Applied and Computational Harmonic Analysis 7, (1999) Article ID acha , available online at

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1 Applied and Computational Harmonic Analysis 7, 0 35 (999) Article ID acha , available online at on Detection of Edges in Spectral Data Anne Gelb Department of Mathematics, Arizona State University, Tempe, Arizona ag@math.la.asu.edu and Eitan Tadmor Department of Mathematics, UCLA, Los Angeles, California tadmor@math.ucla.edu Communicated by Charles K. Chui Received January 8, 998; revised June, 998 We are interested in the detection of jump discontinuities in piecewise smooth functions which are realized by their spectral data. Specifically, given the Fourier coefficients, {fˆk a k ib k } k, we form the generalized conjugate partial sum S f x a k k sin kx b k cos kx. The classical conjugate partial sum, k S [ f ](x), corresponds to and it is known that log S f x converges to the jump function [ f ](x) : f ( x) f ( x); thus, log S f x tends to concentrate near the edges of f. The convergence, however, is at the unacceptably slow rate of order (/log ). To accelerate the convergence, thereby creating an effective edge detector, we introduce the so-called concentration factors, k, k. Our main result shows x that an arbitrary C [0, ] nondecreasing () satisfying / dxo leads x 3 to the summability kernel which admits the desired concentration property, S f xo 3 f x, with convergence rate, S f x Const log for x s away from the jump discontinuities. To improve over the slowly convergent conjugate Dirichlet kernel corresponding to the admissible x log, we demonstrate the examples of two families of concentration functions (depending on free /99 $30.00 Copyright 999 by Academic Press All rights of reproduction in any form reserved.

2 0 GELB AD TADMOR parameters p and ): the so-called Fourier factors, F x sin x, and polynomial Si factors, p (x) px p. These yield effective detectors of (one or more) edges, where both the location and the amplitude of the discontinuities are recovered. 999 Academic Press Key Words: Fourier expansion; conjugate partial sums; piecewise smoothness; concentration factors.. ITRODUCTIO Smooth functions can be accurately represented by their spectral data. For example, given the Fourier coefficients, fˆk a k ib k, the -term truncated Fourier expansion, S f x a k cos kx b k sin kx, (.) k0 provides a highly accurate representation for smooth f s. The situation is different, however, in the case of piecewise smooth functions, and experience has led to two complementing points of view. In the first approach, one sees the smooth pieces of f separated by edges of jump discontinuities. The straightforward Fourier expansion in this case experiences the Gibbs phenomenon: locally, S [ f ](x) suffers () oscillations in the neighborhoods of the jumps, and globally, there is a slow convergence throughout the smooth pieces. It is still possible to recover a piecewise smooth f from its spectral coefficients and to retain the superior spectral accuracy; such spectrally accurate recovery is obtained by filtering S [ f ](x) and could be carried out either on the Fourier side, e.g., [5, 9], or in physical space (consult [0,, 9] and the references therein). As an example of the latter, one introduces a C 0 (, ) bump function, B( x), such that B(0), and with D denoting the usual Dirichlet kernel of degree,, we set the mollifier ( x) : B( x) D ( x). Then, replacing S [ f ] with S f x yields a spectrally accurate approximation of f ( x) for all x s which are at least -away from the set of jump discontinuities [0]. Observe that x x is a two-sided mollifier supported on (, ) with spectrally small moments. In a series of works (reviewed in []), Gottlieb and Shu used one-sided mollifiers to recover a piecewise smooth fuptothe discontinuous edges. All these recovery procedures require a priori knowledge of the location of the underlying jump discontinuities. Thus detection of the edges in this approach remains a critical issue. In the second approach, one is directly interested in seeing the edges of f, edges which are viewed as being separated by pieces of smoothness. Detection of edges in this context is fundamental in a variety of computational algorithms, from spectrally accurate schemes for capturing shock discontinuities, e.g., [4, 8], to image compression (consult [, 6] and the references therein). Of course, wavelet expansions are particularly suitable for edge detection: one traces jump discontinuities by zooming through the dyadic scales (consult [5, 6, 6, 7] and the references therein). In this paper we address the question of edge detection in spectral data. We offer a

3 DETECTIO OF EDGES I SPECTRAL DATA 03 simple and effective procedure to detect edges, based on generalized conjugate partial sums of the form S f x a k k sin kx b k cos kx. k The starting point is the standard conjugate sum, S [ f ](x), corresponding to ( x). The classical result due to Lukács, e.g., [3, Section 4]; [0, Section II, Theorem 8.3], asserts that log S f x converges to the jump function f x : f x f x, and thus, log S f x tends to concentrate near the edges of f. The convergence, however, is at the unacceptably slow rate of order (/log ) (indeed, consult Fig..). To accelerate the convergence, thereby creating an effective edge detector, we introduce the so-called concentration factors, k, arbitrary C [0, ] nondecreasing ( x) satisfying k. Our main result shows that an x dx O x / 3 is admissible, in the sense that the corresponding generalized conjugate sum satisfies the concentration property S f x O f x. 3 To demonstrate our above arguments, we consider the following two examples (on [, ]): x sin, x 0, f a x 3x :sin, 0 x, x cosx f b x : sgn x, x 0, cos 5 x x sgn x, x 0. In both cases, f a ( x) and f b ( x) are recovered from their Fourier coefficients using the Fourier partial sums S [ f ](x). (both the continuous and the discrete cases are considered). The Gibbs phenomenon is depicted in Figs.. (the continuous case) and. (the discrete case). Figure.3 shows the reconstruction of a piecewise smooth function using the one-sided mollifier presented in []. Here f a ( x) and f b ( x) are recovered from their continuous

