Filter Banks for Image Coding. Ilangko Balasingham and Tor A. Ramstad

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1 Nonuniform Nonunitary Perfect Reconstruction Filter Banks for Image Coding Ilangko Balasingham Tor A Ramstad Department of Telecommunications, Norwegian Institute of Technology, 7 Trondheim, Norway ilangko@teleunitno ABSTRACT Unitary nonunitary lter banks with uniform frequency separation have been extensively studied in the past Improvement in coding gain has been achieved when allowing for nonunitary lter banks in image coders Subjective improvements can be achieved by employing nonuniform lter banks The paper presents a method to construct nonuniform nonunitary perfect reconstruction lter banks A polyphase matrix representation of nonuniform parallel lter banks is given for octave b splitting In this case, results show that the parallel lter banks tend to become the treestructured lter banks It is further shown that the perfect reconstruction property can be met only by examining each stage of the treestructured system in a polyphase matrix representation Finally, frequency responses are presented when the lter banks are optimied for coding gain INTRODUCTION Unitary nonunitary lter banks [,, ] with uniform frequency separation are widely employed in subb coding of speech image signals In such systems, the input signal,, is split into subb signals which have equal bwidths Nonuniform lter banks, if correctly optimied, can alleviate some of the typical artifacts experienced in subb coding, notably ringing when the lters' unit sample responses are long, blocking in the case of short responses High resolution at low frequencies lower resolution at higher frequencies represent a good compromise in terms of subjective quality, although such a system constructed on these premises may sometimes lead to reduced coding gain In this paper, we present nonuniform nonunitary perfect reconstruction lter banks as shown in Fig Here the decimation ratios, k, are Pnot the same for all the subbs The relation, (= k= k) =, where is the number of channels, provides minimum representation in the subb domain Careful selection of channel bwidths decimation ratios can make the decimation process act as modulation of each channel down to a baseb representation To guarantee perfect reconstruction through the analysissynthesis system, constrains among the lter coecients have to be enforced The system is then completely free of aliasing, amplitude phase distortions [] Z~ + h Figure Block diagram of the lter bank POLYPHASE REPRESENTATION The invention of polyphase representation by Bellanger et al [5] in multirate signal processing introduced a method of implementing lters The method gives a great deal of simplication of theoretical results also leads to computationally ecient implementations of decimation/interpolation lters in particular of lter banks A detailed description of the method with ne examples can be found in [] Uniform lter bank Any arbitrary lter, H() = P n= h(n) n, can be decomposed as where with H() = p k () = X k= p k ( ) k () X n= p k(n) n ; () p k (n) def = h(n + k); k : () To appear in Proc NORSIG95

2 Then H() can be written as H() = p( )d(); () where p( ) = p ( ) p ( ) : : : p ( ) T ; where ~ d() is reversed (upside down) version of d() in Eq 5 Let us illustrate this with an example A uniform twochannel lter bank with fourtap FIR lters are given as H LP () = + a + a + a ; () d() = h : : : () i T : (5) H HP () = + b + b + b : () In general, if H() represents the transfer function of a lter bank is dened as H() = [ H () H () : : : H () ] T ; () then it can be compactly written as H() = P( )d(): (7) P( ) called a polyphase matrix is dened as h i T P( ) = p T ( ) p T ( ) : : : p T ( ) : P() Q() h + Figure Uniform twochannel lter bank in polyphase form The analysis polyphase matrix, P(), then becomes P() = + a a + a : () + b b + b Figure shows uniform twochannel lter bank in polyphase form The analysis lter bank is described by P() Q() Figure Uniform lter bank in polyphase form Figure shows a general uniform lter bank in polyphase form where P() Q() denote analysis synthesis polyphase matrices, respectively Decimation interpolation are necessary to have a minimum representation By applying the nobel identities [] we obtain a simplied system where the computational complexity is comparatively reduced Perfect reconstruction is obtained if + P()Q() = c k I; (8) where I is an identity matrix, c k are arbitrary constants If the analysis lter bank is dened as in Eq, then the synthesis lter bank, G(), can be written as G() = Q T ( ) ~ d(); (9) H() = [ H LP () H HP () ] T () d() = T : () If we require that the synthesis polyphase matrix, Q(), also represents an FIR lter bank, then detp() must be equal to a pure delay times an arbitrary constant is given by detp() = (b a ) + (b a + a b a b ) + (a b a b ) : (5) There are three possible ways to satisfy the delay condition We choose to set the following terms from Eq 5 to ero: b a + a b a b = a b a b = Then Q() = P () By using Eq 9, G() is given by G() = [ G LP () G HP () ] T () where the lters are given by G LP () = b ( b + b b ) b b a b + b b + b a b ; (7)

