Compactly Supported Tensor Product Complex Tight Framelets with Directionality
aa r X i v : . [ c s . I T ] J u l COMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETSWITH DIRECTIONALITY
BIN HAN, QUN MO, AND ZHENPENG ZHAO
Abstract.
Although tensor product real-valued wavelets have been successfully applied to many high-dimensionalproblems, they can only capture well edge singularities along the coordinate axis directions. As an alternativeand improvement of tensor product real-valued wavelets and dual tree complex wavelet transform, recently tensorproduct complex tight framelets with increasing directionality have been introduced in [8] and applied to imagedenoising in [13]. Despite several desirable properties, the directional tensor product complex tight frameletsconstructed in [8, 13] are bandlimited and do not have compact support in the space/time domain. Since com-pactly supported wavelets and framelets are of great interest and importance in both theory and application, itremains as an unsolved problem whether there exist compactly supported tensor product complex tight frameletswith directionality. In this paper, we shall satisfactorily answer this question by proving a theoretical result ondirectionality of tight framelets and by introducing an algorithm to construct compactly supported complex tightframelets with directionality. Our examples show that compactly supported complex tight framelets with direc-tionality can be easily derived from any given eligible low-pass filters and refinable functions. Several examplesof compactly supported tensor product complex tight framelets with directionality have been presented. Introduction and Motivations
Having better directionality and employing tensor product of a correlated pair of one-dimensional orthogonalwavelet filter banks, dual tree complex wavelet transform in [14, 18] has shown superior performance in ap-plications over the commonly adopted tensor product real-valued wavelets. As alternatives and improvementsto dual tree complex wavelet transform, tensor product complex tight framelets with directionality have beenintroduced in [8]. It has been demonstrated in [13] that tensor product complex tight framelets with improveddirectionality significantly perform better, in terms of PSNR (peak signal-to-noise ratio), than dual tree com-plex wavelet transform in the model problem of image denoising. However, the tensor product complex tightframelets constructed in [8, 13] are only bandlimited, that is, they have compact support in the frequencydomain but they are not compactly supported in the space/time domain. Since compactly supported waveletsand framelets are of importance in both theory and application due to their good space-frequency localiza-tion and computational efficiency desired in many applications, it is a natural and important problem for usto investigate compactly supported tensor product complex tight framelets with directionality. In this paperwe shall satisfactorily resolve this problem by studying and constructing compactly supported tensor productcomplex tight framelets in L ( R d ) with directionality.To explain our motivations, let us first introduce some notation and definitions. For a function f : R d → C and a d × d real-valued matrix U , we shall adopt the following notation: f U ; k ( x ) := [[ U ; k ]] f ( x ) := | det( U ) | / f ( U x − k ) , x, k ∈ R d . For φ, ψ , . . . , ψ s ∈ L ( R d ) with s ∈ N , we define an affine system generated by φ, ψ , . . . , ψ s as follows: AS ( φ ; ψ , . . . , ψ s ) := { φ ( · − k ) : k ∈ Z d } ∪ { ψ ℓ j I d ; k : j ∈ N ∪ { } , k ∈ Z d , ℓ = 1 , . . . , s } , where I d denotes the d × d identity matrix. Recall that { φ ; ψ , . . . , ψ s } is a ( d -dimensional dyadic) tight frameletin L ( R d ) if AS ( φ ; ψ , . . . , ψ s ) is a (normalized) tight frame for L ( R d ), that is, k f k L ( R d ) = X k ∈ Z d |h f, φ ( · − k ) i| + ∞ X j =0 s X ℓ =1 X k ∈ Z d |h f, ψ ℓ j I d ; k i| , ∀ f ∈ L ( R d ) . (1.1) Mathematics Subject Classification.
Key words and phrases.
Finitely supported tight framelet filter banks, complex tight framelets with directionality, tensor product,frequency separation.Research of B. Han and Z. Zhao supported in part by NSERC Canada under Grant RGP 228051. Research of Q. Mo supported inpart by the NSF of China under Grants 10971189 and 11271010, and by the fundamental research funds for the Central Universities.
In particular, if AS ( φ ; ψ , . . . , ψ s ) is an orthonormal basis for L ( R d ), then we call { φ ; ψ , . . . , ψ s } an orthogonalwavelet in L ( R d ). If { φ ; ψ , . . . , ψ s } is a tight framelet in L ( R d ), then { ψ , . . . , ψ s } must be a homogeneoustight framelet in L ( R d ) (see [3, 7]), in other words, k f k L ( R d ) = X j ∈ Z s X ℓ =1 X k ∈ Z d |h f, ψ ℓ j I d ; k i| , ∀ f ∈ L ( R d ) . Consequently, it follows directly from (1.1) and the above identity that every function f ∈ L ( R d ) has thefollowing representations: f = X k ∈ Z d h f, φ ( · − k ) i φ ( · − k ) + ∞ X j =0 s X ℓ =1 X k ∈ Z d h f, ψ ℓ j I d ; k i ψ ℓ j I d ; k = X j ∈ Z s X ℓ =1 X k ∈ Z d h f, ψ ℓ j I d ; k i ψ ℓ j I d ; k (1.2)with the series converging unconditionally in L ( R d ).Due to many desirable properties such as sparsity and good space-frequency localization, wavelet represen-tations in (1.2) have been used in many applications ([1, 3, 14, 18, 19]). To have a fast algorithm to computethe wavelet coefficients in (1.2), wavelets and framelets are often derived from refinable functions and filterbanks. By l ( Z d ) we denote the space of all complex-valued sequences u = { u ( k ) } k ∈ Z d : Z d → C such that k u k l ( Z d ) := ( P k ∈ Z d | u ( k ) | ) / < ∞ . The Fourier series (or symbol) of a sequence u ∈ l ( Z d ) is definedto be b u ( ξ ) := P k ∈ Z d u ( k ) e − ik · ξ , ξ ∈ R d , which is a 2 π Z d -periodic measurable function in L ( T d ) such that k b u k L ( T d ) := π ) d R [ − π,π ) d | b u ( ξ ) | dξ = k u k l ( Z d ) = P k ∈ Z d | u ( k ) | < ∞ .By l ( Z d ) we denote the set of all finitely supported sequences on Z d . If a ∈ l ( Z d ) and b a (0) = 1, then Q ∞ j =1 b a (2 − j ξ ) is convergent for every ξ ∈ R d and it is well known ([3]) that there exists a compactly supporteddistribution φ on R d such that b φ ( ξ ) = Q ∞ j =1 b a (2 − j ξ ) , ξ ∈ R d , where the Fourier transform is defined to be b f ( ξ ) := R R d f ( x ) e − ix · ξ dx for f ∈ L ( R d ). For b , . . . , b s ∈ l ( Z d ), we define ψ , . . . , ψ s by c ψ ℓ ( ξ ) := b b ℓ ( ξ/ b φ ( ξ/ , ξ ∈ R d , ℓ = 1 , . . . , s. Then { φ ; ψ , . . . , ψ s } is a tight framelet in L ( R d ) if and only if { a ; b , . . . , b s } is a tight framelet filter banksatisfying | b a ( ξ ) | + s X ℓ =1 | b b ℓ ( ξ ) | = 1 and b a ( ξ ) b a ( ξ + πω ) + s X ℓ =1 b b ℓ ( ξ ) b b ℓ ( ξ + πω ) = 0 , ∀ ω ∈ Ω \{ } (1.3)for almost every ξ ∈ R d , where Ω := [0 , d ∩ Z d . See [1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 15, 16, 17, 19] and manyreferences therein on tight framelets in L ( R d ) and their applications.High-dimensional wavelets and framelets are often obtained from one-dimensional wavelets and frameletsthrough tensor product. The main advantages of tensor product wavelets and framelets lie in that they havea simple fast numerical algorithm and the construction of one-dimensional wavelets and framelets is oftenrelatively easy. To our best knowledge, almost all successful wavelet-based methods in applications have usedtensor product real-valued wavelets and framelets, partially due to their simplicity and fast implementation.To illustrate the tensor product method, for simplicity, let us only discuss the particular case of dimension twohere. For two one-dimensional functions f, g : R → C , their tensor product f ⊗ g in dimension two is defined tobe ( f ⊗ g )( x, y ) := f ( x ) g ( y ), x, y ∈ R . Similarly, for two sequences u, v : Z → C , their two-dimensional tensorproduct filter u ⊗ v is defined to be ( u ⊗ v )( j, k ) := u ( j ) v ( k ), j, k ∈ Z . Let { φ ; ψ , . . . , ψ s } be a tight frameletin L ( R ) with an underlying tight framelet filter bank { a ; b , . . . , b s } such that b φ (2 ξ ) = b a ( ξ ) b φ ( ξ ) and c ψ ℓ (2 ξ ) = b b ℓ ( ξ ) b φ ( ξ ), ℓ = 1 , . . . , s . Using tensor product, we obtain a tight framelet { φ ; ψ , . . . , ψ s } ⊗ { φ ; ψ , . . . , ψ s } in L ( R ) with an underlying tensor product tight framelet filter bank { a ; b , . . . , b s }⊗ { a ; b , . . . , b s } for dimensiontwo. More precisely, defineΨ := { φ ⊗ ψ , . . . , φ ⊗ ψ s } ∪ { ψ ⊗ φ, . . . , ψ s ⊗ φ } ∪ { ψ ℓ ⊗ ψ m : ℓ, m = 1 , . . . , s } , then we have a two-dimensional tight frame AS ( φ ⊗ φ ; Ψ) for L ( R ) satisfying k f k L ( R ) = X k ∈ Z |h f, ( φ ⊗ φ )( · − k ) i| + ∞ X j =0 X ψ ∈ Ψ X k ∈ Z |h f, ψ j I ; k i| , ∀ f ∈ L ( R ) . OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 3
Moreover, φ ⊗ φ satisfies the tensor product refinement equation \ φ ⊗ φ (2 ξ ) = [ a ⊗ a ( ξ ) \ φ ⊗ φ ( ξ ), a.e. ξ ∈ R andfor each ψ ∈ Ψ, b ψ (2 ξ ) = c b ψ ( ξ ) \ φ ⊗ φ ( ξ ), where { b ψ : ψ ∈ Ψ } := { a ⊗ b , . . . , a ⊗ b s } ∪ { b ⊗ a, . . . , b s ⊗ a } ∪ { b ℓ ⊗ b m : ℓ, m = 1 , . . . , s } . Note that { a ⊗ a ; b ψ , ψ ∈ Ψ } is a two-dimensional tensor product tight framelet filter bank.Though tensor product real-valued wavelets and framelets have been widely used in many applications, theyhave some shortcomings, for example, lack of directionality in high dimensions. For two-dimensional data suchas images, edge singularities are ubiquitous and play a more fundamental role in image processing than pointsingularities. As a consequence, tensor product real-valued wavelets are only suboptimal since they can onlyefficiently capture edge singularities along the coordinate axis directions. For the convenience of the reader,let us explain this point in more detail. When φ and ψ , . . . , ψ s are real-valued functions in L ( R ) such that b φ (0) = 1 and c ψ (0) = · · · = c ψ s (0) = 0, in general b φ concentrates essentially near the origin while c ψ , . . . , c ψ s concentrate largely outside a neighborhood of the origin. Since every ψ ℓ , ℓ = 1 , . . . , s is real-valued, it is trivialto notice that c ψ ℓ ( ξ ) = c ψ ℓ ( − ξ ) and consequently, the magnitude of the frequency spectrum of c ψ ℓ is symmetricabout the origin. For dimension two, it is not difficult to see that all φ ⊗ ψ ℓ have horizontal direction whileall ψ ℓ ⊗ φ have vertical direction for ℓ = 1 , . . . , s . However, all ψ ℓ ⊗ ψ m do not exhibit any directionality for ℓ, m = 1 , . . . , s . The same phenomenon can be said for the associated tight framelet filter bank: all a ⊗ b ℓ exhibit horizontal direction, all b ℓ ⊗ a exhibit vertical direction, but b ℓ ⊗ b m do not exhibit any directionality for ℓ, m = 1 , . . . , s . To see this point better, let us look at the particular example of the Haar orthogonal wavelet { φ ; ψ } with φ = χ [0 , and ψ = χ [0 , ] − χ [ , . Then φ ⊗ φ = χ [0 , and φ ⊗ ψ = χ [0 , × [0 , ] − χ [0 , × [ , , ψ ⊗ φ = χ [0 , ] × [0 , − χ [ , × [0 , , ψ ⊗ ψ = χ [0 , ] ∪ [ , − χ [0 , ] × [ , ∪ [ , × [0 , ] . We can clearly observe that φ ⊗ ψ has horizontal direction, ψ ⊗ φ has vertical direction, but ψ ⊗ ψ does notexhibit any directionality. Note that the above Haar orthogonal wavelet { φ ; ψ } has the underlying orthogonalwavelet filter bank { a ; b } with a = { , } [0 , and b = { , − } [0 , . Then a ⊗ a = (cid:20)
14 1414 14 (cid:21) [0 , , a ⊗ b = (cid:20) − − (cid:21) [0 , , b ⊗ a = (cid:20) − − (cid:21) [0 , , b ⊗ b = (cid:20) −
14 1414 − (cid:21) [0 , . From above, we observe that a ⊗ b has horizontal direction, b ⊗ a has vertical direction, but b ⊗ b does notexhibit any directionality.As one of the most popular and successful approaches to enhance the performance of tensor product real-valued wavelets, the dual tree complex wavelet transform proposed in [14, 18] uses tensor product of a correlatedpair of finitely supported orthogonal wavelet filter banks and offers 6 directions with impressive performance inmany applications. However, horizontal and vertical directions are very common in many two-dimensional datasuch as images. It is also difficult to generalize the approach of dual tree complex wavelet transform to havemore than 6 directions by using dyadic orthogonal wavelet filter banks. To further improve the performanceof and to provide alternatives to dual tree complex wavelet transform, recently [8] introduced tensor productcomplex tight framelets with increasing directionality. Tensor product complex tight framelets not only offeralternatives to dual tree complex wavelet transform but also have improved directionality. In [8, 13], a familyof tensor product complex tight framelets has been constructed in the frequency domain and their performancefor image denoising has been reported in [13]. With more directions and using the tensor product structure, thebandlimited tensor product complex tight framelets constructed in [8, 13] indeed significantly perform betterthan dual tree complex wavelet transform in the area of image denoising. See [13, 14, 18] and many referencestherein on dual tree complex wavelet transform, and see [7, 8, 13] for more details on directional complex tightframelets.This paper is largely motivated by the approach introduced in [8] using tensor product complex tight frameletfilter banks. Let us recall here the tensor product tight framelet filter banks constructed in the frequencydomain in [8, 13]. Let P m ( x ) := (1 − x ) m P m − j =0 (cid:0) m + j − j (cid:1) x j with m ∈ N . Then P m satisfies the identity P m ( x ) + P m (1 − x ) = 1 (see [3]). For c L < c R and two positive numbers ε L , ε R satisfying ε L + ε R c R − c L , BIN HAN, QUN MO, AND ZHENPENG ZHAO we define a bump function χ [ c L ,c R ]; ε L ,ε R on R by χ [ c L ,c R ]; ε L ,ε R ( ξ ) := , ξ c L − ε L or ξ > c R + ε R , sin (cid:0) π P m ( c L + ε L − ξ ε L ) (cid:1) , c L − ε L < ξ < c L + ε L , , c L + ε L ξ c R − ε R , sin (cid:0) π P m ( ξ − c R + ε R ε R ) (cid:1) , c R − ε R < ξ < c R + ε R . (1.4)For simplicity of discussion and presentation, here we only recall a special type of tensor product complex tightframelet filter banks constructed in [8, 13]. We define a real-valued symmetric low-pass filter a ∈ l ( Z ) and twocomplex-valued high-pass filters b p , b n ∈ l ( Z ) by b a := χ [ − c,c ]; ε,ε and b b p := χ [ c,π ]; ε,ε , b b n := χ [ − π, − c ]; ε,ε , (1.5)where c and ε are positive numbers satisfying 0 < ε min( c , π − c ). Then it is easy to directly check that { a ; b p , b n } is a one-dimensional tight framelet filter bank such that a is real-valued and symmetric about theorigin with b a (0) = 1. Define functions φ, ψ p , ψ n on R by b φ ( ξ ) := ∞ Y j =1 b a (2 − j ξ ) , c ψ p ( ξ ) := b b p ( ξ/ b φ ( ξ/ , c ψ n ( ξ ) := b b n ( ξ/ b φ ( ξ/ , ξ ∈ R . Then { φ ; ψ p , ψ n } is a tight framelet in L ( R ). Note that all the functions φ , ψ p , ψ n are bandlimited, that is,their Fourier transforms have compact support. Moreover, φ is real-valued and symmetric about the origin.Note that b b n ( ξ ) = b b p ( − ξ ) and therefore, we have b n = b p and ψ n = ψ p . More importantly, both functions ψ p , ψ n are complex-valued and enjoy the following frequency separation property: c ψ p ( ξ ) = 0 , ∀ ξ ∈ ( −∞ ,
0] and c ψ n ( ξ ) = 0 , ∀ ξ ∈ [0 , ∞ ) . (1.6)In other words, the frequency spectrum of ψ p vanishes on the negative interval ( −∞ ,
0] and concentrates onlyinside the positive interval [0 , ∞ ), while the frequency spectrum of ψ n vanishes on the positive interval [0 , ∞ )and concentrates only inside the negative interval ( −∞ , { φ ; ψ p , ψ n } ⊗ { φ ; ψ p , ψ n } , that is, { φ ⊗ φ } ∪ { φ ⊗ ψ p , φ ⊗ ψ n , ψ p ⊗ φ, ψ n ⊗ φ, ψ p ⊗ ψ p , ψ p ⊗ ψ n , ψ n ⊗ ψ p , ψ n ⊗ ψ n } . (1.7)By (1.6), for f, g ∈ { ψ p , ψ n } , we see that [ f ⊗ g = b f ⊗ b g concentrates on a small rectangle away from the origin.As a consequence, both the real and imaginary parts of f ⊗ g exhibit good directions. We now provide thedetail here. For a complex-valued function f : R d → C , we define f [ r ] ( x ) := Re( f ( x )) and f [ i ] ( x ) := Im( f ( x )) , x ∈ R d . That is, f = f [ r ] + if [ i ] with both f [ r ] and f [ i ] being real-valued functions on R d . Similarly, for u : Z d → C , wecan write u = u [ r ] + iu [ i ] with both sequences u [ r ] and u [ i ] having real coefficients. Define ψ p, [ r ] := Re( ψ p ) , ψ p, [ i ] := Im( ψ p ) , ψ n, [ r ] := Re( ψ n ) , ψ n, [ i ] := Im( ψ n )and similarly b p, [ r ] := Re( b p ) , b p, [ i ] := Im( b p ) , b n, [ r ] := Re( b n ) , b n, [ i ] := Im( b n ) . Then all the above functions and filters are real-valued. It is trivial to check that { φ ; ψ p, [ r ] , ψ n, [ r ] , ψ p, [ i ] , ψ n, [ i ] } (1.8)is a real-valued tight framelet in L ( R ) with the underlying real-valued tight framelet filter bank { a ; b p, [ r ] , b n, [ r ] , b p, [ i ] , b n, [ i ] } . However, we do not apply tensor product to this real-valued one-dimensional tight framelet sinceit shares the same shortcoming as tensor product real-valued wavelets or framelets. Instead, we take tensorproduct of the one-dimensional complex tight framelet first for dimension two as in (1.7), then we separatetheir real and imaginary parts to derive a real-valued tight framelet in L ( R ). If in addition b n = b p andconsequently, ψ n = ψ p since φ is real-valued, then { φ ; √ ψ p, [ r ] , √ ψ p, [ i ] } is a real-valued tight framelet in L ( R )with the underlying tight framelet filter bank { a ; √ b p, [ r ] , √ b p, [ i ] } . Moreover, √ (cid:8) √ φ ⊗ φ ; φ ⊗ ψ p, [ r ] , φ ⊗ ψ p, [ i ] , ψ p, [ r ] ⊗ φ, ψ p, [ i ] ⊗ φ, ψ p, [ r ] ⊗ ψ p, [ r ] − ψ p, [ i ] ⊗ ψ p, [ i ] ,ψ p, [ r ] ⊗ ψ p, [ r ] + ψ p, [ i ] ⊗ ψ p, [ i ] , ψ p, [ r ] ⊗ ψ p, [ i ] − ψ p, [ i ] ⊗ ψ p, [ r ] , ψ p, [ r ] ⊗ ψ p, [ i ] + ψ p, [ i ] ⊗ ψ p, [ r ] (cid:9) (1.9) OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 5 is a two-dimensional real-valued tight framelet in L ( R ) with the following underlying two-dimensional real-valued tight framelet filter bank √ (cid:8) √ a ⊗ a ; a ⊗ b p, [ r ] , a ⊗ b p, [ i ] , b p, [ r ] ⊗ a, b p, [ i ] ⊗ a, b p, [ r ] ⊗ b p, [ r ] − b p, [ i ] ⊗ b p, [ i ] ,b p, [ r ] ⊗ b p, [ r ] + b p, [ i ] ⊗ b p, [ i ] , b p, [ r ] ⊗ b p, [ i ] − b p, [ i ] ⊗ b p, [ r ] , b p, [ r ] ⊗ b p, [ i ] − b p, [ i ] ⊗ b p, [ r ] (cid:9) . (1.10)Now one can check that the derived two-dimensional real-valued tight framelet exhibits four directions:(1) φ ⊗ ψ p, [ r ] and φ ⊗ ψ p, [ i ] have horizontal direction along 0 ◦ ;(2) ψ p, [ r ] ⊗ φ and ψ p, [ i ] ⊗ φ have vertical direction along 90 ◦ ;(3) ψ p, [ r ] ⊗ ψ p, [ r ] − ψ p, [ i ] ⊗ ψ p, [ i ] and ψ p, [ r ] ⊗ ψ p, [ r ] + ψ p, [ i ] ⊗ ψ p, [ i ] have direction along 45 ◦ ;(4) ψ p, [ r ] ⊗ ψ p, [ i ] − ψ p, [ i ] ⊗ ψ p, [ r ] and ψ p, [ r ] ⊗ ψ p, [ i ] + ψ p, [ i ] ⊗ ψ p, [ r ] have direction along − ◦ .As discussed in [8, 13], more