4 04 GELB AD TADMOR FIG... Fourier partial sum, S 40 [ f ](x), of f f a ( x) (left) and f f b ( x) (right). Fourier coefficients (Fig..3) and from their discrete Fourier coefficients (Fig..4). The recovery requires the location of the jump discontinuities. Finally, Fig..5 shows the detection of these jump discontinuities using our proposed generalized conjugate sum. In this case, we use the concentration function ( x) x. Both the location and the amplitude of the jump discontinuities, [ f a ](0) and [ f b ](/), are clearly identified. The paper is organized as follows. The so-called concentration property of the basic conjugate partial sum, S [ f ], is reviewed in Section. In Section 3 we devise our new, more general approach for locating jump discontinuities based on the concentration property of the generalized conjugate partial sums, S [ f ]. Here we provide a systematic study of concentration factors, k, and their improved resolution of the limiting jump function [ f ](x). Finally, in Section 4 we extend our theory to the analogous discrete case. f k FIG... Fourier partial sum of f f a ( x) (left) and f f b ( x) (right) using 40 discrete Fourier modes, x f x j e ikxj, which are based on the given gridvalues at the equidistant grid points x j. j

5 DETECTIO OF EDGES I SPECTRAL DATA 05 FIG..3. Reconstruction of piecewise continuous functions, f f a ( x) (left) and f f b ( x) (right), after filtering S 40 [ f ](x) with one-sided mollifier.. THE COJUGATE FOURIER PARTIAL SUM Let f ( x) bea-periodic piecewise smooth function, with a single jump discontinuity at x, whose associated jump value is defined as f : f f. (.) Given the Fourier coefficients of f( x), a k b k cos kt f t sin ktdt, (.) FIG..4. Reconstruction of piecewise continuous functions, f f a ( x) (left) and f f b ( x) (right), after filtering their discrete Fourier interpolant (of degree 40) with one-sided mollifier.

6 06 GELB AD TADMOR FIG..5. Detection of discontinuous edges using the conjugate sum, S f x i k sgnkfˆke ikx, using 0, 40, and 80 modes, and based on the first-degree polynomial concentration function, p (x) x. our goal is to identify the jump discontinuities, i.e., to locate the jump discontinuities and to accurately evaluate their associated jump values. The key to locating the discontinuities lies in the relationship between the conjugate Fourier partial sum and the jump discontinuities. The conjugate Fourier partial sum is given by S f x : a k sin kx b k cos kx. (.3) k Equivalently, S [ f ](x) can be written as S f x f D x f td x tdt, (.4) where D is the conjugate Dirichlet kernel cos D t t cos t sin kt k sin t. (.5) We recall that the support of the conjugate Fourier partial sum S [ f ](x) approaches the singular support of f(x)as 3, e.g., [3, Section 4; 0, Section, Theorem 8.3]. This will be referred to as the concentration property of S [ f ](x). To illustrate the concentration property of S [ f ](x), we offer the specific example of the saw-tooth function

7 DETECTIO OF EDGES I SPECTRAL DATA 07 x, x x x, x. In this case [ f]() and the conjugate Fourier partial sum is S x cos k x. k k The concentration property of S [ ](x) can be deduced from the following: ASSERTIO.. We have log k cos k x k if x 3 x : 0 otherwise. (.6) The proof is immediate. Let D ( y) denote the usual Dirichlet kernel D y cos ky. (.7) k0 Summation by parts yields Since D y cos k x k k, we have siny/ k D k x D k x k D k x kk D x D 0 x. k cos k x k k 6 sin and (.6) follows for x. Of course, for x we have x, log k cos kx k 3, as asserted. This special case of the saw-tooth function can be generalized to any piecewise smooth function, as told by

8 08 GELB AD TADMOR THEOREM.. (On the Concentration Property). Let f(x) be a -periodic piecewise smooth function with a single discontinuity at x. Then log S f, x, f x 3 f x x 0, otherwise. (.8) We shall offer two proofs for this theorem. The first approach is a straightforward extension of the concentration property of the saw-tooth function asserted in (.6), along the lines of [0, Section II, Theorem 8.3]. f Proof. Consider the function gx fx x, where ( x) is the sawtooth function with a -jump at x. Consequently, g( x) is a C 0 function which vanishes at x. By (.4), the conjugate sum of g( x) equals S gx gtd x tdt. (.9) Applying the standard upper bounds of D (t) in (.5), D t min, t, and the fact that g( x) is a continuous function with g() 0, we obtain S g gt D tdt / gt D tdt gt D tdt / / gtdt 4 / gt dt o olog olog. t By the definition of g( x), Assertion., and the previous estimate it follows that log S f x log S gx f log S x log olog f x o, and we are done.

