IAENG International Journal of Applied Mathematics, 44:1, IJAM_44_1_03 ______________________________________________________________________________________ Numerical Generation of Images for the Gibbs Phenomenon Near a Corner in the Plane Alia Khurram, Member, IAENG, and David W Kammler Abstract—In this paper we have studied the Gibbs ripples corner in the case where the function is doubly periodic. associated with an α-corner, i.e., the indicator function for the Helmberg’s approximation deletes all components that lie R2 smaller region of bounded by two rays from the origin that outside of a square in frequency space. In contarst, we intersect at the angle α, 0 < α < π. (Such corners cannot occur in the univariate case.) We have done this by removing from remove high frequency components by deleting from the the Fourier representation all components that lie outside of a Fourier representation all terms having an index that exceeds disk with radius σ ≥ 0. We have used a technique to rearrange some threshold. the approximating integrals in a form that facilitates efficient computation. We then produced two dimensional images for II. THE GIBBS PHENOMENON FOR THE HEAVISIDE STEP the Gibbs ripples that correspond to the corner angles α = R1 π=3, π=2, 2π=3, π. We observed that the maximum overshoot IN occurs near the corner. A suitably regular function f(x) on R has the Fourier representation Index Terms—Gibbs Phenomena, Fourier Transform, Bivari- Z ate Analysis. 1 f(x) = F (s)e2πisx ds ; (1) s=−∞ I. INTRODUCTION where the Fourier transform F is given by Z T the close of the nineteenth century the physicists A.A. 1 −2πisx Michelson and S.W. Stratton constructed a mechanical F (s) := f(x)e dx : (2) A −∞ harmonic analyzer that could plot graphs for trigonometric x= j j polynomials with up to 80 terms [11]. After producing If we suppress all frequencies s < σ, we obtain the low pass approximation many plots of partial sums for Fourier series of periodic Z σ functions with jump discontinuities, they observed that all 2πisx such approximations exhibit annoying “ripples” near every fσ(x) := F (s)e ds ; σ > 0 : (3) s=−σ point of discontinuity. J.W. Gibbs provided an informal We can use the unit box analysis of this phenomenon [4] which now bears his name ( [5], [6]. An earlier exposition was given by Wilbraham [7] 1 if −1=2 < s < 1=2 in 1841. rect(s) := (4) 0 otherwise, Gibbs ripples are produced by removing the high fre- quency components from the Fourier representation of a to remove the high frequency components from (1). Indeed, since ( function that has a jump discontinuity. If the function is ( ) s 1 if jsj < σ periodic, this involves truncating the corresponding Fourier rect = (5) series. If the function is aperiodic, this involves a similar 2σ 0 otherwise, truncation of the Fourier integral representation. This phe- we see that nomenon has been studied in several dimensions [2], [9]. An Z 1 ( ) evident manifestation of the phenomenon is a rippling effect s 2πisx fσ(x) = F (s) rect e ds : (6) which is undesirable in some applications. For example, in s=−∞ 2σ reconstruction of magnetic resonance imaging (MRI) data, The functions F (s) and rect(s=2σ) are the Fourier transforms methods are being developed to remove the Gibbs ringing of f(x) and 2σ sinc(2σx), respectively, so we can use the artifact that occurs at the boundaries of tissues [1], [8]. convolution rule for Fourier transforms to see that Moving from unvariate to bivariate functions, there are several important new features to consider. If the function has fσ(x) = f(x) ∗ 2σ sinc(2σx) ; a jump discontinuity across a smooth curve in its domain, one or equivalently might reasonably expect to produce Gibbs ripples analogous Z 1 − to those from the univariate case. A less predictable phe- sin[2πσ(x u)] fσ(x) = f(u) − du: (7) nomenon occurs when the function has a jump discontinuity u=−∞ π(x u) across a corner where two smooth curves in the domain We are interested in studying the behavior of such a low intersect at some angle α, 0 < α < π. A recent paper of pass approximation near a point of jump discontinuity, so we G. Helmberg [6] analyzes the Gibbs phenomenon at such a will replace f by the Heaviside step function 8 > Manuscript received July 24, 2013; revised November 30, 2013. <>1 if x > 0 A. Khurram is in the Mathematics Department at Adrian College, Adrian, MI, 49221 USA e-mail: ([email protected]). h(x) = >1=2 if x = 0 (8) David Kammler is professor emiritus at Southern Illinois University :> Carbondale, Carbondale, IL 62901. 