PARAMETRIC OPTIMIZATION OF BIORTHOGONAL WAVELETS AND FILTERBANKS VIA PSEUDOFRAMES FOR SUBSPACES

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1 PARAMETRIC OPTIMIZATION OF BIORTHOGONAL WAVELETS AND FILTERBANKS VIA PSEUDOFRAMES FOR SUBSPACES SHIDONG LI AND MICHAEL HOFFMAN Abstract. We present constructions of biorthogonal wavelets and associated filter banks with optimality using pseudoframes for subspaces (PFFS). PFFS extends the theory of frames in that pseudoframe sequences need not reside within the subspace of interest. In particular, when PFFS is applied to biorthogonal wavelets the underlying flexibility presents opportunities to incorporate optimality, regularity as well as perfect reconstruction into one parametric design approach. This approach reduces certain filter optimization problems to optimization over a free parameter. While past constructions can be reproduced, results with additional optimality are also obtained and presented here with numerical examples. Filter coeffiecients are provided along with graphs are provided and Matlab implementation code is available upon request from the authors.. Introduction: Pseudoframes for subspaces and biorthogonal wavelets Frames and frame variations have generated vast interest for the last two decades, see, e.g., [], [], [5], [], [3], [3], [4], [4], [5], [6], [], [6], [7], [8], [7],and [9] with Pseudoframes first appearing in [5]. Pseudoframes for subspaces (PFFS) is an extension of frames. It is a notion of frame-like expansions for a subspace X of a separable Hilbert space [6]. In the context of frame expansions we define pseudoframes as follows. Let X be a closed subspace of a separable Hilbert space H. Let {x n } H be a Bessel sequence w.r.t. X, and let {x n} be a Bessel sequence in H. We say {x n } is a pseudoframe for the subspace X (PFFS) w.r.t. {x n} if f X, f = f, x n x n. () n The important distinction between PFFS and frames is that none of the sequences {x n } and {x n} are necessarily required to be in X. Consequently, {x n } and {x n} are not generally in the same subspace either.the resulting flexibility is in fact the major advantage provided by PFFS. The purpose of this article is to elaborate on the construction of new biorthogonal wavelets and filter banks utilizing PFFS as a method to obtain certain design criteria. To this end we ll restate the most fundamental characterization of PFFS, first introduced by Li and Ogawa in [6]. Let {x n } H and {x n} H. Assume that {x n } is a Bessel sequence with respect to (w.r.t.) the subspace X. Assume also that {x n} is a Bessel sequence in H. Define U : X l (Z) by and define V : l (Z) H such that f X, Uf = { f, x n }, () c {c(n)} l (Z), V c = n c(n)x n. (3) Then the following characterization of PFFS holds [6]. Theorem. ([6]). Let {x n } and {x n} be two sequences in H (not necessarily in X ). Assume that {x n } is a Bessel sequence w.r.t. the subspace X, and {x n} is a Bessel sequence in H. Let Date: Fall 8. Mathematics Subject Classification. Primary: 4-; Secondary: 4C5. Key words and phrases. Pseudoframes, frames, biorthogonal wavelets, filter banks, compression, filter design.

