FIR Filter Banks for Hexagonal Data Processing

Transcription

1 IEEE TRANS. IMAGE PROC. VOL. 7 NO. 9 SEP. 008 FIR Filter Banks for Hexagonal Data Processing Qingtang Jiang Abstract Images are conventionally sampled on a rectangular lattice. Thus traditional image processing is carried out on the rectangular lattice. The hexagonal lattice was proposed more than four decades ago as an alternative method for sampling. Compared with the rectangular lattice the hexagonal lattice has certain advantages which include that it needs less sampling points; it has better consistent connectivity and higher symmetry; the hexagonal structure is also pertinent to the vision process. In this paper we investigate the construction of symmetric FIR hexagonal filter banks for multiresolution hexagonal image processing. We obtain block structures of FIR hexagonal filter banks with -fold rotational symmetry and -fold axial symmetry. These block structures yield families of orthogonal and biorthogonal FIR hexagonal filter banks with -fold rotational symmetry and -fold axial symmetry. In this paper we also discuss the construction of orthogonal and biorthogonal FIR filter banks with scaling functions and wavelets having optimal smoothness. In addition we present a few of such orthogonal and biorthogonal FIR filters banks. Index Terms Hexagonal lattice hexagonal data -fold rotational symmetry -fold axial symmetry orthogonal and biorthogonal FIR hexagonal filter banks orthogonal and biorthogonal hexagonal wavelets. EDICS Category: MRP-FBNK I. INTRODUCTION Traditional -D data (image) processing is carried out on the rectangular lattice since -D data is conventionally sampled at the sites (points) on a square or rectangular lattice. See a square lattice in the left part of Fig.. The hexagonal lattice (in the right part of Fig. ) was proposed more than four decades ago in as an alternative method for sampling and since then it has been used in numerous applications. Compared with a rectangular lattice a hexagonal lattice has certain advantages see e.g. -9. It was shown in that for functions band-limited in a circular region in the frequency domain the hexagonal lattice needs a smaller number (about.4% smaller) of sampling points to maintain equally high frequency information than the square lattice. The hexagonal structure has better consistent connectivity: each elementary cell of a hexagonal lattice has six neighbors of the same type while an elementary cell of a square lattice has two different types of neighbors. The hexagonal structure possesses higher symmetry: a regular hexagonal lattice has -fold symmetry while a square lattice has 8-fold symmetry. Other advantages of the hexagonal structure over the square structure include that it offers greater angular resolution of images and it is closely related to the human visual system. Hence Q. Jiang is with the Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis MO 6 USA jiangq@umsl.edu web: jiang. Research supported by UM Research Board 0/05 and UMSL Research Award 0/06. Fig.. Rectangular lattice (left) and hexagonal lattice (right) the hexagonal lattice has been used in many areas such as edge detection 0 and pattern recognition -6. The hexagonal lattice has also been applied in Geoscience and other fields. For example in the Soil Moisture and Ocean Salinity (SMOS) space mission led by the European Space Agency the data collected by the Y-shaped antenna of the SMOS space mission is hexagonal data (sampled on a hexagonal lattice) 7 8. A hexagon-based grid has been adopted by the U.S. Environmental Protection Agency for global sampling problems 9 0. Despite numerous advantages of the hexagonal lattice multiresolution (multiscale) hexagonal image processing is a research area with slow pace of activity as pointed out in 7. One probable reason for this could be that most researchers in the Wavelets community are accustomed to the traditional rectangular lattice. Another reason is probably that current approaches have encountered difficulties in the construction and design of desirable hexagonal filter banks to be used for multiresolution image processing. To the author s best knowledge 4-6 are the papers available on the construction/design of hexagonal filter banks with both lowpass and highpass filters constructed. presents a few FIR (finite impulse response) hexagonal filter banks which achieved near orthogonality. provides one 7-channel ( 7-refinement) FIR hexagonal filter bank for image coding. The authors in designed FIR hexagonal filter banks by minimizing the filter bank error and intra-band aliasing error function and applied their filters to image compression and orientation analysis. The highpass filters used in are suitable spatial shifting and frequency modulations of the lowpass filter. FIR hexagonal filter banks was also designed in 4 by the same method as in but with a different filter bank error and intra-band aliasing error function. The FIR filter banks designed in 4 are not perfect reconstruction filter banks. Construction of biorthogonal hexagonal filter banks was fully investigated in 5 and a few biorthogonal FIR filter banks were constructed there. However it is difficult to construct biorthogonal filter banks with smooth wavelets by their approach which also results in filters with large filter lengths. The author in 4 (also in 6) proposed a novel

