z z zA

z and

1 ( )

( ) 1 ( ) ( )p

L

Q Q L = − −

zS

z z z. (8)

The two-channel filter bank designed by Phoong et al. [19] is similar to the 1-D design of [14], where

they used a ladder network in the polyphase domain. This setting as pointed out by Ansari et al. [1] has a

drawback that the values of lowpass (and highpass) filter pair cannot be made the same at / 2ω π= (one is

1 and the other is 0.5 in the normalized 1-D filters of [19]). We will show that the same problem exists

TABLE I

DESLAURIERS-DUBUC FILTERS ( ) ( )NR z

N ( ) ( )NR z

1 1

2 (1 ) 2z+

3 1 3(3 6 ) 2z z−+ −

4 2 1 4( 9 9 ) 2z z z−− + + −

5 2 1 2 7(5 60 90 20 3 ) 2z z z z− −+ + − +

6 3 2 1 2 8(3 25 150 150 25 3 ) 2z z z z z− −− + + − +

Fig. 2. A two-channel multidimensional filter bank using two-step lifting scheme.

M

x

MM

( )Q z r

x

1cz

( )L z ( )Q z ( )L z

1−cz

M 0K

1K

1

0K

−

1

1K

−



8

for the general multidimensional design of two-channel filter banks proposed in [14]. Although in this

paper we focus to address this drawback (dissimilarity between analysis and synthesis filters) in our

design, we will show in Section IV that the designed filters not only are more symmetric, but also are

more regular, have lower frame bounds ratio, and provide better frequency selectivity.

Kovačević and Sweldens showed that by imposing N and N� vanishing moments to the wavelet

analysis and synthesis banks shown in Fig. 1 when using the two-step lifting step depicted in Fig. 2

results in the following solution:

Remark 3 (two-channel two-step lifting design) [14]: In order to have N and N� vanishing moments

in the two-channel wavelet analysis and synthesis banks, ( )L z should be a Neville filter of order N with

shift 10 1M

−=τ c , and *2 ( )Q z should be a Neville filter of order N� with shift 0τ , assuming that N N≥ � .

Remark 4: The lengths of the 1-D filters designed using two-step lifting scheme for N N≥ �

(explained in Remark 3) are

2, 1

length( ) length( )2 1, 1

NG H

N N

== = − >� and

2, 1length( ) length( )

2( ) 3, 1

NH G

N N N

== = + − >

��

.

Proof is straightforward noting that the length of 1-D Neville filters of order N is N . □

The constants 0K and 1K in Fig. 2 are for the appropriate normalization of the filters. Two examples

of the above framework are as follows. We will use these examples in order to compare to our design

Fig. 3. The frequency response of the analysis filters in Example 1.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

X: 0.5Y: 0.7906

Normalized Frequency

Magnitu

de

Analysis Filters

X: 0.5Y: 1.414

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

X: 0.5Y: 0.7906


Magnitu

de

Synthesis Filters

X: 0.5Y: 1.414



9

approach.

Example 1: Consider a 1-D case where 2M = and thus 1/ 2τ = where we can use time-reversed

version of Deslauriers-Dubuc filters. Suppose that 3N = and 2N =� . Therefore, we can obtain the

analysis filters ( )2 2 20( ) 1 ( ) ( ) ( )H z K Q z L z zQ z= − + and ( )2

1( ) ( )G z K L z z= − + , using (3)( ) ( )L z R z= and

1(2)( ) (1/ 2) ( )Q z R z−= from Table I with 0 2K = and 1 2 2K = as

( ) 2 2 1 2( ) 2 32 ( 1) ( 2 2 14 3 )H z z z z z z− −= + − − + − and ( ) 1 3( ) 2 16 (1 ) ( 3)G z z z z−= − + .

We see that at / 2ω π= or z j= the magnitude of H and G are 2 and 10 4 . Fig. 3 shows the

frequency responses of the filters.

Example 2: In this example we consider the quincunx filter bank (QFB) design example in [14] with

4N = and 2N =� . Here, we can find that the filters evaluated at ( , )j j=z give ( , ) 2H j j = and

( , ) 2 2G j j = (see the frequency responses in Fig. 7).

III. THREE-STEP LIFTING DESIGN OF TWO-CHANNEL FILTER BANKS

In this section we address the drawback of the Kovačević’s Design by adding an additional lifting

step to the framework of Fig. 2. The proposed scheme is depicted in Fig. 4. Ignoring the normalization

factors, we can write the analysis and synthesis polyphase matrices in terms of the lifting steps as

1 ( ) 1 0 1 ( )

0 1 ( ) 1 0 1

1 ( ) ( ) ( ) ( ) ( ) ( ) ( ),

( ) 1 ( ) ( )

p

U L

Q

Q U L U L Q U

Q L Q

= −

− + − = − −

z zA

z

z z z z z z z

z z z

(9)

and

1 1 ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) 1 ( ) ( )T

p p

L Q Q

L U L Q U Q U− − = = − − + −

z z zS A

z z z z z z z. (10)

Fig. 4. A two-channel multidimensional filter bank using a three-step lifting scheme.

M

x

MM

( )Q z r

x

1cz

( )L z ( )Q z ( )L z

1−cz

M 0K

1K

1

0K

−

1

1K

−

( )U z ( )U z



10

Now we apply N vanishing moments to the wavelet analysis branch. Using (1), and (9) and we have

( )1( ) ( ) 1 ( ) ( )M M MG Q L Q= − + −cz z z z z .

Thus, by employing (4) we obtain

1 1[ ] [ ] [ ] [ ] [ ] [ ] 0q p M p M q l p M− − ∗ + − − − ∗ − ∗ − =n n n c n n n c ( Np ∈P ). (11)

Suppose that *( )Lk L z , where Lk is a constant, is a Neville filter of order N and shift 10 1M −=τ c , then

we have 1[ ] [ ] [ ]Lk l p M p M− ∗ − =n n c n . Hence, we can rewrite (11) as

1(1 1 ) [ ] [ ] [ ]Lk q p M p M+ − ∗ = −n n n c ( Np ∈P ),

which implies that *( )Qk Q z with

1 1Q Lk k= + , (12)

is a Neville filter of order N and shift 0−τ , or ( )Qk Q z is a Neville filter of order N and shift 0τ .

Now we impose N� vanishing moments on the analysis bank. We can obtain analysis highpass filter

by using (1) and (10) as

( )1( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( )M M M M M M MG L Q U L U Q U−= − − + −c

z z z z z z z z z� .

Applying (5) to the derived ( )G z� we have

1 1

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] 0 ( ).N

u q l p M l p M

u p M p M u q p M p

∗ ∗ ∗ − ∗ −

∗ + − − ∗ ∗ − = ∈

n n n n n n

n n n c n n n c P� (13)

If we assume N N≥ � , since *( )Lk L z and ( )Qk Q z are Neville filters with shift 0τ , we can reduce (13) to

1

1

1 1[ ] [ ] [ ]

1[ ] [ ] [ ] [ ] [ ] 0 ( ),

L Q L

NQ

u p M p Mk k k

u p M p M u p M pk

∗ − − −

∗ + − − ∗ = ∈

n n n c

n n n c n n P�

or

1[ ] [ ] [ ] ( ),U Nk u p M p M p∗ = − ∈n n n c P�

where by using (12) we have



11

2

2

2

1

LU

L

kk

k=

−. (14)

As a result, *( )Uk U z is a Neville filter of order N� and shift 10 1M −=τ c . We summarize the proposed

design in the following theorem.

Theorem 1: A two-channel multidimensional wavelet having three steps of lifting (see Fig. 4) will

have N and N� vanishing moments in the analysis and synthesis banks, respectively if the following

conditions are satisfied:

1) *( )Lk L z and ( )Qk Q z are Neville filters of order N and shift 10 1M −=τ c . Lk is a free parameter and

Qk is given by (12).

2) *( )Uk U z is a Neville filter of order N� and shift 10 1M −=τ c with Uk given in (14). □

As seen, in the proposed design we have a free parameter where we can adjust the filters to have more

symmetry and as a matter of fact, we expect filters with more regularity and other desirable properties

given in Section IV. For the above setting we assumed N N≥ � , which is more useful in compression as

described before. If we need to set N N≥� vanishing moments, we can reverse the directions and also

signs of the lifting steps ( )L z , ( )Q z , and ( )U z in the filter bank shown in Fig. 4.

In the next section we present a few design examples using the proposed scheme.

A. 1-D Filter Banks

In the 1-D case, since 2M = we have 0 1/ 2τ = , and hence, we can use the Deslauriers-Dubuc filters.

Thus, for a filter bank with N vanishing moments in the analysis bank and N� vanishing moments in the

synthesis stage assuming N N≥ � , we have *( )( ) ( )L Nk L z R z= or 1

( )( ) ( )L Nk L z R z−= , and ( )( ) ( )Q Nk Q z R z= ,

and finally 1

( )( ) ( )U N

k U z R z−= � . Using (9), (12), and (14), we derive the highpass analysis filter as

( ) ( )( )2 2 21 ( ) ( ) ( )( ) 1 ( ) 1 1 ( ) ( ) ( )Q N L Q N NG z K k R z z k k R z R z

− = − + −

,

equivalent to



12

( )2 2 21( ) ( ) ( )( ) ( ) 1 ( ) ( )

1L N L N N

L

KG z k R z z k R z R z

k

− = − + + − +, (15)

and the lowpass analysis filter is derived as

2 20 1( ) 1 ( ) ( ) ( )H z K U z G z K zL z = + + ,

or

2 2 2 201 ( )( )2

( ) 2 ( 1) ( ) ( ) 2 ( )2

L L L NN

L

KH z k k R z G z K z k R z

k

− − = + − + � . (16)

Note that we included positive constants 0K and 1K for proper normalization (see Fig. 4). From

Table I, since ( ) (1) 1NR = , we have

0(1) ( 1) / 2L LH K k k= + = , and 1( 1) 2 /( 1) 2L LG K k k− = + = . (17)

Now we can find the parameters by evaluating

( ) ( )H j G j= . (18)

In the case when both vanishing moments are of even order, we have ( ) ( )( 1) ( 1) 0N N

R R− = − =� , hence, (18)

yields 0 1K K= and from (17) we have 0 1 1K K= = , and 2 1Lk = + where this solution gives

( ) ( ) 1H j G j= = . If N or N� , or both are odd then we should solve (17) and (18) to find the parameters.

We can also obtain the synthesis filters using the following equations

1( ) ( )H z z G z−= − −� , and 1( ) ( )G z z H z−= −� .

Remark 5: The lengths of the 1-D filters designed using three-step lifting scheme for N N≥ � are

2, 1

length( ) length( )4 3, 1

NG H

N N

== = − >� and

2, 1length( ) length( )

4 2 5, 1

NH G

N N N

== = + − >

��

.

Here we point out that the design method of Ansari et al. [1] for regular filters is similar to the

proposed 1-D three-step lifting design when N N= � .

Below we provide examples for 1-D filter bank design using the proposed method.

Example 3: Consider the design criteria in Example 1, that is, we desire 3N = and 2N =� vanishing



13

moments. By solving (17) and (18) we obtain the parameters as 2.2686Lk = , 0 0.9816K = , and

1 1.0188K = where we have ( ) ( ) 1.005H j G j= = . Fig. 5 shows the frequency responses of the filters

designed in this example. The symmetry of these filter pairs is clear when compared with the design using

two lifting steps and same number of vanishing moments (see Example 1 and Fig. 3).

Note that to avoid finding a solution for the parameters, we can set 0 1 1K K= = and 2.414Lk = where

they satisfy (17), but we obtain ( ) 1.021H j = and ( ) 0.9919G j = , which are very close together.

Example 4: In this example we design the smallest filter of this kind where 2N = and 1N =� . (Note

that the case 1N N= =� leads to the degenerate Haar filters.) Again we find the parameters by solving (17)

and (18) resulting in 2.1479Lk = , 0 0.9650K = , and 1 1.0363K = . The analysis filters are calculated as

2 1( ) 0.03( 1)( 3.2958 21.372 6.4846 )H z z z z z−= − + + − − , and 2 1( ) 0.0823( 1) ( 6.2958 )G z z z z−= − − + + .

In Fig. 6 we demonstrate the frequency responses of the designed filters.

We see that the design of 1-D filters using the proposed method is easier when we choose both the

order of vanishing moments even. In addition, having linear-phase lifting steps is another advantage of

this setting.

B. Quincunx Filter Banks

Quincunx filter banks (QFB) are 2-D two-channel filter banks where they can provide multiresolution

Fig. 5. The frequency responses of the filters in Example 3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5


Magnitude

Synthesis Filters

X: 0.5Y: 1.005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

X: 0.5Y: 1.005

Normalized Frequecy

Magnitu

de

Analysis Filters



14

analysis under certain regularity conditions. Suppose that we choose the sampling matrix as

1 1

1 1M

= − ,

where we have the nonzero coset vector as 1 (1, 0)T=c and hence 10 1 (1/ 2, 1/ 2)TM −= =τ c . As a result, we

should use Neville filters with shift 0τ . Table II shows the Neville filters with shift 0 (1/ 2, 1/ 2)T=τ using

the de Boor and Ron algorithm [7], [8] (also reported in [14]).

Similar to the 1-D case we choose * 1 1( ) ( ) 1 2( ) ( ) ( , )L N Nk L R R z z− −= =z z , ( )( ) ( )Q Nk Q R=z z , and

1

( )( ) ( )U N

k U R−=z z� . Therefore, the analysis filters are derived as

( )1( ) 1 ( ) ( )( ) ( ) 1 ( ) ( )

1

M M ML N L N N

L

KG k R z k R R

k

− = − + + − +

z z z z , (19)

and

2 201 1 ( )( )2

( ) 2 ( 1) ( ) ( ) / 2 ( )2

M ML L L NN

L

KH k k R G K z k R

k

− − = + − + z z z z� . (20)

We can find the parameters through (1,1) ( 1, 1) 2H G= − − = and ( , ) ( , )H j j G j j= . Note that due to the

symmetry of the filters, H and G on four points ( , )j j± ± and on the points connecting these four vertices

( , )j j± ± (which form a diamond) have same values.

In the next example we use these filters to design the quincunx filters.

Fig. 6. The frequency responses of the filters in Example 4.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

X: 0.5Y: 1.036


Magnitu

de

Analysis Filters

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5


Magnitu

de

Synthesis Filters

X: 0.5Y: 1.036



15

Example 5: Suppose that we desire a QFB with 4N = and 2N =� vanishing moments. Therefore, we

have the filters with order 2N = and 4N = as

2(2) 1 2 1 2 1 2( , ) (1 ) 2R z z z z z z= + + + ,

and

2 2 2 2 1 1 1 1 5(4) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2( , ) 10(1 ) ( ) 2R z z z z z z z z z z z z z z z z z z− − − − = + + + − + + + + + + + .

To find the proper parameters we first evaluate (1,1) ( 1, 1) 2H G= − − = , which leads to

0 1( 1) 2 ( 1) 2L L L LK k k K k k+ = + = ,

and the next requirement ( , ) ( , )H j j G j j= yields 0 1K K= [note that 11 2 1 2( , )M z z z z−=z ]. Therefore, we

have similar result to the one in the 1-D case when we have even orders of vanishing moments, that is,

0 1 1K K= = , and 2 1Lk = + . In Fig. 7 we depicted the frequency responses of the designed analysis filters

as well as the designed quincunx filters using Kovačević’s method given in Example 2.

IV. PROPERTIES OF THE DESIGNED FILTERS

In this section we investigate and compute some of the properties of the designed filters and compare

them to those of the two-step lifting design. We assume N N≤� and use 1 2Lk = + unless otherwise is

mentioned.

TABLE II

2-D NEVILLE FILTERS WITH SHIFT (1/ 2,1/ 2)T , ( ) ( )NR z

N ( ) ( )NR z

2 2

(1)( ) 2r z

4 5

(1) (2)10 ( ) ( ) 2r r − z z

6 9

(1) (2) (3) (4)174 ( ) 27 ( ) 2 ( ) 3 ( ) 2r r r r − + + z z z z

n ( ) ( )nr z

1 1 2 1 21 z z z z+ + +

2 2 2 2 2 1 1 1 11 2 1 2 1 2 1 2 1 2 1 2z z z z z z z z z z z z

− − − −+ + + + + + +

3 2 2 1 2 1 1 2 11 2 1 2 1 2 1 2z z z z z z z z

− − − −+ + +

4 3 3 3 3 2 2 2 21 2 1 2 1 2 1 2 1 2 1 2z z z z z z z z z z z z

− − − −+ + + + + + +



16

A. Symmetry

Symmetry between analysis and synthesis filters is a desirable property. It could be used as a measure

of near-orthogonality, although such measures fail at some cases [9]. In the last section we showed that

the proposed filters have same magnitude at z j= for 1-D case (and ( , )j j=z in the case of quincunx

filters). Here we use 2|| ||h h− � to measure the dissimilarity of the designed filters. Table III shows this

measure for normalized filters of both two-step and three-step lifting designs. As seen, the symmetry of

the proposed filters is higher (dissimilarity is lower) in all cases. Moreover, the symmetry is very high

when both N and N� are even.

We also found the optimal Lk for the proposed filters that minimizes 2|| ||h h− � . Since the calculations

are very cumbersome, we used symbolic toolbox of Matlab to find optimal Lk . The resulting values are

shown in Table IV. Except for a few cases, the symmetry measure is close to those of Table III for the

Fig. 7. The frequency responses of the quincunx analysis filters. Top: Two-step lifting with 4N = and 2N =� in

Example 2. Bottom: The proposed three-step lifting with 4N = and 2N =� in Example 5.

-1

0

1

-1

0

10

0.5

1

1.5X: 0.5Y: -0.5Z: 1


Magnitude

-1

0

1

-1

0

10

0.5

1

1.5

X: 0.5Y: 0.5Z: 1


Magnitu

de

-1

0

1

-1

0

10

0.5

1

1.5

2X: 0.5Y: -0.5Z: 1.414


Magnitu

de

-1

0

1

-1

0

10

0.5

1

1.5X: 0.5Y: 0.5Z: 0.7071


Magnitu

de



17

proposed filters. Interestingly, when N N= � the optimal Lk is 1 2+ which is used as the default value.

B. Regularity

As we observed previously, the proposed filter design yields more symmetric wavelet filters and

hence, we expect they have more regularity. To determine regularity, we measure Sobolev index as

proposed in [11] and [29].

Table V shows the measured Sobolev regularity for both two-step and three-step lifting design

methods for 1-D analysis lowpass filters. Negative values indicate that the wavelet filter is not stable. For

three-step lifting design we always used 0 1 1K K= = and 2 1Lk = + . The results for the three-step lifting

design, shown in boldface, outperform those of the designed filters using two-step lifting for all cases. To

compare the filters with same lengths, consider two-step lifting design with 4N N= =� (to have a fair

comparison we consider same number of vanishing moments of analysis and synthesis filters) which has a

filter length of 13 and the three-step design with 3N N= =� having same filter length (see Remarks 4 and

5). We observe that the proposed design has better regularity (1.61 vs. 1.18 ). Another example showing

better regularity of the three-step design is when 6N N= =� leading to length 21= for the two-step design

as compared to the proposed design with 4N N= =� having a smaller length of 19 (1.77 vs. 1.94 ).

We also compared the regularities for a few designed quincunx filters shown in Table VI using the

TABLE III

DISSIMILARITY MEASURE (3 2

10 || ||h h− � ) FOR BOTH

1-D TWO-STEP AND THREE-STEP

(SHOWN IN BOLDFACE) LIFTING DESIGNS

NN

� 1 2 3 4 5 6

500 1

500

250 93.75 2

42.89 0.46

296.88 105.47 116.58 3

161.63 51.20 86.47

257.81 83.50 89.05 65.05 4

43.00 0.60 6.97 0.34

284.36 91.51 97.04 69.86 74.68 5

110.74 24.18 48.75 22.35 38.96

267.06 82.13 85.26 59.81 63.22 52.63 6

44.27 1.84 6.74 0.11 3.22 0.28

TABLE IV

OPTIMAL Lk FOR MAXIMUM SYMMETRY AND THE DISSIMILARITY

MEASURE (IN BOLDFACE) FOR 1-D THREE-STEP LIFTING DESIGN

NN

� 1 2 3 4 5 6

- 1

500

1.7267 2.4142 2

30.207 0.46

1.0011 2.7357 2.4142 3

116.58 50.61 86.47

1.5914 2.2838 2.1745 2.4142 4

24.697 0.44 6.41 0.34

1.1587 2.3348 2.0365 2.5795 2.4142 5

71.992 24.13 47.93 22.21 38.96

1.5175 2.179 2.0912 2.3489 2.2732 2.4142 6

22.12 1.34 5.77 0.08 3.08 0.28



18

method proposed in [20]. At some instances the regularity algorithm fails to find a solution mask in 2L

which we indicated by “nr” in the table. Again we observe that the proposed approach results in more

regular filters for quincunx filters. A major application of quincunx filters is in the construction of

directional filter banks (DFB) [2]. In [12] we showed that the HWD (Hybrid Wavelets and Directional

filter banks) transform utilizing DFBs constructed with three-step lifting quincunx filters (in comparison

to HWD with DFBs having two-step lifting) demonstrates less ringing artifacts and greater PSNR values

in nonlinear approximation.

C. Frame Bounds

Orthogonality of a basis is desirable in applications such as compression. Energy is preserved in such

transforms and hence, the design of quantizers is not difficult. Relaxing the orthogonality criterion,

however, makes the design of filter banks more flexible. To measure the degree of non-orthogonality, we

use frame bounds which give energy preserving property for bases and dictionaries [9]. For the two-

channel filter bank shown in Fig. 1, the set of { }[ ] [ (2 )], [ ] [ (2 )]m mh n h m n g n g m n= − − = − − form a frame in

2 ( )� � if we have [3], [18]

2 22 2

, ,m mmA x x h x g B x

∞

=−∞≤ + ≤∑

for 0B A≥ > . (This definition is easily extended to multidimensional multi-channel filter banks.)

There are a few approaches to find the frame bounds [3], [9], where we use the one proposed in [3].

TABLE V

REGULARITY OF 1-D LOWPASS ANALYSIS FILTERS FOR BOTH

TWO-STEP AND THREE-STEP (SHOWN IN BOLDFACE) LIFTING DESIGNS

NN

� 1 2 3 4 5 6

0.5 1

0.5

-2.38 0.44 2

-0.92 1.07

-6.49 -1.15 1.20 3

-5.11 0.46 1.61

-10.79 -5.48 -0.98 1.18 4

-9.31 -3.83 0.66 1.94

-15.12 -9.72 -5.29 -0.40 1.77 5

-13.45 -7.93 -3.38 1.50 2.40

-19.44 -14.02 -9.58 -4.68 -0.14 1.77 6

-17.53 -11.95 -7.37 -2.41 2.10 2.63

TABLE VI

REGULARITY OF QUINCUNX FILTERS FOR BOTH

TWO-STEP AND THREE-STEP LIFTING DESIGNS

NN

� 2 4

nr 2

2.150

nr 1.059 4

2.038 2.324



19

Defining the frame operator in the polyphase domain as

* *( ) (1/ ) ( )p pz z z=F A A

where *(.) denotes the conjugate transpose and ( )p zA is the analysis polyphase matrix for the normalized

filters, we can find the frame bounds A and B as the infimum and supremum of the eigen values of F on

the unit circle 2j fz e π= [3]. Hence, we first form 2 * 2 2( ) ( ) ( )j f j f j fp pe e eπ π π=F A A (using symbolic toolbox

of Matlab) and then evaluate 2( )j fe πF for several frequency points uniformly distributed in 0 1f≤ ≤ . At

each frequency we find eigen values of 2( )j fe πF . Then A and B are found as the minimum and

maximum of all the eigen values computed in the last step, respectively.

In Table VII we show the frame bounds as well as their ratio B A for the proposed and two-step filter

bank designs. As seen, B A for the proposed design are significantly less than those for the two-step

lifting filters (even when filters with same length are considered) and are close to 1 at most cases which

indicates that the designed filters are near-orthogonal.

D. Energy of the Error

The last property we investigate for the designed filters is the frequency selectivity. In particular, we

measure the energy of the error between the designed normalized (lowpass) filter and the ideal one. We

define the total energy of the error as

TABLE VII

FRAME BOUNDS AND THEIR RATIO FOR 1-D LOWPASS ANALYSIS FILTERS FOR BOTH


NN

�

1

A B B/A

2

A B B/A

3

A B B/A

4

A B B/A

5

A B B/A

6

A B B/A

1 1 1 1

1 1 1

0.38 2.62 6.85 0.5 2 4 2

0.66 1.51 2.28 0.95 1.05 1.10

0.57 1.77 3.12 0.46 2.16 4.68 0.59 1.69 2.85 3

0.78 1.28 1.64 0.85 1.18 1.39 0.95 1.05 1.10

0.38 2.62 6.85 0.5 2 4 0.46 2.16 4.68 0.5 2 4 4

0.66 1.51 2.28 0.95 1.06 1.12 0.81 1.23 1.51 0.95 1.05 1.10

0.48 2.07 4.29 0.48 2.09 4.38 0.56 1.77 3.15 0.48 2.09 4.38 0.55 1.82 3.31 5

0.74 1.35 1.83 0.89 1.13 1.27 0.91 1.09 1.20 0.89 1.12 1.26 0.95 1.05 1.10

0.38 2.62 6.85 0.5 2 4 0.46 2.16 4.68 0.5 2 4 0.48 2.09 4.38 0.5 2 4 6

0.66 1.51 2.28 0.91 1.11 1.22 0.81 1.23 1.51 0.98 1.03 1.05 0.86 1.17 1.36 0.95 1.05 1.10



20

2 2/ 2

0 / 21 ( ) ( )

j jE H e d H e d

π πω ω

πω ω= − +∫ ∫ .

From Tables VIII and IX we observe that the proposed designed filters for both 1-D and quincunx

settings provide better frequency selectivity when compared to the two-step lifting design. This is valid

even if we consider filters with same length.

V. MULTI-CHANNEL FILTER BANKS

In this section we extend our analysis to a general perfect reconstruction multi-channel filter bank.

Fig. 8 illustrates a critically-sampled M -channel filter bank where ( )H z and ( )H z� are the lowpass filter

pair and ( )iG z (1 1i≤ ≤ −M ) along with ( )iG z� represent highpass filters in the analysis and synthesis

banks, respectively and det( )M=M . Again, we can express the filters in the polyphase domain as

0 1

1,0 1, 1

1,0 1, 1

( ) ( )

( ) ( )( )

( ) ( )

p

H H

G G

G G

−

−

− − −

=

z z

z zA z

z z

M

M

M M M

�

�

�

and

0 1

1,0 1, 1

1,0 1, 1

( ) ( )

( ) ( )( )

( ) ( )

p

H H

G G

G G

−

−

− − −

=

z z

z zS z

z z

M

M

M M M

� ��

� ��

� ��

, (21)

The analysis and synthesis lowpass filters are expressed as 1

0( ) ( )j M

jjH H

−

==∑

cz z z

M and

1

0( ) ( )j M

jjH H

− −

==∑

cz z z

M� � , whereas 1

0( ) ( )j M

i ijjG G

−

==∑

cz z z

M and

1

0( ) ( )j M

i ijjG G

− −

==∑

cz z z

M� � (1 1i≤ ≤ −M )

show how we can derive the highpass filters from the polyphase components. Here 0 1{ }j j≤ ≤ −cM

are cosets

TABLE VIII

ERROR OF 1-D LOWPASS ANALYSIS FILTERS FOR BOTH


NN

� 1 2 3 4 5 6

0.100 1

0.100

0.284 0.150 2

0.097 0.066

0.208 0.138 0.087 3

0.090 0.057 0.051

0.294 0.136 0.124 0.104 4

0.089 0.055 0.050 0.045

0.246 0.131 0.095 0.099 0.076 5

0.086 0.050 0.046 0.042 0.039

0.301 0.130 0.120 0.098 0.093 0.084 6

0.085 0.049 0.046 0.041 0.038 0.036

TABLE IX

ERROR OF QUINCUNX ANALYSIS FILTERS FOR BOTH

TWO-STEP AND THREE-STEP LIFTING DESIGNS

NN

� 2 4

0.2372 2

0.18034

0.22723 0.23164 4

0.18632 0.19332



21

of the sampling matrix M assuming 0 =c 0 .

Similar to what we observed for designing a two-channel filter bank in the last section, we can design

multi-channel filter banks by imposing the desired number of vanishing moments to the highpass

channels. Applying N and N� vanishing moments to the filter bank depicted in Fig. 8 is equivalent to

conditioning the highpass filters as

( ) *( ) ( ) 0iM G P↓ =z z (1 1i≤ ≤ −M ), for ( ) NP ∈z P , (22)

and

( ) ( ) ( ) 0iM G P↓ =z z� (1 1i≤ ≤ −M ), for ( )N

P ∈z P� . (23)

Next we briefly review the multi-channel two-step lifting scheme design by Kovačević and Sweldens

[14] and propose our three-step lifting design subsequently.

A. Multi-Channel Two-Step Lifting Design by Kovačević

Fig. 9 shows a multi-channel filter bank using two-step lifting scheme which is used as a framework

to build filter banks with any number of vanishing moments in [14]. Ignoring the normalization factors,

we can form the analysis polyphase matrix as

1

1 1 1 11

1 1

1 1

1 0 01 1

1 00 1 0 1 0( )

0 10 0 1 0 1

i ii

p

Q Q Q L Q Q

L L

L L

−− −=

− −

− − − = = − −

∑A z

M

M M

MM

��

��

, (24)

and the synthesis matrix is found to be

Fig. 8. A single-level multi-channel filter bank.

M

x

M

1( )G z

H

M M

M M 1( )G − zM

H

( )H z

H

1( )G − zM�

1( )G z�

( )H z�

rx



22

1 2 1

1 1 1 1 2 1 1

2 2 1 2 2 2 1

1 1 1 1 1

1

1

( ) 1

1

p

L L L

Q Q L Q L Q L

Q Q L Q L Q L

Q Q L Q L

−

−

−

− − − −

− − − − = − − − − − − −

S z

M

M

M

M M M M

�

�

�

. (25)

Remark 6 (multi-channel two-step lifting design) [14]: The authors in [14] proved that if ( )iL z

(1 1i≤ ≤ −M ) are Neville filters with shift 1i iM −=τ c and order N , and *( )iQ zM (1 1i≤ ≤ −M ) are

Neville filters with shift 1i iM −=τ c and order N� , then the filter bank will have N and N� vanishing

moments in the analysis and synthesis banks.

Using a similar strategy as in the case of two-cannel filter banks, we extend the above design to have

a more flexibility as explained next.

B. Multi-Channel Three-Step Lifting Design

Having three steps of lifting in a multi-channel filter bank as shown in Fig. 10 provides more degrees

of freedom in developing the filters.

In this setting we can find the analysis polyphase matrix as

1 1 1 1

1

1

1 1 1

1 1 1 1 1 11 1 1

1 1 1 1 2 1 1

2 2 1 2 2 2

1 0 01 1

1 00 1 0 0 1 0( )

0 10 0 1 0 0 1

1

1

1

p

i i i i i ii i i

U U L L

Q

Q

U Q L U L U Q L U L U Q

Q Q L Q L Q L

Q Q L Q L Q L

− −

−

− − −− − −= = =

−

−

−

= −

− + − + −− − − −

= − − − −

∑ ∑ ∑

A z

M M

M

M M M

M M M

M

M

��

��

�

�

� 1

1 1 1 1 2 1 11Q Q L Q L Q L− − − − −

− − − − M M M M M

�

, (26)

and similarly the synthesis matrix is obtained as

Fig. 9. A multi-channel multidimensional filter bank using a two-step lifting scheme.

M

x MM

1( )Q z

rx1c

z

1( )L z

1−cz

M 0K

1K

1

0K

−

1

1K

−

MM

1( )Q − z

M

1−cz M

1( )L − z

M

1−−c

z M 1K −M

1

1K

−−M

1( )Q z

1( )L z

1( )L − z

M

1( )Q − z

M



23

1

1 2 111

1 1 1 1 1 1 2 1 1 11

1

1 1 1 1 1 2 1 1 11

1

1( ) ( ( ))

1

i ii

T i iip p

i ii

L Q Q Q Q

L U U L Q Q U Q U Q U

L U U L Q Q U Q U Q U

−−=

−− −=

−− − − − − − −=

− − − + − − −= = − − + − − −

∑∑

∑

S z A z

M

M

M

M

M

M M M M M M M

�

�

�

. (27)

As a result, for the three-step lifting framework of Fig. 10 we can express the analysis highpass filters

using (26) as

1

0

1

0

( ) ( ) (1 1)

( ) ( ) ( ) ,

j

j i

Mi ijj

M M Mi i jj

G G i

Q Q L

−

=

−

=

= ≤ ≤ −

= − − +

∑

∑

c

c c

z z z

z z z z z

M

M

M

(28)

and using (27) the synthesis filters are obtained as

1

0

1

0

1

0

( ) ( ) (1 1)

( ) ( ) ( ) ( ) ( )

( ) ( ).

j

ji

Mi ijj

M M M M Mi i i j jj

M Mi jj

G G i

L U U L Q

U Q

− −

=

−

=

− −−=

= ≤ ≤ −

= − − +

+ −

∑

∑

∑

c

cc

z z z

z z z z z

z z z z

M

M

M

M� �

(29)

Now to design the filters we would like to have N vanishing moments in the analysis bank and N�

vanishing moments in the synthesis bank. For the former one we employ (22) to (28)

1

1[ ] [ ] [ ] [ ] [ ] [ ] 0i i j j ij

q p M q l p M p M−

=− − ∗ − − ∗ − ∗ − + − =∑n n n n n c n c

M (1 1i≤ ≤ −M , Np ∈P ). (30)

Assuming that [ ]L ik l −n (1 1i≤ ≤ −M ) are Neville filters of order N and shift 1i iMτ −= c , or

[ ] [ ] [ ]L i ik l p M p M− ∗ = +n n n c (1 1i≤ ≤ −M , Np ∈P ), (31)

we can write (30) as

( )1 [ ] [ ] [ ]L i ik q p M p M+ − ∗ = −n n n cM ,

Fig. 10. A multi-channel multidimensional filter bank using a three-step lifting scheme.

M

x MM

1( )U z

rx1c

z

1( )Q z

1−cz

M 0K

1K

1

0K

−

1

1K

−

MM

1( )U − z

M

1−c

z M

1( )Q − z

M

1−−cz M 1

K −M 1

1K

−−M

1( )U z

1( )Q z

1( )Q − z

M

1( )U − z

M

1( )L z

1( )L − z

M

1( )L z

1( )L − z

M



24

or

[ ] [ ] [ ]Q i ik q p M p M∗ = +n n n c (1 1i≤ ≤ −M , Np ∈P ). (32)

Consequently, [ ]Q ik q n (1 1i≤ ≤ −M ) are Neville filters of order N and shift 1i iMτ −= c , where

1Q Lk k= +M . (33)

Now we need to find the filters [ ]iu n (1 1i≤ ≤ −M ) to conclude our multi-channel filter design. This

is accomplished through imposing N� vanishing moments as stated in (23) to the synthesis highpass filters

given in (29):

1

1

1

1

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] 0

i i i j jj

i i j jj

l p M u p M u l q p M

p M u q p M

−

=

−

=

− ∗ − ∗ − ∗ ∗ ∗

+ − − ∗ ∗ − =

∑

∑

n n n n n n n n

n c n n n c

M

M (1 1i≤ ≤ −M ,

Np ∈P� ). (34)

Using (31) and (32) and the fact that N N≥ � , we can reduce (34) to

1 1[ ] [ ] [ ] [ ] [ ]

1[ ] [ ] [ ] 0,

i i i

L Q L

i i

Q

p M u p M u p Mk k k

p M u p Mk

−− − − ∗ − ∗

−+ − − ∗ =

n c n n n n

n c n n

M

M

and after applying (33) it turns out that

[ ] [ ] [ ]U i ik u p M p M− ∗ = +n n n c (1 1i≤ ≤ −M , N

p ∈P� ), (35)

where

2

21 (2 )

LU

L L

kk

k k=

− + − −

M

M M. (36)

We summarize our multi-channel design in the next theorem.

Theorem 2: A multi-channel multidimensional filter bank with three steps of lifting (see Fig. 10)

will have N and N� vanishing moments in the analysis and synthesis banks, respectively if the following

conditions are satisfied:

1) *( )L ik L z (1 1i≤ ≤ −M ) and ( )Q ik Q z are Neville filters of order N and shift 1i iM −=τ c . Similar to

the two-channel case, Lk is a free parameter, and Qk is given by (33).



25

2) *( )U ik U z are Neville filters of order N� and shift 1i iM −=τ c with Uk given in (36). □

Therefore, we have a free parameter Lk that can be employed in designing the wavelet filters. In

contrast to the two-channel wavelets, setting this parameter does not always simply lead to more

symmetric filters in the multi-channel case. We can, however, change the filters responses using Lk . In

the following we present an example demonstrating this fact.

C. Design Example

Example 6: In this example we design filters of a four-channel filter bank with sampling matrix

2 0

0 2M

= . The nonzero coset vectors are 1 (1,0)T=c , 2 (0,1)T=c , and 3 (1,1)T=c . Hence, the

corresponding shifts are 1 (1/ 2,0)=τ , 2 (0,1/ 2)=τ , and 3 (1/ 2,1/ 2)=τ . For 1τ and 2τ we can use 1-D

Deslauriers-Dubuc filters (see Table I) in 1z and 2z , and for the shift 3τ the Neville filters are given in

Table II. Here we design wavelet filters for both two-step and three-step lifting schemes with 4N = and

2N =� vanishing moments in the analysis and synthesis banks.

For the two-step lifting scheme as mentioned in Section V-A (Remark 6), we have 1 (4) 1( ) ( )L R z=z ,

2 (4) 2( ) ( )L R z=z , and 3 (4)( ) ( )L R=z z , and likewise the update filters are expressed as 11 (2) 1( ) ( ) / 4Q R z−=z ,

12 (2) 2( ) ( ) / 4Q R z−=z , and 1

3 (2)( ) ( ) / 4Q R −=z z . Now we can form the analysis filters ( )H z and ( )iG z

(1 3i≤ ≤ ) using the polyphase matrix given in (24) and the constants iK . To find these constants, we set

(1,1) 2H = =M and ( 1, 1) 2iG − − = . Since ( )iL z ( 0 3i≤ ≤ ) and 4 ( )iQ z are one at (1,1)=z , we have

Fig. 11. The frequency response of the highpass analysis filters of Example 6 using a two-step lifting scheme.

-1

0

1

-1

0

10

1

2

ω1

G1

ω2

Magnitude

-1

0

1

-1

0

10

1

2

ω1

G2

ω2

Magnitude

-1

0

1

-1

0

10

1

2

ω1

G3

ω2

Magnitude



26

0(1,1)H K= and ( 1, 1) 2i iG K− − = resulting in 0 2K = and 1iK = (1 3i≤ ≤ ). Fig. 11 shows the frequency

responses of the resultant analysis highpass filters.

Now we provide a possible wavelet design using our proposed three-step lifting for this example.

Using Theorem 2, we choose the first set of lifting filters as 11 (4) 1( ) ( )Lk L R z−=z , 1

2 (4) 2( ) ( )Lk L R z−=z , and

13 (4)( ) ( )Lk L R −=z z . And the second-step filters are expressed as 1 (4) 1( ) ( )Qk Q R z=z , 2 (4) 2( ) ( )Qk Q R z=z , and

3 (4)( ) ( )Qk Q R=z z . The third set of filters are Neville filters of order 2N =� defined as 11 (2) 1( ) ( )Uk U R z−=z ,

12 (2) 2( ) ( )Uk U R z−=z , and 1

3 (2)( ) ( )Uk U R −=z z .

We can derive the analysis filters using the corresponding polyphase matrix given in (26) and

incorporating the constants iK ( 0 3i≤ ≤ ). In addition to these constants we have a free parameter that we

can set to obtain our desirable filter set. One possible constraint is to equate the analysis (or synthesis)

filter magnitudes at ( , )j j=z , which yields 0iK K= (1 3i≤ ≤ ). By setting (1,1) 2H = and 3 ( 1, 1) 2G − − = ,

and using the relations between constants Lk , Qk , and Uk given in (33) and (36), we obtain 1iK =

( 0 3i≤ ≤ ) and 1Lk = − which accordingly we can find the filters.

In Fig. 12 we illustrate the magnitude responses of the resulting highpass analysis filters. The filters

in this design provide a frequency partitioning similar to the separable wavelets and since they form

disjoint passbands, the proposed design would be advantageous to the two-step design.

Fig. 12. The frequency response of the highpass analysis filters of Example 6 using the proposed three-step lifting

scheme.

-1

0

1

-1

0

10

1

2

ω1

G1

ω2

Magnitude

-1

0

1

-1

0

10

1

2

ω1

G2

ω2

Magnitude

-1

0

1

-1

0

10

1

2

ω1

G3

ω2

Magnitude



27

VI. CONCLUSION

In this work we proposed a design method of multidimensional wavelet filters with any order of

vanishing moments based on a triplet of lifting steps. Our design approach provides one more degree of

freedom when compared to the wavelet filter design using two steps of lifting scheme. One can use this

degree of freedom in better shaping the frequency responses of the wavelet filters. For two-channel

wavelets we employed this parameter in order to obtain more symmetry between analysis and synthesis

filters. The proposed filter design in comparison to the lifting design with two steps provides higher

regularity, lower frame bounds ratio, and lower energy of the error leading to better frequency selectivity.

Further, we extended our design for multidimensional multi-channel filter banks of any order of vanishing

moments. Again, using three steps of lifting provides more flexibility in filter design.

ACKNOWLEDGMENT

The authors would like to thank Professors A. Ron, Z. Shen, and K. Toh for providing us with the

Sobolev regularity analysis algorithm code of [20]. We also thank H. Ojanen for the software to measure

Sobolev regularity of 1-D filters.

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design of regular wavelets using a three-step lifting scheme

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