9 DETECTIO OF EDGES I SPECTRAL DATA 09 Theorem. says that f log D x O 3 f x x. We point out that the scaled conjugate Dirichlet kernel, log D, is just one example for a broader class of admissible conjugate kernels which induce the concentration property. This brings us to the following: DEFIITIO. (Admissible Kernels). We say that a conjugate kernel, K, is admissible if it satisfies the following four properties: K is odd; (.0) lim K xdx 3 ; (.) 30 3 cos x K x C sin x R x, R L Const (.) 4 lim 3 sup R x 3 0, fixed 0. (.3) x0 Clearly, the scaled conjugate Dirichlet kernel, log D x, is admissible: indeed, in this case properties (3) and (4) hold with K R log D and C 0. Properties (3) and (4) are motivated by the fact that unlike the scaled Dirichlet kernel, the generalized conjugate kernels we shall meet later on, are not uniformly integrable. Our second proof of concentration property applies to general admissible kernels. Of course, the result applies to piecewise smooth functions with more than just a single discontinuity. We now need to specify our precise notion of piecewise smoothness, making ASSUMPTIO. (Piecewise Smoothness). f(x) is piecewise smooth in the sense of having finite number of jump discontinuities where [ f ](x) 0, and such s, f x t f x t f x t L 0,. (.4) Thus, piecewise smooth f s with smooth pieces which are Hölder of any order 0 will suffice. (To be precise, we may allow appropriate Besov regularity, yet in actual computation we cannot resolve but a finite number of discontinuities.)

10 0 GELB AD TADMOR THEOREM. (The Concentration Property Revisited). Let f (x) be a piecewise smooth function, (.4), and let J {} denote the set of its jump discontinuities. Consider the generalized conjugate partial sum S K f : f K f tk x tdt, where K is an admissible kernel satisfying properties () (4) in (.0) (.3). Then f x J, S K f x 3 f x J x 0 otherwise. (.5) Remark. ote that the convergence asserted in (.5) need not be uniform. Proof. Since by property () in (.0) K is odd, we can rewrite the corresponding conjugate partial sum S as S K f x 0 f x t f x tk tdt. (.6) Define the local variation j x (t) : [ f ](x) ( f ( x t) f ( x t)), and split (.6) into four contributions, S K f x 0 f x t f x tk tdt f x 0 K tdt C 0 j x t sin t cos tdt 0 j x tr tdt j x tr tdt : I II III IV. Property () in (.) yields that the first term approaches the jump [ f ](x), I f x 0 K tdt 3 f x, 3. By piecewise smoothness, j x (t)/sin t/ L [0, ] and by Riemann Lebesgue II O 0. Given an, we can find () such that sup j x t, and since R L 0, 3

11 DETECTIO OF EDGES I SPECTRAL DATA is bounded by (3) in (.), it follows that the third term III can be made as small as we please independently of, III 0 j x tr tdt R L sup j x t Const. 0, And finally, property (4) in (.3) implies that sup R t can be made arbitrarily small t0 for large enough, 0 (()), and hence IV j x tr tdt Const sup R t O 0. t0 3 Thus the convolution of f with any admissible kernel K satisfies the concentration property. The following example illustrates Theorem. for the conjugate Fourier partial sum, S D f x log S f x. In this case, K log D corresponds to the canonical conjugate kernel given in Theorem.. Clearly, it is an odd kernel with unit mass over (0, ), so () and () hold. The estimate D t min, t implies that K log D also satisfies properties (3) and (4); indeed with R x log D x we find R (t)dt [ / 0 / ] R (t)dt Const, yielding (3) in (.); and R t log D t, satisfying (4) in (.3). tlog We close this section with EXAMPLE.. Here 0 and [ f ](). x sin, x 0 f x 3x sin, 0 x. It is clear from Fig.. that S D f x log S f x does in fact locate the singularity point and approximate the jump value there. Furthermore, in agreement with Theorems. and., the numerical convergence rate is at the discontinuity point and away from it. In particular, the slow convergence is exhibited in Fig.. log both where 80 modes do not recover the correct amplitude of the jump, [ f ](0). The improvement of this slow logarithmic convergence rate will occupy our discussion in the remaining sections. ote that naive straightforward smoothing does not improve the

12 GELB AD TADMOR FIG... The conjugate Fourier partial sum S D f x log S f x computed with 40 modes. Here, f f a ( x) given in Example. experiences a single jump, [ f ](0) : before smoothing (left) and after smoothing (right). convergence rate. In fact, the resolution of the smoothed conjugate Dirichlet kernel at the discontinuity is less sharp, as shown in Fig.., where an exponential smoothing filter is used by premultiplying fˆk filters. 3 exp k k fˆk. The results are similar for other smoothing 3.. Introduction 3. COCETRATIO FACTORS Consider a piecewise smooth function f ( x) with a single discontinuity at x. We introduce a generalized conjugate partial sum of the form S f x k, a k sin kx b k cos kx. (3.) k Here { k, } are free summability parameters to be determined so that the concentration property similar to (.8) holds: S f x 3 f x x. (3.) For example, k, corresponds to the canonical conjugate Fourier partial sum log S f x log S f x. In this case, (3.) holds in view of Theorem.. It is clear We use the notation S K and S to indicate the dependence on both the concentration kernel K and the concentration factor k,. This abuse of notation will be clarified in Section 3.3.

