Ladder structures for Multidimensional Linear Phase Perfect. Reconstruction lter banks and Wavelets. Philips Research Laboratories, Abstract

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1 Ladder structures for Multidimensional Linear Phase Perfect Reconstruction lter banks and Wavelets A.A.C. Kalker I.A.Shah Philips Research Laboratories, Eindhoven, The Netherlands Abstract The design of multidimensional lter banks and wavelets have been areas of active research for use in video and image communication systems. At the same time ecient structures for the implementation of such lters are of importance. In -D, the well known lattice structure and the recently introduced ladder structure [] are attractive. However, their extensions to higher dimensions (m-d) have been limited. In this paper we reintroduce the ladder structure, with the purpose of transforming the structure into m-d using the McClellan transform. Introduction Recently the ladder structure was introduced for the design and ecient implementation of -D perfect reconstructing lter banks (PRFB) []. These structures have the advantage of not only being insensitive to coecient quantization but, under fairly general conditions, also to quantization of intermediate results. In [6] it is shown that the class of -D -channel PRFB's which can be realized by a ladder structure coincides with the class of -D -channel bi-orthogonal lter banks. It will be shown in this paper that for higher dimensions every -channel ladder structure PRFB continues to constitute a bi-orthogonal lter bank, but that the converse is no longer true. This negative result is caused by the fact that the Euclidean algorithm is not applicable for polynomials with two or more variables. In a previous paper the authors showed how -D bi-orthogonal PRFB's can be transformed into m-d bi-orthogonal PRFB's using the McClellan transform [4]. In this paper we will extend the scope of this transform. Not only perfect reconstruction (PR) is retained, but also the possibility to implement these transformed lter banks with an m-d ladder structure. This expands the scope of ladder structures from -D to m-d. After dening the notion of a canonical ladder structure for -D PRFB's we show that the transformation retains this canonical ladder structure. In particular the m-d ladder has the same number of steps as the -D canonical ladder. Canonical ladders and their transforms have some extra advantages over ladders as dened in []. First of all, the maintenance of linear phase is guaranteed under quantization and numerical inaccuracies. Secondly, it speeds up the polynomial factorization resulting in ladders of reduced height. Section 3 gives a short overview of the theory of ladder structures. We introduce bi-orthogonal lter banks and show that every ladder gives rise to a bi-orthogonal lter bank. In Section 4 we recall the McClellan transform and its application to PRFB design. We show how transformed PRFB's can be realized with a ladder structure. As a particular application we dene the concept of canonical ladder structures, show their behavior under transforms, and indicate their advantages. Section 5 presents a design example. We end with some concluding remarks. Notations In this paper, digital signals (be it -D or m-d) are represented in the z-domain. We will generally not distinguish between -D and m-d. If the dimensionality of is unimportant or clear from the context, we use the notation or (z). If it is important to stress that the signal is m-d we will use the notation (~z). Complex conjugation of a signal will be denoted by?. Expressions ; M; : : : denote sampling rasters. Non-singular integer matrices L; M; : : : are used for the representations of rasters. Downsampling on a sampling raster with representation L is denoted by \(:) # L ". If the representation L is implicit or not relevant, the notation \(:) # " is used. Moreover, in case is clear from the

2 V + W W V f f V W W V Figure : Ladder step. Figure : Inverse ladder step. V f g h l W V W Figure 3: Ladder with 4 steps. context we will use \(:) #". For upsampling we use the notations \(:) " L ", \(:) " " or \(:) "". If not otherwise specied, up- and downsampling in the -D case will mean up- and downsampling by. For -channel sampling rasters the notation even and odd denotes the parts of living on and o the sampling raster respectively. A signal is even (odd) if even = ( odd = ). In the -D case odd and even are dened with respect to the lattice of even integers. One easily shows that the parity of signals behaves nicely under multiplication, e.g. if is odd and Y is even then Y is odd. These parity properties will be used without mentioning. Filters are represented by their impulse response. In particular when talking about a lter H(z) we either mean its impulse response or we interpret H(z) as the function (z)! H(z)(z). From the context it should be clear what interpretation is meant. Given an m-d variable z and a vector n = (n ; : : :; n m ) t Z m we dene z n by z n = my i= z ni i () Given a rational m m matrix D the notation z D denotes a variable transformation which is completely determined by the equality (z D ) n = z Dn () where n ranges over Z m. Component-wise multiplication of m-d vectors is denoted by juxtaposition. A signal (z) is linear phase with phase Q m and symmetry! f?; g if the equality (z) =!z? (z?i ) (3) holds where I is the m m identity matrix. If! = (?) the signal will be called symmetric (antisymmetric). The signal is called zero-phase if is symmetric with phase = ~0. 3 Ladder structures In [] ladder structures are introduced as a particular form of PRFB's, similar to lattice structures. A ladder structure is a cascade of ladder steps. Each ladder step performs a simple, invertible operation on a two dimensional vector. Cascading ladder steps into a ladder structure allows one to build complex transformations which are guaranteed to be invertible. In this section we rst recount the basic denitions concerning ladders. Next we show how to use