2 IEEE TRANS. IMAGE PROC. VOL. 7 NO. 9 SEP. 008 block structure of orthogonal/biorthogonal FIR hexagonal filter banks with certain symmetry. However a rigorously mathematical proof of the symmetry and the (bi)orthogonality of the filter banks in 4 is desired and the issue of how to select the free parameters in these filter banks needs to be addressed. Though the hexagonal filter banks designed in are not perfect reconstruction filter banks experimental results on their applications to image compression and orientation analysis carried out in and their applications to digital mammographic feature enhancement and the recognition of complex annotations in are appealing. Therefore the construction/design of hexagonal filter banks deserves further investigation. The main objective of this paper is to construct orthogonal and biorthogonal FIR hexagonal filter banks with certain symmetry which is pertinent to the symmetry structure of the hexagonal lattice. This paper is organized as follows. In Section II we first briefly show that the problem of filter construction along the hexagonal lattice can be transformed into that along the square lattice of Z. After that we discuss the symmetry of filter banks and review some basic results on orthogonal/biorthogonal filter banks. In Section III we present block structures of orthogonal and biorthogonal FIR filter banks with -fold rotational symmetry. In Section IV we provide block structures of orthogonal and biorthogonal FIR filter banks with -fold axial symmetry. These structures include that in 4. In both Sections III and IV we also discuss the construction of orthogonal and biorthogonal FIR filter banks with scaling functions and wavelets having optimal smoothness. In this paper we use the following notations. For x = x x T y = y y T x y denotes their dot (inner) product x T y. For a function f on IR ˆf denotes its Fourier transform: ˆf(ω) = IR f(x)e ix ω dx. For a matrix M we use M to denote its conjugate transpose M T and for a nonsingular matrix M M T denotes (M ) T. For ω = ω ω T let z = e iω z = e iω. () II. PRELIMINARIES In this section after showing that the problem of filter construction along the hexagonal lattice can be transformed into that along the square lattice we provide some basic results on the symmetry and the orthogonality/biorthogonality of filter banks. A. Transforming the hexagonal lattice to the square lattice Z Most multiresolution analysis theory and algorithms for image processing are developed along the square lattice with sites k Z though they could be established along general lattices (see e.g. 7 for the shift-invariant theory along general lattices). To design hexagonal filter banks here we transform the hexagonal lattice to the square lattice so that we can use the well-developed integer-shift multiresolution analysis theory and methods. Let G be the regular unit hexagonal lattice defined by G = {n v + n v : n n Z} () v v U U Fig.. Regular unit hexagonal lattice (left) and square lattice Z (right) Fig.. S v Symmetric axes S S S in hexagonal lattice where v = 0 T v = T. Let U be the matrix defined by U =. () v 0 Then U transforms the regular unit hexagonal lattice into the square lattice Z. See Fig.. For a hexagonal filter H(ω) = 4 g G H ge ig ω with (real) impulse response H g (in this paper a factor 4 is added for convenience) by the transformation with the matrix U we have a corresponding filter h(ω) = 4 k Z h ke ik ω for square data (squarely sampled data) with its impulse response h k = H U k. Conversely corresponding to a square filter (filter for square data) h(ω) = 4 k Z h ke ik ω we have a hexagonal filter H(ω) = 4 g G h Uge ig ω. The matrix U also transforms the scaling functions and wavelets along the hexagonal lattice to those along the square lattice Z. For example if Φ is the scaling function associated with a lowpass hexagonal filter H(ω) = 4 g G H ge ig ω namely it satisfies S S Φ(x) = g G H g Φ(x g) x IR then φ defined by φ(x) = Φ(U x) is the scaling function associated with square filter h(ω) = 4 k Z h ke ik ω where h k = H U k. Therefore to design hexagonal filters we need only to construct filters along the traditional lattice Z. Then the matrix U will transform the filters scaling functions and wavelets along the lattice Z into those along the hexagonal lattice. B. Symmetry of filter banks Since the hexagonal lattice has the highest degree of symmetry it is desirable that hexagonal filter banks designed also have certain symmetry pertinent to the symmetric structure of

3 JIANG: FIR FILTER BANKS FOR HEXAGONAL DATA PROCESSING the hexagonal lattice. In this paper we consider two types of symmetry: -fold rotational symmetry and -fold axial symmetry which are defined below. Definition : A hexagonal filter bank {PQ () Q () Q () } is said to have -fold rotational symmetry if its lowpass filter P(ω) is invariant under the π and 4 π rotations and its highpass filters Q () and Q () are the π and 4 π (anticlockwise) rotations of highpass filter Q () resp. Let S S and S be the lines in the regular unit hexagonal lattice shown in Fig.. Definition : A hexagonal filter bank {PQ () Q () Q () } is said to have -fold axial symmetry if P(ω) is symmetric around S S and S and Q () is symmetric around the axis S and Q () and Q () are the π and 4 π (anticlockwise) rotations of Q () resp. Let R R Ñe W and S e be the matrices defined by R = Ñ e = S e = R = W =. 