13 DETECTIO OF EDGES I SPECTRAL DATA 3 that these s influence the convergence rate associated with the concentration property of S [ f ](x). We refer to as the concentration factors of S [ f ](x). As a preliminary step, we begin by estimating the Fourier coefficients to their leading order. Integration by parts yields a k k f sin k k, b k k f cos k k. (3.3) Substituting the leading order terms of (3.3) into (3.) yields S f x f k k, cos k x. (3.4) k Therefore, the desired concentration property of S [ f ](x) in (3.) amounts to ASSERTIO 3.. Let { k, } be the concentration factors with the corresponding generalized conjugate partial sum S f x k, a k sin kx b k cos kx. Then the concentration property (3.) requires k k, cos k x k k 3 x. (3.5) Before turning to our general discussion on concentration factors, we note the following. Remarks.. The scaled conjugate Fourier partial sum log S f x f log D corresponds to the concentration factors k,. We thus denote log D k, log (3.6) as the Dirichlet concentration factors and note that they are independent of k. In this case, Assertion. states that (3.5) holds with an error term of order log, yielding the concentration statement of Theorem. and in agreement with Assertion 3... As a consequence of the leading order expansion in (3.3), the highest accuracy that can be obtained in (3.5) for locating the jump discontinuity x is first order,. Faster convergence of (3.5) may be achieved by further expanding the Fourier coefficients in terms of higher derivatives. This is considered for the particular methods examined in [, 7] and is also suitable for our general method. Here, we are concerned with improving

14 4 GELB AD TADMOR the first-order convergence rate, and we note that higher orders can be handled in a similar manner. 3. The concentration factors to be determined,, must show an overall improved accuracy for (3.). More specifically, we seek concentration factors which, beyond improving the convergence rate, will lead to S [ f ](x) having better resolution of the singular support of f ( x). This will be clarified by the differences between the various concentration factors outlined below. 4. Although only functions with a single point of discontinuity are considered here, our results are easily extended to include any piecewise smooth functions (along the lines of Theorem.), as will be seen in Example Concentration Factors Determined by Regularization One possible approach to improving the convergence rate of (3.) is to (weakly) regularize the partial sums in (3.5) by defining the regularized indicator function x :, x 0, x. (3.7) Observe that ( x) has an even Fourier expansion in ( x ), whose Fourier partial sum is given by S xx cos k x sin k. (3.8) k k Comparing it with Assertion 3., we can identify the summation on the right of (3.8) with concentration factors of the form k, csink ; here we consider vanishing, which depends on a fixed free parameter. The scaling coefficient, c, should be k, determined so that the concentration characterization in (3.5) holds, 3. It k k follows that c /Si() with Si() denoting the usual Si sin x 0 dx. x In summary, we arrive at the family of concentration factors (depending on ) F k, Si sin k,. 3.9 We refer to these as the Fourier concentration factors, denoted { F k, }, since they are in fact (proportional to) the Fourier coefficients of ( x). For this choice of Fourier concentration factors, (3.5) holds with a convergence rate of order ( ). The results for Example. using the Fourier concentration factors, F k,, are shown in Fig. 3.. Compared with the concentration-free conjugate Dirichlet kernel in Fig.., the improved resolution of the discontinuity at x 0 is evident. In this context, we recall an alternative approach to locating jump discontinuities as suggested by Banerjee and Geer []. As described below, the method in [] is based on

15 DETECTIO OF EDGES I SPECTRAL DATA 5 FIG. 3.. Jump value obtained by applying the Fourier concentration factors, (3.9 ),, to Example.. The exact jump is [ f ](0). estimating the Gibbs overshoots which occur exactly at the points of discontinuity. We shall see that the method in [] in fact leads to a particular set of Fourier concentration factors. We briefly describe the method given in []. Starting with Fourier partial sum S fx size k0 where Si 0 follows that a k cos kx b k sin kx, it yields the familiar Gibbs overshoot at x, of lim S f S f Si f, 3 sin u u S f x S f x Si du.7898 accounts for 8% Gibbs overshoot. It f for x 3 0 otherwise. (3.0) Thus, the (scaled) differences of the Gibbs picks at x concentrate at the discontinuity. In [], the location of and an approximation of [ f ]() were recovered by direct evaluation of (3.0). How can this procedure based on (3.0) be interpreted within our general frame-

16 6 GELB AD TADMOR FIG. 3.. Jump value obtained by applying the Gibbs concentration factors to Example. with 0, 40, and 80 modes. The exact jump is [ f ](0). work? After inserting the leading order terms of the Fourier coefficients into (3.3), it follows that S f x S f x Si f Si k sin k cos kx k log, (3.) with an error term on the right of the form k k sin k log. Compared with the characterization of the concentration property in (3.5), one recognizes the summation on the right of (3.) as a generalized conjugate Fourier partial sum associated with the concentration factors k, sink/. In fact, these fall into the special category of Si Fourier concentration factors in (3.9 ), corresponding to. Thus the approach in [] given in (3.0) concurs with the so-called Gibbs concentration factors The results are depicted in Fig. 3.. G k, k sin Si. (3.) 3.3. Concentration Factors Revisited Bearing in mind the concentration factors determined thus far, we revisit (3.) to determine general criteria that will guarantee the concentration property (3.). We start by