3 ladders for PRFB construction. We prove that ladders build bi-orthogonal lter banks and that in the -D case this classication is exhaustive. 3. Basics The basic idea behind ladders is very simple. If we assume that addition and subtraction can be implemented in a mathematically precise manner then the following identity holds: (a + b)? b = a; (4) i.e. addition and subtraction are inverse operations. A ladder step is now dened as an operation which transforms a two dimensional vector V into a new two dimensional vector W according to a formula of the form w v f(v W = = ) v ; or (5) w v v f(v ) where V = v v and where f is an arbitrary function. This is graphically shown in Fig.. The function f will be called the step function of the ladder step. A ladder step is concisely described by its step matrix S dened by where denotes matrix multiplication. So allowed step matrices take the form f ; 0 W = S V (6) 0 f : (7) In this matrix notation the symbol \f" is interpreted as function application and the \0"and \" are interpreted classically, i.e. as multiplication operators. The following proposition is now easily veried if one assumes Eq. 4 to hold: Proposition The inverse of a ladder step is also a ladder step. The ladder matrix of the inverse diers only in the sign of the step function (see Fig. ). We now proceed to dene ladders. Denition A ladder is a cascade of ladder steps. A ladder is completely characterized by an ordered sequence of step matrices fs ; S ; : : :; S n g with input/output relation Fig. 3 shows a typical ladder. And naturally we have: W = S n : : : S V (8) Proposition The inverse of a ladder fs ; S ; : : :; S n g is the ladder fs n? ; : : :; S? g. 3. Ladders in digital signal processing This paper discusses ladder structures in the context of digital signal processing. To be precise we assume that our vectors V and W are vectors of signals represented in the z-domain. In this context application of a step function amounts to multiplication by a lter response. Therefore we can view step matrices as linear transformations operating on a two dimensional space of signals. It follows that to every ladder we can associate a linear transformation, viz. the product of all the step matrices. With simple linear algebra one proves: Proposition 3 The linear transformation associated with a ladder has determinant equal to. The converse of this proposition is also true. a b Theorem Every matrix such that: c d

4 . a, b, c, d are polynomials in one variable,. ad? bc = can be realized as a ladder. The proof of this theorem is given in [6]. The proof relies on the fact that for polynomials in one variable one can use the Euclidean algorithm. A slight extension of this theorem takes the parity of the polynomials into account. a b Theorem Every matrix such that: c d. a; b; c; d are polynomials in one variable, and. a and d are even, 3. b and c are odd, 4. ad? bc = can be realized as a ladder with odd step polynomials. The word \polynomial" in the last two theorems can be replaced by the words \nite length lters". Theorem 3 The assertions of Thm. and Thm. are also valid if one replaces polynomial by nite length lter. The extension of this result to m-d, m, is false. A counter example is given by the matrix D, where + xy?y D = x? xy (9) In [6] it is proved that though D had determinant, it can not be realized as a ladder. 3.3 Ladders and bi-orthogonal lter banks Ladders can be used to generate -channel PRFB's. In fact, it is shown in Thm. 4 that every -D bi-orthogonal lter bank can be generated by the ladder structure. Analyzing the ladder structure yields various well known PRFB identities. The construction of the PRFB starts with a -channel sampling raster and an odd monomial m(z). Given a signal we split it into its two polyphase components, resulting in a vector, ~ ~ = () # (m(z)) # : (0) Transforming the vector ~ with a ladder into a vector ~ Y builds the analysis part of a PRFB. The PR property is easily shown if one realizes that the analysis ladder is invertible. This inverse ladder builds the synthesis part of the PRFB. We now perform the analysis of the PRFB constructed in this way. Letting L denote the ladder matrix we have: ~Y = L ~ = L () # (m(z)) # () The ladder structure corresponding to Eq. is shown in Fig. 4. We refer to this form as the ladder in the downsampled domain: the input signal is split into its polyphase components and downsampled before passing through the ladder. However, the conventional way of representing lter banks is in the upsampled domain. The derivation below shows how we can move the downsamplers and the odd monomial through the ladder. The result of this process is a ladder

5 L Y m(z) Y Figure 4: Ladder structure in the downsampled domain. L Y m(z) Y Figure 5: Moving the downsamplers across the ladder. L ~ m(z) Y Y Figure 6: Ladder structure in the upsampled domain. structure in the upsampled domain. This latter structure will be shown to implement a certain class of lter banks well-known from literature. We have: 0 ~Y = ((L ") ) # () 0 m(z) = ( 0 m(z) 0 m(z)? (L ") ) # 0 m(z) 0 = ( 0 m(z) ~ L {z } ~L ) # : (3) Fig. 5 shows the intermediate step of Eq., where it should be noted that all the entries of (L ") are even. The ladder in the upsampled domain, as in Eq. 3, is shown in Fig. 6. The matrix ~ L is of the form E O ~L = O E where E i is even and O i is odd. Writing H i = E i + O i, and ~Y = Y Y (4) the following proposition is easily proved.

6 Proposition 4 With notations as above we have Filter banks with the properties above are well known from literature. (H ) # = Y (5) (m(z)h ) # = Y (6) (E E? O O ) = (7) Denition Any -channel lter bank with analysis lters fh (z); m(z)h (z)g having an input/output relation as in Eq. 5 and Eq. 6 and satisfying Eq. 7, is called a bi-orthogonal lter bank. The pair fh (z); H (z)g characterizes a bi-orthogonal lter bank. One can easily verify that the synthesis lters in the above dened bi-orthogonal case are given by fg (z); m(z)? G (z)g where Remark Eq. 7 is equivalent to G (z) = E (z)? O (z) and G (z) = E (z)? O (z): (H G ) #= : (8) Remark Yet another way of phrasing Eq. 7 makes use of the modied polyphase matrix A H;H, which is dened by With this denition Eq. 7 is equivalent to Applying Thm. 3 the following theorem is proved. Theorem 4 The following properties hold: (H ) A H;H = even (H ) odd : (9) (H ) odd (H ) even det(a H;H ) = (0). A -channel lter bank constructed from a ladder structure, as above, constitutes a bi-orthogonal lter bank.. Every -D -channel bi-orthogonal lter bank can be realized with a ladder structure. 4 Transformations This section recalls the main results from [4] where it is shown how -D PRFB's can be transformed into m-d PRFB's. The technique which is used was introduced by McClellan []. In [4] the restrictions on this transformation in order to maintain PR are stated. We will give an overview of the main results restricted to the -channel case. 4. Ingredients of the McClellan transform In order to apply the McClellan transform we need:. a -D zero-phase lter H(z),. an m-d kernel lter K(~z), such that K is bounded between? and + on the m-d unit circle. 4. Steps of the McClellan transform The McClellan transform consists of two stages: an extraction stage and a substitution stage. In the extraction stage we rewrite the impulse response of the -D lter H(e j! ) as function a of cos(!). In z-domain representation this amounts to H(z) = H S (S(z)) () where S(z) = (z + z? ). In the substitution step we replace cos(!) by the m-d kernel K. H K (~z) = H S (K(~z)): () By choosing appropriate kernels K, one can design useful m-d lters whose shape is determined by the kernel K and whose quality is determined by the quality of the -D lter H.

7 4.3 Transforming bi-orthogonal lter banks Now suppose we are given a -D bi-orthogonal lter bank fh (z); H (z)g as in the previous section. Moreover assume that these lters are zero-phase. As the synthesis lters are modulated versions of the analysis lters, they are also zero-phase and so we are allowed to extract a kernel S(z) = (z + z? ) from all 4 lters. Substitution with an m-d kernel K(~z) yields 4 m-d lters. Given an m-d sampling raster, we can build in a natural way an m-d -channel lter bank. All we have to do is to replace the up and downsampling by, by up and downsampling on, and secondly replace the -D delay z by some m-d delay m(~z) (see Fig. 5). The obvious question to ask is: will this transformed lter bank be perfect reconstructing? In [4] the following theorem is proved. Theorem 5 The transformed lter bank is an m-d bi-orthogonal lter bank if and only if the following two conditions are satised:. The kernel K(~z) is odd (with respect to ),. the delay m(~z) is odd (with respect to ). 4.4 Ladders for m-d bi-orthogonal lter banks By Prop. 3 every -D bi-orthogonal lter bank can be realized as a ladder. For higher dimensions a similar assertion is not true. However, if the m-d lter bank is designed using the McClellan transform as above, a ladder decomposition is possible. Theorem 6 Every m-d bi-orthogonal lter bank which can be designed using the McClellan transform is realizable with a ladder structure. Proof (sketch) We are given a -D bi-orthogonal lter bank fh (z); H (z)g. Writing H i = E i + O i as before, bi-orthogonality is equivalent to E E? O O = : (3) It is easy to show that the extraction of S(z) = (z + z? ) from H i distributes over the odd and even components of H i. Performing the extraction we get polynomials h i (x), e i (x) and o i (x) such that. H i (z) = h i (S(z)),. E i (z) = e i (S(z)), 3. O i (z) = o i (S(z)), 4. h i (x) = e i (x) + o i (x), 5. e i (x) is even, 6. o i (x) is odd and 7. e e? o o =. Applying Thm. we nd a ladder decomposition e o = s o e : : : s n (4) with odd step polynomials. We call the ladder of Eq. 4 the basic ladder structure. By inserting the kernel K in both sides of the equality of Eq. 4 we nd a ladder decomposition for the transformed lter bank. Denition 3 Choosing K to be S(z) and inserting it into the basic ladder structure, we nd a ladder decomposition for the original -D lter bank. We call such a ladder structure a canonical ladder structure. Proposition 5 The number of steps in the transformed m-d lter bank is equal to the number of steps in a canonical ladder structure of the -D lter bank.

8 Remark 3 For a (transformed) canonical ladder, the symmetry of the total lter bank is now controlled by the individual step lters. As long as the step lters are symmetric, total symmetry is guaranteed. It is therefore easy to guarantee the symmetry of the total lter bank, even if one allows quantization of lter coecients. In Section 5 we will present a design example making the above theory more explicit. 4.5 Advantages of ladder structures Realizing a lter bank with a ladder structure has an advantage over a direct implementation if one takes hardware limitations into account. Due to restricted precision, an implementation of a lter bank uses lter coecients which are approximations of the correct coecients. These approximations may destroy the PR property. This can be overcome in a ladder structure as long as one guarantees that addition and subtraction associated with a ladder step are implemented mathematically correct (e.g. by introducing a truncator in a step function). When m-d ladder structures are designed using the procedure described above, there are some extra advantages. Firstly, note that in the extraction process, S(z) = (z + z? )= x the symmetry is removed from the problem. The substitution x K(z ; z ) re-inserts the symmetry into the m-d ladder structure.this process of removal of symmetry, factorization and re-insertion of symmetry guarantees preservation of linear phase as corresponding coecients are treated the same way by construction of the synthesis procedure. Secondly, the resulting x-domain polynomial has a degree half that of the original z-domain polynomial. This second feature will, in general, speed up the ladder decomposition (factorization) processing substantially. Moreover, a ladder with reduced height will emerge. As the height of the ladder structure does not change in the insertion process a reduced height m-d ladder structure will result, which has advantages with respect to error propagation 5 Implementation Example A linear phase factorization of a maximally at polynomial, called lagrangian half band lters in [9], yields a biorthogonal perfect reconstruction lter bank pair. Since each lter contains zeros at, they are conjectured to yield regular wavelets [9]. We will synthesize a m-d ladder structure by using the above lters. The two analysis lters from [9] are: H (z) = 04 ( (z + z? )? 3(z + z? )? 78(z 3 + z?3 ) + 34(z 4 + z?4 ) + 0(z 5 + z?5 )? 5(z 6 + z?6 )) (5) H (z) = (? z? z? ) (6) It is easy to see that the modied polyphase matrix A H;H (z), see Eq., has determinant equal to. General lter bank theory tells that H and H make up the analysis part of a PRFB [8] [7]. Even a ladder decomposition is allowed ([6], Thm. 3 above). We will transform the above -D PRFB ladder structure into a -D diamond-shape PRFB ladder structure suitable for quincunx subsampling. We begin with the extraction S(z) = (z + z? )= transforming A H;H (z)into a h;h (x) given by:? 9 a h;h (x) = 6 x + x 4? 5 6 x6 + 9x? 6 x x5 (7)?x One computes that a h;h (x) has determinant equal to as expected from the proof of Thm. 6. a h;h (x) into a ladder we obtain ([6], Thm. 6 above) 9 a h;h (x) = x? 6 x x5 0 0?x Decomposing Note that all the step polynomials of the ladder decomposition above are odd. To transform the ladder structure of Eq. 8 into -D lter bank with a diamond-shape lowpass lter, we perform a substitution x K(z ; z ), where (8)