0 0 Then one can easily show that {PQ () Q () Q () } has -fold rotational symmetry if and only if for all g G P Rg = P Rg = P g Q () g = Q () R Q() g = Q () g R ; (4) g and that {PQ () Q () Q () } has -fold axial symmetry if and only if for all g G { PÑeg = P Wg = P Seg = P g Q () = Ñ Q() g Q () g = Q () eg R Q() g = Q () g R. (5) g Observe that R = WÑe R = S e Ñ e. Thus if {PQ () Q () Q () } satisfies (5) then it satisfies (4). Therefore if a hexagonal filter bank has -fold axial symmetry then it has -fold rotational symmetry. The -fold rotational symmetry is considered in 5 where it is called the hexagonal symmetry. Both the -fold rotational symmetry and the -fold axial symmetry are closely related to the symmetry structure of the hexagonal lattice. In this paper we consider filter banks with these two types of symmetry. Compared with filter banks with -fold axial symmetry filter banks with -fold rotational symmetry have less symmetry but they provide more flexibility for the construction of filters. It should be up to one s specific application to choose filter banks with -fold rotational or axial symmetry. For a hexagonal filter bank {PQ () Q () Q () } let {pq () q () q () } be the corresponding square filter bank after the transformation by the matrix U in (). Let R R N e W and S e denote the matrices U R U U R U UÑeU U WU and U S e U resp. namely R = 0 R = 0 (6) N e = 0 0 W = 0 0 S e =. (7) Then one can show that PQ () Q () Q () satisfy (4) if and only if p Rk = p Rk = p k q () k = q() R k q() k = q() R k k Z ; (8) and that PQ () Q () Q () satisfy (5) if and only if { pnek = p Wk = p Sek = p k q () N = ek q() k q() k = q() R k q() k = q() R k k Z. To summarize we have the following proposition. Proposition : Let {PQ () Q () Q () } be a hexagonal filter bank and {pq () q () q () } be its corresponding square filter bank. Then {PQ () Q () Q () } has -fold rotational symmetry if and only if {pq () q () q () } satisfies (8); and {PQ () Q () Q () } has -fold axial symmetry if and only if {pq () q () q () } satisfies (9). In the following for the convenience we say a square filter bank {pq () q () q () } has -fold rotational symmetry (-fold axial symmetry resp.) if it satisfies (8) ( (9) resp.). C. Biorthogonality sum rule order and smoothness In this subsection we review some results on orthogonal/biorthogonal (square) filter banks. Denote η 0 = 00 T η = ππ T η = π0 T η = 0π T. (0) FIR filter banks {pq () q () q () } and { p q () q () q () } are said to be biorthogonal or they are perfect reconstruction filter banks if p(ω + η k ) p(ω + η k ) = () 0 k 0 k 0 k (9) p(ω + η k ) q (l) (ω + η k ) = 0 () q (l ) (ω + η k ) q (l) (ω + η k ) = δ l l () for ll ω IR where δ l is the kronecker-delta sequence. A filter bank {pq () q () q () } is said to be orthogonal if it satisfies ()-() with p = p q (l) = q (l) l. Let φ and φ be the scaling function associated with p and p resp. Then () is the necessary condition for φ and φ to be biorthogonal duals: IR φ(x) φ(x k) dx = δ k δ k (4) for all k = k k T Z. We say φ is orthogonal if it satisfies (4) with φ = φ. Under certain mild conditions the condition () is also sufficient for the biorthogonality of φ and φ (see e.g. 8-0 for the details). For biorthogonal FIR filter banks {pq () q () q () } and { p q () q () q () } if the associated scaling functions φ and φ are biorthogonal duals then ψ (l) ψ (l) defined by ψ (l) (ω) = q (l) ( ω ) φ( ω ) ψ(l) (ω) = q (l) ( ω ) φ( ω )

4 4 IEEE TRANS. IMAGE PROC. VOL. 7 NO. 9 SEP. 008 are biorthogonal wavelets namely {ψ (l) jk : l j Zk Z (l) } and { ψ jk : l j Zk Z } are biorthogonal bases of L (IR ) where ψ (l) jk (x) = j ψ (l) ( j x k) ψ(l) jk (x) = j ψ(l) ( j x k). Similarly for an orthogonal filter bank {pq () q () q () } if the associated scaling function φ is orthogonal then ψ (l) defined above are orthogonal wavelets namely {ψ (l) jk : l j Zk Z } is an orthogonal basis of L (IR ). The reader is referred to and for the multiresolution analysis theory and its applications. For a (lowpass) filter p(ω) = 4 that p(ω) has sum rules of order m if k p k = 4 and (k ) α (k ) α p (kk ) k = k = k = k (k + ) α (k ) α p (k+k ) (k ) α (k + ) α p (kk +) k Z p ke ik ω we say (k + ) α (k + ) α p (k+k +) for all nonnegative integers α α with 0 α + α < m. Under certain mild conditions sum rule order of p(ω) is equivalent to the approximation order and accuracy of the scaling function φ associated with p(ω). The reader may see in for the details. High sum rule order of p(ω) is also a necessary condition for the high smoothness order of φ under certain conditions such as the stability of φ. For example for a stable φ if it is in the Sobolev space W n (IR ) (see the definition of the Sobolev space below) then its associated lowpass filter p(ω) must have sum rules of order at least n+. In the following two sections we obtain block structures of orthogonal/biorthogonal FIR filter banks with -fold rotational symmetry and with -fold axial symmetry. These orthogonal/biorthogonal filter banks are given by some free parameters. When a family of filter banks is available (given by free parameters) one can design the filters with desirable properties for one s specific applications. In this paper we consider the filters based on the smoothness of the associated scaling functions φ. In the consideration of smoothness we will compute the Sobolev smoothness of scaling functions. For s 0 denote by W s (IR ) the Sobolev space consisting of functions f(x) on IR with IR ( + ω ) s ˆf(ω) dω <. If f W s (IR ) with s > k+ for some positive integer k then f C k (IR ). We use the smoothness formula in 4 to compute the Sobolev smoothness order of scaling functions. See 5 for the detailed formulas for the Sobolev smoothness of scaling functions/vectors and 6 for algorithms and Matlab routines to find the Sobolev smoothness order. For an FIR lowpass filter p(ω) given by some free parameters the procedures to construct the scaling function φ with the (locally) optimal Sobolev smoothness are described as follows: () Solve the linear equations for the sum rule orders such that p(ω) has the desired sum rule order. The resulting p(ω) is still given by some (but less) free parameters. () Adjust the free parameters for the resulting p(ω) by applying the algorithms/software in 5/6 to achieve the optimal Sobolev smoothness for φ. III. ORTHOGONAL AND BIORTHOGONAL FIR FILTER BANKS WITH -FOLD ROTATIONAL SYMMETRY In this section we consider the construction of orthogonal and biorthogonal filter banks with -fold rotational symmetry. The -fold rotational symmetry of filter banks is discussed in III. A and block structures of orthogonal and biorthogonal FIR filter banks with -fold rotational symmetry are presented in III. B and III. C resp. A. FIR filter banks with -fold rotational symmetry Suppose p(ω)q () (ω)q () (ω)q () (ω) are FIR filters. Then filter bank {pq () q () q () } has -fold rotational symmetry namely it satisfies (8) if and only if { p(ω) = p(r T ω) = p(r T ω) q () (ω) = q () (R T ω) q () (ω) = q () (R T ω). This together with the facts that R = RR = I leads to the following proposition. Proposition : A filter bank {pq () q () q () } has -fold rotational symmetry if and only if it satisfies p q () q () q () T (R T ω) = where M 0 p(ω) q () (ω) q () (ω) q () (ω) T M 0 = (5) Next we consider the filter bank {pq () q () q () } to be given by the product of block matrices. Assume that we can write p(ω)q () (ω)q () (ω)q () (ω) T as A(ω)p 0 (ω)q () 0 (ω)q() 0 (ω)q() 0 (ω)t where A(ω) is a 4 4 matrix with trigonometric polynomial entries and {p 0 q () 0 q() 0 q() 0 } is another FIR filter bank. If both {pq () q () q () } and {p 0 q () 0 q() 0 q() 0 } have -fold rotational symmetry then Proposition leads to that A(ω) satisfies A(R T ω) = M 0 A(ω)M 0 (6) where M 0 is the matrix defined by (5). Clearly {e i(ω+ω) e iω e iω } has -fold rotational symmetry and it could be used as the initial symmetric filter bank while both diag(e i(ω+ω) e iω e iω ) and diag(e i(ω+ω) e iω e iω ) satisfy (6) and they could be used to build the block matrices. In the following we denote I 0 (ω) = e i(ω+ω) e iω e iω T (7) D(ω) = diag(e i(ω+ω) e iω e iω ). (8)

5 JIANG: FIR FILTER BANKS FOR HEXAGONAL DATA PROCESSING 5 Next we use AD(ω) and AD( ω) as the block matrices where A is a 4 4 (real) constant matrix. One can verify that for AD(±ω) satisfies (6) if and only if A has the form: A = a a a a a a a a 4 a a 4 a a a a a 4 a. (9) From the above discussion we have the following result. Theorem : If {pq () q () q () } is given by p(ω)q () (ω)q () (ω)q () (ω) T = (0) A nd(±ω) A D(±ω)A 0 I 0 (ω) for some n Z + where I 0 (ω) and D(ω) are defined by (7) and (8) resp. each A k is a constant matrix of the form (9) then {p(ω)q () (ω)q () (ω)q () (ω)} is an FIR filter bank with -fold rotational symmetry. In the next two subsections we show that the block structure in (0) will yield orthogonal and biorthogonal FIR filter banks with -fold rotational symmetry. B. Orthogonal filter banks with -fold rotational symmetry In this subsection we provide a block structure of orthogonal filter banks with -fold rotational symmetry. For an FIR filter bank {pq () q () q () } denote q (0) (ω) = p(ω). Let U(ω) = q (l) (ω + η k ) 0 lk where η k 0 k are given in (0). Then {pq () q () q () } is orthogonal if and only if U(ω) satisfies Write q (l) (ω)0 l as U(ω)U(ω) = I 4 ω IR. () q (l) (ω) = (q(l) 0 (ω) + q(l) (ω)ei(ω+ω) + q (l) (ω)e iω + q (l) (ω)e iω ). Let V (ω) denote the polyphase matrix of {p(ω)q () (ω) q () (ω)q () (ω)}: V (ω) = (ω). () Clearly q (l) j 0 lj p(ω)q () (ω)q () (ω)q () (ω) T = V (ω)i 0(ω). Furthermore one can show that () is equivalent to V (ω)v (ω) = I 4 ω IR. () Therefore to construct orthogonal {pq () q () q () } we need only to construct V (ω) such that it satisfies (). If {pq () q () q () } is given by (0) then V (ω) = A n D(±ω)A n D(±ω) A D(±ω)A 0. Furthermore since D(ω)D(ω) = I 4 we have that if the constant matrices A k 0 k n are orthogonal then V (ω) satisfies (). One can obtain that if a constant matrix A k of the form (9) is orthogonal then it can be written as diag(s s s s )C k diag(s s 4 s 4 s 4 ) where s l = ± l 4 and α k β k β k β k C k = β k γ k η k ζ k β k ζ k γ k η k (4) β k η k ζ k γ k with α k = t k + t β k = t k k + t γ k = α k η k ζ k k η k = ( ζ k α k ± αk + 4β k ζ k ζ kα k ). (5) Thus an orthogonal matrix C k of the form (9) is given by two free parameters t k and ζ k. Theorem : If {pq () q () q () } is given by (0) with each A k being diag(s s s s )C k diag(s s 4 s 4 s 4 ) for some C k given in (4) then {pq () q () q () } is an orthogonal FIR filter bank with -fold rotational symmetry. Transforming {pq () q () q () } given in Theorem with the matrix U to filter banks on the hexagonal lattice we have a family of orthogonal FIR hexagonal filter banks with -fold rotational symmetry given by a block structure. For this family of orthogonal filter banks given by free parameters t k ζ k 0 k n one can design the filters with desirable properties for one s specific applications. Here we consider the filters based on the smoothness of the associated scaling functions φ. Next we consider two examples based on this structure. Example : Let {pq () q () q () } be the orthogonal filter bank with -fold rotational symmetry given by C diag(e i(ω+ω) e iω e iω )C 0 I 0 (ω) (6) where I 0 (ω) is defined by (7) C 0 and C are given by (4) for some t 0 ζ 0 t ζ. The lowpass filter p(ω) is given by three free parameters t 0 ζ 0 and t. With the choice of + in ± for η 0 in (5) and t 0 = + ζ 0 = 5 4 t = 4 + the corresponding p(ω) has sum rule order and the scaling function φ is in W (IR ). For p(ω) given by (6) the maximum order of sum rules it can have is. From the numerical calculations we also find that is almost the highest Sobolev smoothness order φ can gain. Example : Let {pq () q () q () } be the orthogonal filter bank with -fold rotational symmetry given by C D(ω)C D( ω)c 0 I 0 (ω) where I 0 (ω) and D(ω) are defined by (7) and (8) resp. C 0 C C are matrices defined by (4) with free parameters t k ζ k k = 0. With the choice + in ± for η k in (5) and t 0 = ζ 0 = t = ζ = t = (ζ is a free parameter) we get the smoothest φ with φ W.88 (IR ).