2 SHIDONG LI AND MICHAEL HOFFMAN U be defined by (), and V be defined by (3). Suppose that P is any projection from H onto X. Then {x n } is a pseudo frame for X w.r.t. {x n} if and only if V UP = P. (4) Therefore, the constructions of PFFS all start from (4) which basically requires the pseudoframes to preserve projections onto the subspace in question. The construction of PFFS has typically two (non-symmetrical) directions. One corresponds to finding the left inverse V from a given U; the other relates to finding the U from a given V, all according to equation (4). We refer to [6] for details of the constructions. Let {x n} be a Bessel sequence in H such that sp{x n} X. All PFFS-dual sequences {x n } can be constructed as follows. Let {x n} H be such that x n = (V ) e n, where (V ) denotes the pseudoinverse of (V ), and c l, (V ) c = n c(n)x n, (5) and let {y n } be an arbitrary Bessel sequence in H. We have then the following: Corollary.. [6] Let {x n} be a Bessel sequence in H such that sp{x n } X. Assume further that R(V ) is closed. Let {x n} be defined in (5) and {y n } be a Bessel sequences in H. All dual PFFS sequences {x n } for X w.r.t. {x n} are given by x n = P x n + y n m x n, x m P y m, n Z, (6) where P is the traditional adjoint operator of the projection P. Applying this construction to a shift-invariant subspace X such that sp{τ n φ} X and where {τ n φ} L (R) is a Bessel sequence it was shown in [6] that PFFS allows us to characterize the entire subspace of duals of translates as follows: φ n = τ n φ n Z, where φ P φ + y P m φ, τ m φ τ m y in L (R), (7) and φ V e n is the dual corresponding to the pseudo-inverse of V in sp{x n }, and y L (R) is such that {τ n y} is a Bessel sequence. Here, we have chosen P as an orthogonal projection. PFFS applied to biorthogonal wavelets. In a special case, let us assume that {τ n φ} is an exact frame of sp{τ n φ} X, and translate the result of Corollary. into this special setting. With the given assumption that {τ n φ } is the unique biorthogonal dual frame to {τ n φ}, τ n φ, τ m φ = δ nm. Therefore, τ n φ = P τn φ + τ n y P τ n y = τ n φ + (I P )τ n y. (8) As one can see that if y / sp{τ n φ} = X, the second term is nonzero, yielding non-unique biorthogonal dual sequences {τ n φ}. We shall show that biorthogonal wavelet construction can be favorably carried out by using (8) with the off-subspace component (I P )τ n y. Since PFFS builds new biorthogonal duals from existing pairs by adding an off-subspace component it opens up opportunities for design optimization without disrupting any important features of the original pair such as symmetry, compact support, improved regularity or vanishing moments. Here, we will show that maximum attenuation and desired filter response can be incorporated with the design of FIR bi-filter banks while maintaining the vanishing moments, symmetry etc.

3 BI-WAVELETS WITH PFFS 3. Construction of biorthogonal wavelets and filters via PFFS Assume that φ L (R) and that {τ n φ} forms a biorthogonal basis of V sp{τ n φ}. Assume also that {φ, V j } generates a (biorthogonal) MRA of L (R). By the theory of PFFS [6], all biorthogonal PFFS-dual scaling functions { φ n = τ n φ} are given by φ = φ + φ, (9) where φ V is the standard dual function of φ and φ V. In general, it is evident that if φ is any biorthogonal (PFFS-dual) function of φ, then so is φ = φ + φ for any φ V. With a slight abuse of notation, we shall be considering the Equation (9) with φ being any biorthogonal dual. Assume that we are only interested in sufficiently regular and refinable φ such that φ = h n φn. Considerable studies on the conditions for sequences such as h can be found in []. n Then the following relationship holds: h n = φ, φ n = φ + φ, φ n h n + h n, where we have assumed φ = φ + φ + with φ j W j V j+ \ V j, and h n φ, φ n. What does it mean to have { h n } φ W? In the context of biorthogonal wavelets and multiresolution analysis, the add-on filter sequence component { h n } is solely relevant to information in the subspace W. We know that the regularity and symmetry of compactly supported biorthogonal wavelets and the corresponding linear phase and regularity properties of FIR filters are solely the consequence of having a biorthogonal dual φ outside of the subspace V [9]. We have thus observed that these nice properties come with the add-on components from information in the complement W = V \ V. From the sub-band processing point of view, since these nice properties are demanded in practical applications [7], [9],[4], [8], [3], [8], [], [], [3] we now understand the need to bring some information in the high-pass band (W ) back to the low-pass band to off-set some of the draw backs due to a biorthogonal dual that is not quite as nice. This has to be done in such a way that the perfect reconstruction principle is not violated. Recall that a set of four filter sequences {h n }, {g n } { h n } and { g n } are said to form a biorthogonal sub-band system (or perfect reconstruction filter bank (PRFB)) if ) (h n k hn l + g n k g n l = δ kl, () n where { h n } is often termed the dual filter sequence to {h n } and { g n } the dual filter to {g n }. The following are the basic constructions of PFFS focusing on FIR biorthogonal filters. Theorem.. Let {h n } and {g n } be a set of filter bank filters, and let {h n} and {gn} be the corresponding biorthogonal dual filters satisfying (). Let H(γ) be the Fourier series of { h n }, and H(γ) be the Fourier series of {h n }. Then { h n = h n + h n } is a biorthogonal PFFS-dual filter if and only if where ˆq(γ) is trigonometric polynomial satisfying H(γ) = H(γ + ) ˆq(γ) () H(γ)H(γ + ) (ˆq(γ) + ˆq(γ + )) = () Moreover, the other two corresponding biorthogonal filters are given by G(γ) = e πiγ ( H (γ + ) + H(γ)ˆq(γ + ) ) G(γ) = e πiγ H(γ + )

