Lattice Structure for Regular Paraunitary Linear-Phase ... - CiteSeerX

THIS PAPER APPEARED IN THE PROCEEDINGS OF SECOND INTERNATIONAL WORKSHOP ON TRANSFORMS AND FILTER BANKS, BRANDENBURG AN DER HAVEL, GERMANY, MARCH 5–7, 1999.

Lattice Structure for Regular Paraunitary Linear-Phase Filter Banks Soontorn Oraintara

Peter Heller

Trac Tran

Truong Nguyen

Boston University, ECE Dept., Boston MA oraintar, nguyent @bu.edu Aware Inc., Bedford MA [email protected] John Hopkins University, ECE Dept. [email protected]

Abstract Orthogonal M -channel uniform linear-phase filter banks (GenLOT) can be designed and implemented using lattice structure. This paper discusses the conditions of the lattice structures that yield regular GenLOTs. For one regularity, the relations among the plane rotation angles in the lattice structure will be derived, and can be used to design regular GenLOTs. Since the conditions for higher degree of regularity are much more complex, in the paper, we shall study these conditions for two regularities. Some examples of filter bank designs are presented, and applied to image coding. The simulation results show that filter bank with higher degree of regularity provides smoother reconstruction images.

1

Introduction

x(n)

M

-1

M

F 1 (z) -1

M E(z)

-1

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Processing Block

y(n)

M

(a)

M -1

R(z)

M

-1

-1

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F 0 (z)

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HM-1 (z)

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Perfect reconstruction (PR) filter bank is an invertible linear-time frequency representation. Wavelets are even more recent approach in which the filter bank is iterated to represent information at multiple scales as well as at distinct time and frequency bins. These new tools for signal processing and communications have a rich mathematical structure, and are introducing an entirely new collection of filter design constraints and approaches. Figure 1(a) and (b) show the structure of an M -channel filter bank in the regular form and its polyphase representation respectively. For PR systems, the product between the analysis and synthesis

M

y(n)

(b)

Figure 1: The block diagram of M -channel filter banks: (a) filter structure and (b) its equivalent polyphase structure.

1

polyphase matrices R(z)E(z) is equal to I or its pseudo-circulant version for the more general case [1]. The filter bank is paraunitary (PU) if R(z) = ET (z −1 ) since E(z) becomes an orthogonal matrix on the unit circle. The synthesis filters Fi (z) now become the time-reverse (complex conjugate) versions of the analysis filters Hi (z). Recently PU linear-phase filter banks together with the zero-tree coding give significant improvement in still image compression to the literature [2, 3, 4]. Linear phase (LP) property of the filters is also needed in processing limited length signal which provides even localization, and thus one uses symmetric extension at the boundaries of the image. Such filter banks (PULP) are also known as the lapped orthogonal transform (LOT) and its generalized version GenLOT. These terms will be used interchangeably in the paper. PULP filter banks are often designed and implemented using lattice structure since the PR condition is structurally imposed. Moreover the structure is shown to be robust, efficient and complete. A complete and minimal factorization of PULP filter banks have been first presented in [5]. In [6], an equivalent modular factorization, GenLOT, is presented and it covers the DCT and LOT as special cases. The number of channels is assumed to be even and the filters have equal length being multiple of M . Theory and lattice structure for filter banks with odd number of channels and unequal length filters can be found in [7] and [3] respectively. Smoothness of the scaling function which is obtained from infinite iteration of the lowpass channel of the filter bank is important in signal approximation and interpolation. The smoothness of the scaling function is closely related to regularity which is defined to be the number of zeros at mirror frequencies 2kπ M of the lowpass filter [8]. The number of zeros at each mirror frequency of the lowpass filter determines the degree of regularity of the filter bank. System with higher degree of regularity has smoother scaling function. Note that the degree of regularity of the filter banks is defined from the scaling function view point. From the wavelets perspective, this property is also known as vanishing moment because the corresponding wavelets are orthogonal to any piecewise polynomial of degree R − 1 where R is the number of vanishing moments (degree of regularity). From the iterative construction of wavelets, it has been proven that one vanishing moment is a necessary condition for the convergence of the scaling function and the mother wavelets [9]. The equivalent condition 2π on the lowpass filter is H0 (W k ) = 0 for all k > 0 where W = ej M . Consequently, the scaling function φ(t) can exactly represent a (piecewise) constant signal. Moreover if the lowpass filter has R-regularity, the scaling function φ(t) can exactly represent a polynomial up to degree R − 1, and thus this property is very important in smooth signal approximation application such as image coding. Nevertheless the lattice structure for regular PULP filter banks have not been reported in the literature. In the case when M = 2, it is known that PU and LP are exclusive properties (except for the Haar wavelet), and each of them has its own lattice structure [10, 11]. For PU 2-channel case, the filter bank has one regularity if all the rotation angles in its lattices are summed up to π4 [2]. This paper aims to investigate the lattice structure of M -channel regular PULP filter banks. Throughout this paper, the number of channel M is assumed to be even and the length of all filters is an integer multiple of M . It has been proven that when M is even, an M -channel filter bank with linear-phase consists of an equal number of symmetric and antisymmetric filters [5]. Hence one expects that for the first vanishing moment, the condition can be imposed into the M/2 symmetric filters. Two conventional approaches in designing regular filter banks have been suggested. The first method is to impose the number of zeros at mirror frequencies into the lowpass filter [8, 12]. Having constructed the first filter, the remaining M − 1 filters have to be designed such that the resulting filter bank is PR. For 2-channel case, the highpass channel can be uniquely determined from the lowpass filter, while for M > 2, there are more freedoms in choosing the bandpass and highpass filters. This step can be done using a Gram-Schmidt process [8, 13]. However the approach does not guarantee the linear-phase property of the filters. Another method is to use time-domain constraint. A PR filter bank is first constructed, and often this is done by using the lattice structure. The number of vanishing moments is used as the time-domain side-constraints. The disadvantage of this approach is that the time spent in the optimization is much longer than when there is no side-constraint and the regularity can only be approximately imposed. 2

In this paper, we present a novel approach of imposing two vanishing moments into the lattice structure of M -channel PULP filter banks. The next section discusses the lattice structure of GenLOT. In section 3, the conditions for R-regular M -channel filter banks is given. The first and second vanishing moments are discussed in terms of the lattice component. The equivalent relation on the rotation angles of the lattice structure is derived so that the resulting filter bank is guaranteed to have R-regularities as well as the PR property. The designed filters are then applied in image coding and their results are presented in section 4. Section 5 concludes the paper.

2

Lattice structure for PULP filter banks (GenLOT)

Lattice structure is an attractive structure in the design and implementation of PULP filter banks by which the PU and LP properties are simultaneously imposed into the filters’ impulse responses. It is assumed that the number of channel M ≥ 4 is even, and all the filters have equal length ` = KM where K is an integer. M It has been proven that when the number of channel is even, there are M 2 symmetric and 2 anti-symmetric filters [4]. The polyphase matrix E(z) is a polynomial matrix of degree K − 1. Under the assumptions on M , K and on the filter symmetry, define the lattice elements as followed:         1 Ui IL IL IL IL ˜ Φi = , W= √ , Λ= , and I = Vi z −1 IL JL 2 IL −IL and J is the matrix. Ui and Vi are orthonormal matrices of size L × L, and each can  reversal  L be parameterized using rotation angles. The polyphase matrix E(z) of a PULP filter bank of degree 2 K − 1 can then be factorized as a product of PU matrices with degree one [6], i.e. where L =

M 2

E(z) = G0 (z)G1 (z)...GK−2 (z)E0

(1)

where Gi (z) = Φi WΛW and E0 = Φ0 W˜ I. Figure 2 shows the complete and minimal lattice structure of PULP filter banks. 1/2

1/ 2 1/ 2 1/ 2

1/2

UK

1/2 1/2

1/ 2 -

1/ 2 1/ 2 1/ 2

-

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-

Λ

GK-2

1/2 1/2 1/2

VK-1

1/2

... ... ... ... ... ... ... ...

G0

ΦK-1

W GK-1

Figure 2: Lattice structure for PULP filter banks (GenLOT)

Having characterized the PULP filter banks this way, one can view the DCT (Discrete-Cosine-Transform) and LOT (Lapped-Orthogonal-Transform) as special cases with K = 1 and K = 2 respectively. This is the reason why the structure is called GenLOT (Generalized-LOT). Since Ui and Vi are orthonormal matrices,   L one can completely parameterize each using rotation angles. Note that the order of these rotation 2 angles is not unique and hence there are many possible ways to parameterize an orthonormal matrix. Figure 3 show two examples of the lattice elements which will be used as preferred orders of the rotation angles in the discussion of the paper. 3

. . .

. . .

. . .

. . .

. . .

. . .

(a) Type I

. . .

. . .

(b) Type II

Figure 3: Examples of different orders of the rotation angles in the lattice structure of orthonormal matrices.

3

The condition for R-regular filter banks

Theorem 1 The following two statements are equivalent for an M -channel filter bank: 1. F0 (z) has R zeros at aliasing frequencies

2kπ M

for k = 1, ..., M − 1.

2. Hi (z) has R zeros at zero frequency for k = 1, ..., M − 1. Proof: For convenience, we will prove by induction on R using the modulation matrix Hm (z). The PR condition yields Fm T (z)Hm (z) = diagonal matrix Fm (z)Hm T (z) = diagonal matrix

(2) (3)

P (a) R = 0. Assume that F0 (W j ) = 0 for every j > 0. Substituting z = 1 into (2) yields j F0 (W j )Hj (W i ) = 0 for all i. Hence Hi (1) = 0 for i > 0. The converse can be proved in the same way by using (3). (b) R = N. Assume that the theorem is true for R < N . From (2) and (3), we have  N  X N Fm T (z)(n) Hm (z)(N −n) n n=0  N  X N Fm (z)(n) Hm T (z)(N −n) n

= diagonal matrix

(4)

= diagonal matrix

(5)

n=0

By substituting z = 1 into (4), one concludes that property 1 implies property 2 Consequently, using (5), one concludes that property 2 implies property 1.2 Collorary 2 For PU filter banks, Hi (z) has R zeros at z = 1 for i = 1, 2, ..., M − 1, if and only if H0 (z) has R zeros at z = W j for j = 1, 2, ..., M − 1. Equivalently,   dn  E(z M )  n dz 

1

z −1 .. .

z −M+1

    

   = 

∗ 0 .. . 0

z=1

where ∗ is a non-zero element. 4

    

for n = 0, 1, ..., R − 1

(6)

3.1

Imposing one vanishing moment

Consider the polyphase matrix E(z) of an M -channel PULP filter bank: I. E(z) = Φ0 WΛWΦ1 WΛW · · · ΦK − 1 WΛWΦK W˜

(7)