3 G HP () = b + b b + b a b b b a b + b b + b a b + a b b a + a b b b b a b + b b + b a b : (8) The parameters a a have been eliminated in the example This means that once H HP () is xed only one free parameter a can be selected for H LP () Nonuniform lter bank Nonuniform lter banks are generaliations of discrete wavelets where the signal can be decomposed to enhance some of the frequency regions make cruder decompositions in the other regions, as desired The parallel lter bank is the most general form of the lter banks where the treestructured lter bank is a special case In the following subsections both types of lter banks are introduced as examples Parallel lter bank h + Figure Nonuniform threechannel lter bank Assume that we have a threechannel lter bank as shown in Fig To ensure the perfect reconstruction property by polyphase matrix representation, we formulate the problem as a fourchannel uniform lter bank as shown in Fig 5 By applying the nobel identities property [] we show that the analysis lter, h (n), is equal to h (n ) Similarly, the synthesis lter, g (n), is equal to g (n + ) Z~ + h 7 Figure 5 Uniform fourchannel lter bank This shift property is necessary to dene the polyphase matrix as a square matrix in order to nd its exact inverse Let us assume that the lters h (n) h (n) both have lter length ve h (n) has lter length three Hence, the analysis polyphase matrix, P(), can be written as P() = a + a a a a b + b b b b c c c c c c 7 5 : (9) The analysis lter bank, H(), is then dened as H() = H () H () H () H () T : () Following the same procedure as given in Section, we nd the synthesis polyphase matrix, Q(), by setting appropriate terms to ero of the detp() Furthermore, to ensure the shift property introduced in the analysis lter bank to have the same property in the synthesis lter bank, we impose constrains to the synthesis polyphase matrix's, Q(), elements This means that in the synthesis polyphase matrix, the fourth row is the shifted version of the third row This assures the preservation of the same structure as in the analysis lter bank Basically, we started with thirteen free parameters after having imposed the above constrains, the number of free parameters has been reduced to eight The synthesis lter bank can be written as G() = G () G () G () G () T ; () the lters are given by G () = a (c b b c b c ) a (b c + b c b c ) ; () G () = a (c a a c + a c ) + a (a c + a c a c ) ; () where G () = (a c a c ) c (a c a c + a c ) + a c a c c (a c a c + a c ) ; () = a b c a b a c a c a b +a b c a + b a a c b a c : (5) Here k, dened as in Eq 8, is equal to ero

4 Treestructured lter bank A A Figure Treestructured analysis lter bank Figure shows a twostage treestructured analysis lter bank with subanalysis lters H A () H A () dened as H A () = [ H ALP () H AHP () ] T ; () H A () = [ H ALP () H AHP () ] T : (7) Let us assume that the lters are dened as H ALP () = e + f + g ; (8) H AHP () = c + c + c ; (9) H ALP () = p + q ; () H AHP () = r + s : () The parameters in Eq 9 are deliberately chosen equal to the parameters in the highpass lter in the parallel system in Section Following the same procedure as mentioned in Section, each stage of the treestructured lter bank A A can be formulated as subanalysis polyphase matrices, P A () P A (), respectively Then the subanalysis polyphase matrices can be written as e + g f P A () = ; () c + c c p P A () = r q s : () Setting appropriate terms to ero of detp A () detp A () to ensure that the inverse of P A () P A () to become pure FIR lters, the synthesis polyphase matrices Q A () Q A () can be derived thereof By multiplying out the lters in cascade form, the analysis polyphase matrix, P(), can be written as P() = ep + gq fp eq + gp fq er + gs fr es + gr fs c c c c c c 7 5 : () The synthesis polyphase matrix, Q(), can be found by the same procedure as mentioned above The synthesis lter banks obtained by both methods are identical, the synthesis lters de ned as in Eq can be written as G () = c r rc (c s c r) pc es + fpsc + rc eq frqc c s sc ; (5) pc es + fpsc + rc eq frqc G () = c p pc (c q c p) pc es + fpsc + rc eq frqc c q + qc ; () pc es + fpsc + rc eq frqc G () = c e fc + c f : (7) c (c e + fc ) In this way, we have eliminated the computationally cumbersome matrix inversion of a matrix reduced to only two sets of matrices By only imposing the determinants to be pure delays times a constant, the number of free parameters are reduced from ten to nine This means we have one extra free parameter in the treestructured lter bank compared to the parallel lter bank as stated in Section By carefully examining both systems, the lter coecients of the treestructured lter bank can be expressed as a function of the parallel lter bank's lter coecients: p = a f ; e = f(c a + c a ) c a ; q = a f ; r = b f ; s = b f : (8) If the parameter, f, in Eq 8 is a constant not equal to ero, both systems then have equal number of free parameters therefore are identical This is also evident from the results obtained in Section that the extra parameter did not in uence to enhance the coding gain both lter banks become identical have the same coding gain OPTIIZATION OF THE FILTER COEFFICIENTS BASED ON CODING GAIN Coding gain is used as a measure of data compression [] atto Yasuda [7] have derived a compact formula to evaluate the generalied subb coding gain, is given by G SBC = Q (A k= kb k k ) = k (9)

5 where A k = h T k R xx h k ; () B k = k g T k g k : () It is a common practice to assume that the underlying statistics of the image is an AR() process Then the autocorrelation matrix, R xx, is given by R xx = : : : N k : : : N k : : : N k N k N k N k : : : () where denotes the correlation coecient of the input signal N k is the analysis lter length A typical value for in the case of image signals is 95 RESULTS Equation 9 is a function of the lter coecients To obtain the maximum coding gain, we maximie Eq 9 for both types of lter banks The \optimiation toolbox" in ATLAB simulated annealing are used to obtain the lter coecients Z~ + h Figure 7 Nonuniform fourchannel lter bank The two lter banks, shown in Figs 7, have been optimied for coding gain The nonuniform threechannel lter bank, shown in Fig, called FB 5 5 has three bs where the rst two channels, h (n) h (n), have ve taps each, the highpass channel, h (n), has three taps The nonuniform fourchannel lter bank, shown in Fig 7, called FB has four bs where the rst two channels h (n) h (n) have fteen taps each, the third channel, h (n) has seven taps, the highpass channel, h (n), has three taps The analysis synthesis responses of FB 5 5 FB are shown in Figs 8, 9,,, respectively Coding gains of FB 5 5 FB are 79 db 95 db, respectively It is interesting to note that the lowpass channel of the analysis lter bank tends to suppress dcleakage the bpass highpass channels Amplitude response in db 5 FB 5_5_ Figure 8 Analysis frequency responses Amplitude response in db 5 FB 5_5_ Figure 9 Synthesis frequency responses 5 FB 5_5_ analysis 5 5 FB 5_5_ synthesis 5 5 Figure Impulse responses of the lter bank; lowpass at top of the lter bank attenuate frequencies centered around 5 5 The lift in the lowpass response is important to enhance energy packing This lift is more evident in Fig is referred to as the half whitening property of the nonunitary lter banks [, ] To reduce the blocking eects at low bit rates, the synthesis unit sample responses as shown in Figs 9 should smoothly decay without any ripples [8] Impulse responses are shown in Figs for FB 5 5 FB 5 5 7, respectively From this it is also obvious that the lters do not have linear phase 5

6 Amplitude response in db Amplitude response in db 5 FB 5_5_7_ Figure Analysis frequency responses 5 FB 5_5_7_ Figure Synthesis frequency responses FB 5_5_7_ analysis FB 5_5_7_ synthesis Figure Impulse responses of the lter bank; lowpass at top 5 CONCLUSION A new approach to construct nonuniform nonunitary lter banks is presented By employing the nobel identities setting appropriate terms to ero in the determinant of the analysis polyphase matrix imposing the shift property as introduced in the analysis polyphase matrix to the synthesis polyphase matrix, we obtain nonuniform nonunitary perfect reconstruction lter banks Two independent investigations done on parallel treestructured lter banks when optimiing the lter coecients based on coding gain, show that both lter banks converge become identical when decimation/interpolation ratios are integers powers of two It is also interesting to note that each stage of the treestructured lter bank can be formulated as subpolyphase matrices By setting appropriate terms to ero of the determinants of the subanalysis polyphase matrices, we construct nonuniform nonunitary perfect reconstruction systems By adopting this approach, we eliminate the computationally cumbersome matrix inversion of large matrices The frequency responses of the lter banks indicate that they have the half whitening property Coding gains obtained by the lter bank examples in Section are comparable to those mentioned in [8] Due to the nonlinear phase of the lters, caution is needed to avoid transient eects at the edges of the ltered image REFERENCES [] T A Ramstad, S O Aase, J H Husy, Subb Compression of Images { Principles Examples North Holl: ELSEVIER Science Publishers BV, To be published, 995 [] S O Aase T A Ramstad, \On the optimality of nonunitary lter banks in subb coders," IEEE Trans Image Processing, 995 To be published [] P P Vaidyanathan, ultirate Systems Filter Banks Englewood Clis: Prentice Hall, 99 [] PQ Hoang P P Vaidyanathan, \Nonuniform multirate lter banks theory design," Proc Int Symp on Circuits Systems (ISCAS), pp 7{7, 989 [5] Bellanger, G Bonnerot, Coudreuse, \Digital ltering by polyphase network: application to sample rate alteration lter banks," IEEE ASSP againe, vol, pp 9{, 97 [] N S Jayant P Noll, Digital Coding of Waveforms, Principles Applications to Speech Video Englewood Clis, New Jersey: PrenticeHall, Inc, 98 [7] J atto Y Yasuda, \Performance evaluation of subb coding optimiation of its lter coecients," in Proc SPIE's Visual Communications Image Processing, pp 95{, Nov 99 [8] S O Aase, Image Subb Coding Artifacts: Analysis Remedies PhD thesis, The Norwegian Institute of Technology, Norway, ar 99

Filter Banks with Variable System Delay. Georgia Institute of Technology. Atlanta, GA Abstract

A General Formulation for Modulated Perfect Reconstruction Filter Banks with Variable System Delay Gerald Schuller and Mark J T Smith Digital Signal Processing Laboratory School of Electrical Engineering