4 4 SHIDONG LI AND MICHAEL HOFFMAN Proof: Following a similar proof in [], the Fourier transform of () shows Since H(γ) = H (γ) + H(γ), (3) implies that H(γ) H(γ) + H(γ + ) H(γ + ) =. (3) H(γ) H(γ) + H(γ + ) H(γ + ) =. (4) This in turn implies that H(γ) = H(γ + ) ˆq(γ) for some -periodic trig polynomial ˆq such that H(γ)H(γ + ) (ˆq(γ) + ˆq(γ + )) =, This finishes the proof of the first half of the assertion. The proof of the filter relationships for the complementary filters is similar. a.e. or We comment that the conditions we have just derived are essentially to require that h n hn k = δ k, k (5) n h n h n k =, k. (6) n Note that (5) is the standard biorthogonal principle of the perfect reconstruction filter banks, and (6) is a direct consequence of (5), which corresponds to the fact that { h n } φ W V. 3. Symmetric biorthogonal wavelets and linear phase FIR filters We demonstrate how to construct new linear phase and symmetric FIR biorthogonal filters that maintain a minimum number of vanishing moments in thier wavelets as well as preserve perfect reconstruction while introducing the flexibility of parametric design. Theorem 3.. Let H be a symmetric biorthogonal filter such that H(γ) for all γ except for γ = and perhaps a set of measure zero. Assume that a dual filter H is symmetric with l zeros at γ =. Let ˆq(γ) be the trig polynomial in Theorem. satisfying ˆq() = ˆq( ) =. Then a set of symmetric ˆq function with l zeros at γ = is given by ˆq(γ) = ( cos 4πγ) l ˆq (γ) cos πnγ, N = odd, (7) where ˆq (γ) = ˆp(cos 4πγ) is a trig polynomial s.t. corresponding sequence {q n } of ˆq(γ) is given by where q n = τ N l b n + τ N l b n, ( )/ l ( ) b n = m ( ) m l, n = m, n = m +. ˆq ( ). In particular, let ˆq =, the For such choices of ˆq, new biorthogonal PFFS-duals H remain symmetric, and H has at least the same number of zeros at as that of H. Notice that there are obviously other choices of ˆq, depending on the filter H to begin with. It is interesting to observe that if H(γ) = χ [, the choice of ˆq would be free for any trigonometric ), polynomial. However, this is not of interest here because these do not correspond to FIR filters. Proof (of Theorem 3.): By the assumption of the theorem, () holds if and only if ˆq(γ) = ˆq(γ + ), (8)

5 BI-WAVELETS WITH PFFS 5 implying that ˆq is -periodic. Viewing the symmetry requirement and the number of zeros needed at, we see that ˆq is a trig polynomial of cos(4πγ) modulated by a factor of cos πnγ for some odd integer N. Hence, ˆq(γ) = ( cos 4πγ) l ˆq (cos 4πγ) cos πnγ, where ˆq should have no zeros at. A simple trig identity simplification will show that such a ˆq will indeed have at least l zeros at. Let us find the filter sequence associated with ˆq. For ˆq =, where Therefore, ˆq(γ) = l sin l πγ ˆq cos πnγ = ( ) l l ( e πiγ e πiγ) l cos πnγ l ( = ( ) l l ( cos πnγ) e πi(l)γ ( ) k l k 4l+ = cos πnγe πi(l)γ n= b n e πinγ k= ( )/ l ( ) b n m ( ) m l, n = m, n = m +. ) e πi(k)γ ˆq(γ) = ( e πinγ + e πinga) 4l+ e πi(l)γ = 4l+ n= n= (τ N l b n + τ N l b n ) e πinγ Here τ k b n b n k. We have therefore proved the assertion. b n e πinγ Letting ˆq be a scalar parameter λ. In Theorem 3. the choice was made to let ˆq =. This is valid since the only condition placed on the trig polynomial ˆq is that this function must have no zeros at. In the implementations discussed below and in Proposition 3. The scalar parameter λ is a simple selection of the ˆq = λ. While ˆq could be any polynomial in cos4πγ (see discussions at the end of this article), the choice of the scalar parameterization λ minimizes the length of PFFS bi-filters. Combining the above result of Theorem 3. with (), we have obtained the following proposition. Proposition 3.. Let {h n }be a pair of symmetric biorthogonal filters with l zeros at γ =.Let { h n } be any bi-dual filter with l zeros at γ =, then ( h n = h n + λ ( ) k ( ) ) h k τn l b n+k + τ N l b n+k (9) is a biorthogonal PFFS-dual with at least l zeros at. k The proof of this result amounts to a deconvolution, plus the fact that any scalar multiple of ˆq will not alternate the principle (). In other words, (6) is always satisfied whenever δh is equal to the second term of (9).

6 6 SHIDONG LI AND MICHAEL HOFFMAN 4. Shift-symmetric bi-wavelets and linear phase FIR bi-filters As seen in CDF s construction [], among symmetric biorthogonal wavelets, there are also ones that are (without loss of generality) symmetric after shifting by a time index t =. These, in relevance to linear phase filters, translate into the fact that H( γ) = e πiγ H(γ). () We shall term those filters shift-symmetric. For this class of PFFS-duals, the ˆq as in () will be slightly different. We have the following construction. Theorem 4.. Let H and H be a pair of biorthogonal filters, both satisfying (). Assume that H(γ) for all γ except for γ = and perhaps a set of measure zero. Assume further that the dual filter H has l + zeros at γ = (l =,, ). Then a class of H(γ) satisfying the shifted-symmetry property () with l + zeros at is given by H(γ) = (cos πγ) l+ (sin πγ) l+ cos πγ ˆq (γ)h(γ + )e πinγ, () where ˆq (γ) = ˆq (cos 4πγ) γ=, and N is an odd integer. The proof of Theorem 4. is similar to that of Theorem 3.. We can derive the expression of the filter sequence { h n } with a straightforward calculation. Proposition 4.. Let H be given in Theorem 4. and let h n be abi-dual filter. Then ( h n = h n + λ ( ) k ( ) ) h k τn l c n+k + τ N l c n+k () k is a PFFS biorthogonal dual, where the numbers N and l are given in Theorem 4., and {c n } is given by ( )/ l + ( ) c n = m ( ) m l, n = m +, n = m. The proof of this proposition is very similar to that of Theorem 3. and Proposition Design Opportunities provided by the parametric construction In this section we provide examples of construction of bi-filters optimized to three different criteria: () Targeting a desired filter response in order to keep the filter length short. () Maximize stopband attenuation to any given stopband. In each case the problem is reduced to optimization over a free parameter. We conclude the section with a discussion on adding integer vanishing moments to a given bi-filter, which was the traditional focus of known constructions. 5.. Targeting a desired filter. In filter design it is common to attempt to emulate the frequency response characteristics of some target filter H t. Here we characterize this problem as finding min H{ H H t } subject to the condition that H is dual to a B-spline filter H. In the PFFS context, this problem is reduced to unconstrained linear optimization over a freeparameter while maintaining perfect reconstruction and a given number of vanishing moments, namely min λ R { Ht (H + λ H) } (3)

7 BI-WAVELETS WITH PFFS 7 Proposition 5.. Let H t L ( ˆR) be a frequency response function with desirable characteristics. Let H and H be as in Theorem. where H has p zeros at γ =. If λ t = Re ( H t H ) dγ + Re ( H H ) dγ H (4) then H t (H + λ t H) = min λ R { Ht (H + λ H) } and H = H + λ t H will have at least p zeros at γ =. Proof. With the given assumptions we can expand the square of the norm from 3: So our objective function is H H t = H t (H + λ H) (5) Γ(λ) = = H t (H + λ H) dγ (6) H t (H + λ H) dγ Expanding the integrand, the objective function becomes a quadratic in λ. Γ(λ) = λ [ H t Re ( H t H )] dγ + λ [ Re ( H t H ) Re ( H H )] dγ H dγ Taking the derivative with respect to λ we get d Γ(λ) = dλ Re ( H t H ) dγ + Setting this to zero and solving for λ we will arrive at (4) Re ( H H ) dγ + λ H dγ 5... Similar Performance with Shorter Filters. We can now apply this fact to design bi-duals of spline wavelets that have frequency characteristics similar to those of longer length. We begin by writing out one of the B-spline bi-filters from [] as a sum of appropriately weighted out of subspace components. Let H be a spline dual filter as constructed in [] with p zeros at γ =. From our previous discussions a filter with an additional 4 zeros (and eight more filter taps) will be given by H = H + λ H + λ H (7) where H and H are the appropriate out of subspace components from Theorem. and λ, λ are the corresponding parameter values from Proposition 5.4. H will then correspond to one of the filters in [] with p + 4 zeros at γ =. Applying Proposition 5. we can now construct a new spline bi-filter with only p + zeros at γ = by adding λt H to H where λ t is from (4). The following corollary states this result in detail and can be easily verified by substituting (7) for H t in to (5.) with H = H + λ H. Corollary 5.. Let H be a spline bi-filter with p zeros at γ = and let H, H,λ, λ and the corresponding H are as in (7) For H = H + λ H then the choice of ( ) Re λ = λ + λ H H dγ H gγ

8 8 SHIDONG LI AND MICHAEL HOFFMAN will minimize H H while H will maintain at least p vanishing moments and the time sequence h will have 4 fewer taps than h. Figures () and () show two examples of this construction. Notice that the frequency responses as well as the wavelets and scaling functions are nearly identical..5 W avelets.4. Scaling (i) (ii)..8.6 Response.8.7 C oefficients (iii) (iv) Figure. 6-tap PFFS Dual-filters (solid) versus CDF -tap filter (dashed). Shown are the (i) Wavelets, (ii) Scaling Functions, (iii) Frequency Response and (iv) Filter coefficients dual to those of the first-order B-Spline 5.. Maximum stopband attenuation. In certain applications it is desirable that the filter response have a sharp decay at a specific frequency. In this section we show how the parametric PFFS construction can produce dual filters with optimal attenuation for any given stopband. We also show that this implementation reproduces the filters of the CDF construction as γ s The Parametric Formula. One stopband optimization problem (also seen in [7]) can be formulated as follows. Let our objective function J be the energy of the frequency response of some dual filter H between a chosen stopband γ s and. We can minimize J over all possible dual filters by simply manipulating the λ parameter. min H J min H γ s = min λ γ s H(γ) dγ (8) H (γ) + λ H(γ) dγ. Where we minimize only over real λ to keep the sequence { h n } real valued.

9 BI-WAVELETS WITH PFFS 9.5 W avelets.4 Scaling (i).5 Response (ii).4 C oefficients (iii) (iv) Figure. 3-tap PFFS Dual-filters (solid) versus CDF 7-tap filter (dashed). Shown are the (i) Wavelets, (ii) Scaling Functions, (iii) Frequency Response and (iv) Filter coefficients dual to the second-order B-Spline To derive this formula, first note that H + λ H = ( H + λ H ) (H + λ H) ( = H H + λ H H + H H Thus our objective function becomes J = γ s = H + λre ( H H ) + λ H ) + λ H H H dγ + λ Re ( H H ) dγ + λ H dγ γ s γ s The minimization of J with respect to λ can now be accomplished by simply differentiating with respect to λ, setting this derivative to zero and solving for λ. This yields equation (9): the λ-value for which the area of the filter response and between the stop band and is minimal. Given the quadratic nature of J, one can show that the minimum is achieved at ( ) λ d γ s Re H (γ) H(γ) dγ =. (9) γ s H(γ) dγ This formula thus provides a means to optimize the bi dual filters to any γ s ( 4, ]. Examples with maximum stopband attenuation for given stopbands are shown in Figures (3) and (4). We see that the stopband attenuation of the new bi-filter frequency responses are indeed greater than that of CDF examples of the same length. Meanwhile, the smoothness of the scaling and wavelet functions have also been improved.

10 SHIDONG LI AND MICHAEL HOFFMAN.5 W avelets.4 Scaling (i).5 Response (ii).4 C oefficients (iii) (iv) Figure 3. PFFS Duals optimized to a given stopband frequency (solid) are compared with those of CDF (dashed). Shown are the (i) Wavelets, (ii) Scaling Functions, (iii) Frequency Response and (iv) Filter coefficients associated with the length 7 filter, dual to the second-order B-Spline 5... Limiting behavior of λ d (γ s ). By letting γ s the value of λd from (9) converges to λ which adds precisely two zeros at γ = to the original bi-dual H. See also Proposition 5.4 Proposition 5.3. Let H be a biorthogonal dual of a spline function with l vanishing moments (or l + vanishing moments in the shift-symmetric case) as constructed in [] and let H be as in Theorem. then as γ s λ d λ where λ is such that H = H + λ H has two additional integer vanishing moments. Proof Part:Symmetric Filters. We start by simply expanding the general form of the filters as given by (3.) into (9). Since the functions in the integrands are bounded and integrable, as γ s the integrals must go to zero and we are justified in using L Hopital s rule to evaluate the limit. The fundamental theorem of calculus then allows us to evaluate the limit: lim λ d = γ s = lim γ s = k n= ( k k + n n= n ( k + n n 3 l+ ) (cos πγ s ) 4 l(sin πγ s ) n (sin πγ s ) k cos πγ s 3 l+ (sin πγ s ) 4k (cos πγ s ) 4 l(cos πγ s ) ) Plugging this value into (3.) and setting γ s = shift-symmetric case is similar. will confirm the result. The proof of the