directions can be achieved by using more high-pass filters. For simplicity, weonly discuss the particular case { φ ; ψ p , ψ n } in this paper, which plays a critical role for obtaining general finitelysupported tensor product complex tight framelets with increasing directionality.Although the derived two-dimensional real-valued tight framelet in (1.9) and its underlying real-valued tightframelet filter bank in (1.10) no longer have the tensor product structure, it is not difficult to see that theycan be obtained through a simple transform using a constant unitary matrix from { φ ; √ ψ p, [ r ] , √ ψ p, [ i ] } ⊗{ φ ; √ ψ p, [ r ] , √ ψ p, [ i ] } and its underlying real-valued tight framelet filter bank { a ; √ b p, [ r ] , √ b p, [ i ] }⊗{ a ; √ b p, [ r ] , √ b p, [ i ] } . Therefore, similar to dual tree complex wavelet transform in [14, 18], the algorithm using the tensorproduct complex tight framelets in (1.9) with their filter banks in (1.10) can be implemented using the tensorproduct discrete framelet transform employing the tight framelet filter bank { a ; √ b p, [ r ] , √ b p, [ i ] } , followed bysimple linear combinations of the wavelet/framelet coefficients.However, the filters a, b p , b n constructed in (1.5) (see [8, 13] for more detail) have infinite support in the timedomain. Since compactly supported wavelets and framelets have great interest and importance in both theoryand application, this naturally leads us to ask the following question:Q1: Is it possible to construct compactly supported one-dimensional complex tight framelets { φ ; ψ p , ψ n } withfinitely supported tight framelet filter banks { a ; b p , b n } such that c ψ p almost vanishes on the negativeinterval ( −∞ ,
0] and c ψ n almost vanishes on the positive interval [0 , ∞ )?By c ψ p (2 ξ ) = b b p ( ξ ) b φ ( ξ ) and c ψ n (2 ξ ) = b b n ( ξ ) b φ ( ξ ), since generally b φ ≈ χ [ − π,π ] , to satisfy the condition in(1.6), it is very natural to require that b b p should be relatively small on the negative interval [ − π,
0) so that b b p concentrates largely on the positive interval [0 , π ), while b b n should be relatively small on the positive interval[0 , π ) so that b b n concentrates largely on the negative interval [ − π, b p and b n must have good frequency separationproperty. A natural quantity to measure the quality of frequency separation (and therefore, the directionalityof tensor product tight framelets) is B b p ,b n ( ξ ) := | b b p ( ξ + π ) | + | b b n ( ξ ) | , ξ ∈ [0 , π ] . (1.11)That is, the smaller the quantity B b p ,b n on the interval [0 , π ], the better the frequency separation of the twohigh-pass filters b p and b n in the frequency domain and consequently, the better the directionality of theirassociated high-dimensional tensor product tight framelets. More precisely, if we can construct a tight frameletfilter bank { a ; b p , b n } such that the quantity B b p ,b n ( ξ ) is relatively small for all ξ ∈ [0 , π ], then the high-passfilters b p and b n have good frequency separation and thus, the resulting tensor product tight framelet filter bank { a ; b p , b n } ⊗ { a ; b p , b n } and its associated real-valued tight framelet by separating real and imaginary parts in { φ ; ψ p , ψ n } ⊗ { φ ; ψ p , ψ n } will have four directions: 0 ◦ (horizontal), ± ◦ , and 90 ◦ (vertical) in dimension two.In addition to Q1, we are interested in the following two problems:Q2: For filters a, b p , b n ∈ l ( Z ) such that { a ; b p , b n } is a tight framelet filter bank, can we achieve B b p ,b n ( ξ ) ≈ ξ ∈ [0 , π ]? More precisely, given a low-pass filter a ∈ l ( Z ), we want to find a sharp theoreticallower bound which is a function A : [0 , π ] → [0 , ∞ ) depending only on the given filter a such that (i) | b b p ( ξ + π ) | + | b b n ( ξ ) | > A ( ξ ) a.e. ξ ∈ [0 , π ] for any tight framelet filter bank { a ; b p , b n } . (ii) There existsa tight framelet filter bank { a ;˚ b p , ˚ b n } derived from the filter a such that | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | = A ( ξ )a.e. ξ ∈ [0 , π ]. BIN HAN, QUN MO, AND ZHENPENG ZHAO
Q3: From every given real-valued low-pass filter a ∈ l ( Z ) such that 1 − | b a ( ξ ) | − | b a ( ξ + π ) | > a ), can we construct a finitelysupported tight framelet filter bank { a ; b p , b n } such that its associated tensor product complex tightframelet exhibit almost best possible directionality? More precisely, is it possible to construct finitelysupported high-pass filters b p , b n such that { a ; b p , b n } is a tight framelet filter bank and | b b p ( ξ + π ) | + | b b n ( ξ ) | ≈ A ( ξ ) on [0 , π ]? Here A is the sharp theoretical lower bound for frequency separation in Q2.We shall satisfactorily and positively answer all the above questions in this paper. We shall provide a sharptheoretical lower bound for frequency separation using the natural quantity | b b p ( ξ + π ) | + | b b n ( ξ ) | . More precisely,we shall prove in Section 2 the following result which completely answers Q2: Theorem 1.
Let a, b p , b n ∈ l ( Z ) such that { a ; b p , b n } is a tight framelet filter bank. Then | b b p ( ξ + π ) | + | b b n ( ξ ) | > A ( ξ ) , a.e. ξ ∈ [0 , π ] , (1.12) where the frequency separation function A associated with the filter a is defined to be A ( ξ ) := 2 − | b a ( ξ ) | − | b a ( ξ + π ) | − q (cid:0) − | b a ( ξ ) | − | b a ( ξ + π ) | (cid:1) + ( | b a ( ξ ) | − | b a ( ξ + π ) | ) . (1.13) Moreover, the inequality in (1.12) is sharp in the sense that there exist ˚ b p , ˚ b n ∈ l ( Z ) such that { a ;˚ b p , ˚ b n } isa tight framelet filter bank satisfying | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | = A ( ξ ) a.e. ξ ∈ [0 , π ] . If in addition the filter a is real-valued, that is, b a ( ξ ) = b a ( − ξ ) a.e. ξ ∈ R , then the tight framelet filter bank { a ;˚ b p , ˚ b n } can satisfy theadditional property: b ˚ b n ( ξ ) = b ˚ b p ( − ξ ) a.e. ξ ∈ R , that is, ˚ b n = ˚ b p . Interestingly, as demonstrated by the following result, the frequency separation function A in (1.13) is oftenvery small for most known low-pass filters in the literature. Theorem 2.
Let A be the frequency separation function defined in (1.13) associated with a filter a ∈ l ( Z ) satisfying | b a ( ξ ) | + | b a ( ξ + π ) | for almost every ξ ∈ R . Then A ( ξ ) min( | b a ( ξ ) | , | b a ( ξ + π ) | ) , a.e. ξ ∈ R . (1.14) In particular, (i) A ( ξ ) = 0 a.e. ξ ∈ [0 , π ] if and only if b a ( ξ ) b a ( ξ + π ) = 0 a.e. ξ ∈ R . (ii) A ( ξ ) = min( | b a ( ξ ) | , | b a ( ξ + π ) | ) a.e. ξ ∈ [0 , π ] if and only if | b a ( ξ ) | + | b a ( ξ + π ) | = 1 for almost every ξ ∈ R satisfying min( | b a ( ξ ) | , | b a ( ξ + π ) | ) = 0 . In particular, if | b a ( ξ ) | + | b a ( ξ + π ) | = 1 a.e. ξ ∈ R (thatis, a is an orthogonal filter), then A ( ξ ) = min( | b a ( ξ ) | , | b a ( ξ + π ) | ) a.e. ξ ∈ [0 , π ] . (iii) If a is the B-spline filter a Bm of order m given by c a Bm ( ξ ) := cos m ( ξ/ with m ∈ N , then − m sin m ( ξ ) A ( ξ ) − m sin m ( ξ ) , ∀ ξ ∈ [0 , π ] . (1.15)To answer Q1 and Q3 and to construct tight framelet filter banks with directionality, in Section 3 we shallinvestigate the structure of all finitely supported tight framelet filter banks { a ; b p , b n } derived from a givenfilter a . More precisely, from any given finitely supported filter a ∈ l ( Z ), in Theorem 4 and Algorithm 1 weshall construct all possible finitely supported tight framelet filter banks { a ; b p , b n } derived from a given low-passfilter a . For prescribed filter lengths of b p and b n , such a result enables us to find the best possible complextight framelet filter bank { a ; b p , b n } having the best possible frequency separation, that is, having the smallestpossible R π (cid:2) | b b p ( ξ + π ) | + | b b n ( ξ ) | (cid:3) dξ . Finally, in Section 4 we shall provide an algorithm for constructing finitelysupported complex tight framelet filter banks { a ; b p , b n } having the smallest possible R π (cid:2) | b b p ( ξ + π ) | + | b b n ( ξ ) | (cid:3) dξ among all high-pass filters b p and b n with prescribed filter supports. Several examples will be presented toillustrate the results and algorithms in this paper.This paper mainly concentrates on the construction of a particular family of finitely supported tensor productcomplex tight framelet filter banks with directionality (more precisely, in the terminology of [13], TP- C TF having four directions in dimension two). We shall leave the construction of general finitely supported tensorproduct complex tight framelet filter banks with increasing directionality (that is, tensor product complex tightframelets TP- C TF n with n >
4) and their possible applications as a future work.
OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 7 A Sharp Lower Bound for Directionality of Tight Framelet Filter Banks
In this section, we shall prove the sharp theoretical lower bound stated in Theorem 1 for the best possiblefrequency separation of a tight framelet filter bank { a ; b p , b n } derived from a given low-pass filter a . Then weshall prove Theorem 2 showing that the frequency separation function A in (1.13) is often small for many knownlow-pass filters. As a contrast to the result in Theorem 1 for complex-valued tight framelet filter banks, at theend of this section we provide a result showing that all real-valued tight framelet filter banks cannot have goodfrequency separation. Proof of Theorem 1.
