A Spectral-Factorization Combinatorial-Search Algorithm Unifying the Systematized Collection of Daubechies Wavelets. Abstract.

Transcription

1 A Sectral-Factorization Combinatorial-Search Algorithm Unifying the Systematized Collection of Daubechies Wavelets Carl Taswell UCSD School of Medicine, La Jolla, CA Comutational Toolsmiths, Stanford, CA Abstract A sectral-factorization combinatorial-search algorithm has been develoed for unifying a systematized collection of Daubechies minimum length maximum flatness wavelet filters otimized for a diverse assortment of criteria. This systematized collection comrises real and comlex orthogonal and biorthogonal wavelets in families within which the filters are indexed by the number of roots at z = 1. The main algorithm incororates sectral factorization of the Daubechies olynomial with a combinatorial search of sectral factor root sets indexed by binary codes. The selected sectral factors are found by otimizing the desired criterion characterizing either the filter roots or coefficients. Daubechies wavelet filter families have been systematized to include those otimized for time domain regularity, frequency domain selectivity, time frequency uncertainty, and hase nonlinearity. The latter criterion ermits construction of the orthogonal least and most asymmetric real and least and most symmetric comlex filters. Biorthogonal symmetric sline and balanced length filters with linear hase are also comutable by these methods. 1 Introduction Since the discovery of comact orthogonal and biorthogonal wavelets by Daubechies [1], various discussions of the general theory and secific arameterizations of her wavelets have also been ublished (cf. [6] for a literature review). These comact Daubechies wavelets, which have the maximal number of vanishing moments for their finite length, can be imlemented as discrete filters that are iterated or auto-convolved to generate aroximations of the continuous functions. A systematic treatment collecting and evaluating all of the Daubechies real and comlex orthogonal and biorthogonal wavelets constructed with a single unifying comutational algorithm has not yet aeared in the literature. Such an effort was begun by Taswell [5] focusing on wavelets with varying degrees of asymmetry or symmetry that can be derived by sectral factorization of the Daubechies olynomial. Significant advantages of the sectral factorization aroach include its generalizability to many different classes and tyes of wavelets, its suitability for easily interretable visual dislays, and thus its racticality in edagogy. Comact asymmetric and symmetric wavelets include the original orthogonal extremal hase and least asymmetric families as well as the biorthogonal sline and sline variations families described by Daubechies [1]. As introduced by Taswell [5], these families can be extended and systematized with a generalized flexible yet automated algorithm that ermits consistent selection of alternative choices and the identification of filters with otimized arameters. As further develoed in this reort, these arameters now include hase nonlinearity, time domain regularity, frequency domain selectivity, and time frequency uncertainty, but can be readily extended to include other arameters as otimization criteria. In all cases, a combinatorial search algorithm incororating a binomial subset selection [4, 5] is used to choose the sectral factors satisfying the required objective criterion defined for each family. 2 Methods Methods resented here focus on the sectralfactorization combinatorial-search algorithm which requires a searate algorithm for comuting the roots of the roduct filter P D (z) or its related form the quotient filter Q D (z) =(z +1) 2(D+1) P D (z) (1) which is the roduct filter divided by the roots at z = 1. Refer to [6] for details. 2.1 Factorization Rules When considering sectral factorization, the roduct filter olynomial P D (z) with N =4D + 3 coefficients and K =2D + 2 roots at z = 1 is factored into the analysis and synthesis filter olynomials A(z) and S(z) with N a and N s coefficients, and K a and K s roots at z = 1, resectively. This factorization yields the constraints N = N a + N s 1 and K = K a + K s on the lengths of the three filters and their numbers of roots at z = 1. Each family of filters in the collection has been named with an identifying acronym followed by the arameters (N a,n s ; K a,k s ) in the biorthogonal