17 DETECTIO OF EDGES I SPECTRAL DATA 7 considering the concentration factors k, k, where ( x) ( x) isaconcentration function which is yet to be determined. ote that we still allow ( x) ( x) to depend on. In the generic case, however, ( x) is independent of, (e.g., F ( x) sin(x) for the Fourier concentration factors in (3.9 )), and so we omit the subindex. We start by summing S f x : k a k k sin kx b k cos kx which leads to generalized conjugate kernels of the form f(t) k k x tdt, sin k (3.3) K t k kt. (3.4) sin k We ask ourselves when such kernels are admissible in the sense of satisfying the four properties outlined in Definition., so that by Theorem. the concentration property holds: S f x f K x 3 f x x. (3.5) In the language of Assertion 3., one focuses here on the Heaviside function fx H x : sgnx, where (3.5) boils down to (3.5), S H x H K k/ cos k x 3 k x, k x, H x : (3.6) x. Clearly, the K (t) are odd so property () holds; moreover, with the minimal requirement of bounded concentration factors, K L (0,) is bounded if K L (/,) is. amely we start with COROLLARY 3.. Consider the conjugate kernel k sin kt Const and hence k K t k kt, sin k

18 8 GELB AD TADMOR associated with bounded concentration factors, k. Then K is an admissible kernel (and hence the concentration property (3.5) holds), if the following conditions are fulfilled: lim 3 k k/ k k ; (3.7) 3 cos K t t C sin t R t, / R tdt Const.; (3.8) 4 lim 3 sup R t 0, fixed 0. (3.9) t0 ext, we provide easily checkable characterizations of properties () (4). We summarize our results (adding minimal requirements on the smoothness of the concentration function ( x)) in the following two assertions. The first deals with the total mass of the concentration kernel, K. ASSERTIO 3.. Assume that the concentration function (x) (x) C [0, ] satisfies x dx 3, x / j j O 0. (3.0) j 3 Then property () and hence () hold, i.e., lim 0 K tdt. 3 Remark. If x is integrable then the summation encountered in property () is, in x fact, the Riemann sum of k/ k k / k j0 j j And thus we find that if x L 0,, dx satisfies x x dx. x 0 0 x dx 3, (3.) x then property () in (3.7) holds. The (slight) refinement asserted in (3.0) extends to L -weak kernels which are excluded by (3.).

19 DETECTIO OF EDGES I SPECTRAL DATA 9 Proof. Set x j : j. By continuity, x j x x j x Summing such terms we find / x x / x j x dx x j x j dx x j x j xj x dx. / dx x j j j j k/ k k / x k j j log, and the result follows. Our next assertion, dealing with properties (3) (4), provides a sharp upper bound cos on the amplitude of the conjugate kernel, R t: K t t sin t. ASSERTIO 3.3. Consider the conjugate kernel K t k kt, with concentration function ( ) C [0, ]. Then the following estimate k sin holds: R cos t K t t sin t 3 C Const. t t. (3.) Remark. Thus, (3.) shows that if / Const. then both properties log (3) and (4) hold. Proof. 4 sin t k Twice summation by parts leads to the identity recall the notation x k : k x k sin kt x k x k sink t sin kt k sin t cos t x sin t cos t x k x k x k sink t k x sin t x x sin t sin t cos t x sin t cos t. (3.3)

20 0 GELB AD TADMOR By the C smoothness of ( )wefind K t C C t t. (3.4) To conclude the proof we consider the special example of ( x) x: the corresponding k sin kt and it coincides with the conjugate kernelwith k, k reads K x t k differentiated Dirichlet kernel in (.7), D t, K x t k k sin kt cos t sin t cos t sin t 4 sin t. 3.5 ow we decompose K t K t K x t K x t. The first difference on the right is a conjugate kernel associated with concentration function ( x) : ( x) ()x. Application of (3.4) to K (t) implies the upperbound cos asserted in (3.). Also, by (3.5), K t x t sin t does not exceed /t, and the result follows. We summarize our last two assertions, by stating our main THEOREM 3. (Main Theorem). Consider the conjugate kernel K t k sin kt associated with a C [0,] concentration function (x), such that Const log. Assume x dx O, (3.6) x / 3 j j O 0. (3.7) j 3 (i) The concentration property: K (t) is an admissible kernel, so that S [ f ](x) f K satisfies the concentration property,

21 DETECTIO OF EDGES I SPECTRAL DATA f f, at jump discontinuities x, S f x O f x 0, otherwise. 3 (3.8) (ii) The convergence rate: Assume, further, that f ( ) is piecewise C function. Then, at any C regularity point x of f, the vanishing rate of S [ f ](x) is upper bounded by S f x Const log. (3.9) Remarks.. One can relax the assumption of piecewise C regularity, requiring, instead, that at any fixed x away from the jump discontinuities, f () admits a local Zygmund regularity ( f ( x t) f ( x t))/t BV(0, ).. In the generic cases, (x) Const x. In these cases, the error estimate (3.9) shows that away from the jump discontinuities of f the conjugate kernel decay is at least first order, log, in agreement with the (log /) decay we found earlier for the Fourier concentration factors in (3.). We note, however, that (3.9) does not imply uniform convergence rate up to the jump discontinuities. Indeed, the polynomial concentration factors introduced in Section 3.4, p (x) px p, admit first-order convergence rate (and exhibit even faster convergence rate of order p p at selected gridpoints away from the jumps), yet they fail to maintain this rate at the proximity of the jumps. The further smoothness of p (x) at x 0 does not seem to improve the convergence rate beyond the first-order error bound stated in (3.9). Proof. We address the error estimate (3.9). The local smoothness of f(x) is measured by the modulus of continuity x (t) : f(x t) f(x t). Since K is odd we can rewrite the corresponding conjugate partial sum as S [ f ](x) ( / 0 / ) x (t)k (t)dt, and cos /t utilizing property (3) ofk in (3.8), K t R t, we sint/ decompose S f x 0 / x tk tdt / x t sint/ cos tdt / x tr tdt. (3.30) Since f has (more than) Lipschitz regularity at x, x (t) Const t. This, together with K (t) Const, implies that the first integral on the right of (3.30) does not exceed

22 GELB AD TADMOR 0 / x tk tdt Const. Moreover, f () is assumed to have C regularity at x. This implies the C smoothness of x (t)/t (as a function of t), and hence the (/ ) decay of its Fourier coefficients. It follows that the second integral on the right of (3.30) is upper bounded by / x t sint/ cos tdt Const. And finally, using Assertion 3.3 to upper bound the amplitude of R (t), we find that the third integral on the right of (3.30) does not exceed / x tr tdt Const / t t t t dt Const log. The last three bounds imply the convergence rate estimate (3.9). Remark. If ( x) is nondecreasing, then necessarily, Const log for (3.7) to hold, and in this case (3.7) is fulfilled. If, in addition, x x L 0,, then admissibility requires only the scaling condition 0 x dx. (3.3) x It is easy to see from the above discussion that D k,, F k, concentration factors: in the first case, the Fourier concentration factors x satisfy (3.7) log (and, in particular, G k, ) are admissible (and note that in this case x is only weak L so that we need the refinement of (3.0)); in x the second case of Fourier concentration factors, F x sin x satisfies (3.3). Si 3.4. Polynomial Concentration Factors Guided by the results of Theorem 3. we define a family of what we refer to as polynomial concentration factors, based on concentration functions, p ( x) px p. The first two members in this family yield first-degree polynomial concentration factors (( x) x), p k, k ; (3.3)

23 DETECTIO OF EDGES I SPECTRAL DATA 3 second-degree polynomial concentration factors (( x) x ), p k, k. (3.33) Clearly p ( x) px p are admissible by Theorem 3. and hence the concentration property holds. We note in passing that the generalized conjugate sums associated with the polynomial concentration factors, p, coincide with the differentiated Fourier partial sums, p S p f x p k p a k sin kx b k cos kx k The corresponding concentration property then reads p d p p p dx p S f x. p p p S p f x 3 f x p The special case p 0 was already referred to in the proof of Assertion 3.3, where we made use of the identity K x t D t. The corresponding concentration property, (3.34 p ) with p 0, goes back to Fejér, [0, Section III, Theorem 9.3] and to Golubov [9, 3] for higher p s. To gain better insight into their overall improved accuracy, we analyze the behavior of S p [H ](x) H K p ( x) for x s away from the assumed jump discontinuity of the Heaviside H x sgn x at x ; consult (3.5). To this end, we let : x p and rewrite the sum (3.5) corresponding to k, as S p H x cos k k sin cos sin, x. Substituting in the discrete values l l for l,...,, yields S p H x l l l sin cos l sin 0 l is even l l is odd. The uniform convergence is clearly depicted in Fig. 3.3 by the oscillatory behavior between the odd and even gridpoints. It is important to clarify that the convergence at

24 4 GELB AD TADMOR FIG Example. with 0, 40, and 80 modes. The exact jump is [ f ](0). Jump value obtained by applying the first-order polynomial concentration factors, p k,to x away from the point of discontinuity does not depend on the value of x, but rather on its distance from an x l with an odd or even index. It follows that the convergence rate for (3.) corresponding to the first-order polynomial factors, p k, is the same for all odd (even) x l, regardless of proximity to the points of discontinuity. For the second-order polynomial factors, p k, we rewrite (3.5) as S p H x k cos k k sin k k cos k sin cos k k cotk/ sin k k k sin k cos k, k x. Using the closed formulas sin k cos k k cos k sin, sin k k cos cos sin, sin

25 DETECTIO OF EDGES I SPECTRAL DATA 5 FIG Jump value obtained by applying the second order polynomial concentration factors to Example. with 0, 40, and 80 modes. The exact jump is [ f ](0). we substitute the discrete values l obtain S p H x l l cot l / and apply a fair amount of algebra to if l even if l odd. Thus the second-degree polynomial factors attain second-order convergence (but only at the even discretization points). It is also oscillatory, and due to the added error term cot l/ at the odd points, it is dependent on the proximity of the jump discontinuity to the discretized value of x l. This error implies that the convergence is worse near the points of discontinuity, and the lack of uniform convergence is depicted in Fig On a positive note, S p [ f ](x) 3 0 more rapidly outside the immediate proximity of the discontinuity point, which may be helpful in identifying jump discontinuities for functions TABLE 3. Error Comparison for Example. with 40 Modes D [ f ](x) k, F k, G k, p k, p k, at x E-0 6.5E-03.4E-0.E-0 at x E E-0 6.7E-0 ote. First row: Absolute error for [ f ](0). Second row: average error for [ f ](x 0) 0.

26 6 GELB AD TADMOR FIG Jump value obtained by different values of F k with and (left) and p k with p and p (right), when applied to Example 3. with 40 modes. The exact solution is f, and f x 0. with stronger variation, as will be seen in Example 3.. We note that there are higher order polynomial factors corresponding to admissible kernels K that may work as well. Table 3. compares different concentration factors for Example., with the first row showing the magnitude of error for [ f ](0) and the second row comparing the average error for [ f ](x 0). As expected, the worst case is with the Dirichlet concentration factors. The results are F comparable for k, and G k,. Overall, the polynomial concentration factors work better p than the Fourier concentration factors, and it is not surprising that k, produces a slower convergence rate averaged over x s 0, due to the contribution of order cot l/ which prevents uniform convergence near the point of discontinuity. Until now we have only discussed functions with one discontinuity. Example 3. demonstrates the detection of edges for a function with two discontinuities. EXAMPLE 3.. We consider x f x cos, x 3x, x cos 7x, x. Here,, and [ f ]( ) [ f ]( ). Figure 3.5 displays the results for Example 3. using different concentration factors. The polynomial concentration factors work better than the Fourier concentration factors, p and the fast convergence of S [ f ](x) for { k, } is more evident than in the first example.

27 DETECTIO OF EDGES I SPECTRAL DATA 7 TABLE 3. Absolute Error for Example 3. at x D k, F k, G k, p k, p k, E E-0 5.4E E-0.6E-0 3.E-0 6.0E E E-03.7E-03.5E-0 Table 3. compares the different concentration factors for [ f ]( ), while Table 3.3 compares the average error of [ f ](x, ). The tables indicate that the polynomial concentration factors yield better results than their Fourier counterparts. Here we see that p p k, yields better average accuracy away from the points of discontinuity than k,.in this case, the smooth pieces in Example 3. exhibit stronger variation than before, and p the faster convergence of S [ f ](x) corresponding to k, away from the discontinuities plays a more dominant role. We mention again that while the estimates above are at best first order, they can be improved to (/ ) by substituting the results of [ f ]() back into (3.3) and applying another integration by parts. Finally, we emphasize that the possibilities for are not exhausted and that other concentration factors may provide better results. 4. DISCRETE FOURIER EXPASIO Suppose we are given the discrete grid values f ( x j ) defined at the equidistant points, x j :( j )x, with x:. The discrete Fourier expansion approximation is given by T f x k cos kx k sin kx, k0 where the corresponding discrete Fourier coefficients based on those equidistant grid values are defined as TABLE 3.3 Average Error for Example 3. Away from the Discontinuities D k, F k, G k, p k, p k, E E-0 4.8E E-0 8.9E-0 4.5E-0.E-0

28 8 GELB AD TADMOR FIG. 4.. Jump values obtained by applying different continuous concentration factors to the discrete Fourier coefficients for Example 3. with 40 modes. The exact solution experiences two jumps f. k x f x j cos kx j, j 0 k k x f x j sin kx j, k. (4.) j The discrete conjugate Fourier partial sum is therefore T f x k sin kx k cos kx. (4.) k In the discrete case, every grid value experiences a jump discontinuity. The jumps that are of order (x) are acceptable, but the () jumps indicate a jump discontinuity in the underlying function f ( x). Hence, in the discrete case we identify a jump discontinuity at by its enclosed grid cell, [ x j, x j ], which is characterized by the asymptotic statement f x j f x j f x for j j : x j, x j x for other js j. (4.3) Of course, this asymptotic statement, (4.3), may serve as an edge detector based the given grid values, { f(x j )} j. We now seek alternative edge detectors based on the discrete Fourier coefficients, { k, k } k, analogous to our study of the continuous case in Section 3. As a starting point, we point out the inadequacy of the concentration factors studied in Section 3, k, in the present context of discrete Fourier expansion (4.). Figure 4. shows the results for the discrete data of Example 3.. The discrepancy in Figs. 3.5 and 4. clearly

29 DETECTIO OF EDGES I SPECTRAL DATA 9 indicates that the concentration factors determined in Section 3 are not applicable here. A separate (but closely related) study is required for discrete concentration factors. To analyze the discrete case, we follow our framework in Section 3. We introduce concentration factors k, and consider the (discrete) generalized conjugate sums T r f x : k, k sin kx k cos kx. (4.4) k Summing by parts the discrete Fourier coefficients we find x k sin kx sin kx/ j/ f x j f x j, j x k cos kx sin kx/ j/ f x j f x j. j Let j/ x j / denote the midpoint of the cell [x j, x j ] which encloses the discontinuity at x. Applying (4.3) to the discrete Fourier coefficients in (4.) gives x k sin kx/ fsin k j/ sin x, k and substituting (4.5) into (4.4) leads to x sin kx/ fcos k j/ x, (4.5) x T f x f k, sin kx/ cos k x j/ x. (4.6) k Observe that as x 3 0, the discrete conjugate sum T [ f ](x) approaches the corresponding continuous conjugate sum S [ f ](x). In fact, by comparing (4.6) with S [ f ](x) in (3.4), S f x f k, k cos k x, k we see that the concentration property of the discrete conjugate Fourier partial sum is a direct analogue of the continuous case. Of course, in the discrete case, we do not identify the exact location of the underlying discontinuity at x, but rather the location of the