Since { a ; b p , b n } is a tight framelet filter bank, it follows from the definition in (1.3) with d = 1 that " b b p ( ξ ) b b n ( ξ ) b b p ( ξ + π ) b b n ( ξ + π ) b p ( ξ ) b b p ( ξ + π ) b b n ( ξ ) b b n ( ξ + π ) = (cid:20) − | b a ( ξ ) | − b a ( ξ ) b a ( ξ + π ) − b a ( ξ + π ) b a ( ξ ) 1 − | b a ( ξ + π ) | (cid:21) . (2.1)Since the determinant of the matrix on the right-hand side of (2.1) is 1 − | b a ( ξ ) | − | b a ( ξ + π ) | , it follows directlyfrom (2.1) that we must have 1 − | b a ( ξ ) | − | b a ( ξ + π ) | > ξ ∈ R .We also notice from (2.1) that { a ; b p , b n } is a tight framelet filter bank if and only if for almost every ξ ∈ [0 , π ],the following three equations hold: | b a ( ξ ) | + | b b p ( ξ ) | + | b b n ( ξ ) | = 1 , (2.2) | b a ( ξ + π ) | + | b b p ( ξ + π ) | + | b b n ( ξ + π ) | = 1 , (2.3) b a ( ξ ) b a ( ξ + π ) + b b p ( ξ ) b b p ( ξ + π ) + b b n ( ξ ) b b n ( ξ + π ) = 0 . (2.4)In the rest of the proof, we always assume ξ ∈ [0 , π ]. Note that (2.2) and (2.3) imply | b b p ( ξ ) | = q − | b a ( ξ ) | − | b b n ( ξ ) | , | b b n ( ξ + π ) | = q − | b a ( ξ + π ) | − | b b p ( ξ + π ) | . (2.5)Using (2.5), we deduce from (2.4) that | b a ( ξ ) b a ( ξ + π ) | (cid:16) | b b p ( ξ ) b b p ( ξ + π ) | + | b b n ( ξ ) b b n ( ξ + π ) | (cid:17) = (cid:16) | b b p ( ξ + π ) | q − | b a ( ξ ) | − | b b n ( ξ ) | + | b b n ( ξ ) | q − | b a ( ξ + π ) | − | b b p ( ξ + π ) | (cid:17) (cid:16) | b b p ( ξ + π ) | + | b b n ( ξ ) | (cid:17)(cid:16) − | b a ( ξ ) | − | b a ( ξ + π ) | − ( | b b p ( ξ + π ) | + | b b n ( ξ ) | ) (cid:17) , where we used Cauchy-Schwarz inequality in the last inequality. Define B ( ξ ) := | b b p ( ξ + π ) | + | b b n ( ξ ) | . Thenthe above inequality can be rewritten as f ( B ( ξ )) > f ( x ) := − x + (cid:0) − | b a ( ξ ) | − | b a ( ξ + π ) | (cid:1) x − | b a ( ξ ) b a ( ξ + π ) | . (2.6)Since f is a polynomial of degree two, by calculation, we see that f has two real roots: A ( ξ ) and 2 − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) , where A is defined in (1.13). Rewrite A ( ξ ) in (1.13) as A ( ξ ) = 2 − | b a ( ξ ) | − | b a ( ξ + π ) | − p C ( ξ )2 (2.7)with C ( ξ ) := 4 (cid:0) − | b a ( ξ ) | − | b a ( ξ + π ) | (cid:1) + ( | b a ( ξ ) | − | b a ( ξ + π ) | ) . (2.8)Note that we can also rewrite the function C ( ξ ) as follows: C ( ξ ) = (2 − | b a ( ξ ) | − | b a ( ξ + π ) | ) − | b a ( ξ ) b a ( ξ + π ) | (2 − | b a ( ξ ) | − | b a ( ξ + π ) | ) . (2.9)From the expression of A in (2.7) and the above inequality, we see that A ( ξ ) > A ( ξ ) − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) . (2.10)In particular, we see that f ( x ) > A ( ξ ) < x < − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ). Therefore, since f ( x ) < x < A ( ξ ), we conclude from f ( B ( ξ )) > B ( ξ ) > A ( ξ ). Thus, we proved inequality (1.12). BIN HAN, QUN MO, AND ZHENPENG ZHAO
We now show that the inequality in (1.12) is sharp by explicitly constructing a tight framelet filter bank { a ;˚ b p , ˚ b n } satisfying | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | = A ( ξ ) for all ξ ∈ [0 , π ]. In the following, we shall construct such2 π -periodic measurable functions b ˚ b p and b ˚ b n by defining b ˚ b p ( ξ ) , b ˚ b p ( ξ + π ) , b ˚ b n ( ξ ) , b ˚ b n ( ξ + π ) on the interval ξ ∈ [0 , π ].For ξ ∈ [0 , π ], we define b ˚ b p ( ξ + π ) = , if C ( ξ ) = 0, s A ( ξ ) (cid:18) − | b a ( ξ ) | −| b a ( ξ + π ) | √ C ( ξ ) (cid:19) , otherwise (2.11)and b ˚ b n ( ξ ) = , if C ( ξ ) = 0, s A ( ξ ) (cid:18) | b a ( ξ ) | −| b a ( ξ + π ) | √ C ( ξ ) (cid:19) , otherwise. (2.12)We first show that both b ˚ b p ( ξ + π ) and b ˚ b n ( ξ ) are well defined nonnegative functions for ξ ∈ [0 , π ]. By thedefinition of C ( ξ ) in (2.8), it is straightforward to see that p C ( ξ ) > (cid:12)(cid:12)(cid:12) | b a ( ξ ) | − | b a ( ξ + π ) | (cid:12)(cid:12)(cid:12) for ξ ∈ [0 , π ].Consequently, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | b a ( ξ ) | − | b a ( ξ + π ) | p C ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since A ( ξ ) >
0, we now see that both b ˚ b p ( ξ + π ) in (2.11) and b ˚ b n ( ξ ) in (2.12) are well defined nonnegativefunctions for ξ ∈ [0 , π ]. Let β ( ξ ) denote the phase of b a ( ξ ) b a ( ξ + π ), that is, β is a real-valued measurablefunction on [0 , π ] such that b a ( ξ ) b a ( ξ + π ) = e iβ ( ξ ) | b a ( ξ ) b a ( ξ + π ) | , ξ ∈ [0 , π ] . (2.13)If b a ( ξ ) b a ( ξ + π ) = 0, then we simply define β ( ξ ) = 0. For ξ ∈ [0 , π ], we define b ˚ b p ( ξ ) = − e iβ ( ξ ) q − | b a ( ξ ) | − | b ˚ b n ( ξ ) | (2.14)and b ˚ b n ( ξ + π ) = − e − iβ ( ξ ) q − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | . (2.15)We now prove that b ˚ b p ( ξ ) and b ˚ b n ( ξ + π ) are well defined by proving that for ξ ∈ [0 , π ],1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | > − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | > | b a ( ξ ) b a ( ξ + π ) | = | b ˚ b p ( ξ ) b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) b ˚ b n ( ξ + π ) | . (2.17)We prove (2.16) and (2.17) by considering four cases.Case 1: C ( ξ ) = 0. Since C ( ξ ) = 0, it follows from (2.11) and (2.12) that b ˚ b p ( ξ + π ) = b ˚ b n ( ξ ) = . By C ( ξ ) = 0,it follows from the definition of C ( ξ ) in (2.8) that 1 − | b a ( ξ ) | − | b a ( ξ + π ) | = 0 and | b a ( ξ ) | − | b a ( ξ + π ) | = 0.Hence, we must have | b a ( ξ ) | = | b a ( ξ + π ) | = . Consequently, 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | = 1 − − = > − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | = 1 − − = >
0. Hence, (2.16) holds. Now by the definition of b ˚ b p ( ξ ) in (2.14)and b ˚ b n ( ξ + π ) in (2.15), we have b ˚ b p ( ξ ) = − e iβ ( ξ ) / b ˚ b n ( ξ + π ) = − e − iβ ( ξ ) /
2. Thus, it is trivial to check that(2.17) holds.Case 2: C ( ξ ) = 0 and A ( ξ ) = 0. By the definition of b ˚ b p ( ξ + π ) in (2.11) and b ˚ b n ( ξ ) in (2.12), we have b ˚ b p ( ξ + π ) = b ˚ b n ( ξ ) = 0. Clearly, (2.16) holds since 1 − | b a ( ξ ) | − | b a ( ξ + π ) | >
0. It is also easy to see that A ( ξ ) = 0implies b a ( ξ ) b a ( ξ + π ) = 0. Therefore, (2.17) is obviously true.Case 3: C ( ξ ) = 0, A ( ξ ) = 0, and | b a ( ξ ) | − | b a ( ξ + π ) | = p C ( ξ ) or − p C ( ξ ). Without loss of any generality,we only consider | b a ( ξ ) | − | b a ( ξ + π ) | = p C ( ξ ), from which we deduce that1 − | b a ( ξ ) | − | b a ( ξ + π ) | = 0 , b ˚ b p ( ξ + π ) = 0 , b ˚ b n ( ξ ) = p A ( ξ ) . OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 9
It follows from 1 − | b a ( ξ ) | − | b a ( ξ + π ) | = 0 and the definition of A ( ξ ) in (1.13) that A ( ξ ) = −| b a ( ξ ) | + | b a ( ξ + π ) | = | b a ( ξ + π ) | . Now we see that (2.16) is satisfied, since 1 − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | = 1 − | b a ( ξ + π ) | = | b a ( ξ ) | > − | b a ( ξ ) | − | b ˚ b n ( ξ ) | = 1 − | b a ( ξ ) | − A ( ξ ) = 1 − | b a ( ξ ) | − | b a ( ξ + π ) | = 0 . Consequently, we deduce from the above identity and the definition of b ˚ b p ( ξ ) in (2.14) that b ˚ b p ( ξ ) = 0. Since b ˚ b p ( ξ + π ) = 0 and A ( ξ ) = | b a ( ξ + π ) | , from the definition of b ˚ b n ( ξ + π ) in (2.15) we deduce that | b ˚ b n ( ξ + π ) | = 1 − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | = 1 − | b a ( ξ + π ) | = | b a ( ξ ) | . Therefore, by b ˚ b p ( ξ ) = b ˚ b p ( ξ + π ) = 0, b ˚ b n ( ξ ) = p A ( ξ ), and | b ˚ b n ( ξ + π ) | = | b a ( ξ ) | , we see that | b ˚ b p ( ξ ) b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) b ˚ b n ( ξ + π ) | = | b ˚ b n ( ξ ) b ˚ b n ( ξ + π ) | = p A ( ξ ) | b a ( ξ ) | = | b a ( ξ ) b a ( ξ + π ) | , where we used the identity A ( ξ ) = | b a ( ξ + π ) | in the last identity. Hence, (2.17) holds.Case 4: C ( ξ ) = 0, A ( ξ ) = 0, and | b a ( ξ ) | − | b a ( ξ + π ) | = ± p C ( ξ ). Note that the last two conditions implythat b ˚ b p ( ξ + π ) = 0 and b ˚ b n ( ξ ) = 0. From the definition of b ˚ b p ( ξ + π ) in (2.11) and b ˚ b n ( ξ ) in (2.12), we see that | b ˚ b p ( ξ + π ) | | b ˚ b n ( ξ ) | = p C ( ξ ) − ( | b a ( ξ ) | − | b a ( ξ + π ) | ) p C ( ξ ) + ( | b a ( ξ ) | − | b a ( ξ + π ) | ) = 1 − | b a ( ξ ) | − A ( ξ )1 − | b a ( ξ + π ) | − A ( ξ ) , (2.18)where we used the relation p C ( ξ ) = 