0 if x < 0: (Advance online publication: 13 February 2014) IAENG International Journal of Applied Mathematics, 44:1, IJAM_44_1_03 ______________________________________________________________________________________ 1.2 where the bivariate Fourier transform is given by Z Z 1 1 1 −2πi(ux+vy) 0.8 F (u; v) := f(x; y)e dydx: x=−∞ y=−∞ 0.6 (14) 2 0.4 We will suppress all frequencies u; v outside the disk u + 2 2 0.2 v ≤ σ to obtain the low pass approximation ZZ 0 f (x; y) := F (u; v) e2πi(ux+vy) dvdu: (15) −0.2 σ −10 −5 0 5 10 u2+v2≤σ2 x We will use the unit diameter cylinder Fig. 1. The low pass approximation g(x) = hσ( 2σ ) for the Heaviside step (8) ( 1 if u2 + v2 < 1=4 cyl(u; v) := (16) 0 otherwise, The low pass approximation (7) for this function is Z 1 to write sin[2πσ(x − u)] ZZ ( ) hσ(x) = du: (9) u v 2πi(ux+vy) π(x − u) fσ(x; y) = F (u; v) cyl ; e dvdu : u=0 2σ 2σ We shall find it convenient to work with the dilate uv Z (17) 1 − x sin[π(x 2σu)] The radially symmetric function cyl(x; y) is the hσ( ) = − 2σdu 2σ u=0 π(x 2σu) Fourierp transform of the radially symmetric function Z 1 sin[π(x − u)] jinc( x2 + y2), where = du ; π(x − u) ( u=0 1 if r = 0 which is independent of σ. With this in mind, we define the jinc(r) := univariate Gibbs function J1(π r)=2r if r > 0, Z 1 x sin[π(x − u)] and where, J1 is the first order Bessel function [3], p.359. g(x) := h ( ) = du: (10) Z σ 2σ π(x − u) 1 u=0 −2πix ir sin 2πx J1(r) := e e dx; The oscillating integrand in (10) decreases like 1=u 0 as u ! 1. Thus the improper integral converges, but it u v Since the functions F (u; v) and cyl( 2σ ;p2σ ) are the Fourier converges much too slowly to evaluate by direct numerical transforms of f(x; y) and (2σ)2 jinc (2σ x2 + y2), respec- integration. We take x > 0 and use the known definite tively, we can use the two dimensional convolution rule to integral [10] p. 62 Z Z see that 0 1 p sin(πu) sin(πu) 1 2 2 2 du = du = ; (11) fσ(x; y) = f(x; y) ∗ (2σ) jinc (2σ x + y ); u=−∞ πu u=0 πu 2 or equivalently to rewrite (10) in the form ZZ p Z Z − 2 − 2 x − 1 − J1[2πσ (x p) + (y q) ] sin[π(x u)] sin[π(x u)] fσ(x; y) = σf(p; q) p dqdp: g(x) = − du + − du (x − p)2 + (y − q)2 u=0 π(x u) u=x π(x u) pq Z Z 1 x sin(πu) sin(πu) (18) = du + du We are interested in studying the behavior of such a low u=0Z πu u=0 πu 1 x pass approximation near “edges”and “corners” where f has = + sinc(u) du: 2 a jump discontinuity. With this in mind, we select a corner u=0 angle α with 0 < α < 2π and define the α-corner function (12) 8 We can use numerical quadrature to evaluate the remaining > >1 if −α=2 < arg(x; y) < α=2 integral in (12). > > ± <>1=2 if arg(x; y) = α=2 The Gibbs function g is shown in Fig. (1). The character- 6 Cα(x; y) := > and (x; y) = (0; 0) (19) istic Gibbs ripples are evident. The function g has maxima > > of 1:089; 1:033; 1:020; ::: at x = 1; 3; 5; ::: respectively and >α=2π if (x; y) = (0; 0) :> minima of 0:951, 0:974, 0:983; ::: at x = 2; 4; 6; :::, respec- 0 otherwise. tively. The maximum Gibbs overshoot is a bit less than 9% of the unit jump in h. Here arg(x; y) is the smaller angle between the ray joining (0; 0) to (x; y) and the positive x-axis as shown in Fig. 2. 2 2 C (x; y) R III. THE GIBBS PHENOMENON FOR AN α-CORNER IN R Thus α is the indicator function of a wedge from that has the corner angle α (with suitable average values on We will now extend the analysis from the previous section the edge and vertex, cp.(8).) to a two dimensional context. A suitably regular bivariate It can be seen from (19) and Fig. 2 that two such corner f R2 function on has the Fourier representation functions with vertex angles α and 2π − α can be combined Z 1 Z 1 2πi(ux+vy) as f(x; y) = F (u; v)e dvdu (13) − u=−∞ v=−∞ Cα(x; y) + C2π−α( x; y) = 1: (20) (Advance online publication: 13 February 2014) IAENG International Journal of Applied Mathematics, 44:1, IJAM_44_1_03 ______________________________________________________________________________________ IV.