6 6 IEEE TRANS. IMAGE PROC. VOL. 7 NO. 9 SEP. 008 We have considered orthogonal filters with more nonzero impulse response coefficients by using more blocks A k D(±ω) in (0). Unfortunately in term of the smoothness of the scaling functions using a few more blocks A k D(±ω) does not yield orthogonal scaling functions with significantly higher smoothness order. In the next section we consider - fold rotational symmetric biorthogonal filter banks which give us more flexility for the construction of perfect reconstruction filter banks. C. Biorthogonal filter banks with -fold rotational symmetry Let {pq () q () q () } and { p q () q () q () } be two filter banks V (ω) and Ṽ (ω) be their polyphase matrices defined by (). Then one can shows as in III.B that {pq () q () q () } and { p q () q () q () } are biorthogonal to each other if and only if V (ω) and Ṽ (ω) satisfy V (ω)ṽ (ω) = I 4 ω IR. If {pq () q () q () } is the FIR filter bank given by (0) for some 4 4 real nonsingular matrix A k then V (ω) = A n D(±ω)A n D(±ω) A D(±ω)A 0. Hence (V (ω) ) = A T n D(±ω) A T D(±ω)A T 0. One can easily show that if A k has the form of (9) then so dose A T k D(±ω)A T n. Thus by Proposition { p q() q () q () } with its polyphase matrix Ṽ (ω) = (V (ω) ) also has -fold rotational symmetry. Therefore we have the following result. Theorem : Let {pq () q () q () } be the FIR filter bank given by (0) for some nonsingular A k of the form (9). Suppose { p q () q () q () } is given by p(ω) q () (ω) q () (ω) q () (ω) T = (7) n D(±ω) A T D(±ω)A T 0 I 0 (ω) A T where I 0 (ω) and D(ω) are defined by (7) and (8) resp. Then { p q () q () q () } is an FIR filter bank biorthogonal to {pq () q () q () } and it has -fold rotational symmetry. Theorem provides a family of biorthogonal FIR filter banks with -fold rotational symmetry. Compared with the orthogonal filter banks given in Theorem this family of biorthogonal filter banks has more flexibility for the design of desired filters. Example : Let {pq () q () q () } and { p q () q () q () } be the biorthogonal filter banks with -fold rotational symmetry given by Theorem with n = : p(ω)q () (ω)q () (ω)q () (ω) T = A D( ω)a 0 I 0 (ω) p(ω) q () (ω) q () (ω) q () (ω) T = A T D( ω)a T 0 I 0 (ω). where A 0 and A are nonsingular matrices of the form (9). We can choose the free parameters for A 0 and A such that the resulting scaling functions φ W.860 (IR ) φ W 0.5 (IR ) and the lowpass filters p(ω) and p(ω) have sum rules of order and order resp. Since smoothness order of the scaling functions is still low the selected parameters are not provided here. Here and in the next example we intently construct the scaling functions with one smoother than the other so that one filter bank can be used as the analysis filter bank and the other can be used as the synthesis filter bank which requires smoother scaling function and wavelets. Example 4: Let {pq () q () q () } and { p q () q () q () } be the biorthogonal filter banks with -fold rotational symmetry given by Theorem with n = : p(ω)q () (ω)q () (ω)q () (ω) T = A D(ω)A D( ω)a 0 I 0 (ω) p(ω) q () (ω) q () (ω) q () (ω) T = A T D(ω)A T D( ω)a T 0 I 0 (ω) where A 0 A and A are nonsingular matrices of the form (9). In this case we can select the free parameters for A 0 A and A such that the resulting scaling functions φ W.594 (IR ) φ W (IR ) and the lowpass filters p(ω) and p(ω) have sum rules of order and order resp. The selected parameters are provided in Appendix A. IV. ORTHOGONAL AND BIORTHOGONAL FIR FILTER BANKS WITH -FOLD AXIAL SYMMETRY In this section we study the construction of orthogonal and biorthogonal filter banks with -fold axial symmetry. The - fold axial symmetry of filter banks is discussed in VI.A and block structures of orthogonal and biorthogonal FIR filter banks with -fold axial symmetry are presented in VI.B and VI.C resp. A. FIR filter banks with -fold axial symmetry Let {pq () q () q () } be an FIR filter bank. Then it has -fold axial symmetry that is it satisfies (9) if and only if { p(ω) = p(ne ω) = p(w T ω) = p(s e T ω) q () (ω) = q () (N e ω) = q () (R T ω) = q () (R T ω). (8) Before we derive a block structure of filter banks with -fold axial symmetry we first have the following proposition about the -fold axial symmetry property of a filter bank. Proposition : A filter bank {pq () q () q () } has -fold axial symmetry if and only if it satisfies p q () q () q () T (Ne ω) = (9) T M p(ω) q () (ω) q () (ω) q (ω) () p q () q () q () T (W T ω) = (0) T M p(ω) q () (ω) q () (ω) q (ω) () p q () q () q () T T (Se ω) = () M p(ω) q () (ω) q () (ω) q () (ω) T