2 − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) (derived from the definition of A ( ξ ) in(1.13)) in the last identity. Since C ( ξ ) = 0, we deduce from the definition of b ˚ b p ( ξ + π ) in (2.11) and b ˚ b n ( ξ ) in(2.12) that | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | = A ( ξ ). Now it follows directly from (2.18) that | b ˚ b p ( ξ + π ) | | b ˚ b n ( ξ ) | = 1 − | b a ( ξ ) | − A ( ξ )1 − | b a ( ξ + π ) | − A ( ξ ) = 1 − | b a ( ξ ) | − A ( ξ ) + | b ˚ b p ( ξ + π ) | − | b a ( ξ + π ) | − A ( ξ ) + | b ˚ b n ( ξ ) | = 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | . That is, we proved | b ˚ b p ( ξ + π ) | | b ˚ b n ( ξ ) | = 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | . (2.19)From the identity in (2.19), we further deduce that | b ˚ b p ( ξ + π ) | A ( ξ ) = | b ˚ b p ( ξ + π ) | | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | = 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | (1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | ) + (1 − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | )= 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) . In other words, we proved | b ˚ b p ( ξ + π ) | A ( ξ ) = 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) . (2.20)Similarly, we can prove that | b ˚ b n ( ξ ) | A ( ξ ) = 1 − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) . (2.21)By our assumption A ( ξ ) >
0, we see from (2.10) that 2 − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) > A ( ξ ) >
0. Since b ˚ b p ( ξ + π ) = 0 and b ˚ b n ( ξ ) = 0, we deduce from (2.20) that we must have 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | >
0. By the sameargument, we deduce from (2.21) that 1 − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | >
0. Hence, we proved (2.16). Therefore, b ˚ b p ( ξ ) and b ˚ b n ( ξ + π ) are well defined. It now follows from (2.19) that | b ˚ b p ( ξ + π ) | | b ˚ b n ( ξ ) | = 1 − | b a ( ξ ) | − | b ˚ b n ( ξ ) | − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | = | b ˚ b p ( ξ ) | | b ˚ b n ( ξ + π ) |
20 BIN HAN, QUN MO, AND ZHENPENG ZHAO from which we see that the vector ( | b ˚ b p ( ξ + π ) | , | b ˚ b n ( ξ ) | ) is parallel to the vector ( | b ˚ b p ( ξ ) | , | b ˚ b n ( ξ + π ) | ). Consequently,we must have | b ˚ b p ( ξ ) b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) b ˚ b n ( ξ + π ) | = q | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | q | b ˚ b p ( ξ ) | + | b ˚ b n ( ξ + π ) | . By the definition of b ˚ b p ( ξ + π ) in (2.11) and b ˚ b n ( ξ ) in (2.12) and by the definition of b ˚ b p ( ξ ) in (2.14) and b ˚ b n ( ξ + π )in (2.15), we conclude that | b ˚ b p ( ξ ) b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) b ˚ b n ( ξ + π ) | = q | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | q | b ˚ b p ( ξ ) | + | b ˚ b n ( ξ + π ) | = p A ( ξ )(2 − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ )) = | b a ( ξ ) b a ( ξ + π ) | , where in the last identity we used the fact that A ( ξ ) and 2 − | b a ( ξ ) | − | b a ( ξ + π ) | − A ( ξ ) are the two roots of f in (2.6) and f (0) = −| b a ( ξ ) b a ( ξ + π ) | . Thus, we proved (2.17).By our construction, it is now trivial to see that | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | = A ( ξ ) for all ξ ∈ [0 , π ] such that C ( ξ ) = 0. If C ( ξ ) = 0, as discussed in Case 1, then we have A ( ξ ) = and we still have | b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) | = + = = A ( ξ ). To complete the proof, we now show that { a ;˚ b p , ˚ b n } is a tight framelet filter bank. By ourconstruction of b ˚ b p and b ˚ b n , it is trivial to see that (2.2) and (2.3) are satisfied with b p and b n being replaced by˚ b p and ˚ b n , respectively. To check (2.4), we have b a ( ξ ) b a ( ξ + π ) + b ˚ b p ( ξ ) b ˚ b p ( ξ + π ) + b ˚ b n ( ξ ) b ˚ b n ( ξ + π )= e iβ ( ξ ) | b a ( ξ ) b a ( ξ + π ) | − e iβ ( ξ ) ( | b ˚ b p ( ξ ) b ˚ b p ( ξ + π ) | + | b ˚ b n ( ξ ) b ˚ b n ( ξ + π ) | ) = 0 , where in the last identity we used (2.17). Therefore, { a ;˚ b p , ˚ b n } is indeed a tight framelet filter bank.If the filter a is real-valued, then b a ( ξ ) = b a ( − ξ ) a.e. ξ ∈ R . Consequently, we have | b a ( − ξ ) | = | b a ( ξ ) | and C ( − ξ ) = C ( ξ ) = C ( π − ξ ) , A ( − ξ ) = A ( ξ ) = A ( π − ξ ) . (2.22)We now prove that b ˚ b p ( − ξ ) = b ˚ b n ( ξ ) a.e. ξ ∈ R , which is equivalent to verify that b b p ( − ξ ) = b ˚ b n ( ξ ) and b b p ( π − ξ ) = b ˚ b n ( ξ − π ) , a.e. ξ ∈ [0 , π ] . (2.23)By (2.22) and the definition of b ˚ b p ( ξ + π ) in (2.11) and b ˚ b n ( ξ ) in (2.12), we see that b ˚ b p ( − ξ ) = b ˚ b p (( π − ξ ) + π ) = b ˚ b p (( π − ξ ) + π ) = b ˚ b n ( ξ ) , ξ ∈ [0 , π ] , which is the first identity in (2.23). Similarly, we have b b p ( π − ξ ) = − e − iβ ( π − ξ ) q − | b a ( π − ξ ) | − | b ˚ b n ( π − ξ ) | = − e − iβ ( π − ξ ) q − | b a ( ξ + π ) | − | b ˚ b p ( ξ + π ) | = e i ( β ( ξ ) − β ( π − ξ )) b ˚ b n ( ξ + π ) , where we used (2.15) and the first identity in (2.23). If we can prove that e i ( β ( ξ ) − β ( π − ξ )) = 1 , ξ ∈ [0 , π ] , (2.24)then the second identity in (2.23) holds and therefore, we proved b b p ( − ξ ) = b b n ( ξ ) a.e. ξ ∈ R .We now prove (2.24). Replacing ξ by π − ξ in the definition of β ( ξ ) in (2.13) and using (2.22), we have b a ( π − ξ ) b a (2 π − ξ ) = e iβ ( ξ − π ) | b a ( π − ξ ) b a (2 π − ξ ) | = e iβ ( ξ − π ) | b a ( ξ ) b a ( ξ + π ) | . Since b a ( ξ ) = b a ( − ξ ), we have b a ( π − ξ ) b a (2 π − ξ ) = b a ( ξ − π ) b a ( − ξ ) = b a ( ξ + π ) b a ( ξ ) = b a ( ξ ) b a ( ξ + π ) . Consequently, comparing with (2.13), we conclude that for ξ ∈ [0 , π ] such that b a ( ξ ) b a ( ξ + π ) = 0, we musthave e iβ ( π − ξ ) = e iβ ( ξ ) , which is simply (2.24). For the case that b a ( ξ ) b a ( ξ + π ) = 0, (2.24) is trivially true since β ( ξ ) = β ( π − ξ ) = 0. This completes the proof of Theorem 1. (cid:3) As demonstrated by Theorem 2, the frequency separation function A in (1.13) is often small for many knownlow-pass filters. OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 11
Proof of Theorem 2.
Define x := | b a ( ξ ) | and y := | b a ( ξ + π ) | . Then 0 x, y x + y
1. In termsof x and y , the function A ( ξ ) in (1.13) can be rewritten as A ( ξ ) = A ( x, y ) with A ( x, y ) := 2 − x − y − p − x − y ) + ( x − y ) . (2.25)By a simple direct calculation, we have A ( x, y ) = x − x (1 − x − y ) g ( x, y ) x, (2.26)where g ( x, y ) := 2 − x − y + p − x − y ) + ( x − y ) > − x − y + ( x − y ) = 2(1 − x − y ) > . If g ( x, y ) >
0, by the symmetry between x and y in A ( x, y ), then it follows from (2.26) that A ( ξ ) = A ( x, y ) min( x, y ) = min( | b a ( ξ ) | , | b a ( ξ + π ) | ). Note that g ( x, y ) = 0 if and only if x + y = 1 and x > y . If g ( x, y ) = 0,then we also have A ( ξ ) = A ( x, y ) = y = min( x, y ) = min( | b a ( ξ ) | , | b a ( ξ + π ) | ). Therefore, we proved theinequality (1.14).Item (i) follows directly from the definition of A ( ξ ) and the relation in (2.8). Item (ii) follows directly from(2.26). For item (iii), by the definition of the function A in (1.13) with a = a Bm , we have A ( ξ ) = A ( x, y ) andsin m ( ξ ) = 2 m sin m ( ξ/
2) cos m ( ξ/
2) = 4 m xy . Note that A ( x, y ) = (2 − x − y ) − p (2 − x − y ) − xy = 4 xy (2 − x − y ) + p (2 − x − y ) − xy . Since 0 x, y
1, we obviously have 0 p (2 − x − y ) − xy − x − y . Therefore, we conclude that xy − x − y A ( x, y ) xy − x − y . Consequently, by 0 x, y x + y
1, we deduce that xy xy − x − y A ( x, y ) xy − x − y xy. This completes the proof of (1.15). (cid:3)
The following result shows that for a tight framelet filter bank { a ; b p , b n } , if the high-pass filters b p and b n arereal-valued (but the filter a can be complex-valued), then its frequency separation between b p and b n cannotbe good. Moreover, the best possible frequency separation between two real-valued high-pass filters b p and b n in a tight framelet filter bank { a ; b p , b n } is achieved when a is an orthogonal filter. On the other hand,Theorem 2 tells us that the frequency separation between two complex-valued high-pass filters b p and b n in acomplex-valued tight framelet filter bank { a ; b p , b n } is the worst when a is an orthogonal filter. Theorem 3.
