Multiresolution Inverse Wavelet Reconstruction from a Fourier Partial Sum

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1 City University of New York (CUNY) CUNY Academic Works Publications and Research Kingsborough Community College -200 Multiresolution Inverse Wavelet Reconstruction from a Fourier Partial Sum Nataniel Greene CUNY Kingsborough Community College How does access to this work benefit you? Let us know! Follow this and additional works at: Part of the Analysis Commons, and the Numerical Analysis and Computation Commons Recommended Citation N. Greene, Multiresolution Inverse Wavelet Reconstruction from a Fourier Partial Sum, Journal of Concrete and Applicable Mathematics, Vol. 8, No. 2, 200, pp This Article is brought to you for free and open access by the Kingsborough Community College at CUNY Academic Works. It has been accepted for inclusion in Publications and Research by an authorized administrator of CUNY Academic Works. For more information, please contact AcademicWorks@cuny.edu.

2 JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS, VOL.8, NO.2, , 200, COPYRIGHT 200 EUDOXUS PRESS, 353LLC Multiresolution Inverse Wavelet Reconstruction from a Fourier Partial Sum Nataniel Greene Department of Mathematics and Computer Science Kingsborough Community College, CUNY 200 Oriental Boulevard, Brooklyn, NY ngreene.math@gmail.com March 6, 2009 The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approimations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. In previous work [,2] we described a numerical procedure for overcoming the Gibbs phenomenon called the Inverse Wavelet Reconstruction method (IWR). The method takes the Fourier coe cients of an oscillatory partial sum and uses them to construct the wavelet coe cients of a non-oscillatory wavelet series. However, we only described the method standard wavelet series and not for the continuous wavelet transform. We also only described the method for so-called crude wavelets: wavelets which do not admit a scaling function, and therefore lack a multiresolution analysis. Here we describe a version of IWR based on a quadrature discretization of the continuous inverse wavelet transform. We also describe the multiresolution potential of the Inverse wavelet reconstruction method, which is the ability to decompose a function into a coarse trend plus ner details. Key Words: Gibbs phenomenon, Fourier Series, Inverse polynomial reconstruction, Wavelets, Inverse wavelet reconstruction. Introduction Fourier and orthogonal polynomial series are known for their highly accurate epansions for smooth functions. In fact it is known that the more derivatives a function has, the faster the approimation will converge. However, when a function possesses jump-discontinuities the approimation will fail to converge uniformly. In addition, spurious oscillations will cause a loss of accuracy throughout the entire domain. This lack of uniform convergence is

3 354 GREENE: WAVELET RECONSTRUCTION known as the Gibbs phenomenon. Methods for post-processing approimations which su er from the Gibbs phenomenon include the Gegenbauer reconstruction method of Gottlieb and Shu [7, 9], the method of Pade approimants due to Driscoll and Fornberg [2], the method of spectral molli ers due to Gottlieb and Tadmor [8] and Tadmor and Tanner [2, 22], the Inverse polynomial reconstruction (IPR) method of Shizgal and Jung [4,5,6,9,20], and the Freund polynomial reconstruction method of Gelb and Tanner [6]. These reconstruction methods can be combined with an e ective method for edge-detection developed by Gelb and Tadmor [3, 4, 5], to yield an eponentially accurate reconstruction of the original function. In previous work we described a numerical method for overcoming the Gibbs phenomenon following the work of Shizgal and Jung, called the Inverse Wavelet Reconstruction method (IWR) [0,,2]. The IWR method was shown numerically to perform at least as well as inverse polynomial reconstruction. However, our main justi cation for choosing wavelets over other orthogonal or non-orthogonal bases, lies in ability of wavelets to decompose a function into a coarse trend and ner details. This decomposition is essentially what is known as a multiresolution analysis. Our previous papers on the IWR method focused on so-called crude wavelets: wavelets which do not admit a scaling function and which therefore do not o er a multiresolution analysis The present paper addresses the notion of multiresolution reconstruction from a Fourier partial sum. Our previous papers described the method using standard wavelet series approimations. The present paper also addresses how the method can be implemented using the continuous wavelet transform (CWT). The CWT is truncated and discretized using a uniform Trapezoidal rule or a Gauss quadrature grid and a non-standard wavelet series approimation is obtained. We begin with a brief review of the essential de nitions of wavelets which we will need. Recall that a wavelet is a function 2 L 2 (R) satisfying: Z () d = 0 () and Z j ()j d < ; (2) jj where here is the Fourier transform of. To avoid a con ict of notations we will use a capital letter for the Fourier transform and a hat above the letter for a Fourier coe cient. The function is known as a mother wavelet or an analyzing wavelet since any function f 2 L 2 (R) can be epressed as a continuous sum of translations and dilations involving according to the continuous wavelet transform. The continuous wavelet transform (CWT) is given by F (b; a) = jaj Z =2 f () b d a 2

4 GREENE: WAVELET RECONSTRUCTION 355 and the inverse CWT is given by f () = C Z Z F (b; a) a 2 b;a () dadb (3) where and b;a () = jaj C = Z =2 b a j ()j d: jj A discrete wavelet series is given for ed constants a 0 ; b 0 by f () = X X m= l= where the discrete wavelet series coe cients are given by b0 l d m;l = F ; and m;l () in this case has a slightly di erent meaning: a m 0 d m;l m;l () (4) a m 0 m;l () = a m=2 0 (a m 0 b 0 l). In general the wavelet functions m;l do not constitute an orthonormal basis. Instead they constitute what is known as a frame. The family of functions m;l is a frame if X X A kfk 2 f; m;l 2 B kfk m= l= for positive constants A and B: When A = B the functions j;k are known as a tight frame. For more information see Daubechies []. The inverse wavelet reconstruction method makes use of wavelet bases and frames, as well as the CWT. Much of the present paper focuses on wavelets which are known to have what is known as a multiresolution analysis. A scaling function, sometimes called a father wavelet, is function () which satis es the two-scale relations: () = X l= h l (2 l) and X () = g l (2 l) : l= 3

5 356 GREENE: WAVELET RECONSTRUCTION The functions m;l () = 2 m=2 (2 m l) are orthonormal to each other and they are also orthonormal to m;l () = 2 m=2 (2 m l). The functions and allow for a decomposition of a function into the sum of a coarse approimation and ner details as follows: X X X f () = c M0;l M0;l () + d m;l m;l () which is truncated as f M;L () ' l= l= L c M0;l M0;l () + m=m 0 l= MX m=m 0 l= L d m;l m;l () : The c M0;l are known as approimation coe cients for the function at resolution scale M 0 and the d m;l are detail coe cients for the ner scales. This particular decomposition of a function into trend and details is known as a multiresolution analysis. 2 The Multiresolution Inverse Wavelet Reconstruction Method 2. Inverse Wavelet Reconstruction We begin with a Fourier partial sum and assume that the function f () can also be epressed as a discrete wavelet series: X X f () = d m;l m;l () where and d m;l = m= l= Z f () m;l ()d m;l () = a m=2 0 (a m 0 b 0 l) : We now derive a formula epressing the Fourier coe cients in terms of wavelet coe cients. bf (n) = 2 = 2 = = Z Z X f () e in d X X m= l= X m= l= X X m= l= d m;l 2 d m;l m;l () e in d Z m;l () e in d d m;l b m;l (n) (5) 4

6 GREENE: WAVELET RECONSTRUCTION 357 The inverse wavelet reconstruction method is obtained by truncating the doubly in nite sum described above and solving for the wavelet coe cients. We suggest solving the following system of equations. bf (n) = MX m= M l= L d m;l b m;l (n) (6) for the wavelet coe cients d m;l where n = N:::N and N = 2ML + M + L: One then computes the IWR approimation S M;L f () = MX m= M l= L d m;l m;l () : (7) The terms b m;l (n) are the nth Fourier coe cients of m;l () : Since we are solving a system of 2N + equations in (2M + ) (2L + ) unknown wavelet coe cients, in order for the system to be invertible we must have 2N + = (2M + ) (2L + ), or N = 2ML + M + L: Numerical eperiments indicate that the invertibility condition can be relaed and fewer wavelet coe cients can be computed if one solves the overdetermined system in a least squares sense using Matlab s backslash operator. This reconstruction is valid for both crude and non-crude wavelets. 2.2 Multiresolution IWR: Wavelet and Scaling Function Approach Here we describe the IWR method for wavelets which allow for a multiresolution reconstruction. A formal derivation of the method proceeds again as follows. Write bf (n) = 2 = 2 = = Z Z X l= + X f () e in d X l= c M0;l 2 m=m 0 l= X l= X c M0;l M0;l () + Z d m;l 2 c M0;l b M0;l (n) + X X m=m 0 l= M0;l () e in d Z X m;l () e in d X m=m 0 l= d m;l m;l ()! e in d c m;l b m;l (n) (8) 5

7 358 GREENE: WAVELET RECONSTRUCTION The IWR method in this case is obtained by truncating the sums and solving the system of equations for the approimation and detail coe cients bf (n) = l= L c M0;l b M0;l (n) + MX m=m 0 l= L d m;l b m;l (n) (9) where n = N:::N. Numerical results again indicate that the system can be solved in a least squares sense and one need not be concerned with meeting an invertibility condition eactly. One then constructs the multiresolution IWR approimation S M0 M;Lf () = AM0 L () + DM0 M;L () where the coarse trend is given by A M0 L () = X L and the ner details are given by l= L M D M0 M;L () = X m=m 0 l= L c M0;l M0;l () d m;l m;l () : 2.3 Multiresolution IWR: Pure Wavelet Approach Net we describe an alternative multiresolution reconstruction for non-crude wavelets which does not make eplicit use of the scaling function. It is simply a modi cation of the original IWR approimation. We write where the coarse trend is given by S M0 M;Lf () = AM0 M;L () + DM0 M;L () M0 A M0 M;L () = X and the ner details are given by D M0 M;L () = m= M l= L M X m=m 0 l= L d m;l m;l () d m;l m;l () : The derivation is simply to begin with the original IWR reconstruction and write MX S M;L f () = d m;l m;l () = m= M l= L MX 0 m= M l= L d m;l m;l () + = A M0 M;L () + DM0 M;L () 6 MX m=m 0 l= L d m;l m;l ()

8 GREENE: WAVELET RECONSTRUCTION 359 The Pure Wavelet Approach will only provide a good decomposition into trend and details for wavelets which admit a scaling function, even though the method does not make eplicit use of the scaling function. This is because the two approaches are approimately equivalent, that is: A M0 L () ' AM0 M;L () : In fact, but also A M0 L () = X L l= L c M0;l M0;l () Since A M L M0 A M0 L () = X = m= l= L XM m= l= L = A M L d m;l m;l () d m;l m;l () + () + A M0 M;L () : MX 0 () is the tail of a convergent series, A M L m= M l= L d m;l m;l () ()! 0 as M! since 3 A Version of IWR Based on the Continuous Wavelet Transform Finally, we describe here a related method for implementing IWR which is based on a quadrature discretization of the inverse continuous wavelet transform. It produces results which are comparable to the standard wavelet series approach described previously. Our chief reason for considering it is allows us to consider any wavelet family as a candidate for use in reconstruction. The derivation is as follows. bf (n) = 2 = 2 = C = Z Z f () e in d C Z Z Z Z Z Z C F (b; a) a 2 F (b; a) a 2 Z 2 b a b a dadb e in d e in d dadb F (b; a) a 2 b b;a (n) dadb: (0) Note that in this section we use b b;a (n) to denote the nth Fourier coe cient of b : a 7

9 360 GREENE: WAVELET RECONSTRUCTION Truncate the limits of integration and write bf (n) ' Z B Z A B A C F (b; a) a 2 b b;a (n) dadb: () Net, discretize the integrations using a quadrature rule such as the trapezoidal rule or Gauss quadrature. The resulting equations become bf (n) ' MX l=0 m=0 C F (b l ; a m ) b a 2 bl ;a l (n) (2) m which can be epressed in simpler form by letting d m;l = C writing MX bf (n) ' m=0 l=0 F (b l ;a m) a 2 m and d m;l b bl ;a l (n) : (3) Finally, solve the system of equations, possibly in a least squares sense, for the coe cients d m;l and construct the nonstandard wavelet series approimation: f () ' MX m=0 l=0 4 Numerical Results d m;l a m bl : (4) Numerical eperiments show that the IWR method based on a quadrature discretization of the CWT yields uniformly converging approimations for all wavelet families we have tested. We display the results of the Haar, Shannon, and Meican hat wavelets in the gures below. The Haar and Shannon systems are some of the earliest eamples of what later became wavelets. The Haar wavelet is apparently able to resolve an interior discontinuity without prior knowledge as to its location, as well as the endpoints. Therefore the Haar wavelet is able to provide a global reconstruction, albeit a low order accuracy one. The low order of convergence is due to the fact that the wavelet is not well-localized in Fourier space. Shannon wavelets will resolve the Gibbs phenomenon at the endpoints of the interval and Meican hat wavelets will do it even better. Numerical eperiments also show that the Multiresolution IWR method is able to decompose a function into a course trend plus ner details, given a function s Fourier coe cients. We provide numerical eamples using the both the Pure Wavelet Approach to reconstruction and the Wavelet and Scaling function Approach described above. We implemented the inverse wavelet reconstruction method by computing Fourier coe cients for the test functions in question using trapezoidal rule quadrature. This makes our Fourier series a pseudospectral series. For the method to be accurate we compute the coe cients b m;l (n) using the same trapezoidal rule quadrature. 8

10 y error GREENE: WAVELET RECONSTRUCTION 36 Inverse Haar Wavelet Reconstruction 0 0 Absolute Error Figure : Inverse Haar Reconstruction of 4 tan based on 64 Fourier coe - cients using the wavelet series approach. 9

11 y error 362 GREENE: WAVELET RECONSTRUCTION Inverse Shannon Wavelet Reconstruction 0-6 Absolute Error Figure 2: Inverse Shannon reconstruction of 4 tan using the CWT approach from 28 Fourier coe cients. The limits of integration are A = B = 20 and 64 Gauss-Legendre quadrature nodes in a and b are employed. 0

12 Absolute Error GREENE: WAVELET RECONSTRUCTION Figure 3: Converge of Inverse Shannon reconstruction of 4 tan using the CWT approach from 6, 32, 64, and 28 Fourier coe cients. The limits of integration are A = B = 20 and the number of Gauss-Legendre quadrature nodes in a and b are equal to the number of Fourier coe cients.

13 y error 364 GREENE: WAVELET RECONSTRUCTION Inverse Haar Wavelet Reconstruction 0 0 Absolute Error Figure 4: Inverse Haar reconstruction of sin(cos(4)) sgn() utlilizing the CWT approach with A = B = 20, and Gauss-Legendre quadrature with 32 quadrature nodes in a and 32 quadrature nodes in b. The method resolves both endpoint and interior discontinuties, albeit with low accuracy. 2

14 y error GREENE: WAVELET RECONSTRUCTION 365 Inverse Haar Wavelet Reconstruction, 64 Coefficients Absolute Error Figure 5: Inverse Haar wavelet reconstruction using the pure wavelet approach of the function f () = 4 tan + 0 sin 30 based on 64 Fourier coe cients. We consider the test function f () = 4 tan () + 0 sin (30) to observe the ability of various wavelets to etract the trend, mostly due to tan (); and smaller scale oscillations due to 0 sin (30). We display here the results of the Haar wavelet: the simplest wavelet which admits a multiresolution analysis. We also consider the piecewise linear Franklin wavelets generated by the second order splines. 5 Conclusions Numerical results indicate that the IWR method based on the continuous wavelet transform works as well as the IWR method based on a standard wavelet series. It yields an accurate and rapidly converging approimation. The multiresolution IWR methods allows one to separate out from the Fourier coe cients a course trend and ner details regardless of any Gibbs phenomenon. Work in progress includes a numerical study of the multiresolution IWR method for a wider variety of wavelets, multiresolution reconstruction from series other than Fourier series, a two-dimensional implementation and a proof of convergence. References [] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia,

15 y y 366 GREENE: WAVELET RECONSTRUCTION Haar Trend for 64 Coefficients Details Trend Error Figure 6: Inverse Haar decomposition of f () = 4 tan + 0 sin 30 into a trend and details using the pure wavelet approach. 4

16 y error GREENE: WAVELET RECONSTRUCTION 367 Inverse Franklin Wavelet Reconstruction, 64 Coefficients Absolute Error

17 y y 368 GREENE: WAVELET RECONSTRUCTION Franklin Trend for 64 Coefficients Details Trend Error Figure 7: Inverse Franklin decomposition of f () = 4 tan + 0 sin 30 into a trend and details. 6

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