7 JIANG: FIR FILTER BANKS FOR HEXAGONAL DATA PROCESSING 7 where M = M = M = The proof of Proposition is given in Appendix B. Next we consider the filter bank {pq () q () q () } which can be given by the product of block matrices. Assume that p(ω)q () (ω)q () (ω)q () (ω) T can be written as B(ω)p 0 (ω)q () 0 (ω)q() 0 (ω)q() 0 (ω)t where B(ω) is a 4 4 matrix with trigonometric polynomial entries and {p 0 q () 0 q() 0 q() 0 } is another FIR filter bank. If both {pq () q () q () } and {p 0 q () 0 q() 0 q() 0 } have -fold axial symmetry then Proposition implies that B(ω) satisfies { B(Ne ω) = M B(ω)M B(W T ω) = M B(ω)M B(S e T ω) = M B(ω)M. I 0 (ω) defined in (7) has -fold axial symmetry and it could be used again as the initial filter bank. For D(ω) defined by (8) since both D(ω) and D( ω) satisfy () they could be used to build the block matrices. Next we will use BD(ω) and BD( ω) as the block matrices where B is a 4 4 (real) constant matrix. One can verify that BD(±ω) satisfies () if and only if B has the form: B = b b b b b b b b b b b b b b b b (). () Based on the above discussion we reach the following theorem on the filter banks with -fold axial symmetry. Theorem 4: If {pq () q () q () } is given by p(ω)q () (ω)q () (ω)q () (ω) T = (4) B nd(±ω) B D(±ω)B 0 I 0 (ω) for some n Z + where I 0 (ω) and D(ω) are defined by (7) and (8) resp. and each B k is a 4 4 constant matrix of the form () then {pq () q () q () } has -fold axial symmetry. B. Allen s orthogonal filter banks In this subsection we show that the block structure in (4) will yield -fold axial symmetric orthogonal FIR filter banks which were studied in 4 6. For an FIR filter bank {pq () q () q () } let V (ω) denote its polyphase matrix defined in (). If {pq () q () q () } is given by (4) then V (ω) = B n D(±ω)B n D(±ω) B D(±ω)B 0. Thus if the constant matrices B k are orthogonal then V (ω) satisfies () and hence (4) gives a family of orthogonal filter banks. One can check that if a constant matrix B k of the form () is orthogonal then it can be written as (refer to 4) g g g g G = g + g g g + g g g + g g (5) g g g + g or as diag(± )G where g IR. Theorem 5: If {pq () q () q () } is given by (4) where each B k is G k or diag(± )G k with G k given by (5) for g k IR then {pq () q () q () } is an orthogonal filter bank with -fold axial symmetry. Transforming {pq () q () q () } given in Theorem 5 with the matrix U to hexagonal filter banks we have a family of orthogonal hexagonal filter banks with -fold axial symmetry given by a block structure. This structure is the one given by Allen in 4 and it is referred here as Allen s structure. 4 and 6 constructed several orthogonal filter banks based on the compaction of filters. Here we consider the filters based on the smoothness of the associated scaling functions. Example 5: Let {p(ω)q () (ω)q () (ω)q () (ω)} be the orthogonal filter bank given by G diag(e i(ω+ω) e iω e iω )G 0 I 0 (ω) with G 0 G given by (5) for some g 0 g IR. The hexagonal filter bank corresponding to this filter bank is called the L-Trigon of the R-Trigon in 4. Then with the choices of g 0 = 9 ( ) g = (4 ) the resulting lowpass filter p(ω) has sum rule order and the scaling function φ is in W (IR ). Actually this resulting p(ω) is the lowpass filter of sum rule order in Example. The non-zero impulse response coefficients p k q (l) k of the filters are p 00 = + 6 p = p 0 = p 0 = 9 + p = p 0 = p 0 = 6 p 0 = p 0 = p = 6 p = p = p = p = p = p = 5 + q () 00 = + q () 6 = q () 0 = q() 0 = q() q () 0 = q() q () 0 = q() q () = q() = 6 0 = + 5 q () = = = 9 + = q () = q() = q() = q() and q () k q() k are given by (8).

8 8 IEEE TRANS. IMAGE PROC. VOL. 7 NO. 9 SEP. 008 Fig. 4. Non-zero impulse response coefficients of filters with Allen s structure (left) and those with new structure (right) We have also considered orthogonal filters with more nonzero impulse response coefficients by using more blocks B k D(±ω) in (4). Again we find that using a few more blocks B k D(±ω) does not yield orthogonal scaling functions with significantly higher smoothness order. Let {pq () q () q () } be the FIR filter bank given by (4) where B k 0 k n are 4 4 nonsingular constant real matrices of the form (). Then { p(ω) q () (ω) q () (ω) q () (ω)} given by B T n D(±ω) B T D(±ω)B T 0 I 0 (ω) is biorthogonal to {pq () q () q () } and it has -fold axial symmetry. Therefore by choosing nonsingular matrices B k of the form () one has a family of biorthogonal FIR filter banks given by free parameters. Compared with the orthogonal filter banks of -fold axial symmetry given in Theorem 5 these biorthogonal filter banks give us more flexibility for the design of desired filters. However in terms of the smoothness of the scaling functions φ and φ they do not yield smooth φ φ with reasonable supports. Because of this we introduce in the next subsection another family of biorthogonal filter banks with - fold axial symmetry. C. Biorthogonal filter banks with -fold axial symmetry In this subsection we introduce another family of biorthogonal filter banks with -fold axial symmetry which is based on the following block matrix: E(ω) = (6) a b + cz z b + cz b + cz t t 5 + t z z t + t 4 z t + t 4 z t t + t 4 z z t 5 + t z t + t 4 z t t + t 4 z z t + t 4 z t 5 + t z where abct j j 5 are real numbers and z z are given by (). One can verify that E(ω) satisfies (). E(ω) yields primal filter banks {pq () q () q () } with denser non-zero impulse response coefficients (and hence they will produce smoother scaling functions and wavelets). For example the black dots in the left part of Fig. 4 indicate the non-zero impulse response coefficients of p(ω) from B D( ω)b 0 e iω e iω e iω e iω T while the non-zero impulse response coefficients of p(ω) from E (ω)e 0 (ω)e iω e iω e iω e iω T are shown in the right part of Fig. 4 as the black dots. We can choose some special t j such that det(e(ω)) is a constant and hence Ẽ(ω) = (E(ω) ) is a matrix with each entry being a (Laurent) polynomial of z z. The possible choices are or t = t 5 = bt a (7) t = t 4 = ct a. (8) In these two cases det(e(ω)) is 6 (t t 4 ) (at + at 4 ct ). With t t 5 given in (7) Ẽ(ω) = (E(ω) ) is where Ẽ(ω) = (t t 4 )(at + at 4 ct ) (9) ã b b a (z z + z + b b z ) bz z z z ẽ b a ( cz z + d z + d z ) cz z d z ẽ b a ( dz z + c z + d z ) dz z c z ẽ b a ( dz z + d z + c z ) dz z d z ã = (t + t 4 )(t t 4 ) b = t (t t 4 ) d z d z c z c = at + at 4 ct d = ct at 4 ẽ = c(t t 4 ) while if t t 4 are given by (8) then Ẽ(ω) = (E(ω) ) is where Ẽ(ω) = (t 5 t )(at 5 + at bt ) (40) ã c b a (z z + z + z ) b b b ẽ c a ( c z z + d z + d z ) c d d ẽ c a ( d z z + c z + d z ) d c d ẽ c a ( d z z + d z + c z ) d d c ã = (t 5 + t )(t 5 t ) c = at + at 5 bt b = t (t 5 t ) d = bt at ẽ = b(t 5 t ). Theorem 6: Let {pq () q () q () } be the filter bank: p(ω)q () (ω)q () (ω)q () (ω) T = (4) E n(±ω) E 0 (±ω)i 0 (ω) where I 0 (ω) is defined by (7) E k (ω)0 k n are given by (6) with parameters satisfying (7) (or (8)). If { p q () q () q () } is the filter bank given by p(ω) q () (ω) q () (ω) q () (ω) T = (4) Ẽn(±ω) Ẽ0(±ω)I 0 (ω) where Ẽk(ω) = (E k (ω) ) are given by (9) (or by (40)) then {pq () q () q () } and { p q () q () q () } are biorthogonal FIR filter banks with -fold axial symmetry. Again with a family of symmetric biorthogonal FIR filter banks available in Theorem 6 one can design biorthogonal filter banks for one s particular applications. Here we provide two filter banks based on the smoothness of the associated scaling functions φ and φ.

9 JIANG: FIR FILTER BANKS FOR HEXAGONAL DATA PROCESSING 9 Example 6: Let p(ω)q () (ω)q () (ω)q () (ω) T = E ( ω)e 0 (ω)i 0 (ω) p(ω) q () (ω) q () (ω) q () (ω) T = Ẽ( ω)ẽ0(ω)i 0 (ω) be the biorthogonal FIR filter banks given in Theorem 6 with n = and the parameters satisfying (7). Then we can choose free parameters such that the resulting scaling functions φ W.505 (IR ) φ W (IR ) and the lowpass filters p(ω) p(ω) have sum rules of order and order resp. The impulse response coefficients of these two resulting filter banks are provided in the long version of this paper downloadable at author s web site. Example 7: Let p(ω)q () (ω)q () (ω)q () (ω) T = E (ω)e ( ω)e 0 (ω)i 0 (ω) p(ω) q () (ω) q () (ω) q () (ω) T = Ẽ(ω)Ẽ( ω)ẽ0(ω)i 0 (ω) be the biorthogonal FIR filter banks given in Theorem 6 with n = and the parameters satisfying (7). Then we can choose free parameters such that the corresponding scaling functions φ W.7055 (IR ) φ W (IR ) and the lowpass filters p(ω) p(ω) have sum rules of order and order resp. Again the corresponding impulse response coefficients are provided in the long version of this paper. V. CONCLUSION In this paper we obtain block structures of FIR filter banks with -fold rotational symmetry and FIR filter banks with -fold axial symmetry. These block structures yield orthogonal/biorthogonal hexagonal filter banks with -fold rotational symmetry and orthogonal/biorthogonal hexagonal filter banks with -fold axial symmetry. The construction of orthogonal and biorthogonal FIR filters with scaling functions having