Let a, b p , b n ∈ l ( Z ) such that { a ; b p , b n } is a tight framelet filter bank and the two high-pass filters b p and b n are real-valued (but the filter a may be complex-valued). Then Z π (cid:2) | b b p ( ξ + π ) | + | b b n ( ξ ) | (cid:3) dξ = 12 Z π (cid:2) − | b a ( ξ ) | − | b a ( ξ + π ) | (cid:3) dξ > π , (2.27) where the equal sign holds if and only if a is an orthogonal filter (that is, | b a ( ξ ) | + | b a ( ξ + π ) | = 1 a.e. ξ ∈ R ).Proof. Define B ( ξ ) := | b b p ( ξ + π ) | + | b b n ( ξ ) | . Note that a general filter u has real coefficients if and only if b u ( ξ ) = b u ( − ξ ). Therefore, we have b b p ( ξ + π ) = b b p ( ξ − π ) = b b p ( π − ξ ). Hence, B ( ξ ) = | b b p ( π − ξ ) | + | b b n ( ξ ) | .By | b a ( ξ ) | + | b b p ( ξ ) | + | b b n ( ξ ) | = 1, we have | b a ( π − ξ ) | + | b b p ( π − ξ ) | + | b b n ( π − ξ ) | = 1. Therefore, B ( ξ ) + B ( π − ξ ) = | b b p ( π − ξ ) | + | b b n ( ξ ) | + | b b p ( ξ ) | + | b b n ( π − ξ ) | = 2 − | b a ( ξ ) | − | b a ( π − ξ ) | . (2.28)Note that1 = | b a ( − ξ ) | + | b b p ( − ξ ) | + | b b n ( − ξ ) | = | b a ( − ξ ) | + | b b p ( ξ ) | + | b b n ( ξ ) | = 1 + | b a ( − ξ ) | − | b a ( ξ ) | , from which we must have | b a ( − ξ ) | = | b a ( ξ ) | . Therefore, it follows from (2.28) that B ( ξ ) + B ( π − ξ ) = 2 − | b a ( ξ ) | − | b a ( ξ + π ) | , from which we have Z π (cid:2) − | b a ( ξ ) | − | b a ( ξ + π ) | (cid:3) dξ = Z π (cid:2) B ( ξ ) + B ( π − ξ ) (cid:3) dξ = 2 Z π B ( ξ ) dξ. Since | b a ( ξ ) | + | b a ( ξ + π ) | ξ ∈ R , we conclude from the above identity that (2.27) holds. (cid:3) Structure of Finitely Supported Complex Tight Framelet Filter Banks
In order to design finitely supported complex tight framelet filter banks { a ; b p , b n } with good directionality,we have to investigate the structure of all possible finitely supported complex-valued high-pass filters b p , b n suchthat { a ; b p , b n } is a tight framelet filter bank. More precisely, from any given finitely supported filter a ∈ l ( Z ),we are interesting in finding all possible finitely supported complex tight framelet filter banks { a ; b p , b n } derivedfrom a given low-pass filter a . For prescribed filter lengths of b p and b n , such a result enables us to find thebest possible complex tight framelet filter bank { a ; b p , b n } with the best possible frequency separation, that is, | b b p ( ξ + π ) | + | b b n ( ξ ) | ≈ A ( ξ ) , ξ ∈ [0 , π ].To construct finitely supported tight framelet filter banks, it is convenient to use Laurent polynomials insteadof 2 π -periodic trigonometric polynomials. Recall that l ( Z ) denotes the linear space of all finitely supportedsequences on Z . For a sequence u = { u ( k ) } k ∈ Z ∈ l ( Z ), its z -transform is a Laurent polynomial defined to be u ( z ) := X k ∈ Z u ( k ) z k , z ∈ C \{ } . (3.1)Let u : Z → C r × s be a sequence of r × s matrices. We define u ⋆ to be its associated adjoint sequence by u ⋆ ( k ) := u ( − k ) T , k ∈ Z . In terms of Fourier series, we have c u ⋆ ( ξ ) = b u ( ξ ) T and b u ( ξ ) = u ( e − iξ ). Using Laurentpolynomials, we have u ⋆ ( z ) := [ u ( z )] ⋆ := X k ∈ Z u ( k ) T z − k , z ∈ C \{ } . (3.2)In terms of Laurent polynomials, for a, b , b ∈ l ( Z ), { a ; b , b } is a tight framelet filter bank if (cid:20) a ( z ) b ( z ) b ( z ) a ( − z ) b ( − z ) b ( − z ) (cid:21) (cid:20) a ( z ) b ( z ) b ( z ) a ( − z ) b ( − z ) b ( − z ) (cid:21) ⋆ = I (3.3)for all z ∈ C \{ } , where I is the 2 × × U of Laurent polynomials, we say that U is paraunitary if U ( z ) U ⋆ ( z ) = I for all z ∈ T := { ζ ∈ C : | ζ | = 1 } , or equivalently, U ( e − iξ ) U ( e − iξ ) T = I forall ξ ∈ R .For a Laurent polynomial u , we shall use the notation u ≡ u is identically zero, and the notation u u is not identically zero. We say that u is an orthogonal filter if u ( z ) u ⋆ ( z )+ u ( − z ) u ⋆ ( − z ) = 1for all z ∈ C \{ } .The main result in this section is as follows: Theorem 4.
Let a, b , b , b p , b n ∈ l ( Z ) such that { a ; b , b } is a tight framelet filter bank and the filter a is notidentically zero. Suppose that | a ( z ) | + | a ( − z ) | , ∀ z ∈ T . (3.4) Then the following are equivalent: (i) { a ; b p , b n } is a finitely supported tight framelet filter bank and b p ( z ) b n ( − z ) − b p ( − z ) b n ( z ) = λz k [ b ( z ) b ( − z ) − b ( − z ) b ( z )] (3.5) for some k ∈ Z and λ ∈ T . Remove condition (3.5) if a is an orthogonal filter. (ii) There exists a × paraunitary matrix U of Laurent polynomials such that (cid:2) b p ( z ) b n ( z ) (cid:3) = (cid:2) b ( z ) b ( z ) (cid:3) U ( z ) , ∀ z ∈ C \{ } . (3.6)From (3.3), we see that { a ; b , b } is a tight framelet filter bank if and only if (cid:20) b ( z ) b ( z ) b ( − z ) b ( − z ) (cid:21) (cid:20) b ( z ) b ( z ) b ( − z ) b ( − z ) (cid:21) ⋆ = M a ( z ) (3.7)with M a ( z ) := (cid:20) − a ( z ) a ⋆ ( z ) − a ( z ) a ⋆ ( − z ) − a ( − z ) a ⋆ ( z ) 1 − a ( − z ) a ⋆ ( − z ) (cid:21) . (3.8)We define a Laurent polynomial d b ,b by d b ,b ( z ) := z [ b ( z ) b ( − z ) − b ( − z ) b ( z )] . (3.9)Note that d b ,b is a well-defined Laurent polynomial. Then it follows from (3.7) that | d b ,b ( z ) | = det( M a ( z )) = 1 − | a ( z ) | − | a ( − z ) | , ∀ z ∈ T . (3.10) OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 13 If a is an orthogonal filter, then we must have d b ,b ≡
0. For d b ,b
0, by Fej´er-Riesz lemma, we see that upto a monomial factor there are essentially only finitely many Laurent polynomials d b ,b satisfying (3.10). Aswe shall discuss in Section 4, all finitely supported complex-valued tight framelet filter banks { a ; b , b } havingthe shortest possible filter supports can be derived from the low-pass filter a by solving a system of linearequations. Consequently, Theorem 4 allows us to obtain all finitely supported complex-valued tight frameletfilter banks { a ; b , b } with the low-pass filter a being given in advance. Using Theorem 4, we shall discussin Section 4 how to find the best possible complex tight framelet filter bank { a ; b p , b n } with the best possiblefrequency separation for any prescribed filter lengths of the high-pass filters b p and b n .To prove Theorem 4, we need several auxiliary results. Let us first introduce some definitions. We say that u is a trivial factor if it is a nonzero monomial, that is, u ( z ) = λz k for some λ ∈ C \{ } and k ∈ Z . For twoLaurent polynomials u and v , by gcd( u , v ) we denote the greatest common factor of u and v . In particular, weuse the notation gcd( u , v ) = 1 to mean that u and v do not have a nontrivial common factor. Lemma 5.
Let p , p , p , p be Laurent polynomials. Define P ( z ) := (cid:20) p ( z ) p ( z ) p ( z ) p ( z ) (cid:21) . (3.11) Then the following are equivalent: (1) det( P ( z )) = 0 for all z ∈ C \{ } . (2) p ( z ) p ( z ) − p ( z ) p ( z ) = 0 for all z ∈ C \{ } . (3) There exist Laurent polynomials q , q , q , q such that p ( z ) = q ( z ) q ( z ) , p ( z ) = q ( z ) q ( z ) , p ( z ) = q ( z ) q ( z ) , p ( z ) = q ( z ) q ( z ) . (3.12)(4) There exist Laurent polynomials q , q , q , q such that P ( z ) = (cid:20) q ( z ) q ( z ) (cid:21) (cid:2) q ( z ) q ( z ) (cid:3) . Proof. If P is identically zero, then all claims hold obviously. Hence, we assume that at least one of p , p , p , p is not identically zero. It is trivial that (1)= ⇒ (2) and (3)= ⇒ (4)= ⇒ (1). To complete the proof, it suffices toprove (2)= ⇒ (3).If both p and p are identically zero, then the claim in item (3) obviously holds by taking q = p , q = p , q = 0 and q = 1. Now we assume that either p p
0, that is, at least one of p and p is notidentically zero. Define q := gcd( p , p ) and q := p / q , q := p / q . (3.13)Since q is not identically zero, all q , q , q are well-defined Laurent polynomials and at least one of q and q are not identically zero. Moreover, p = q q , p = q q , and gcd( q , q ) = 1, which means that q and q haveno nontrivial common factor. By item (2), we have0 = p p − p p = q ( q p − q p ) . Since q is not identically zero, from the above identity we must have q p = q p . Because at least one of q and q is not identically zero, without loss of generality, we may assume that q is not identically zero. Bygcd( q , q ) = 1 and q p = q p , we must have q | p . Then we define q = p / q , which is a well-definedLaurent polynomial. By q p = q p , we see that p = q q . Using (3.13), now one can directly check that(3.12) holds. Therefore, we complete the proof of (2)= ⇒ (3). (cid:3) Proposition 6.
Let Q and V be × matrices of Laurent polynomials. If V ( z ) Q ( z ) = (cid:20) c ( z ) 00 d ( z ) (cid:21) , (3.14) then there exist Laurent polynomials u , u , u , u , v , v , v , v such that V ( z ) = (cid:20) v ( z ) 00 v ( z ) (cid:21) (cid:20) u ( z ) − u ( z ) u ( z ) u ( z ) (cid:21) , Q ( z ) = (cid:20) u ( z ) u ( z ) − u ( z ) u ( z ) (cid:21) (cid:20) v ( z ) 00 v ( z ) (cid:21) (3.15) and c ( z ) = v ( z ) v ( z ) (cid:0) u ( z ) u ( z ) + u ( z ) u ( z ) (cid:1) , d ( z ) = v ( z ) v ( z ) (cid:0) u ( z ) u ( z ) + u ( z ) u ( z ) (cid:1) . (3.16) If c = 1 , then we can particularly take v = v = 1 so that u ( z ) u ( z ) + u ( z ) u ( z ) = 1 and d ( z ) = v ( z ) v ( z ) . Proof.
By our assumption in (3.14), we have [ V ( z ) Q ( z )] , ( z ) = V , ( z ) Q , ( z ) + V , ( z ) Q , ( z ) = 0 for all z ∈ C \{ } . By Lemma 5, there exist Laurent polynomials u , u , v , v such that (cid:20) V , ( z ) V , ( z ) − Q , ( z ) Q , ( z ) (cid:21) = (cid:20) v ( z ) − v ( z ) (cid:21) (cid:2) u ( z ) − u ( z ) (cid:3) . Similarly, by our assumption in (3.14), we have [ V ( z ) Q ( z )] , ( z ) = V , ( z ) Q , ( z ) + V , ( z ) Q , ( z ) = 0 for all z ∈ C \{ } . By Lemma 5, there exist Laurent polynomials u , u , v , v such that (cid:20) V , ( z ) V , ( z ) − Q , ( z ) Q , ( z ) (cid:21) = (cid:20) v ( z ) v ( z ) (cid:21) (cid:2) u ( z ) u ( z ) (cid:3) . Now we can directly check that both (3.15) and (3.16) are satisfied.If c = 1, then it follows from (3.16) that all v , v and u u + u u must be monomials. Now it follows directlyfrom (3.15) that V ( z ) = (cid:20) v ( z ) / v ( z ) (cid:21) (cid:20) u ( z ) v ( z ) − u ( z ) v ( z ) u ( z ) v ( z ) u ( z ) v ( z ) (cid:21) and Q ( z ) = (cid:20) u ( z ) v ( z ) u ( z ) v ( z ) − u ( z ) v ( z ) u ( z ) v ( z ) (cid:21) (cid:20) v ( z ) / v ( z ) (cid:21) . Redefine u , u , u , u , v , v as u v , u v , u v , u v , v / v , v / v , respectively. We now see that the claim holdsfor the particular case of c = 1. (cid:3) As a direct consequence of Proposition 6, we have the following two corollaries.
Corollary 7.
Let P be a × matrix of Laurent polynomials defined in (3.11) . Then P is paraunitary, thatis, P ( z ) P ⋆ ( z ) = I for all z ∈ C \{ } , if and only if p ( z ) = − λz k p ⋆ ( z ) , p ( z ) = λz k p ⋆ ( z ) , p ( z ) p ⋆ ( z ) + p ( z ) p ⋆ ( z ) = 1 with λ ∈ T , k ∈ Z . (3.17) Proof.