2 2 Taswell: Unifying Algorithm for Daubechies Wavelets cases and by (N; K) in the orthogonal cases which require N N a = N s and K K a = K s. See Table 1 for a list summarizing the names and designs of the filter families. With regard to the resective cases of real biorthogonal, real orthogonal, and comlex orthogonal, various additional constraints must be imosed. If K a, K s, and K = K a + K s are the numbers of roots at z = 1 for A(z), S(z), and P(z), then the corresonding filters have coefficient lengths N a = K a +4n cq a +2n rd a + 1 (2) N s = K s +4n cq s +2n rd s + 1 (3) N = 2K 1 (4) where n cq a, n cq s, n rd a, and n rd s are the numbers of comlex quadrulets {z,z 1, z, z 1 } and real dulets {r, r 1 } for A(z) and S(z). Both n cq and n rd may be whole or half integer. If half integer, then half a comlex quadrulet denotes a comlex dulet while half a real dulet denotes a real singlet. For K a and K s necessarily both odd or both even, then K is always even and K = K /2 a whole integer determines n cq = n cq a +n cq s and n rd = n rd a +n rd s accord- = (K 1)/2 and n rd =(K 1) mod 2. If K a and K s are given, then K and K yield n cq and slit into {n cq a,n rd a } and {n cq s,n rd s } and the roots ing to n cq n rd are factored accordingly. For real coefficients, a root z must be aired with its conjugate z. For symmetric coefficients, a root z must be aired with its recirocal z 1. For 2-shift orthogonal coefficients, a root z must be searated from its conjugate recirocal z 1. Thus, in the real biorthogonal symmetric case, each comlex quadrulet {z, z,z 1, z 1 } and real dulet {r, r 1 } must be assigned in its entirety to either A(z) or S(z). In the real orthogonal case, each comlex quadrulet is slit into two conjugate airs {z, z} and {z 1, z 1 }, while each real dulet is slit into two singlets {r} and {r 1 }, with one factor assigned to A(z) and the other to S(z). The comlex orthogonal case is analogous to the real orthogonal case excet the comlex quadrulets are slit into recirocal airs {z,z 1 } and { z, z 1 } instead of conjugate airs. The comlex orthogonal symmetric case requires use of comlex quadrulets without real dulets. All orthogonal cases require K = K a = K s = K /2, n cq a = n cq s = n cq /2, and n rd s = n rd /2 with N = N a = N s = 2K. Note that n rd can only equal 0 or 1. Therefore, in biorthogonal cases, either {n rd a = 0, n rd s = 1} or {n rd a = 1, n rd s = 0}. However, in orthogonal cases, either {n rd s =0} or {n rd s =1/2} with 1/2 of a dulet denoting a singlet. For all real orthogonal cases as well as those comlex orthogonal cases not involving symmetry criteria, K can be any ositive integer. For the comlex orthogonal least-asymmetric and most-asymmetric cases, K must be a ositive even integer. For the comlex orthogonal least-symmetric and most-symmetric cases, K must be a ositive odd integer. For the real biorthogonal symmetric cases, K a and K s must be both odd or both even. In the biorthogonal symmetric sline case, all additional roots (other than those at z = 1 with assignment determined by K a and K s ) are assigned to the analysis filter leaving the synthesis filter as the sline filter. All other biorthogonal symmetric cases incororate a root assignment constraint that balances the lengths of the analysis and synthesis filters such that N a N s as much as ossible. For K a =2i 1and K s =2j 1both odd with i, j {1, 2, 3,...}, balancing of equal filter lengths is ossible. In fact, requiring both K a = K s and N a = N s is also ossible when N = N a = N s =2K with K = K a = K s for {K =1+4k k =1, 2, 3...}. However, for K a =2i and K s =2j both even, equal balancing of filter lengths N a and N s is not ossible. The additional unbalanced roots are assigned to the analysis filter such that N a >N s leaving the synthesis filter as the shorter filter. 2.2 Selection Criteria Selection criteria to be otimized include the hase nonlinearity nl(a), time domain regularity tdr(a), frequency domain selectivity fds(a), and time frequency uncertainty tfu(a) [5, 3]. For all of the biorthogonal symmetric balanced-length families (which excludes the biorthogonal symmetric sline family) and for all of the orthogonal families, the selection criterion is otimized for the analysis filter. However, for the biorthogonal balanced-length balanced-regular family, the selection criterion is otimized for both analysis and synthesis filters by maximizing a balancing measure B defined as B(tdr( ), A, S) = tdr(a) + tdr(s) tdr(a) tdr(s) (5) when alied to tdr( ) for A and S. Phase nonlinearity nl(a) does not aly to real biorthogonal filters with linear hase. However, it does aly to the real and comlex orthogonal filters. Minimizing or maximizing nl(a) for real filters defines the least and most asymmetric families, resectively. In addition, if the arity of K is ignored, then minimizing or maximizing nl(a) for comlex filters defines the least and most nonlinear families, resectively. See Table 1 for a summary of the filter designs. Note that the DROLD and DROMA families are comuted via different factorization and selection methods but should ideally be equivalent. Also note that the DCOLN family is the union of the even-indexed DCOLA and odd-indexed DCOMS