optimal smoothness is discussed and several orthogonal/biorthogonal filters banks are presented. Our future work is to apply these hexagonal filter banks for hexagonal image processing applications such as image enhancement and edge detection. In this paper we just consider the hexagonal filters with the dyadic refinement. We will also consider the hexagonal filter banks with the and 7 refinements and study the construction of idealized hexagonal tight frame filter banks which contain the idealized highpass filters with nice frequency localizations. APPENDIX A Selected parameters in Example 4: selected a ij for A 0 are a = a = a = a = a = a 4 = ; select parameters a ij for A are a = a = a = a = a = a 4 = ; and select parameters a ij for A are a = a = a = a = a = a 4 = APPENDIX B Proof of Proposition. Using R = WN e R = S e N e one can easily show that (9)-() imply (8). On the other hand using the following facts (i)-(v) about relationship among the matrices N e WS e R and R one can show that (8) leads to (9)-(): (i). N e = R T N e R T ; (ii). R T = N e W T ; (iii). R T = N e S T e ; (iv). R T W T = N e R T ; (v). R T W T = N e R T. Here we just show the formulas in (9)-() for q (). The proof of other formulas in (9)-() for pq () and q () is similar. For q () we have q () (N e ω) = q () (R T N e ω) = q () (N e R T ω)(by (i)) = q () (R T ω) = q () (ω); q () (W T ω) = q () (R T W T ω) = q () (N e R T ω)(by (iv)) = q () (R T ω) = q () (ω); q () (S T e ω) = q () (R T S T e ω) = q () (N e R T ω)(by (iii)) = q () (R T ω) = q () (ω) as desired. ACKNOWLEDGMENT The author thanks James D. Allen and Xiqiang Zheng for providing preprints 4 and 9. The author is very grateful to the anonymous reviewers for making many valuable suggestions that significantly improve the presentation of the paper. REFERENCES D.P. Petersen and D. Middleton Sampling and reconstruction of wavenumber-limited functions in N-dimensional Euclidean spaces Information and Control vol. 5 no. 4 pp. 79- Dec. 96. R.M. Mersereau The processing of hexagonal sampled two-dimensional signals Proc. IEEE vol. 67 no. 6 pp Jun R.C. Staunton and N. Storey A comparison between square and hexagonal sampling methods for pipeline image processing In Proc. of SPIE Vol. 94 Optics Illumination and Image Sensing for Machine Vision IV 990 pp J.D. Allen Coding transforms for the hexagon grid Ricoh Calif. Research Ctr. Technical Report CRC-TR-985 Aug G. Tirunelveli R. Gordon and S. Pistorius Comparison of squarepixel and hexagonal-pixel resolution in image processing in Proceedings of the 00 IEEE Canadian Conference on Electrical and Computer Engineering vol. May 00 pp D. Van De Ville T. Blu M. Unser W. Philips I. Lemahieu and R. Van de Walle Hex-splines: a novel spline family for hexagonal lattices IEEE Tran. Image Proc. vol. no. 6 pp Jun L. Middleton and J. Sivaswarmy Hexagonal Image Processing: A Practical Approach Springer X.J. He and W.J. Jia Hexagonal Structure for Intelligent Vision in Proceedings of the 005 First International Conference on Information and Communication Technologies Aug. 005 pp

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Allen Perfect reconstruction filter banks for the hexagonal grid In Fifth International Conference on Information Communications and Signal Processing 005 Dec. 005 pp C. Cabrelli C. Heil and U. Molter Accuracy of lattice translates of several multidimensional refinable functions J. Approx. Theory vol. 95 no. pp. 5 5 Oct R.Q. Jia Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets In Advances in Wavelets Springer- Verlag Singapore 999 pp W. Lawton S.L. Lee and Z.W. Shen Stability and orthonormality of multivariate refinable functions SIAM J. Math. Anal. vol. 8 no. 4 pp Jul C.K. Chui and Q.T. Jiang Multivariate balanced vector-valued refinable functions in Mordern Development in Multivariate Approximation ISNM 45 Birhhäuser Verlag Basel 00 pp M. Vetterli and J. Kovacevic Wavelets and Subband Coding Prentice Hall 995. G. Strang and T. Nguyen Wavelets and Filter Banks Wellesley- Cambridge Press 996. R.Q. Jia Approximation properties of multivariate wavelets Math. Comp. vol. 67 no. pp Apr R.Q. Jia and S.R. Zhang Spectral properties of the transition operator associated to a multivariate refinement equation Linear Algebra Appl. vol. 9 no. pp May R.Q. Jia and Q.T. Jiang Spectral analysis of transition operators and its applications to smoothness analysis of wavelets SIAM J. Matrix Anal. Appl. vol. 4 no. 4 pp Q.T. Jiang and P. Oswald Triangular -subdivision schemes: the regular case J. Comput. Appl. Math. vol. 56 no. pp Jul. 00. PLACE PHOTO HERE Qingtang Jiang received his B.S. and M.S. degrees from Hangzhou University Hangzhou China in 986 and 989 resp. and his Ph.D. degree from Peking University Beijing China in 99 all in mathematics. He was with Peking University Beijing China from 99 to 995. He was an NSTB post-doctoral fellow and then a research fellow at the National University of Singapore from 995 to 999. Before joined University of Missouri - St. Louis in 00 he held visting positions at Unversity of Alberta Canada and West Virginia University USA. He is now a full professor in the Department of Math and Computer Science at University of Missouri - St. Louis. Dr. Jiang s current research interests include time frequency analysis wavelet theory and its applications filter bank design signal classification image processing and surface subdivision.