11 BI-WAVELETS WITH PFFS.5 W avelets.4. Scaling (i).5 Response.4 (ii) C oefficients (iii) (iv) Figure 4. PFFS Duals optimized to a given stopband (solid) are compared to those of CDF (dashed) with the same length filter. Shown are the (i) Wavelets, (ii) Scaling Functions, (iii) Frequency Response and (iv) Filter coefficients associated with the length filter, dual to the third-order B-Spline 5.3. Other Optimization Potentials. We demonstrated in the previous sections a few possibilities for optimizing the bi-filter construction over a scalar λ. This is quite effective and the major rationale for working with a scalar is to keep the bi-filter length as small as possible. We also mentioned only three criteria that are all signal independent. So, we mention here two other possible optimization problems that could be carried out with PFFS Non-scalar ˆq polynomials. We can choose the trig polynomial ˆq (cos 4πγ) to have more than one parameter to work with so as to enhance the optimization potentials. For instance, let ˆq = λ + ξ( cos 4πγ). (3) These two parameters could be used to tune the filter to meet two different criteria at the same time. For Example, one would be able to increase the number of zeros at γ = by a choice of λ since the second term is zero at, ξ could be adjusted to enhance uniform Lipschitz-α regularity. The only sacrifice is that such out of subspace components add a significant number of nonzero coefficients to the filters (up to eight taps in the spline context). Choices such as (3) with two or more parameters open up a vast degree of flexibiliy in filter design and are believed to have further applications to signal-dependent methods not discussed here Ability to maximize the coding gain. Maximum coding gain is often a design goal to maximize the energy compaction after the sub-band decomposition and to enhance the coding/compression efficiency. In [] the PFFS construction methodology is used to construct filters with maximum coding gain for common signal models. This is a signal-dependent design approach and it is thus not discussed further here.

12 SHIDONG LI AND MICHAEL HOFFMAN 5.4. Ability to increase wavelet vanishing moments. The number of vanishing moments of a wavelet is directly related to the regularity of the bi-scaling and bi-wavelet functions, while the number of vanishing moments is related to the number of zeros of the filter H and H at γ =. We shall demonstrate how zeros at γ = of a PFFS-dual filter can be easily increased by the parametric approach in the general setting, then show how results for the setting of [] can be reproduced usinng PFFS. Assume that H and H are a pair of dual biorthogonal filters. Then according to our earlier discussions, any new PFFS-dual biorthogonal filter can be written as H(γ) = H (γ) + H(γ + ) ˆq(γ) where ˆq is a trig polynomial satisfying ˆq(γ) + ˆq(γ + ) =. One can verify that to keep at least the same number of zeros at (same number of vanishing moment in ψ), ˆq can be of the following form ˆq(γ) = H (γ)h (γ + )ˆq (cos 4πγ) cos πnγ for some odd integer N. Hence, ( ) H(γ) = H (γ) + H (γ + )H(γ + )ˆq (cos 4πγ) cos πnγ = H (γ)f (γ) where F (γ) + H (γ + )H(γ + )ˆq (cos 4πγ) cos πnγ. Evidently, the new PFFS-dual biorthogonal filter has at least the same number of zeros at γ =. Observe however, F ( ) = H ()H() ˆq (), which can easily yield another zero at γ = for ˆq () = /H ()H() since H ()H(). In the context of the spline bi-wavelet system constructed in [] the following Proposition can easily be verified and shows that the PFFS approach reproduces the results of []. Proposition 5.4. Suppose H is the filter corresponding to a spline function with l zeros at for the symmetric case or l + zeros at for the shift-symmetric case and that the biorthogonal dual filter H has l zeros at for the symmetric case or l+ zeros at for the shift-symmetric case. If H is given by () and k ( ) k + n n λ n= = 3 l+ where k = l + l or for the shift-symmetric case: k ( ) k + n n λ n= = 3 l+3 where k = l + l + the PFFS dual given by H = H + λ H will have an additional zeros at. References [] A. Aldroubi. Portraits of frames. Proc. Amer. Math. Soc., Vol. 3: pp , 995. [] J. J. Benedetto. Frame decompositions, sampling, and uncertainty principle inequalities. Wavelets: Mathematics and Applications, J. J. Benedetto and M. W. Frazier, editors, CRC Press Inc., Boca Raton, FL, Chapter 7, 994. [3] J. J. Benedetto and W. Heller. Irregular sampling and the theory of frames. Mat. Note, (suppl. ): pp. 3 5, 99. [4] J. J. Benedetto and S. Li. The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal., 5: pp389 47, 998. [5] J.J. Benedetto and D. F. Walnut. Gabor frames for L and related spaces. Wavelets: Mathematics and Applications, J. J. Benedetto and M. W. Frazier, editors, CRC Press Inc., Boca Raton, FL, Chapter 3, 994.

13 BI-WAVELETS WITH PFFS 3 [6] P. G. Casazza and O. Christensen. Hilbert space frames containing a Riesz basis and Banach spaces which have no subspace isomorphic to C. J. Math. Anal. Appl., (3): pp. 94, 996. [7] O. Christensen. Frames and pseudo-inverses. J. Math. Anal. Appl., 95: pp. 4, 995. [8] O. Christensen and C. Heil. Perturbations of Banach frames and atomatic decompositions. Math. Nach., 85: pp , 997. [9] C. K. Chui. Wavelets: a tutorial in Theory and Applications. Academic Press, Boston, 99. [] A. Cohen, I. Daubechies, and J.-C. Feauveau. Biorthogonal bases of compactly supported wavelets. Comm. Pure and Appl. Math., XLV: pp485 56, 99. [] Zoran Cvetkovic and Martin Vetterli. Oversampled filter banks. IEEE Trans. Signal Processing, 46:45 55, 998. [] I. Daubechies. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Information Theory, 36(5): pp. 96 5, 99. [3] I. Daubechies. Ten lectures on wavelets. 99. [4] I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 7: pp. 7 83, 986. [5] R. Duffin and A. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 7: pp , 95. [6] H. G. Feichtinger and K. Grochenig. Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. Wavelets: A Tutorial in Theory and Applications, C. K. Chui ed., Academic Press, Boston, : pp , 99. [7] H. G. Feichtinger and G. Zimmermann. A Banach space of test functions for Gabor analysis. in Gabor Analysis and Algorithms: Theory and Applications, H. G. Feichtinger and T. Strohmer eds., Birkhäuser, Basel, 997. [8] R. A. Gopinath and C. S. Burrus. Wavelet transforms and filter banks. Wavelets: A Tutorial in Theory and Applications, C. K. Chui, editor, Academic Press, Boston, pp63-654, 99. [9] K. Gröchenig. Describing functions: atomic decompositions versus frames. Monatsh. Math., :pp. 87 4, 99. [] C. Heil and D. Walnut. Continuous and discrete wavelet transforms. SIAM Review, 3: pp68 666, 989. [] M. Herman and S. Li. Biorthogonal wavelets of maximum coding gain through pseudoframes for subspaces. In Mathematics of Data/Image Pattern Recognition, Compression, and Encryption with Applications IX., volume 635, page 635, 6. [] N. S. Jayant and P. Noll. Digital Coding of Waveforms. Prentice Hall, Englewood Cliffs, NJ 763, 984. [3] D. Larson. Frames and wavelets from an operator-theoretical point of view. Contemporary Math, 8: pp 8, 998. [4] D. Larson and D. Han. Frames, bases and group representations. Memoirs American Math. Society, 47, No. 697,. [5] S. Li and H. Ogawa. Pseudoframes in separable Hilbert spaces. in preparation,. [6] S. Li and H. Ogawa. Pseudoframes for subspaces with applications. J. Fourier Anal. Appl., June,, no. 4: pp49 43, 4. [7] G. Strang and T. Nguyen. Wavelets and Filter Banks. Welesley-Cambridge Press, Wellesley MA, 996. [8] P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice Hall, Englewood Cliffs, New Jersey, 993. [9] M. Vetterli and C. Herley. Wavelets and filter banks: theory and design. IEEE ASSP, 4, No. 9: pp7 3, 99. [3] M. Vitterli. Filter banks allowing perfect reconstruction. Signal Processing, : pp. 9 44, 986.

