A Wavelet-based Method for Overcoming the Gibbs Phenomenon

Transcription

1 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '08), Harvard, Massachusetts, USA, March 4-6, 008 A Wavelet-based Method for Overcoming the Gibbs Phenomenon NATANIEL GREENE Department of Mathematics Computer Science Kingsborough Community College, CUNY 00 Oriental Boulevard, Brooklyn, NY 35 UNITED STATES ngreene.math@gmail.com Abstract: The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coefcients of an oscillatory partial sum uses them to construct the wavelet coefcients of a non-oscillatory wavelet series. Key Words: Gibbs phenomenon, wavelets, Gegenbauer reconstruction, inverse polynomial reconstruction. Introduction Fourier orthogonal polynomial series are known for their highly accurate expansions for smooth functions. In fact it is known that the more derivatives a function has, the faster the approximation will converge. However, when a function possesses jumpdiscontinuities the approximation 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 known as the Gibbs phenomenon. Methods for post-processing approximations which suffer from the Gibbs phenomenon include the Gegenbauer reconstruction method of Gottlieb Shu [7,9,0], the method of Pade approximants due to Driscoll Fornberg [], the method of spectral molliers due to Gottlieb Tadmor [8] Tadmor Tanner [7, 8], the inverse polynomial reconstruction method of Shizgal Jung [3,4,5,6], the Freund polynomial reconstruction method of Gelb Tanner [6]. These reconstruction methods can be combined with an effective method for edge-detection developed by Gelb Tadmor [3,4,5,6], to yield an exponentially accurate reconstruction of the original function. In this paper we describe a new numerical method for overcoming the Gibbs phenomenon following the work of Shizgal Jung, called the inverse wavelet reconstruction method. We begin with a brief review of the essential denitions of wavelets which we will need. Recall that a wavelet is a function L (R) satisfying: Z Z (x) dx 0 () b () d < ; () jj b here is the Fourier transform of. The function is also known as an analyzing wavelet or a mother wavelet since any function f L (R) can be expressed as a continuous sum of translations dilations involving according to the continuous wavelet transform. The continuous wavelet transform is given by Z t b (W f) (b; a) jaj f (t) dt a the inverse continuous wavelet transform is given by f (x) C Z Z b;a (t) jaj C Z [(W f) (b; a)] t b a b () d: jj b;a (t) a dadb (3) ISSN: ISBN:

2 A discrete wavelet series is given by f (x) X X j k c j;k j;k (x) (4) the discrete wavelet series coefcients are given by k c j;k (W f) j ; j AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '08), Harvard, Massachusetts, USA, March 4-6, 008 j;k (x) j j x k. For more information see, for example, Chui []. We will make use of wavelet series for our reconstruction technique further on. The Fourier coefcients b f (n) can be expressed in terms of Gegenbauer- coefcients as follows. bf (n) Z Z X X f (x) e inx dx X bf (m) bf (m) C m (x) e inx dx Z Cm (x) e inx dx bf (m) c m;n (7) The connection coefcients c m;n are given by The Inverse Wavelet Reconstruction Method. Inverse Polynomial Reconstruction One numerical technique for dealing with the problem of the Gibbs phenomenon is the inverse polynomial reconstruction method of Shizgal Jung [3, 4, 5, 6]. Their inverse method is itself an alternative approach to the original direct Gegenbauer reconstruction method of Gottlieb Shu [7, 9, 0]. In the inverse method, one solves a system of linear equations for the Gegenbauer polynomial reconstruction coefcients in terms original Fourier coefcients, whose expansion suffers from the Gibbs phenomenon. The inverse polynomial reconstruction method begins with a Fourier partial sum S N f (x) NX n N bf (n) e inx : (5) The function also has a Gegenbauer- expansion given by bf (m) h m f (x) Z X bf (m) C m (x) h m (m + ) m! () f (x) C m (x) x dx (6) c m;n Z C m (x) e inx dx (8) can be computed numerically. An explicit formula for c m;n was derived by this author, Greene [, ], is given for the cases n 6 0 n 0 respectively by: c m;n c m;0 [m] X j0 8 < : ( i) m j p J n m j+ (n) (9) j () 3 m j m j j! 3 m j m j ( ) m () m j!( 3 ) m ; m even 0; m odd. (0) The inverse polynomial reconstruction procedure is obtained by truncating the innite series above solving the system of equations bf (n) NX bf (m) c m;n, n N:::N: () One then computes the Gegenbauer reconstruction approximation S Nf (x) NX bf (m) C m (x) : () + () (m + ) :. Inverse Wavelet Reconstruction We explore here an analogous approach which seeks to reconstruct the original function in terms of ISSN: ISBN:

3 wavelets. We begin with a Fourier partial sum as before assume that the function f (x) can also be expressed as a discrete wavelet series: f (x) X X m l Z m;l (x) f (x) m;l (x)dx m;l (x) m ( m x l) : We now derive a formula expressing the Fourier coef- cients in terms of wavelet coefcients. bf (n) Z Z X f (x) e inx dx X X m l X m l X X m l m;l (x) e inx dx Z m;l (x) e inx dx b m;l (n) (3) The inverse wavelet reconstruction method is obtained by truncating the doubly innite sum described above solving for the wavelet coefcients. We suggest solving the following system of equations. AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '08), Harvard, Massachusetts, USA, March 4-6, 008 bf (n) m M l L b m;l (n) (4) n N:::N ; N ML (5) for the wavelet coefcients : One then computes the wavelet reconstruction approximation S M;L f (x) m M l L m;l (x) : (6) The terms b m;l (n) are the nth Fourier coefcients of m;l (x) : Since we are solving a system of N equations in (M) (L) 4ML unknown wavelet coefcients, in order for the system to be invertible we must have N 4ML or N ML: It is this convention which we will use for our numerical implementation below, owing to the simplicity of the relation N ML: Alternatively, one may solve the system bf (n) m M l L b m;l (n) (7) n N:::N; N ML + M + L: (8) then compute S M;L f (x) m M l L m;l (x) : (9) The condition in (8) is due to the fact that we are solving a system of N + equations in (M + ) (L + ) unknown wavelet coefcients. In order for the system to be invertible we must have N + (M + ) (L + ) or N ML+M +L: A detailed proof of convergence is the subject of concurrent work. However, we illustrate the apparent convergence for certain wavelets with some numerical examples. 3 Numerical Results Numerical experiments show that the inverse wavelet reconstruction approach does yield accurate uniform approximations for a variety of wavelet families. We illustrate this with three wavelet families: Poisson wavelets given by (x) Mexican hat wavelets given by (x) p 3 4 ( + x ) ; (0) x e x ; () Morlet wavelets given by (x) cos x e x : () For each case we begin with a Fourier series of N 6; 36; 64; 00 expansion coefcients of an analytic non-periodic test function f(x) 4 tan (x) whose partial sum suffers from the Gibbs phenomenon. We then compute the corresponding wavelet series reconstructions. We chose the number of Fourier coefcients such that N ML M L: The absolute error of these reconstructions, jf(x) S M;L f (x)j, are displayed in the gures provided. For the three wavelets shown the Morlets perform the best for a xed number of Fourier coefcients, the Mexican hat wavelets perform comparably, though slightly less well, the Poisson wavelets perform the least well. ISSN: ISBN:

4 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '08), Harvard, Massachusetts, USA, March 4-6, 008 Figure : The graphs illustrate the Gibbs phenomenon for a 6 term Fourier partial sum. The 6 Fourier coefcients are used to reconstruct a 6 term Morlet wavelet series accurate to four decimal places. Figure 3: The absolute error of a Poisson wavelet reconstruction based on 6, 36, 64, 00 Fourier coefcients. Figure : The absolute error of a Morlet wavelet reconstruction based on 6, 36, 64, 00 Fourier coefcients. Figure 4: The absolute error of a Mexican hat wavelet reconstruction based on 6, 36, 64, 00 Fourier coefcients. ISSN: ISBN:

5 4 Conclusions The numerical results indicate that the inverse wavelet reconstruction method yields an accurate uniformly converging reconstruction approximation for a variety of wavelets. Current work underway includes a study of the technique for a broader spectrum of wavelets, reconstruction from series other than Fourier series, such as orthogonal polynomials or wavelets, comparison with other methods, an analytic estimation of error proof of convergence. Acknowledgements: This research was supported in part by a PSC-CUNY Research Award (grant No ). References AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '08), Harvard, Massachusetts, USA, March 4-6, 008 [] C. K. Chui, An Introduction to Wavelets, Academic Press, San Diego, 99. [] T. A. Driscoll, B. Fornberg, A Pade-based Algorithm for Overcoming the Gibbs Phenomenon, Numerical Algorithms, 6, 00, pp [3] A. Gelb, E. Tadmor, Detection of Edges in Spectral Data, Appl. Comp. Harmonic Anal., 7, 999, pp [4] A. Gelb, E. Tadmor, Detection of Edges in Spectral Data II. Nonlinear Enhancement, SIAM J. Numer. Anal., Vol. 38, No. 4, 000, pp [5] A. Gelb, E. Tadmor, Spectral Reconstruction of Piecewise Smooth Functions from their Discrete Data, Mathematical Modeling Numerical Analysis, 36:, 00, pp [6] A. Gelb, J. Tanner, Robust Reprojection Methods for the Resolution of the Gibbs Phenomenon., Applied Computational Harmonic Analysis, Vol. 0,, 006, pp [7] D. Gottlieb, C.-W. Shu, A. Solomonoff, H. Veven, On the Gibbs's Phenomenon I: Recovering Exponential Accuracy from the Fourier Partial Sum of a Nonperiodic Analytic Function, J. Comput. Appl. Math., 43, 99, pp [8] D. Gottlieb, E. Tadmor, Recovering Pointwise Values of Discontinuous Data within Spectral Accuracy, in: Progress Supercomputing in Computational Fluid Dynamics, Proceedings of a 984 U.S.-Israel Workshop, Progress in Scientic Computing, Vol. 6 (E. M. Murman S. S. Abarbanel eds.), Birkhauer, Boston, 985, pp [9] D. Gottlieb C.-W. Shu, On the Gibbs Phenomenon its Resolution, SIAM Rev., Vol. 39, 997, pp [0] D. Gottlieb C.-W. Shu, A General Theory for the Resolution of the Gibbs Phenomenon, in: Tricomi's Ideas Contemporary Applied Mathematics, National Italian Academy of Science, 997. [] N. Greene, On the Recovery of Piecewise Smooth Functions from their Integral Transforms Spectral Data, Ph.D. Dissertation, SUNY Stony Brook, 004. [] N. Greene, Fourier Series of Orthogonal Polynomials, in: Proceedings of the American Conference on Applied Mathematics, Cambridge, Massachusetts, USA, March 4-6, 008. [3] J.-H. Jung, B. D. Shizgal, Generalization of the Inverse Polynomial Reconstruction Method in the Resolution of the Gibbs Phenomenon, J. Comp. Appl. Math., v. 7, n., 004, pp.3-5. [4] J.-H. Jung, B. D. Shizgal, Inverse polynomial reconstruction of two dimensional Fourier images, J. Scientic Computing, v.5, n.3, 005, pp [5] B. D. Shizgal J.-H. Jung, Towards the resolution of the Gibbs Phenomena, J. Comput. Appl. Math., 6, 003, pp [6] R. Pasquetti, On Inverse methods for the Resolution of the Gibbs Phenomenon, Journal of Computational Applied Mathematics, Vol. 70, no., 004, pp [7] E. Tadmor J. Tanner, Adaptive Molliers - High Resolution Recovery of Piecewise Smooth Data from its Spectral Information, J. Foundations of Comp. Math., 00, [8] E. Tadmor J. Tanner, Adaptive Filters for Piecewise Smooth Spectral Data, IMA J. Numerical Anal., Vol. 5, 4, 005, pp ISSN: ISBN: