Optimal Design of FIR Triplet Halfband Filter Bank and Application in Image Coding

586 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 2, FEBRUARY 2011

Correspondence

Optimal Design of FIR Triplet Halfband Filter Bank andApplication in Image Coding

H. H. Kha, H. D. Tuan, and T. Q. Nguyen

Abstract—This correspondence proposes an efficient semidefinite pro-gramming (SDP) method for the design of a class of linear phase finite im-pulse response triplet halfband filter banks whose filters have optimal fre-quency selectivity for a prescribed regularity order. The design problem isformulated as the minimization of the least square error subject to peakerror constraints and regularity constraints. By using the linear matrix in-equality characterization of the trigonometric semi-infinite constraints, itcan then be exactly cast as a SDP problem with a small number of vari-ables and, hence, can be solved efficiently. Several design examples of thetriplet halfband filter bank are provided for illustration and comparisonwith previous works. Finally, the image coding performance of the filterbank is presented.

Index Terms—Image coding, perfect reconstruction, regularity, semi-definite programming (SDP), triplet halfband filter bank.

I. INTRODUCTION

Two-channel filter banks have found important applications in imageprocessing, speech processing, communications, and the constructionof wavelet bases [11], [13], [16]. The filter bank design is commonlyformulated as a highly nonlinear optimization problem because of theperfect reconstruction condition. As a result, high complexity algo-rithms are required to obtain a good solution, and the globally optimalsolution is not guaranteed. To reduce the computational complexity ofthe design, finding filter bank structures that structurally satisfy perfectreconstruction is of great interest.

Lifting structures are very attractive for the construction and imple-mentation of filter banks and wavelets because the perfect reconstruc-tion property can be structurally imposed. In addition, they are robustto quantization errors and efficient to jointly realize filters. A class oflifting scheme with two lifting steps, namely halfband pair filter bankintroduced by Phoong et al. [8] offers a filter bank with low implemen-tation complexity. However, there are certain restrictions on the fre-quency responses of filters [2], [4], [14]. To overcome this restriction,

Manuscript received December 02, 2009; revised June 10, 2010; acceptedJune 10, 2010. Date of publication July 19, 2010; date of current version Jan-uary 14, 2011. This correspondence was presented in part at the Second Interna-tional Conference on Communications and Electronics (ICCE 2008), Hoi An,Vietnam, June 2008. This work was partially supported by the Australia Re-search Council (ARC) under Discovery Project 0772548. The associate editorcoordinating the review of this manuscript and approving it for publication wasProf. Ljubisa Stankovic.

H. H. Kha and H. D. Tuan are with the School of Electrical Engineering andTelecommunications, University of New South Wales, UNSW Sydney, NSW2052, Australia (e-mail: [email protected]; [email protected]).

T. Q. Nguyen is with the Department of Electrical and Computer Engineering,University of California, San Diego, La Jolla, CA 92093-0407 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2010.2059450

Ansari et al. [2] introduced a structure with three lifting steps, knownas triplet halfband filter bank. In addition to the advantage of simpledesign, the triplet filter bank can offer the flexible control of filter fre-quency responses. Two approaches for designing the triplet halfbandfilter banks were presented in [2]. The first approach using the Remezalgorithm leads to the equiripple filters with narrow transition band-widths. However, the regularity constraints cannot be straightforwardimposed on the filters. The second one employs Lagrange halfband fil-ters which result in maximally flat filters with poor frequency selec-tivity. Another method was introduced to design the triplet halfbandfilters with arbitrary regularity order using Bernstein polynomial [14].An advantage of using Bernstein polynomial interpolation is that theregularity constraint is structurally imposed. However, the Bernsteinpolynomial is suitable for nearly maximally flat responses rather thanfor ripple responses with sharp transition band [3], [14]. Alternatively,the design of triplet halfband filter banks with regularity using SDPwas presented in [4]. This method is flexible to incorporate the regu-larity constraint into the optimization problem. However, the semi-in-finite constraints in [4] were approximated by a large finite number ofconstraints. As a result, this method is not efficient for the design ofhigh-order filters, and its solution can violate the specifications at be-tween grid frequencies.

In this correspondence, an efficient SDP formulation for designingthe triplet halfband filter bank with optimal frequency selectivity fora given regularity order is presented. First, the design of linear phasesubfilters is formulated as a semi-infinite programming (SIP) problemin which the objective is to minimize the energy error, and the con-straints are peak ripples in the passband and stopband. Moreover, theregularity condition expressed as a linear equation can be straightfor-ward incorporated into the design problem. Then, by employing linearmatrix inequality (LMI) characterizations of trigonometric semi-infi-nite constraints, we can precisely cast the SIP problem as an SDP formwith small number of variables. Consequently, the design problem canbe efficiently solved for filters with high order and prescribed regularityorder. Design examples of optimized triplet filter banks with differentregularity are presented to illustrate the performance of our method.Moreover, we provide an example of image compression using the filterbank designed by the proposed method.

Notations: Bold-faced characters denote matrices and column vec-tors, with upper case used for the former and lower case for the latter.The inner product between vectors �� and �� is defined as ��

��. For a given set � � � its convex hull (conic hull), denotedby �� , is the smallest convex set (cone) in � thatcontains �. The polar set of � is the cone ��

��.

II. PROBLEM FORMULATION

Triplet halfband filter bank is the lifting scheme with threelifting steps, as shown in Fig. 1, where two-channel filter bank isparameterized by a set of three halfband filters [2], [4], [14]. Sincethe filter bank is structurally perfect reconstruction for arbitrarychoices of ��, a good choice of constant parameters is given in[4]: ��

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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 2, FEBRUARY 2011 587

Fig. 1. Structurally perfect reconstruction two-channel triplet halfband filter bank. (a) Analysis bank. (b) Synthesis bank.

and �� . From Fig. 1, the analysis and synthesis filtersare given by

��

��

��

�

� �� (1)

��

� � ��

��

��

� � ��

�� (2)

�� (3)

The design of the triplet halfband filter bank is to find subfilters��

�� , such that the analysis and synthesis filters havegood frequency selectivity and desired regularity. It is well-known thatlinear phase filters are desirable in certain applications such as imageprocessing due to the fact that linear phase filters preserve phase andhandle the boundary extension efficiently [11]. Therefore, we focus onsubfilters with symmetric responses, that is

��

�

��

�� with ��

(4)

It can be shown that only the symmetric subfilters �� of odd ordersare suitable for linear phase analysis filters. Subsequently, the ampli-tude response of ��, denoted by ��, can be expressed as acosine polynomial

��

�

��

��

� �� (5)

where ��

�� , and �� is a constant matrix of appropriate

dimension.As discussed earlier, the Remez algorithm and Lagrange formulation

yield filters with extreme characteristics as the former yields equiripplefilters without regularity and the latter yields maximally flat subfilterswith poor frequency selectivity. In this correspondence, we present amore general approach that can design the optimal filter frequency re-sponses for a given regularity order. Moreover, to provide a tradeoffbetween the mean square error and the maximum peak error, our filterdesign is formulated as a peak constrained least square (PCLS) filterdesign which can be precisely cast into a semidefinite programming(SDP) problem. Theoretically, the joint optimization of three subfilterscan yield the analysis filters with improved performance comparing tofilters obtained from separate optimization of the subfilters. However,

due to nonlinear relation of analysis filters with subfilters, the joint opti-mization problem then is highly nonlinear and, hence, finding the glob-ally optimal solutions is extremely difficult. Therefore, like the pre-vious methods [2], [4] we focus on separate designs of the subfiltersto control the frequency responses of analysis filters. Given the edgefrequencies �� and �, the design of �� is to control the passbandresponses of both �� and ��. The desired frequency responseof �� is given by [2] and [4]

� � �� for � � �� (6)

Comparing the phase of �� with that of � � �� in (6) yields

� � ��. The error between the amplitude response of ��and its desired amplitude response in the passband is defined as

�� for � � ��

(7)Once �� is found, we design ��

�� to control the stopband re-sponse of ��. The frequency response of the filter�� in (1) canbe rewritten as

��

� � �

��

��

��

To guarantee that �� is linear phase, we choose � � ��

�� . Then, the error function of the amplitude response of ��in the stopband is given by

��

� ��

�for � � �� (8)

where�� is a known constant matrix. With �� and �� known,we design ��

�� to handle the stopband attenuation of ��. Thefilter �� in (2) can be expressed in frequency domain as

��

��

� � ��

��

� � ��

To guarantee that�� is linear phase, we choose � � ��. Then, the error between the amplitude response of�� andits ideal amplitude response in the stopband can be expressed as

��

� ��

for � � ��

where�� and ��, respectively, are a known constant matrix and vector.


Fig. 2. (a) Magnitude responses of the analysis filters, solid line for 2-regularity, dash-dot line for 4-regularity. (b) Analysis scaling function (top) and waveletfunction (bottom) with 2-regularity.

We define the objective function as the error energy in the interestfrequency band �� given by

��

��

��

� ��

�� constant (9)

Here �� is a positive definite matrix, and �� is a constant vector. Theminimization of the mean squared error in general yields the filter withlarge ripples near passband and stopband edge frequencies [1], [16].Therefore, in order to reduce the maximum peak error, the error shouldbe bounded by a given constant

�� for � � �� (10)

It is important to note that the error functions �� are cosine polynomials. Therefore, we can rewrite the inequality con-straints (10) as nonnegative cosine polynomials

�� for � � �� (11)

where ��

�

��

�� and �� are, respectively, a vector and a

constant matrix of appropriate dimensions.On the other hand, the wavelets and filter banks with regularity are

highly desirable for the efficient presentation of piecewise polynomialsignals. It was shown in [4] and [14] that the desired order of regularitycan be imposed on the subfilters. It can be verified that if the halfbandfilter � � �� has �� zeros at � � ��, that is

��

��

��

��

� �� (12)

then the analysis filters �� and �� will have the regularity or-ders of �� , and �� , respectively.It can be shown that the condition (12) is structurally satisfied for �odd. Therefore, the regularity order �� of the linear phase filters mustbe even, and then the regularity condition (12) can be expressed as alinear equation

�� (13)

TABLE IPERFORMANCE COMPARISON

where ��

��

� � � � � �

��

� �� ...

.... . .

...� ��

� ��

� �

The maximum ripple of the error can be minimized by the followingproblem:

� ��

� �� (14)

whose the resulting filter is an equiripple filter with the ripple � ��.It is shown in [1], the equirriple filter has the largest mean square error.To provide the general method that can trade off between the meansquare error and peak error, we use the PCLS design method. For agiven � � � ��, the PCLS design problem is defined by

� ��

�� (15)

It can be seen that (14) and (15) can be expressed as an SDP if andonly if the constraint (11) can be cast into LMIs. Constraints (11) mustbe satisfied for a band of continuous frequencies, i.e., there are infinitenumber of linear inequality constraints on finite number of variablesand, therefore, (15) is called the SIP problem. The approach in [4] forsolving the SIP problem is to approximate the semi-infinite constraintsby discretizing them in frequency. For high filter order, this methodcan become prone to numerical difficulties because a large number ofconstraints can be produced, and the optimal solution cannot guaranteesatisfaction of the semi-infinite constraints.


III. SDP FORMULATION

In this section, we will employ the LMI characterizations of a non-negative trigonometric polynomial to precisely express the semi-infi-nite constraints as LMI constraints [5]–[7], [9]. Let us define trigono-metric curve �� and and its polar cone �� as follows:

��

��

In other words, �� is a set of all possible coefficient vectors �� such that the cosine polynomial � �� is non-

negative for � � �� . Therefore, the semi-infinite constraints ofnonnegative cosine polynomial (11) can be compactly rewritten as

�� and � � �� (16)

The SIP problem (15) can be rewritten in terms of polar cones asfollows:

��

�� (17a)

�� (17b)

�� (17c)

It was shown in [5], [7], [9], and [15] that each constraint�� can be exactly parameterized by a set of linear equations andtwo LMI constraints of two symmetric positive semidefinite matricesand, hence, (17) can be expressed an SDP in the equality form withtwo additional symmetric matrix variables. In order to cast (17) intoanother SDP in inequality form with the minimum number of additionalvariables, we employ convex duality. The dual problem associated with(17) is defined as

��

��

��

��

�

�� (18)

which then reduces to

��

��

�

��

�

��

��

�

��

�

� ��

� ��

�

�� (19)

The optimal solution �� of problem (17) can be achieved from the op-timal solutions ��

�� by the following equation:

��

��

��

�

�

��

�� (20)

Like the polar cone �� , the conic hull �� of the trigono-metric curve �� can also be characterized by LMIs. The followingtheorem will show that �� can be described by LMIswithout introducing any additional variable [15].

Theorem 1: Let �� be the conic hull of the trigonometriccurve �� , and let �� . Then �� if and onlyif the following inequality constraints hold:

��

��

��

�� (21)

Fig. 3. Magnitude responses of the analysis filters obtained by the methods in[14] (dash-dot line), in [4] (dashed line) and our method (solid line).

where the matrices �� are, respectively, createdfrom ��

��

�� by making the change of variables �� .

It is straightforward to show that �� and �� canbe expressed an affine function of variable ��, hence, (21) is two LMIconstraints. Applying Theorem 1 and using Schur’s complement, theoptimization problem (19) can be rewritten as

��

��

�

��

��

��

�

��

��

�

�

��

�

��

��

��

��

(22)

The optimization problem (22) is an SDP and, hence, its optimal solu-tion can be efficiently computed by available SDP software packages[12].

IV. DESIGN EXAMPLES AND APPLICATION TO IMAGE COMPRESSION

In this section, we provide several design examples of triplet half-band filter bank to illustrate the performance of our proposed method.In all examples, we chose � �

� � so that the magnitudes of anal-

ysis filters at � � �� are equal to �� [4], [14].

Example 1: For comparison, we design the triplet halfband filterbank with the same specifications as the example in [2], i.e., � � �� . However, all subfiltershere are imposed to have the regularity order 2 and 4. Moreover, themaximum peak ripples are specified to be � � �� for the regularity order of 2, and to be � � �� for the regularity order of 4. The mag-nitude responses of the analysis filters are shown in Fig. 2(a), and itcan be observed that the frequency responses �� and�� . Moreover, the corresponding scaling and waveletfunctions are shown in Fig. 2(b) for the regularity order of 2. It can beseen that scaling and wavelet functions are symmetric. Table I showsthe design results of using the proposed method and the method of [2]for comparison. It can be seen that although our filters have the regu-larity order of 2 and 4, the passband ripple and stopband attenuation ofour resulting filters are comparable to those of filters without regularityproperty reported in [2]. It should be emphasized that while the Remez


TABLE IIPERFORMANCE COMPARISON

Fig. 4. PSNR versus bit rate of the filters obtained by: Remez algorithm (�o line), Lagrange formula (�� line), the proposed method (�� line). (a) Lena image.(b) Barbara image.

Fig. 5. Reconstructed image Barbara at 0.5 bpp by different filters. (a) Filters obtained by Remez algorithm. (b) Filters obtained by Lagrange formula. (c) Filtersdesigned by the proposed method. (a) PSNR: 31.58 dB. (b) PSNR: 31.58 dB. (c) PSNR: 32.14 dB.

algorithm in [2] cannot impose any regularity condition on filters, ourproposed method can design the filters with a given regularity order.

Example 2: In this example, we illustrate the effectiveness of ourmethod as compared to methods in [4] and [14] which can control thefrequency selectivity and regularity. We adopt the design parametersfrom Example 2 in [14]. In particular, the lengths of analysis lowpassand highpass filters are 29 and 39, respectively. The regularity order offilters is 4. However, the chosen passband and stopband frequencies are�� and �� while the maximum peak ripples arespecified to be �� , and �� . Fig. 3shows the magnitude responses of the analysis filters which are de-signed by the Bernstein polynomial and least squares method [14], theleast squares with peak error constraints [4] with 200 samples of dis-

cretized frequencies in the band of interest, and our proposed method.The performances of the analysis filters are presented in Table II. Wecan see that our method can offer better frequency selectivity as com-pared to the method in [14]. On the other hand, it should be empha-sized that our method can exactly parameterize the semi-infinite peakripple constraints while discretization method in [4] highly relies ona heuristic number of the samples. To illustrate the dependence of thefilter performance on the number of discretized frequencies, we con-sider the method in [4] with different number of discrete samples. Itcan seen from Table II that the performance of discretization methodwith small number of samples is worse than that of our method. How-ever, these two methods offer the same performance as the number ofsamples of the discretization method is large enough.


Example 3: This example illustrates the impact of the frequency se-lectivity and regularity of filters on image compression performance.Like an example in [2], the triplet halfband filter bank chosen forthe experiment has analysis lowpass and highpass filters of lengths17 and 31, respectively. The triplet halfband filter banks in compar-ison are designed by three different methods: Lagrange halfband filterformulation [2], Remez algorithm [2], and our proposed method forfilters with regularity order of 2. The test images in experiment are512� 512 8-bit grayscale images Lena and Barbara which representtypical images with smooth regions and plentiful textures, respec-tively. We choose the set partitioning in hierarchial trees (SPIHT) [10]without entropy coding to compare the relative performance of thefilter banks. Fig. 4 compares the PSNRs of the reconstructed imagesobtained from different filter banks at various bit rates. For both im-ages, the filter bank obtained by Remez algorithm offers the worst ob-jective performance. On the other hand, for the smooth image (Lena),our resulting filter bank has similar objective performance as the filterbank obtained from Lagrange halfband filters, while for the textureimage (Barbara) our resulting filter bank has highest objective perfor-mance. These results indicate that the image coding performance canbe improved by tradeoff between the regularity and stopband attenu-ation. To measure the subjective visual quality, reconstructed imagesof Barbara at 0.5 bits per pixel (bpp) with the different filter banksare shown in Fig. 5. It can be observed that the filter banks withoutregularity obtained by Remez algorithm causes a checkerboarding ar-tifact on the reconstructed image.

V. CONCLUSION

This correspondence has presented a computationally tractable SDPapproach for the design of a class of linear phase finite impulse re-sponse (FIR) triplet halfband filter banks with filters having optimalfrequency responses for a given regularity order. While the method of[4] is not efficient due to the approximation of constraints, our methodcan precisely handle the semi-infinite constraints by LMIs with smallnumber of variables. As compared to the methods of [2] which cannotdesign the filters with an arbitrary regularity order, our method caneasily incorporate the regularity condition into SDP formulation. Ad-ditionally, since the proposed method can flexibly control the filterfrequency responses and regularity, our resulting filters can achieve abetter image coding performance comparing with the filters obtainedby Remez algorithm and Lagrange halfband filter formulation [2], es-pecially for highly textured images.

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Optimal Design of FIR Triplet Halfband Filter Bank and Application in Image Coding