446 SCIENCE IN CHINA (Series F) Vol. 46 introduced in refs. [6, ]. Based on this inequality, we add normalization condition, symmetric conditions and

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1 Vol. 46 No. 6 SCIENCE IN CHINA (Series F) December 003 Construction for a class of smooth wavelet tight frames PENG Lizhong (Λ Π) & WANG Haihui (Ξ ) LMAM, School of Mathematical Sciences, Peking University, Beijing 007, China Correspondence should be addressed to Peng Lizhong ( lzpeng@pku.edu.cn) Received December 5, 00 Abstract From the inequality jp (z)j + jp ( z)j 6, assuming that both of the low-pass filters and high-pass filters are unknown, we design compactly supported wavelet tight frames. The unknowing of low-pass filters allows the design more freedom, and both the low-pass filters and highpass filters have symmetries or anti-symmetries. We give the algorithm for filters with odd and even lengths separately, some concrete examples of wavelet tight frames with the length 4, 5, 6, 7, and at last we give the result of decomposing Lena image with them. Keywords wavelet tight frame, compact support, smoothness, symmetry (anti-symmetry). DOI 0.360/0ye06 Recently, wavelet tight frames have attracted more and more attention, just because they have good time-frequency localization property, shift-invariance, and more design freedom. Wavelet tight frames have been widely used in denoising (cf. refs. [, ]), and have certain applications in image processing (cf. ref. [3]). Refs. [4 ] studied the design of wavelet tight frames, built a systematic theory. Chui and He [] mainly constructed wavelet tight frames with multiresolution analysis, gave the existence of wavelet functions for the given B-splines and the systematic construction of tight frames on the splines with small support and high approximations. Selesnick [0] constructed the filters of wavelet tight frames with minimum length, which satisfies certain polynomial properties, and the method was to solve the nonlinear equations with Gröbner basis. Ron and Shen in ref. [6] gave a general theory and some examples of wavelet tight frames in one dimension and multi-dimension. All of the scaling functions in the above papers are B-splines, which have some advantages in applied and computational mathematics, such as construction of geometric modelling, solution of differential equations (numerical analysis), and noise reduction in image processing. But there should be some other scaling functions except B-splines, which may have other advantages. Under the conditions that both of the low-pass filters and high-pass filters are unknown, in this paper we first construct the low-pass filters, and then construct the high-pass filters. The greater freedom widens the selecting range for both low-pass and high-pass filters. To be more precise, this paper constructs first a class of new low-pass filters and then the corresponding high-pass filters from the inequality jp (z)j + jp ( z)j 6, which was first

2 446 SCIENCE IN CHINA (Series F) Vol. 46 introduced in refs. [6, ]. Based on this inequality, we add normalization condition, symmetric conditions and proper vanishing moments for the two cases of odd and even lengths separately, we get the parameterization formulas of the low-pass filters and the parametric interval. Then we select the low-pass filters corresponding to more smooth scaling functions by optimization in the parametric interval. These low-pass filters include not only B-splines in refs. [6,, 0, ], but also some other smooth low-pass filters. From the resulting low-pass filter, we design the corresponding high-pass filters. There are two cases one case has two high-pass filters, one being symmetric, the other anti-symmetric the other case has three high-pass filters, one being symmetric, two others anti-symmetric. The latter is better for the smoothness and the simplicity of the algorithm. We give the concrete examples with lengths of 4, 5, 6, 7. At last we give the decomposing result for Lena image with them. Notations Our discussion refers to dyadic system. Let us recall some basic notations. Let L (R) be square integrable function space, whose inner product is defined as follows Z hfgi = f(y)μg(y)dy The Fourier transform of f L (R) is defined to be ^f(w) = Z R R f(y)e iwy dy For a given function ψ L (R), we can define function family ψ jk y 7! j ψ( j y k) by dilation and shift. The following definitions come from refs. [, 9]. Definition.. Let H be a Hilbert space. fh k g kz ρ H is called a frame if there exists two positive numbers A and B such that j< fh k >j 6 B k f k A k f k 6 kz where A and B are called lower bound and upper bound separately when A = B, we call it tight frame. Definition.. A family ψ = fψ ψ ψ N g ρ L (R) is called a tight frame of L (R) if it satisfies N i= jkz j hfψ i jki j = kfk () There we have normalized ψ i that k ψ i k=, which makes it analogous with orthonormal wavelet. Definition.3. Let ffi L (R), ^ffi L ^ffi be continuous at 0, and ^ffi(0) =. It is called a refinable function if it generates the nested subspace fv j g in the sense of V j = fffi jk g kz. A finite family of functions ψ = fψ ψ N g ρ V is called a minimum-energy (wavelet) frame associated with ffi, if j hfffi k i j = kz j hfffi 0k i j + N kz kz j hfψ i 0ki j f L (R) ()

3 No. 6 CONSTRUCTION FOR A CLASS OF SMOOTH WAVELET TIGHT FRAMES 447 It is obvious that (.) is equivalent to the formulation N k iffi k = hfffi 0k iffi 0k + hfψ0kiψ kzhfffi i i 0k f L (R) (3) kz i= kz Analogous with the multiresolution analysis of orthogonal wavelet, there is corresponding multiresolution analysis of wavelet tight frame (see ref. [7]). In this paper, every tight frame is built on one scaling function, two or three wavelet functions. Here the scaling space V j and the wavelet spaces W "j are defined as follows V j = fffi( j t ng nz (4) W "j = fψ " ( j t n)g nz "= (or and 3) (5) Corresponding to ffi and ψ ", there are one scaling filter h 0 (n) and two (or three) wavelet filters h i (n), i=, (or and 3). Orthogonal wavelet can be regarded as a special wavelet tight frame with one scaling function and one wavelet function. An important property of wavelet system is its degree of vanishing moments of wavelet function. It decides the smoothness of wavelet tight frame, whose definition is as follows Definition.4. We say that ψ = fψ ψ ψ N g has vanishing moments of order m if Z R t k ψ l (t)dt = 0 l = N k = 0 m Parameterization of wavelet tight frame and examples In order to get symmetric wavelet tight frames with finite length, we only consider compactly supported scaling function ffi and wavelet functions Ψ = fψ ψ ψ 3 g ρ V. All of their symbols P (z), Q (z)q (z) and Q 3 (z) are Laurent polynomials. Because when the wavelet tight frame is composed of one scaling function and three wavelet functions (among these filters, two are symmetric), the result is better, so we will mainly give the construction with three high-pass filters. Lemma. (Theorem [] ). A compactly supported refinable function ffi L, with ^ffi(0) = and two -scale Laurent polynomial symbol P (z) has an associated minimum-energy frame ψ with compact support, if and only if P (z) satisfies jp (z)j + jp ( z)j 6 for all jzj = (.) Now we give two theorems which show the concrete algorithm of constructions of compactly supported wavelet tight frames with n = N and n = N +separately. After the theorem, we will give the concrete formulations and examples for the given n by MATLAB. Theorem.. There exists the parametric formulations for the low-pass filters and highpass filters with odd length, which is symmetric and has one parametric variable, and a closed interval for parametric variable. Every parameter in this interval will give a wavelet tight frame. Proof. Let h 0 h N be low-pass filter coefficients with symmetry and odd length. Then they should satisfy normalized condition N h i =