9 K(z ; z ) = (z + z? + z + z? )=4. One easily checks that K is odd as required by Thm. 5. This substitution transforms Eq. 8 into: 9 a h;h (K(z ; z )) = K(z 6 ; z )? K(z ; z ) K(z 6 ; z ) 5 0 (9) 0?K(z ; z ) Eq. 9 is the ladder decomposition of a -D diamond-shape lter bank suitable for quincunx subsampling. This structure is in the upsampled domain. However, from implementation considerations it is desirable to have a realization in the downsampled domain. In [5] it is proved that such a realization is always possible. 6 Conclusions In this paper we have shown how ladder structures can be used to build bi-orthogonal m-d lter banks and that this construction is complete in the -D case. In the m-d case not every bi-orthogonal lter bank can be realized as a ladder and we presented a counter example. In a previous paper [4] we have shown that the McClellan transform can be used to design m-d lter banks by transforming -D lter banks. If certain conditions are satised, the transform will preserve the PR property of the -D lter bank. This paper extends the scope of the McClellan transform. We have shown that not only PR is preserved but also the -D canonical ladder structure. The advantages of ladder structures for implementing lter banks are mentioned. Firstly, the PR property of ladders is insensitive to coecient and data corruption. Secondly, the zero-phase property of the lters in the lter bank can be totally incorporated in the step functions, thereby preserving symmetry if proper restrictions are placed on coecient quantization. Thirdly, by transforming lter banks to the x-domain a reduction of the height of the ladders can be achieved. References [] A.A.M.L. Bruekers, A.W.M. van den Enden, New Networks for Perfect Inversion and Perfect Reconstruction, IEEE Journal on Selected Areas in Communications, vol. 0, no., January 99. [] J.H. McClellan, The Design of Two-Dimensional Digital Filters by Transformations, 7 th Annual Princeton Conf. Inf. Sci. and Syst., Proc., pp. 47{5,973. [3] I. Shah, A. Kalker, Theory and Design of Multidimensional QMF Sub-Band Filters From -D Filters Using Transforms, Proc. of the 4 th Int. Conf. on 'Image Processing and its Applications', 99 [4] I. Shah, A. Kalker, Generalized Theory of Multidimensional M-Band lter bank Design to appear in Proc. of Eusipco, Aug. 99. [5] A. Kalker, I. Shah, Bi-orthogonal lter banks and ladder structures: an algebraic approach, In preparation. [6] L.Tolhuizen, H.Hollmann, Which polynomial pairs allow a ladder decomposition?, preprint 99. [7] M.J.T.Smith, T.P.Barnwell, A procedure for designing exact reconstruction lter banks for tree structured sub-band coders, Proc. IEEE ICASSP, San Diego, March 984. [8] M.Vetterli, lter banks Allowing Perfect Reconstruction, Signal Processing, Vol. 0, No.3, April 986, pp [9] R.Ansari, C.Guillemot, J.Kaiser, Wavelet Construction Using Lagrange Halfband Filters, IEEE Trans. on Circuits and Systems. Vol. 38, No. 9, September 99, pp 6-8. [0] J. Kovacevic and M. Vetterli, Nonseparable Multidimensional Perfect Reconstruction lter banks and Wavelet Bases for R n, Columbia University Tech. Report, CU/CTR/TR -90-0, October 99.