Let Q and V be the 2 × V ( z ) := P ( z ) and Q ( z ) := P ⋆ ( z ) = (cid:20) p ⋆ ( z ) p ⋆ ( z ) p ⋆ ( z ) p ⋆ ( z ) (cid:21) . If P is paraunitary, then V ( z ) Q ( z ) = I . By Proposition 6 with c = 1, we see that (3.17) must hold.Conversely, if (3.17) is satisfied, then we can directly check that P is a paraunitary matrix. (cid:3) Corollary 8.
Let Q , V , ˚ Q , ˚ V be × matrices of Laurent polynomials. If V ( z ) Q ( z ) = (cid:20) d ( z ) (cid:21) = ˚ V ( z )˚ Q ( z ) (3.18) and det(˚ V ( z )) = λz k det( V ( z )) for some λ ∈ C \{ } , k ∈ Z . (3.19) Then there exists a × matrix U of Laurent polynomials such that det( U ( z )) = λz k and ˚ V ( z ) = V ( z ) U ( z ) . (3.20) Proof.
By Proposition 6 with c = 1, we see that V ( z ) = (cid:20) V ( z )) (cid:21) U ( z ) , ˚ V ( z ) = (cid:20) V ( z )) (cid:21) U ( z ) , where U , U are 2 × U ( z )) = det( U ( z )) = 1. Therefore,[ U ( z )] − is also a matrix of Laurent polynomials. Define U ( z ) := [ U ( z )] − (cid:20) λz k (cid:21) U ( z ) . Now it is trivial to check that (3.20) holds and det( U ( z )) = λz k is a nontrivial monomial. (cid:3) Now we have the following result about the essential uniqueness of factorization of a positive semidefinite2 × OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 15
Theorem 9.
Let P be a × matrix of Laurent polynomials given in (3.11) such that det( P ( z )) (thatis, the determinant of P is not identically zero) and gcd( p , p , p , p ) = 1 . If V and ˚ V are × matrices ofLaurent polynomials satisfying V ( z ) V ⋆ ( z ) = P ( z ) = ˚ V ( z )˚ V ⋆ ( z ) (3.21) and det(˚ V ( z )) = λz k det( V ( z )) for some λ ∈ C \{ } , k ∈ Z , (3.22) then there exists a × paraunitary matrix U of Laurent polynomials such that ˚ V ( z ) = V ( z ) U ( z ) , det( U ( z )) = λz k , and U ( z ) U ⋆ ( z ) = I for all z ∈ C \{ } .Proof. It is a basic result in linear algebra that there exist two 2 × A and B of Laurent polynomialssatisfying det( A ( z )) = det( B ( z )) = 1 and A ( z ) P ( z ) B ( z ) = (cid:20) c ( z ) 00 d ( z ) (cid:21) with c , d being Laurent polynomials satisfying c | d . The above result can be proved using elementary matrixforms and Euclidian division of Laurent polynomials. The diagonal matrix diag( c , d ) is called the Smith normalform of P and such Laurent polynomials c , d are essentially unique. See [20] for a detailed proof of the aboveresult. Moreover, one can directly verify that c = gcd( p , p , p , p ) = 1 and d = det( P ) / c
0. Consequently,by (3.21), we have ( A ( z ) V ( z ))( V ⋆ ( z ) B ( z )) = (cid:20) d ( z ) (cid:21) = ( A ( z )˚ V ( z ))(˚ V ⋆ ( z ) B ( z )) . Note that det( A ( z )˚ V ( z )) = det( A ( z )) det(˚ V ( z )) = det(˚ V ( z )) = λz k det( V ( z )) = λz k det( A ( z ) V ( z )) . Consequently, it follows from Corollary 8 that there exists a 2 × U of Laurent polynomials such thatdet( U ( z )) = λz k and A ( z )˚ V ( z ) = A ( z ) V ( z ) U ( z ), from which we have ˚ V ( z ) = V ( z ) U ( z ) since det( A ( z )) = 1.Therefore, it follows from (3.21) that V ( z ) V ⋆ ( z ) = ˚ V ( z )˚ V ⋆ ( z ) which leads to V ( z ) (cid:0) U ( z ) U ⋆ ( z ) − I (cid:1) V ⋆ ( z ) = 0 . By (3.21), we have det( V ( z )) det( V ⋆ ( z )) = det( P ( z )) V ( z ))
0. Thus, V ( z ) is invertiblefor all z satisfying det( V ( z )) = 0. Now we deduce from the above identity that we must have U ( z ) U ⋆ ( z ) = I for all z ∈ C \{ } . (cid:3) We are now ready to prove Theorem 4.
Proof of Theorem 4. (ii)= ⇒ (i) is trivial. Note that (3.6) is equivalent to (cid:20) b p ( z ) b n ( z ) b p ( − z ) b n ( − z ) (cid:21) = (cid:20) b ( z ) b ( z ) b ( − z ) b ( − z ) (cid:21) U ( z ) , ∀ z ∈ C \{ } . (3.23)Since { a ; b , b } is a tight framelet filter bank and U is paraunitary, it follows directly from (3.7) and (3.23)that { a ; b p , b n } is a finitely supported tight framelet filter bank. Moreover, it follows directly from (3.23) that(3.5) holds with λz k := det( U ( z )).We now prove (i)= ⇒ (ii). For a sequence u : Z → C and γ ∈ Z , its coset sequence u [ γ ] is defined to be u [ γ ] ( k ) := u ( γ + 2 k ) , k ∈ Z . Since both { a ; b , b } and { a ; b p , b n } are finitely supported tight framelet filterbanks, using coset sequences, we see from (3.7) that (cid:20) b p, [0] ( z ) b n, [0] ( z ) b p, [1] ( z ) b n, [1] ( z ) (cid:21) (cid:20) b p, [0] ( z ) b n, [0] ( z ) b p, [1] ( z ) b n, [1] ( z ) (cid:21) ⋆ = N a ( z ) = " b [0]1 ( z ) b [0]2 ( z ) b [1]1 ( z ) b [1]2 ( z ) b [0]1 ( z ) b [0]2 ( z ) b [1]1 ( z ) b [1]2 ( z ) ⋆ , (3.24)where N a ( z ) := (cid:20) − a [0] ( z )( a [0] ( z )) ⋆ − a [0] ( z )( a [1] ( z )) ⋆ − ( a [0] ( z )) ⋆ a [1] ( z ) − a [1] ( z )( a [1] ( z )) ⋆ (cid:21) . Define c ( z ) := gcd([ N a ( z )] , , [ N a ( z )] , , [ N a ( z )] , , [ N a ( z )] , ). By direct calculation, we have 2 det( N a ( z )) = − a [0] ( z )( a [0] ( z )) ⋆ − a [1] ( z )( a [1] ( z )) ⋆ and trace( N a ( z )) = 1 − a [0] ( z )( a [0] ( z )) ⋆ − a [1] ( z )( a [1] ( z )) ⋆ . Therefore, c must be a factor of trace( N a ( z )) − N a ( z )) = 1 /
2. Consequently, we conclude that c = 1. We now consider two cases. We first consider the case that a is not an orthogonal filter. Then det( N a ( z ))
0. By Theorem 9,there must exist a 2 × U of Laurent polynomials such that (cid:20) b p, [0] ( z ) b n, [0] ( z ) b p, [1] ( z ) b n, [1] ( z ) (cid:21) = " b [0]1 ( z ) b [0]2 ( z ) b [1]1 ( z ) b [1]2 ( z ) U ( z ) (3.25)for all z ∈ C \{ } . Since u ( z ) = u [0] ( z ) + z u [1] ( z ) holds for any u ∈ l ( Z ), it is straightforward to deduce from(3.25) that (3.6) holds. Hence item (ii) is proved if a is not an orthogonal filter.We now consider the case that a is an orthogonal filter. Define a filter b by b ( z ) := z a ⋆ ( − z ). Then { a ; b } isa tight framelet filter bank. It suffices to prove item (ii) with b = b and b = 0. Since a is an orthogonal filter,we must have a [0] ( z )( a [0] ( z )) ⋆ + a [1] ( z )( a [1] ( z )) ⋆ = b [0] ( z )( b [0] ( z )) ⋆ + b [1] ( z )( b [1] ( z )) ⋆ = 1 / N a ( z )) = 0. By (3.24) and det( N a ( z )) = 0, we must have b p, [0] ( z ) b n, [1] ( z ) − b p, [1] ( z ) b n, [0] ( z ) = 0. ByLemma 5, there exist Laurent polynomials p , p , p , p such that (cid:20) b p, [0] ( z ) b n, [0] ( z ) b p, [1] ( z ) b n, [1] ( z ) (cid:21) = (cid:20) p ( z ) p ( z ) (cid:21) (cid:2) p ( z ) p ( z ) (cid:3) . Since b = b and b = 0, now (3.24) and (3.26) imply (cid:20) p ( z ) p ( z ) (cid:21) (cid:2) p ( z ) p ( z ) (cid:3) (cid:20) p ⋆ ( z ) p ⋆ ( z ) (cid:21) (cid:2) p ⋆ ( z ) p ⋆ ( z ) (cid:3) = (cid:20) b [0] ( z ) b [1] ( z ) (cid:21) (cid:2) ( b [0] ( z )) ⋆ ( b [1] ( z )) ⋆ (cid:3) . Multiplying (cid:2) b [0] ( z ) b [1] ( z ) (cid:3) T from the right on both sides of the above identity, by (3.26), we see that q ( z ) (cid:20) p ( z ) p ( z ) (cid:21) = (cid:20) b [0] ( z ) b [1] ( z ) (cid:21) with q ( z ) := 2[ p ( z ) p ⋆ ( z ) + p ( z ) p ⋆ ( z )][ p ⋆ ( z ) b [0] ( z ) + p ⋆ ( z ) b [1] ( z )] . (3.27)Since gcd( b [0] , b [1] ) = 1 by (3.26), q must be a nontrivial monomial. Consequently, without loss of any generality,we can assume that p = b [0] and p = b [1] . Then it follows from (3.26) and (3.27) that q = 1 and p ( z ) p ⋆ ( z ) + p ( z ) p ⋆ ( z ) = 1. Consequently, U ( z ) := (cid:20) p ( z ) p ( z ) − p ⋆ ( z ) p ⋆ ( z ) (cid:21) is a paraunitary matrix and it is trivial to check that (3.25) is satisfied, since " b [0]1 ( z ) b [0]2 ( z ) b [1]1 ( z ) b [1]2 ( z ) U ( z ) = (cid:20) b [0] ( z ) b [1] ( z ) (cid:21) (cid:2) (cid:3) U ( z ) = (cid:20) b [0] ( z ) b [1] ( z ) (cid:21) (cid:2) p ( z ) p ( z ) (cid:3) = (cid:20) b p, [0] ( z ) b n, [0] ( z ) b p, [1] ( z ) b n, [1] ( z ) (cid:21) . This proves item (ii) for the case that a is an orthogonal filter. (cid:3) Algorithms and Examples of Finitely Supported Complex Tight Framelet Filter Bankswith Directionality
In this section we shall propose an algorithm to construct finitely supported complex tight framelet filterbanks { a ; b p , b n } with good frequency separation from any given finitely supported low-pass filter a satisfying(3.4). Then we shall provide several examples to illustrate our algorithm.For a finitely supported sequence u = { u ( k ) } k ∈ Z such that u ( k ) = 0 for all k ∈ Z \ [ m, n ] and u ( m ) u ( n ) = 0,we define fsupp( u ) := fsupp( u ) := [ m, n ] to be the filter support of u and define len( u ) := len( u ) := n − m tobe the length of the filter u .In order to employ Theorem 4 to obtain all finitely supported tight framelet filter banks derived from a givenlow-pass filter, we now recall an algorithm, which is a special case of [10, Algorithm 4], to construct all possiblecomplex tight framelet filter banks { a ; b , b } having the shortest filter support, that is, max(len( b ) , len( b )) len( a ). Algorithm 1.
Let a ∈ l ( Z ) be a finitely supported filter on Z satisfying (3.4) . (S1) Define A ( z ) := 1 − a ( z ) a ⋆ ( z ) , B ( z ) := − a ( z ) a ⋆ ( − z ) , and D ( z ) := 1 − a ( z ) a ⋆ ( z ) − a ( − z ) a ⋆ ( − z ) ; (S2) Select ǫ, s , s ∈ { , } and a polynomial d satisfying d ( z ) d ⋆ ( z ) = D ( z ) with ⌈ s + s − ⌉ m d n d ⌊ s + s − ⌋ + n + ǫ , where [ − n , n ] := fsupp( A ) and [ m d , n d ] := fsupp( d ) ; OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 17 (S3)
Parameterize a filter b by b ( z ) = z s P n + ǫj =0 t j z j . Find the unknown coefficients { t , . . . , t n + ǫ } bysolving a system X of linear equations induced by R ( z ) ≡ andcoeff ( b ⋆ , z, j ) = 0 , j = s − n − m d − , . . . , s − and j = s + n + ǫ + 1 , . . . , s + 2 n − n d + ǫ − , where R and b ⋆ are uniquely determined by fsupp( R ) ⊆ [2 m d , n d − and B ( − z ) b ( z ) − A ( z ) b ( − z ) = d ( z ) z b ⋆ ( z ) + R ( z );(S4) For any nontrivial solution to the system X in (S3), there must exist λ > such that λ d ( z ) = z − [ b ( z ) b ( − z ) − b ( − z ) b ( z )] holds. Replace b , b by λ − / b , λ − / b , respectively;Then { a ; b , b } is a finitely supported tight framelet filter bank satisfying max(len( b ) , len( b )) len( a ) + ǫ . We are now ready to present an algorithm to construct finitely supported complex tight framelet filter bankswith frequency separation property.
Algorithm 2.
Let a ∈ l ( Z ) be a finitely supported filter on Z satisfying (3.4) . (S1) Construct a finitely supported tight framelet filter bank { a ; b , b } by Algorithm 1; (S2) Choose a suitable filter length N ∈ N ∪ { } and parameterize filters u and u by u ( z ) := c + c z + · · · + c N z N , u ( z ) := d + d z + · · · + d N z N , where c , . . . , c N , d , . . . , d N are complex numbers to be determined later. We can further assume c ∈ R by normalizing the first filter u ; (S3) Define new high-pass filters b p and b n by b p ( z ) := b ( z ) u ( z ) + b ( z ) u ( z ) , b n ( z ) := z m [ b ( z ) u ⋆ ( z ) − b ( z ) u ⋆ ( z )] , where m is an integer such that the centers of fsupp( b p ) and fsupp( b n ) are close to each other; (S4) If in addition the given filter a is real-valued, then we further require that the initial filters b , b shouldbe real-valued and c , . . . , c N , d , . . . , d N ∈ R . Further replace the filters b p and b n in (S3) by [ b p ( z ) + i b n ( z )] / √ and [ b p ( z ) − i b n ( z )] / √ , respectively; (S5) Find a solution { c , . . . , c N , d , . . . , d N } of the following constrained optimization problem: min u ,u Z π [ | b p ( − e − iξ ) | + | b n ( e − iξ ) | ] dξ under the constraint | u ( e − iξ ) | + | u ( e − iξ ) | = 1 for all ξ ∈ R (such constraint on u , u can be rewrittenas equations using c , . . . , c N , d , . . . , d N ).Then { a ; b p , b n } is a tight framelet filter bank. For a real-valued filter a , in addition we have b n = b p . Using Algorithms 1 and 2, many examples of finitely supported complex tight framelet filter banks with gooddirectionality can be easily constructed. Here we only present several examples to illustrate Algorithms 1 and2. In order to see the improvement of directionality of a tight framelet filter bank { a ; b p , b n } , we shall use thefollowing quantities: d R := 12 Z π [2 − | b a ( ξ ) | − | b a ( ξ + π ) | ] dξ, d A := Z π A ( ξ ) dξ, d B := Z π [ | b b p ( ξ + π ) | + | b b n ( ξ ) | ] dξ, (4.1)where the sharp theoretical lower bound frequency separation function A is defined in (1.13) and the subscript R in d R refers to the case of real-valued high-pass filters. By Theorem 1, we always have d A d B . If both b p and b n are real-valued filters, by Theorem 3 we always have d R = d B . Example 1.
Let a ( z ) = ( z − + 2 + z ) / { , , } [ − , be the B-spline filter of order 2. Using Algorithm 1,we obtain a tight framelet filter bank { a ; b , b } with b ( z ) = √ (1 − z − ) and b ( z ) = √ (1 − z − )(1 + 3 z ).Applying Algorithm 2 with N = 0, we have a finitely supported complex tight framelet filter bank { a ; b p , b n } with b n = b p and b p ( z ) := (1 − z − )[( − √ i ) z + (3 √ i )] . By calculation we have d R = π ≈ . d A ≈ . d B ≈ . N = 2, then b p ( z ) =( − . . i ) z − + (0 . − . i ) z − − (0 . . i ) z − − (0 . . i )+ (0 . − . i ) z + (0 . . i ) z + (0 . . i ) z . By calculation, we have d B ≈ . L ( R ) in (1.9). Figure 4.1.
The first row is for the real part and the second row is for the imaginary part ofthe tight framelet generators in Example 1 with N = 0. The third row is the greyscale image ofthe eight generators: the first four for real part and the last four for imaginary part. Example 2.
Let a ( z ) = z − (1 + z ) /
16 = { , , , , } [ − , be the B-spline filter of order 4. Using Algo-rithm 1, we obtain a tight framelet filter bank { a ; b , b } with b ( z ) = p
34 + 8 √ √ − − z )[65 z + (64 √
14 + 261) z + (40 √
14 + 155) z + 8 √
14 + 31] , b ( z ) = p
34 + 8 √ √ − − z )[10 z − (5 √
14 + 15) z − √ − . Applying Algorithm 2 with N = 0, we have a finitely supported complex tight framelet filter bank { a ; b p , b n } with b n = b p and b p ( z ) =( − . . i ) z − + ( − . . i ) z − − (0 . . i ) + (0 . − . i ) z + (0 . − . i ) z . By calculation we have d R = π ≈ . d A ≈ . d B ≈ . N = 2, then b p ( z ) =(0 . − . i ) z − + (0 . − . i ) z − − (0 . − . i ) z − + (0 . − . i ) z − (0 . . i ) − (0 . − . i ) z − (0 . − . i ) z − (0 . . i ) z − (0 . . i ) z . OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 19
By calculation, we have d B ≈ . L ( R ) in (1.9). Figure 4.2.
The first row is for the real part and the second row is for the imaginary part ofthe tight framelet generators in Example 2 with N = 2. The third row is the greyscale image ofthe eight generators: the first four for real part and the last four for imaginary part. Example 3.
Let a ( z ) = − z − + z − + + z − z = {− , , , , , , − } [ − , be an interpolatoryfilter. Using Algorithm 1, we obtain a tight framelet filter bank { a ; b , b } with b ( z ) = p − √ √ z − ( z − ( z + 2 − √ z + (512 + 57 √ z + (21 + 86 √ z − − √ , b ( z ) = p − √ √ √ z − ( z − ( z + 2 − √ − z + ( √ − z + 2 √ − . Applying Algorithm 2 with N = 0, we have a finitely supported complex tight framelet filter bank { a ; b p , b n } with b n = b p and b p ( z ) =(0 . . i ) z − − (0 . . i ) z − − (0 . . i ) + (0 . . i ) z − (0 . − . i ) z + (0 . − . i ) z . By calculation we have d R = π ≈ . d A ≈ . d B ≈ . N = 2, then b p ( z ) =(0 . . i ) z − − (0 . . i ) z − − (0 . . i ) z − + ( − . . i ) z − − (0 . . i ) + (0 . . i ) z − (0 . − . i ) z − (0 . . i ) z − (0 . e − . i ) z + (0 . − . i ) z . By calculation, we have d B ≈ . L ( R ) in (1.9). Figure 4.3.
The first row is for the real part and the second row is for the imaginary part ofthe tight framelet generators in Example 3 with N = 2. The third row is the greyscale image ofthe eight generators: the first four for real part and the last four for imaginary part. Example 4.
Let a ( z ) = − z − + z − + + z + z − z = {− , , , , , − } [ − , . UsingAlgorithm 1, we obtain a tight framelet filter bank { a ; b , b } with b ( z ) = √ z − ( z − (3203 z + 1921 z − z − , b ( z ) = − √ z − ( z − (248 z + z + 3) . Applying Algorithm 2 with N = 0, we have a finitely supported complex tight framelet filter bank { a ; b p , b n } with b n = b p and b p ( z ) =( − . . i ) z − + (0 . − . i ) z − − (0 . . i ) + (0 . . i ) z − (0 . − . i ) z + (0 . − . i ) z . By calculation we have d R = π ≈ . d A ≈ . d B ≈ . N = 2, then b p ( z ) =(0 . . i ) z − − (0 . . i ) z − (0 . . i ) z − − (0 . . i ) z − (0 . . i ) − (0 . . i ) z (0 . − . i ) z + ( − . . i ) z (0 . − . i ) z + ( − . . i ) z . By calculation, we have d B ≈ . L ( R ) in (1.9). References [1] R. Chan, S. D. Riemenschneider, L. Shen, and Z. Shen, Tight Frame: An efficient way for high-resolution image reconstruction,
Appl. Comput. Harmon. Anal. , (2004), 91–115.[2] C. K. Chui, W. He and J. St¨ockler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl.Comput. Harmon. Anal. (2002), 224–262.[3] I. Daubechies, Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, , SIAM, Philadel-phia, PA, 1992.[4] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. (1986), 1271–1283.[5] I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon.Anal. (2003), 1–46. OMPACTLY SUPPORTED TENSOR PRODUCT COMPLEX TIGHT FRAMELETS WITH DIRECTIONALITY 21
Figure 4.4.
The first row is for the real part and the second row is for the imaginary part ofthe tight framelet generators in Example 4 with N = 2. The third row is the greyscale image ofthe eight generators: the first four for real part and the last four for imaginary part. [6] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. , (1997), 380–413.[7] B. Han, Nonhomgeneous wavelet systems in high dimensions, Appl. Comput. Harmon. Anal. (2012), 169–196.[8] B. Han, Properties of discrete framelet transforms, Math. Model. Nat. Phenom. (2013), 18–47.[9] B. Han, Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters, Appl. Comput.Harmon. Anal. , (2013), 200–227.[10] B. Han, Algorithm for constructing symmetric dual framelet filter banks, Math. Comp. , (2012), to appear.[11] B. Han, G. Kutyniok, and Z. Shen, Adaptive multiresolution analysis structures and shearlet systems,