4 44 SCIENCE IN CHINA (Series F) Vol. 46 and symmetric conditions h i = h N i i = 0 N Then we have N h i + h N = Because wavelet tight frame do not satisfy orthogonal condition, there are N variables and one equation in the above condition. We will add N vanishing moments to it, which satisfies the following equations (see ref. [6]) n ( ) n n k h n = 0 k = N We need to explain why we only use N degree vanishing moments. It aims to guarantee there is a free variable. By putting the above conditions together, we have > N N h i = h N ( ) n [( n) k +(n + N) k ]h i = ( N) k ( ) N h N The above equations have N + variables and N independent equations, so there exists just one free variable. Let h N be the free variable. Then we can write the linear equations in matrix form A 0 h h N where A is the matrix formed by the left coefficients of the equations, and b is the coefficients vectors related to h N. We get h i = h i (h N )i= 0 N, where h N [a N b N ], so we get the parameterization of H as follows C A = b (h 0 (h N ) h N (h N )h N h N (h N ) h 0 (h N )) By experience we know that when h i (h N ) 6 h N (h N ) 6 h N, i = 0 N the properties of the filters are much better, which will be seen in the following examples. As we see from the above, there exist c N d N satisfying h N [c N d N ] T [a N b N ]. By selecting a certain value in this interval, we can get B-spline filters, and of course we also can get other non-spline filters. For a given cost function, we can select the proper h N [c N d N ] T [a N b N ] and get the best low-pass filters. This process is called optimization. Now we are going to construct the corresponding high-pass filters. Let P (z) = N h i (h N )(z i + z N i )+h N z N be the corresponding Laurent polynomial of low-pass filter. From Riesz lemma we know that there exists a Laurent polynomial Q (z) satisfying j P (z) j + j P ( z) j + j Q (z) j + j Q ( z) j = j z j= (.)

5 No. 6 CONSTRUCTION FOR A CLASS OF SMOOTH WAVELET TIGHT FRAMES 449 First we let the high-pass filter Q (z) have the length N + and be anti-symmetric, that is G = (g 0 g N ), where g i = g N i i= 0 N g N = 0. Then we have Q (z) = N g i z i = N g i (z i z N i ) (.3) Substituting (.3) into (.) we get g i = g i (h N )i = 0 N, so the parameterization of high-pass filters is as follows G = (g 0 (h N ) g N (h N ) 0 g N (h N ) g 0 (h N )) (.4) The Laurent polynomials of another two high-pass filters are separately Q (z) = zp( z) Q 3 (z) = zq ( z) So we get the parameterization form of wavelet tight frame as follows > (h 0 (h N ) h N (h N )h N h N (h N ) h 0 (h N )) G = (h 0 (h N ) ( ) N h N (h N ) ( ) N h N ( ) N + h N (h N ) h 0 (h N )) G = (g 0 (h N ) g N (h N ) 0 g N (h N ) g 0 (h N )) (g 0 (h N ) ( ) N g N (h N ) 0 ( ) N + g N (h N ) g 0 (h N )) where h N [a N b N ] T [c N d N ] Next we give concrete parameterization form and examples for n = 5 and n = 7 separately by using Matlab. Example. Let (h 0 h h h 3 h 4 ), where h 0 = h 4 h = h 3. According to Theorem., we can get the parameterization form by using Matlab as follows > where 4 6 h 6 p + 4 h 4 h 4 4 h G = 4 h 4 h 4 4 h r G = 4 h 4 h 4. h h r h h + 6 h 4 r h h r h h + 6 h 4 Let h = 3. We get the wavelet tight frame as follows G = G = > This is a cubic B-spline, which was already included in ref. [], see fig..

6 450 SCIENCE IN CHINA (Series F) Vol. 46 Fig.. ffi is a scaling function, ψ ψ ψ3 are wavelet functions. Let h = 9. We get the wavelet tight frame as follows G = G = p p > p p This is a new and symmetric wavelet tight frame, but it is not generated by B-spline, see fig.. Example. For (h 0 h h h 3 h 4 h 5 h 6 )h 0 = h 6 h = h 5 h = h 4, according to Theorem., we can get the parameterization form by using Matlab as follows h 3 4 h 3 4 h h 3 4 h h h 3 3 G = 3 4 h h 3 4 h h 3 4 h h h 3 3 G = 3 4 h 3bc0 c b 4 h h 3 b c 0 c b 4 h 3 3 r 3 > 9 b = h 3 3 r c c = 6 6 h 3 h + 3 h 3

7 No. 6 CONSTRUCTION FOR A CLASS OF SMOOTH WAVELET TIGHT FRAMES 45 Fig.. ffi is a scaling function, ψ ψ ψ3 are wavelet functions. where h 3 6 p 4. When h 3 = 5, we get the wavelet tight frame as follows G = G = 64 p p p p > p p p p This is just B-spline wavelet tight frame, see its figure in ref. []. When h 3 =, we get the filter bank of another wavelet tight frame as follows G = G = 3 p p p p > p p p p

8 45 SCIENCE IN CHINA (Series F) Vol. 46 This is a new wavelet tight frame, which is not generated by B-spline, see fig. 3. Fig. 3. ffi is a scaling function, ψ ψ ψ3 are wavelet functions. Theorem.. There exist the parameterization formulations for the low-pass filters and high-pass filters with even length, which is symmetric and has one parametric variable, and a closed interval for parametric variable. frame. Proof. Every value in this interval will give a wavelet tight Let h 0 h N be low-pass filter coefficients with symmetry and even length. Then they should satisfy normalized condition and symmetric conditions Then we have N h i = h i = h N i i = 0 N N h i = Because wavelet tight frames do not satisfy orthogonal condition, there are N variables and one equation in the above condition. We will add N degrees of vanishing moments, which satisfies

9 No. 6 CONSTRUCTION FOR A CLASS OF SMOOTH WAVELET TIGHT FRAMES 453 the following equations (see ref. [7]) n ( ) n n k h n = 0 k = N There is one thing we need to explain, that is why we only add N vanishing moments in order to guarantee there is a free variable. By putting the above conditions together, we have > N N h i = h N ( ) n [( n) k +(n + N) k ]h i = (( N) k ( N) k )( ) N h N The above equations have N variables and N equations, so there exists one free variable. Let h N be the free variable. We can write the linear equations in matrix form A 0 h h N where A is the matrix formed by the left coefficients of the equation, and b is the coefficients vector related to h N. Then we get h i = h i (h N )i = 0 N, where h N [d N e N ], C A = b so we get the parameterization of low-pass filter as follows (h 0 (h N ) h N (h N )h N h N h N (h N ) h 0 (h N )) The next step is the same with Theorem.. We get h N [d N e N ] T [m N n N ], so we get the parameterization form of wavelet tight frame as follows > (h 0 (h N ) h N (h N )h N h N h N (h N ) h 0 (h N )) G = (h 0 (h N ) ( ) N h N (h N ) ( ) N h N ( ) N h N h 0 (h N )) G = (g 0 (h N ) g N (h N ) g N (h N ) g 0 (h N )) (g 0 (h N ) ( ) N g N (h N ) ( ) N + g N (h N ) g 0 (h N )) where h N [d N e N ] T [m N n N ] Example 3. Let (h 0 h h h 3 ), where h 0 = h 3 h = h. According to Theorem., we can get the parameterization form by using Matlab as follows > h h h h G = G = h h h h r r r r h h h h h h h h r r r r h h h h h h h h By experience we know that when the values of both sides are less than that in the middle, the

10 454 SCIENCE IN CHINA (Series F) Vol. 46 resulting filter has good properties. Then by we get 4 6 h 6. > h h > 0 h 6 h When h = 7, we get the filter bank of wavelet tight frame as follows G = p p G = > p p p p This is a new and symmetric wavelet tight frame. Its scaling function is not a B-spline and its smoothness is not good (see fig. 4). Fig. 4. ffi is a scaling function, ψ ψ ψ3 are wavelet functions. Example 4. Let (h 0 h h h 3 h 4 h 5 )where h 0 = h 5 h = h 4 h = h 3. According to Theorem., we can get the parameterization form by using Matlab as follows

11 No. 6 CONSTRUCTION FOR A CLASS OF SMOOTH WAVELET TIGHT FRAMES h 5 6 h h h 5 6 h 3 6 h 3 G = 6 h h h h 5 6 h h q h G = + r q h 5 56 h + r r h h 5 56 h h + h r r q q h + 5 h 56 h + h h h + r h h q h + r q h 5 56 h + r r h h h h + h r r q q > h + 5 h 56 h + h h h + r h 5 56 h Let h 3 = 7. We get the wavelet tight frame G = G = 3p p p p p p p p p p > 3p p p p p p p p p p Similar to the above, we can get h 6 5 Fig. 5. ffi is a scaling function, ψ ψ ψ3 are wavelet functions.

12 456 SCIENCE IN CHINA (Series F) Vol. 46 This is a new and symmetric wavelet tight frame with good smoothness. Its scaling function is not B-spline (see fig. 5). Remark. Wavelet tight frames with one low-pass filter and two high-pass filters can be constructed in a similar way. Now we get an example as follows a a a a G = a a a a > G = 0 a a a a 0 Let a =. Then we get the parameterization form of the wavelet tight frame as follows G = > 5 5 G = This is a new and symmetric wavelet tight frame. Its scaling function is not B-spline (see fig. 6). Fig. 6. ffi is a scaling function, ψψ are wavelet functions.

13 No. 6 CONSTRUCTION FOR A CLASS OF SMOOTH WAVELET TIGHT FRAMES 457 Fig. 7. (a) Decomposition with orthogonal wavelet in ref. []. (b) Decomposition with wavelet tight frame constructed in this paper.

14 45 SCIENCE IN CHINA (Series F) Vol Conclusion This paper gives a designing method for constructing wavelet tight frames under the condition that both the low-pass filters and high-pass filters are unknown, and the concrete algorithm and parameterization form with the odd length and even length separately with a closed interval of the parameter. It also gives some examples by optimizing the smoothness. Generally speaking, we can make optimization from the practical aim by certain cost function, and then select the optimal filters we need. Our algorithm can also be used to construct other wavelet tight frames. For example the length of low-pass filter may be odd, and the length of high-pass filter is even. As we know, though wavelet tight frame loses orthogonal property, it allows the filters short length as well as the scaling and wavelet functions more smoothness. Now we give the decomposing result of Lena image with wavelet tight frame constructed in this paper, compared with that decomposed by 4-band orthogonal wavelet in ref. [] (fig. 7), it fully satisfies the perfect reconstruction condition. An interesting question is how is the energy distributed in every block, how can we explain its physical meaning? This question need to be further studied! Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos and ) and 973 Project of China (Grant No. G ). References. Berkner, K., Wells Jr, R. O., A correlation-dependent model for denoising via nonorthogonal wavelet transforms, Technical Report CML TR9-07, Computational Mathematics Laboratory, Rice University, 99.. Selenick, I. W., Sendur, L., Smooth wavelet frames iwth application to denoising, Proc. IEEE Int. Conf. Acoust. Speech, Signal Processing (ICASCP), 000, iong, Z., Orchard, M. T., Zhang, Y. Q., A deblocking algorithm for JPEG compressed images using overcomplete wavelet representations, IEEE Trans. Circuits Syst. Video Technol., 997, Grochenig, K., Ron, A., Tight compactly supported wavelet frames of arbitarily high smoothness, Proceeding of the Amercian Mathematical Society, 99, 6(4) Chui, C. K., He Wenjie, Stöckler, J., Compactly supported tight and sibling frames with maximum vanishing moments, Applied Computational Harmonic Analysis, to appear. 6. Ron, A., Shen, Z., Affine system in L (R d ) The analysis of analysis operator, J. Funct. Anal., 997, Benedetto, J. J., Li, S., The thory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal., 99, Chui, C. K., He Wenjie, Compactly supported tight frames associated with refinable functions, Applied Computational Harmonic Analysis, 000, Daubechies, I., Ten lectures on wavelet, CBMS-NSF series in Applied Math, 6, SIAM Publ.(99). 0. Selenick, I. W., Smooth wavelet tight frames with zero moments, Applied Computational Harmonic Analysis, 000, Daubechies, I., Han Bin, Ron, A. et al., Framelets MRA-Based constructions of wavelets of wavelet frames, Applied and Computational Harmonic Analysis, 003, 4() 46.. Peng, L. Z., Wang, Y. G., Construction of compact supported orthonormal wavelets with beautiful structure, Science in China, to appear.