3 In ress, 1998 ICSSCC, Durban South Africa. 3 families, while the DCOMN family is the union of the even-indexed DCOMA and odd-indexed DCOLS families. Comlete details for the algorithms to comute each of the various selection criteria can be found elsewhere [6, 3]. 2.3 Unifying Algorithm All filter families of the systematized collection are generated by the sectral factorization and otimizing combinatorial search incororated in the following algorithm: 1. Inut the identifying name FiltName for the family of filters and the indexing design arameters K a and K s. 2. Comute K = K a + K s, D = K /2 1, and the n cq = D/2 comlex quadrulet and n rd = D mod 2 real dulet roots of the quotient filter Q D (z). 3. Determine the factorization rules and selection criterion that define the family of filters named FiltName. 4. Comute the slitting number airs {n cq a,n cq s } and {n rd a,n rd s } from {n cq,n rd } for the FiltName filter air indexed by {K a,k s }. 5. Sort the roots in an order convenient for the class of slitting aroriate to the tye of filter. For examle, all roots of a comlex quadrulet should be adjacent with dulets of the quadrulet subsorted according to conjugates or recirocals deending on the filter tye. Assign binary coded labels 0 and 1 to the first and second dulet of each quadrulet. Analogously assign binary codes to the first and second singlet of the real recirocal dulet if resent. 6. Generate the ossible binomial subsets for these binary codes [2] subject to the imosed factorization rules and slitting numbers. For examle, in the orthogonal case, there are a total of n cq a + n rd a binary selections and 2 ncq a +nrd a 1 binomial subsets ignoring comlements. 7. For each root subset selected by the binomial subset codes, characterize the corresonding filter by the otimization criterion aroriate for the FiltName family. These otimization criteria may be any of the numerically estimated filter arameters comuted from the roots or the coefficients [6, 3]. 8. Search all root subsets to find the one with the otimal value of the desired criterion. 9. Include the K a and K s required roots at z = 1 for the selected otimal subset of roots intended for the sectral factor A(z) and for the comlementary subset intended for the synthesis sectral factor S(z). 10. In the orthogonal case, comare the selected (rimary) subset of filter roots and coefficients with its comlementary subset to choose the one with minimax grou delay over the interval ω [0, 3] as the subset for A(z). Full searches of all ossible combinatorial subsets should be erformed for a sufficient number of K indices for the filter family s members to infer the aroriate attern of binary codes characterizing the family. Using such a attern ermits successful artial rather than full combinatorial searches. These artial searches rovide significant reduction in comutational comlexity convenient for larger values of K. 3 Results All filters of all families were demonstrated to meet or surass requirements for orthogonality, biorthogonality, and reconstruction when tested [3] in 2-band wavelet filter banks. In general, reconstruction errors ranged from erfect at O(10 16 ) to near-erfect at O(10 8 )ask ranged from K =1toK = 24 for both orthogonal and biorthogonal classes. All filter families were observed to have the otimal values of their defining selection criterion when comared to the other families. Figures 1 and 2 dislay examles of results for two of the selection criteria, nl(a) and tfu(a), as a function of K for each filter family. Note that the lists in the figure legends order the filter families according to the median values observed for 1 K 24. Refer to [6] for a comlete catalogue of all results for all of the filter families with both numerical tables of arameter estimates and grahical dislays of the filters in the time, frequency, and Z domains. 4 Discussion An algorithm has been develoed to unify all of the diverse families of real and comlex orthogonal and biorthogonal Daubechies wavelets. This automated algorithm is valid for any order K of wavelet and insures that the same consistent choice of roots is always made in the comutation of the filter coefficients. It is also sufficiently flexible and extensible that it can be generalized to select roots for filters otimized by criteria other than those mentioned here in this reort. Systematizing a collection of filters with a mechanism both for

4 4 Taswell: Unifying Algorithm for Daubechies Wavelets generating and evaluating the filters enables the develoment of filter catalogues with tables of numerical arameter estimates characterizing their roerties. Use of these catalogues as a resource enables the investigator to choose an available filter with the desirable characteristics most aroriate to his research roblem or develoment alication. References [1] Ingrid Daubechies. Ten Lectures on Wavelets. Number 61 in CBMS-NSF Series in Alied Mathematics. Society for Industrial and Alied Mathematics, Philadelhia, PA, [2] Edward M. Reingold, Jurg Nievergelt, and Narsingh Deo. Combinatorial Algorithms: Theory and Practice. Prentice-Hall, Inc., Englewood Cliffs, NJ, [3] Carl Taswell. Emirical tests for the evaluation of multi-rate filter bank arameters. IEEE Transactions on Circuits and Systems II. Analog and Digital Signal Processing. Submitted March 1998; available at htt:// [4] Carl Taswell. Algorithms for the generation of Daubechies orthogonal least asymmetric wavelets and the comutation of their Holder regularity. Technical reort, Scientific Comuting and Comutational Mathematics, Stanford University, Stanford, CA, August [5] Carl Taswell. Comutational algorithms for Daubechies least-asymmetric, symmetric, and most-symmetric wavelets. In Proceedings of the International Conference on Signal Processing Alications and Technology, ages Miller Freeman, Setember [6] Carl Taswell. The Systematized Collection of Wavelet Filters Comutable by Sectral Factorization of the Daubechies Polynomial. Comutational Toolsmiths, htt:// Table 1: Summary of Filter Designs for Systematized Collection of Daubechies Wavelets Nonorthogonal P(z) Descrition Construction Index Constraint LRNSI(N ; K ; d) Symmetric Interolating P L(z) coefs K =2L; L 1 DRNSI(N ; K ; d) Symmetric Interolating via Q D(z) roots K =2D +2; D 0 Biorthogonal A(z), S(z) Descrition Factorization Constraint Otimization DRBSS(N a,n s; K a,k s) Symmetric Sline {z k 1} to A(z) even (K a + K s) none DRBLU(N a,n s; K a,k s) Least Uncertain conj reci quads even (K a + K s) min tfu(a) DRBMS(N a,n s; K a,k s) Most Selective conj reci quads even (K a + K s) max fds(a) DRBMR(N a,n s; K a,k s) Most Regular conj reci quads even (K a + K s) max tdr(a) DRBBR(N a,n s; K a,k s) Balanced Regular conj reci quads even (K a + K s) max B(tdr( ), A, S) Orthogonal A(z) Descrition Factorization Constraint Otimization DROLD(N; K) Least Delayed { z k < 1} to A(z) K 1 none DROLU(N; K) Least Uncertain quads conj dus K 1 min tfu(a) DROMR(N; K) Most Regular quads conj dus K 1 max tdr(a) DROLA(N; K) Least Asymmetric quads conj dus K 1 min nl(a) DROMA(N; K) Most Asymmetric quads conj dus K 1 max nl(a) DCOLU(N; K) Least Uncertain quads reci dus K 1 min tfu(a) DCOMR(N; K) Most Regular quads reci dus K 1 max tdr(a) DCOLS(N; K) Least Symmetric quads reci dus odd K 3 max nl(a) DCOMS(N; K) Most Symmetric quads reci dus odd K 3 min nl(a) DCOLA(N; K) Least Asymmetric quads reci dus even K 4 min nl(a) DCOMA(N; K) Most Asymmetric quads reci dus even K 4 max nl(a) DCOLN(N; K) Least Nonlinear quads reci dus K 1 min nl(a) DCOMN(N; K) Most Nonlinear quads reci dus K 1 max nl(a) Product filter P(z) is slit into sectral factors for analysis filter A(z) and synthesis filter S(z). Names are abbreviated with first character L or D for Lagrange or Daubechies, second character R or C for Real or Comlex, and third character N, B, or O for nonorthogonal, biorthogonal, or orthogonal. Filters have length N coefficients and K roots at z = 1. Interolating filters are exact for olynomials of regular degree d. All biorthogonal filters are symmetric. All biorthogonal filters excet DRBSS have balanced length. Comlex conjugate recirocal quadrulets are slit into conjugate dulets or recirocal dulets; real recirocal dulets are slit into real singlets but have been omitted from the table; see Sections 2.1 and 2.2 for further exlanation of factorization rules and selection criteria.

5 In ress, 1998 ICSSCC, Durban South Africa. 5 Orthogonal Filter Banks: Phase NonLinearity DCOMN DROMA DCOMR DROMR 4.24 DROLU 2.21 DCOLU 1.81 DCOLN 1.56 DROLA K Figure 1: Phase nonlinearity of orthogonal filters. Orthogonal Filter Banks: Time Frequency Uncertainty 6 5 DCOMN 3.04 DCOMR 1.59 DROMA 1.49 DROMR 0.97 DCOLN 0.89 DCOLU 0.86 DROLA 0.85 DROLU K Figure 2: Time frequency uncertainty of orthogonal filters.