14 4 SHIDONG LI AND MICHAEL HOFFMAN 6. Similar Performance with Shorter Filters W avelets Scaling Response (a) 6-tap PFFS vs -tap CDF Duals (b) -tap PFFS vs 4-tap CDF Duals Figure 5. Shorter PFFS vs. longer CDF: Duals of the st Order B-Spline

15 BI-WAVELETS WITH PFFS 5 W avelets Scaling Response (a) 9-tap PFFS vs 3-tap CDF Duals (b) 3-tap PFFS vs 7-tap CDF (c) 7-tap PFFS vs -tap CDF Duals Figure 6. Shorter PFFS vs. longer CDF: Duals of the nd Order B-Spline

16 6 SHIDONG LI AND MICHAEL HOFFMAN W avelets Scaling Response (a) 8-tap PFFS vs -tap CDF Duals (b) -tap PFFS vs 6-tap CDF Duals (c) 6-tap PFFS vs -tap CDF Duals (d) -tap PFFS vs 4-tap CDF Duals Figure 7. Shorter PFFS vs. longer CDF: Duals of the 3 rd Order B-Spline

17 BI-WAVELETS WITH PFFS 7 7. Maximum Stopband Attenuation W avelets Scaling Response (a) 6-tap Duals (b) -tap Duals Figure 8. Maximum Attenuation PFFS (solid) vs. CDF (dashed): Duals of st Order B-Spline

18 8 SHIDONG LI AND MICHAEL HOFFMAN W avelets Scaling Response (a) 9-tap Duals (b) 3-tap (c) 7-tap Duals Figure 9. Maximum Attenuation PFFS (solid) vs. CDF (dashed): Duals of nd Order B-Spline

19 BI-WAVELETS WITH PFFS 9 W avelets Scaling Response (a) 8-tap Duals (b) -tap Duals (c) 6-tap Duals (d) -tap Duals Figure. Maximum Attenuation PFFS (solid) vs. CDF (dashed): Duals of 3 rd Order B-Spline

20 SHIDONG LI AND MICHAEL HOFFMAN 8. Similar Performance with Shorter Filters n PFFS /6 CDF /,.5.5, , , , Table. /6 PFFS vs / CDF n PFFS / CDF /4,.5.5, , , , , , Table. / PFFS vs /4 CDF n PFFS 3/9 CDF 3/ , , , , , , Table 3. 3/9 PFFS vs 3/3 CDF n PFFS 3/3 CDF 3/ , , , , , , , , Table 4. 3/3 PFFS vs 3/7 CDF

21 BI-WAVELETS WITH PFFS n PFFS 3/7 CDF 3/ , , , , , , , , , , Table 5. 3/7 PFFS vs 3/ CDF n PFFS 4/8 CDF 4/, , , , , , Table 6. 4/8 PFFS vs 4/ CDF n PFFS 4/ CDF 4/6, , , , , , , , Table 7. 4/ PFFS vs 4/6 CDF n PFFS 4/6 CDF 4/, , , , , , , , , , Table 8. 4/6 PFFS vs 4/ CDF

22 SHIDONG LI AND MICHAEL HOFFMAN n PFFS 4/ CDF 4/4, , , , , , , , , , , , Table 9. 4/6 PFFS vs 4/ CDF

23 BI-WAVELETS WITH PFFS 3 9. Maximum Stopband Attenuation n PFFS / CDF / PFFS /6 CDF /6, , , , , Table. Maximum Attenuation PFFS v.s CDF: /6 and / filters. n PFFS 3/3 CDF 3/3 PFFS 3/9 CDF 3/ , , , , , , Table. Maximum Attenuation PFFS v.s CDF: 3/9 and 3/ filters. n PFFS 3/7 CDF 3/ , , , , , , , , Table. Maximum Attenuation PFFS v.s CDF: 3/7 filters. n PFFS 4/8 CDF 4/8, , , , Table 3. Maximum Attenuation PFFS v.s CDF: 4/8 filters.

24 4 SHIDONG LI AND MICHAEL HOFFMAN n PFFS 4/ CDF 4/, , , , , , Table 4. Maximum Attenuation PFFS v.s CDF: 4/ filters. n PFFS 4/6 CDF 4/6, , , , , , , , Table 5. Maximum Attenuation PFFS v.s CDF: 4/ filters. n PFFS 4/ CDF 4/, , , , , , , , , , Table 6. Maximum Attenuation PFFS v.s CDF: 4/ filters.

25 BI-WAVELETS WITH PFFS 5 San Francisco State University Department of Mathematics address: shidong@sfsu.edu San Francisco State University Department of Mathematics (MA) address: kilgore@sfsu.edu