We derive the condition on the structure such that the resulting PULP filter bank has one vanishing moment in this section. Substituting n = 0 into (6) implies that      √ U0 · · · UK 1L U0 · · · UK 1L = = LaM (8) V0 · · · VK 0L V0 · · · VK 0L where 1L and 0L are vector of elements 1 and 0, respectively, for all indices with lengths L, and a = [1 0 · · · 0]T with the subscription indicating the vector length. The equivalent condition is: √ A0 : U0 · · · UK 1L = LaL . Note that the above condition A0 does not depend on the matrices Vi because of the fact that the L filters HL (z), HL+1 (z), ..., HM−1 (z) are anti-symmetric filters and therefore they have zero response at zero frequency. Let U = U0 · · · UK be the product of the orthonormal matrices Ui . It is obvious that U is also √ orthonormal. Since it rotates (preserves the norm) the vector 1L to LaL , it can be parameterized by L − 1 rotation angles. We choose to impose this condition into the matrix U0 and the reason will become clear when the condition for two vanishing moments in the next subsection is discussed. The condition A0 is equivalent to having identical elements of the first row of the rotation matrix U. Note that the popular DCT used in JPEG and MPEG standards satisfies this condition. Without loss of  matrix  L generality, let the rotation angles of U0 be ordered as in Figure 3(b) (Type II). Let U0 = U01 U02 , 2 the product  of two orthonormal matrices where U02 which operates on the lower L − 1 rows is parameterized  L−1 by the first rotation angles. U01 is defined to be the complementary part of U02 , i.e. it consists 2     L L−1 of the last L − 1 = − rotation angles of the matrix U0 . The following lemma give a 2 2 necessary and sufficient condition on the rotation angles which provides one vanishing moment. Lemma 3 Let A be an orthonormal matrix of size N . Let B be a matrix constructed by cascading the matrix A and N − 1 rotation angles as shown in figure 4. B1,1 = B1,2 = ... = B1,N if and only if   P N Y Ai+1,k θi = tan−1  Pk cos θj  , i = 1, 2, ..., N − 1 (9) k A1,k j=i+1 where θN = 0. The notation Ai,j denotes the (i, j) element of the matrix A. From Lemma 3, it is clear that by letting A = U02 U1 · · · UK = UT 01 U and B = U, the first vanishing moment can be obtained by choosing the L − 1 rotation angles of U01 accordingly as in (9). In the design method, the rotation angles of Ui and Vi , except for the last L−1 rotation angles of U0 , are free parameters. Once they are determined, the last L − 1 rotation angles can be calculated. Design example 1: In this example, a 1-regular 8-channel 24-tap PULP filter bank is designed using the lattice structure. The filters are optimized in order to minimize the stopband attenuation. Figure 5(a) shows the frequency responses of the resulting filters. The zeros of the lowpass filter are plotted in Figure 5(b) showing that the filter bank has one degree of regularity which confirms the theory.

5

θ1 A

B

θ2

.. . θN-1

Figure 4: Embedded structure of lattice structure which yields one regularity of the first channel.

8 channels, 24 tabs, Coding gain=9.3571 dB

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Figure 5: A design example 8-channel PULP filter bank with with one vanishing moment: (a) frequency responses and (b) zero locations of the lowpass filter.

3.2

Imposing two vanishing moments

In this section, the condition on the lattice structure is derived for a PULP filter bank with two vanishing moments. Substituting n = 1 in (6) and sufficient condition for the filter bank to have  yields a necessary  IL −IL two vanishing moments. Let Ψ = . One can show that when n = 1, (6) can be simplified to −IL IL −M [Φ0 ΨΦ1 Φ2 · · · ΦK + Φ0 Φ1 ΨΦ2 · · · ΦK + · · · + Φ0 Φ1 Φ2 · · · ΦK − 1 ΨΦK ] W˜ I1M + Φ0 · · · ΦK W˜ Ib = αaM (10) for some constant α, where b = [0 1 ... M − 1]T . With some manipulation, the above equation can be simplified to     M 1 U0 U1 · · · UK 1L (M − 1)U0 · · · UK 1L √ √ − − = αaM . −V0 · · · VK b0 2 −(V0 U1 · · · UK + V0 V1 · · · UK + V0 V1 · · · VK − 1 UK )1L 2 (11) where b0 = [2L − 1 2L − 3 · · · 3 1]T . The top row is consistent with the condition A0 while the bottom row forms a new condition for the second vanishing moment. Multiplying UT 0 both sides of the bottom row yields:

6

A1 :

˜ = 0L . (U1 U2 · · · UK + V1 U2 · · · UK + · · · + V1 · · · VK − 1 UK )1L + V1 V2 · · · VK b 0

˜ = b Note that the condition A1 does not depend on the matrix U0 and this is why we chose where b M to impose the first vanishing moment on U0 . Keep in mind that in order for the filter bank to have two vanishing moments, both conditions A0 and A1 must be satisfied. The first condition can be taken care by choosing U0 properly (as discussed in section 3.1). The second condition composes of K + 1 vectors forming a loop (zero sum). Taking into account that Ui and Vi are orthonormal matrices, have √ 2 2 the first K2 vectors q q √ 1 +3 +···+(2L−1) M M 2 −1 ˜ equal length k1L k = L = , and the last vector has length kbk = = . 2

M

6M

Therefore in order for the condition A1 to holds, there must be at least three vector, i.e. K ≥ 2. This proves that the minimal length for PULP filter banks with two vanishing moments is 3M . 3.2.1

Minimal length case

In this section, we discuss how to impose the second vanishing moment into the lattice structure. From the previous discussion the minimal length for PULP filter banks is 3M (K = 2) where M is the number of channels. Substituting K = 2 into the condition A1 , we have ˜ = 0L . U1 U2 1L + V1 U2 1L + V1 V2 b

(12)

˜ For convenience √ let x = U1 U2 1L , y = V1 U2 1L and z = V1 V2 b where they form a bilateral triangle with kxk = kyk = L as in Figure 6. Having known the three sides of the triangle, one can determine the angles ψ

y

z

φ

φ

λ = φ+ψ

x

Figure 6: The geometrical interpretation of (12) where the three vectors x, y and z forms a triangle. between each pair of the three vectors. Let φ and ψ be defined as in Figure 6. It is easy to see that ! ! r r 1 M2 − 1 1 M2 − 1 −1 −1 ψ = 2 sin and φ = cos . 2M 3 2M 3  T  p q Define 6 (p, q) to be the angle between the vectors p and q, i.e. 6 (p, q) = cos−1 kpkkqk . Note also that the 6 6 angle between two vectors does not depend on the lengths  of the vectors,i.e. (c1 p, c2 q) = (p, q) where ˜ = 6 U2 1L , V2 b ˜ = λ =φ+ψ c1 and c2 are any scalar constants. Hence 6 (y, z) = 6 V1 U2 1L , V1 V2 b where we have used the fact that V1T V1 = IL . We choose to impose (12) into the matrices U1 and U2 . Fixing the vector z (letting V1 and V2 be free), U2 must tune the angle between y and z to be λ in order for the three vectors to form a triangle. This 7

can be done by reducing one degree of freedom in the vector U2 which will be soon discussed. Having U2 determined, by the law of cosine, ky + zk = kxk which permits U1 to be able to close the triangle. This step reduces the degree of freedom of U1 by L − 1. Define the operator R [p] to be a square matrix that rotates p into the direction of aL , i.e. R [p] p = kpkaL which will be used repeatedly in our later discussion. Such an operator can be constructed by the Householder matrix which is given by [14] R [p] = I − 2

vp vp T kvp k2

(13)

p where vp = kpk − a. Since a has unit length, it can be shown that R [p] p = kpka [14], and now we are ready to proceed.

Fix V2 and find U2 From the above discussion, we have λ = 6 (y, z) = 6

   h i  ˜ = 6 R V2 b ˜ U2 1L , aL . U2 1L , V2 b

(14) 

 L Let Q be an orthonormal matrix which rotates aL by an angle λ. Assume that the rotation angles 2 of the matrix Q are ordered as in Figure 3(a)(Type I). It is easy to show that 6 (QaL , aL ) = λ if and only if Q11 = cos(λ) which can be imposed into the rotation angle of Q as in Figure 7 where cos(θ1 ) cos(θ2 ) · · · cos(θL−1 ) = cos(λ)

(15)

¯ is an orthonormal matrix with the size L − 1 which is free to be chosen. In order to satisfy (15), the and Q degree of freedom of Q by one and hence Q and U2 have a unique correspondence relation. Taking into account that λ = 6 (QaL , aL ) = 6 (QR [1L ] 1L , aL ) ,

(16)

and comparing with (14), we can express U2 in terms of the Q matrix as: h iT ˜ QR [1L ] . U2 = R V2 b

(17)

From (15), without of lost of generality, we choose to impose the condition on θL−1 , i.e.

θ1 θ2 θ3

Q . . .

. . . θL-1

Figure 7: Parameterization of matrix Q

8

cos(λ)

−1

QL−2

θL−1 = cos

j=1

!

cos(θj )

(18)

QL−2 However this choice of θL−1 might not be possible if | j=1 cos(θj )| < | cos(λ)|. Therefore the rotation angles θi have to be chosen properly. The following inductive construction allows us to completely impose (15) without any violation to (18). Procedure 1 Step 0 λ is calculated from M . Step 1 Choose θ1 so that | cos(θ1 )| ≥ | cos(λ)|. cos(λ)| Step 2 Choose θ2 so that | cos(θ2 )| ≥ ||cos(θ . 1 )| .. . | cos(λ)| Step i Choose θi so that | cos(θi )| ≥ Qi−1 | cos(θ j=1

.. . Step L − 1 Choose θL−1 so that cos(θL−1 ) =

j )|

Q

. cos(λ) . cos(θj )

L−2 j=1

Fix V1 and find U1 Having constructed U2 , y is known. What remains in the equation (12) is to choose U1 so that x = −y−z. This takes out L − 1 degrees of freedom from U1 . From (12), U1 U2 1L h i ˜ U1 U2 1L R −(V1 U2 1L + V1 V2 b) h i ˜ U1 U2 1L R −(V1 U2 1L + V1 V2 b)

˜ = −(V1 U2 1L + V1 V2 b)

√ ˜ L = LaL = k − (V1 U2 1L + V1 V2 b)ka √ = LQ0 aL = Q0 R [U2 1L ] U2 1L

(19) (20) (21)

where Q0 is an orthonormal matrix satisfying that Q0 aL = aL . It is obvious that such a matrix Q0 has  L−1 degrees of freedom, and can be parameterized in the same fashion as Q (see Figure 7) except 2 that here θ1 = θ2 = · · · = θL−1 = 0. Since Q0 and U1 share the same degree of freedom, we can write h iT ˜ U1 = R −(V1 U2 1L + V1 V2 b) Q0 R [U2 1L ]

(22)

Procedure 2 Step 1 Given M , compute φ, ψ and λ. Step 2 Solve for θi , i = 1, 2, ..., L − 1 using Procedure 1. ¯ and θi , calculate Q. Step 3 Having known Q Step 4 Having known V2 , calculate U2 using (17). Step 5 Calculate Q0 . Step 6 Having known V1 , calculate U1 using (22). Step 7 From U1 and U2 , calculate U0 using lemma 3. Step 8 Calculate the filters’ coefficients. Alternative parameterization of U0 Having discussed the method in parameterizing U1 and U2 above, one can use the similar technique to impose the first regularity as well. From the condition A0 with K = 2, √ U0 U1 U2 1L = LaL . (23)

9

Therefore U0 can be written as U0 = Q00 R [U1 U2 1L ]

(24)

where Q00 is the same form as Q in Figure 7 except that θi ≡ 0 for i = 1, 2, ..., L − 1. Design example 2: In this example, a 2-regular 8-channel PULP filter bank is designed with minimal length, i.e. all the filters have length ` = 3 × 8 = 24. The frequency responses are presented in Figure 8(a), and the zeros of the lowpass filters are plotted in Figure 8(b). It can be seen that the lowpass filter has (at least) double zeros at each the mirror frequencies which confirms that the bandpass and highpass filters will have (at least) two vanishing moments. In this design example, the stopband attenuation is used to optimize the filters. One observes that frequency response of this filter bank is worse than the one in the design example 1. This is simply because of the fact that a number of free parameters have been used to impose the second vanishing moment. 2.5

8 channels, 24 tabs, Coding gain=9.02 dB 0

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Figure 8: A design example of 2-regular minimal length PULP filter bank with M = 8, i.e. the filter length ` = 24: (a) frequency responses and (b) zero locations of the lowpass filter.

3.2.2

A special case when the filter length is equal to 4M (K = 3)

The same technique can also be applied to the case of filters with arbitrary length. Substituting K = 3 into A1 yields ˜ = 0L . U1 U2 U3 1L + V1 U2 U3 1L + V1 V2 U3 1L + V1 V2 V3 b

(25)

˜ Similar to the minimal length case, let x = U1 U2 U3 1L , y = V1 U2 U3 1L and z = V1 V2 U3 1L + V1 V2 V3 b. Again we choose to impose the condition into U√ 1 and U2 which means that U3 and Vi (for all i) are free ˜ < L which implies that to be chosen. This is permissible because kbk √ √ ˜ + L ≤ 2 L = kxk + kyk |kxk − kyk| = 0 ≤ kzk ≤ kbk (26) and thus z satisfies the triangle inequality. The only difference between the cases of K = 2 and K = 3 is that, based on the notation in Figure 6, the angles φ and ψ are now dependent on z, i.e.     kzk kzk √ √ φ = cos−1 and ψ = 2 sin−1 (27) 2 L 2 L 10

Procedure 3 Step 1 Since Vi and U3 are free, calculate z. Step 2 Compute φ, ψ and λ from z. Step 3 Solve for θi , i = 1, 2, ..., L − 1 using Procedure 1. ¯ and θi , calculate Q. Step 4 Having known Q Step 5 Having known V2 , calculate U2 using (17). Step 6 Calculate Q0 . Step 7 Having known V1 , calculate U1 using (22). Step 8 From U1 and U2 , calculate U0 using lemma 3. Step 9 Calculate the filters’ coefficients. Design example 3: In this design example, a 2-regular 32-tap 8-channel PULP filter bank is designed using the proposed theory. Its frequency response, the zeros of the lowpass filter, and the corresponding scaling and wavelet functions are respectively shown in Figure 9(a), (b) and (c).

4

Image coding result

In this section, the filter banks obtained from the design examples are used in image compression application. The progressive image transmission algorithm (block transform + regrouping coefficients + zero-tree coding) is used to compare the performances of the transforms. For more details on progressive image transmission, the readers are referred to [15]. The PULP filter banks are used as block transforms in the progressive image transmission In the simulation, 8-channel 24-tap PULP filter banks with one and two vanishing moments are compared. The lena, barbara and goldhill images are used in the experiment. Figure 10 shows the PSNR of the reconstruction images compared at various compression ratios ranging from 16:1 to 128:1. The one with one vanishing moment (1VM) has higher PSNR comparing to that with two vanishing moments (2VM). However, the perceptual quality of the 2VM filter bank is slightly better. Figure 4 shows the original and the zoom in part of the barbara image. One can notice that in some smooth region, the 1VM filter bank has blocking artifact while the 2VM filter bank gives smoother reconstruction. Note that length 24 is the minimum length for 8-channel PULP filter bank. Therefore there are only a few degrees of freedom left in optimizing the transform. The PSNR difference between the two transforms (1VM and 2VM) can be reduced by allowing longer filter length. Note that the filter banks used in the compression are optimized to reduce the stopband attenuation only, and thus they are not yet optimal for practical image coding. There are many criteria such as transform coding gain, non-uniform stopband attenuation, mirror frequency, etc, which can improve the coding performance [15]. These will be incorporated in the design procedure in future work.

5

Concluding remarks

In this paper, the theory, design, and lattice structure of PULP filter banks with one and two degrees of regularity are presented. The proposed lattice structure guarantees that the resulting filter banks have all the desired properties (PR, PU, LP, one or two vanishing moments). The number of channel M is assumed to be even and all the filters have the same length KM for some integer K. The conditions for obtaining one and two vanishing moments from the resulting filters are derived in terms of the lattice elements. It turns out that for one vanishing moment, one only have to choose the last M 2 − 1 rotation angles of the orthonormal matrix U0 properly provided that U0 is parameterized using Type II structure (see section 2). Therefore the minimal permissible length for this case is M , and the popular DCT satisfies this condition. For two vanishing moments, a necessary and sufficient condition of the lattice elements is derived. Unlike

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8 channels, 32 tabs, Coding gain=9.2821 dB 0

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(c) Figure 9: A design example of 8-channel 2-regular PULP filter bank with length 4M = 32: (a) frequency responses and (b) zero locations of the lowpass filter, and (c) the scaling function and wavelets.

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38 1 vanishing moment 2 vanishing moment 36

34 lena

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Figure 10: The PSNR of the reconstruction images using 1VM (solid line) and 2VM (dash line) filter banks in the progressive image transmission coding. The transforms are tested on lena, barbara and goldhill images.

the PU and LP filter banks which require the minimum permissible filter length to be 2M , the minimum permissible length in the PULP case is proven to be 3M . The special cases for when ` = 3M and ` = 4M are presented. The filter banks are then used as the transform part of the progressive image transmission coding. With filter length ` = 24 which is the minimum length for 8-channel PULP filter banks, the one with one vanishing moment gives better quality in PSNR (ranging from 0 to 1.5 dB). However, in terms of perceptual quality, the filter bank with two vanishing moments offers smoother reconstruction images.

References [1] P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice-Hall, Englewood Cliffs, NJ, 1993. [2] G. Strang and T. Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge, Wellesley, MA, 1996. [3] Trac D. Tran and T. Q. Nguyen. Generalized lapped orthogonal transform with unequal-length basis functions. In Proc. ISCASS, 1996. [4] Trac D. Tran and T. Q. Nguyen. On m-channel linear-phase fir filter banks and applications in image compression. IEEE Trans. Signal Processing, 40:2175–2187, 1996. [5] A. K. Soman, P. P. Vaidyanathan, and T. Q. Nguyen. Linear-phase orthogonal filter banks. IEEE Trans. SP, 41:3480–3496, December 1993. [6] Ricardo L. de Queiroz, Truong Q. Nguyen, and K. R. Rao. The genlot: Generalized linear-phase lapped orthogonal transform. IEEE Trans. SP, 44(3):497–507, March 1996. [7] C. W. Kok, T. Nagai, M. Ikehara, and T. Q. Nguyen. Structures and factorizations of linear phase paraunitrary filter banks. In Proc. ISCASS, 1996. 13

Figure 11: Coding results at compression rate 32:1. The first column corresponds to the original image. The second and third columns are the images coded using systems with one and two-regularities respectively.

[8] Peter Steffen, Peter N. Heller, Ramesh A. Gopinath, and C. Sidney Burrus. Theory of regular m-band wavelet bases. IEEE Trans. on Signal Processing, 41(12):3497–3510, December 1993. [9] I Daubechies. Ten Lectures on Wavelets. SIAM, 1992. [10] P. P. Vaidyanathan and P.-Q. Hoang. Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction qmf banks. IEEE Trans. on Acoustics, Speech, Signal Proc., ASSP-36:81–94, January 1988. [11] T. Q. Nguyen and P. P. Vaidyanathan. Two-channel perfect reconstruction fir qmf structures which yield linear phase fir analysis and synthesis filters. IEEE Trans. on Acoustics, Speech, Signal Proc., ASSP-37:676–690, May 1989. [12] Y. Wisutmethangoon and T. Q. Nguyen. Nonlinear phase halfband and mth band filter. Submitted to IEEE Trans. on Circuits and Systems, Febuary 1997. [13] P.N. Heller and H.L. Resnikoff. Regular m-band wavelets and applications. In ICASSP, volume III, pages III–229–III–232, 1993. [14] Gilbert Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley, MA, 1993. [15] T.D. Tran and T.Q. Nguyen. A progressive transmission image coder using linear phase uniform filter banks as block transforms. IEEE Trans. on Image Processing. to appear. [16] P.N. Heller, T.Q. Nguyen, H. Signh, and W.K. Carey. Linear-phase m-band wavelets with application to image coding. ICASSP, 1995.

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