30 30 GELB AD TADMOR discrete cell that encloses this discontinuity which is realized here in terms of its midpoint at x j/ (Figs. 4.3 and 4.4). We arrive at the following discrete analogue of our Theorem 3. for detecting edges in spectra of piecewise smooth functions. In this discrete context, piecewise smoothness refers to piecewise C functions; i.e., we refer to f s with finite number of jump discontinuities where [ f ](x) 0, such that (.4) is strengthened into f x t f x t f x t L 0,. (4.7) THEOREM 4.. Let f(x) be a piecewise smooth function, (4.7), and let J {} denote the set of its jump discontinuities. Given the discrete Fourier coefficients, { k i k } k, we consider the generalized discrete conjugate partial sum T f x k, k sin kx k cos kx (4.8) k corresponding to the discrete concentration factors k, k. If k, are related to admissible continuous concentration factors k, in (3.), k, sinkx/ kx/ then T [ f ](x) satisfies the concentration property k,, x, (4.9) T f x 3 f J x. (4.0) Furthermore, the direct analogue to the continuous case offers a more general result: THEOREM 4.. Const Consider a C [0, ] discrete concentration function (x) such that log. Then k, k are admissible and the concentration property is fulfilled, if the following conditions are met: T f x 3 f J x, x dx O ; (4.) sin x/ / 3 j/ j O 0. (4.) j 3

31 DETECTIO OF EDGES I SPECTRAL DATA 3 FIG. 4.. Jump value obtained by applying discrete Fourier and Gibbs concentration factors, F k (corresponding to and ), with 40 modes. The exact solutions exhibit the jump discontinuities [ f a ](0) (left) and f b (right). All of the continuous concentration factors { k, } from Section 3 can therefore be converted into discrete concentration factors { k, } up to a scaling factor of sin k x sin x :. Dirichlet concentration factors k D. Fourier concentration factors (Fig. 4.) k F x sin k k x log ; (4.3) x sin k kxsi sin k ; 4.4 FIG Jump value obtained by applying discrete first- and second-degree polynomial concentration factors with 40 modes. The exact solution exhibit the jump discontinuities [ f a ](0) (left) and f b (right).

32 3 GELB AD TADMOR FIG Jump value for Example 3. obtained by applying the first-order polynomial factors (4.6) (left) and second-order polynomial concentration factors (4.7) (right), using 0, 40, and 80 modes. The exact solution jumps at f. 3. Gibbs concentration factors (Fig. 4.) k G x k sin k sin kxsi ; (4.5) 4. First-order polynomial concentration factors (Fig. 4.3, Fig. 4.4) p k x sin k x ; (4.6) 5. Second-order polynomial concentration factors (Fig. 4.3, Fig. 4.4) k p 4k x sin k x. (4.7) We close this section noting that in the case of f f a ( x) in Example., Tables 4. and 4. indicate a comparable order of resolution for the different concentration factors, k F, k G, k p, and k p, both at the value at the point of discontinuity and at the average convergence away from the point of discontinuity. For f f b ( x) in Example 3., however, k p produces best average errors outside the discontinuities (at x ), and TABLE 4. Absolute Error for Example. at x 0 k D k F k G k p k p E-0.3E-0 9.8E-0 8.8E E-0.9E-0 5.0E-0 3.7E E-0.E-0.5E-0.6E-0

33 DETECTIO OF EDGES I SPECTRAL DATA 33 TABLE 4. Average Error for Example. Away from the Discontinuity, x 0 k D k F k G k p k p E E-0 6.7E E-0 5.4E-0.7E-0 3.5E-0 Fig. 4.4 shows faster convergence for { k p }, T [ f ](x) 3 0 away from the immediate proximity of these points of discontinuity. 5. COCLUDIG REMARKS The theorems provided in Sections 3 and 4 enable us to determine concentration factors for both continuous and discrete Fourier expansion coefficients that improve the overall accuracy of the concentration property of the conjugate Fourier partial sum. FIG. 5.. Jump value obtained by generalized conjugate partial sum, S 40 [ f a ](x), using various Fourier concentration functions F x sin x, with,.5 (top) and, 3 (bottom). The exact solutions Si exhibit the jump discontinuity [ f a ](0).

34 34 GELB AD TADMOR It is important to mention that the choice of an appropriate concentration factor depends on various factors. Consider the following cases: For the one-sided mollifier proposed in [], only the approximate jump location is required to reconstruct a piecewise continuous function, making k p and k p appealing choices due to their rapid convergence away from the discontinuities. Reconstruction methods in [] and [7] require exact knowledge of the jump locations, but in [7], for example, knowledge of the jump locations and the Fourier coefficients is enough to determine the jump discontinuities, implying that locating the jump discontinuities is more important than determining their corresponding amplitudes. This makes k p and k p poor choices because of the strong oscillations they cause near the discontinuities. For highly varying functions, we have seen that k p and k p display better results due to their rapid convergence away from the discontinuities. In the case of several discontinuities, k p and k p produce too many oscillations between the points of discontinuities unless there are sufficiently many modes to resolve the smooth pieces of f. Finally we note that the results for the Fourier concentration factors, F, F (with ), and the first-degree concentration function, k p, k p, bear close similarity which is not shared by the Gibbs concentration factors, k G, k G (corresponding to, with ). Indeed, the sensitivity of the Fourier concentration factors on the free parameter is clearly depicted in Fig. 5. and deserves further study in the future. ACKOWLEDGMETS This research was supported in part by SF Cooperative agreement o. CCR (A.G.) and by SF Grant DM and OR Grant J-076 (E.T.). REFERECES. L. Alvarez and J.-M. Morel, Formulation and computational aspects of image analysis, Acta umer. (994), S. Banerjee and J. Geer, Exponentially Accurate Approximations to Piecewise Smooth Periodic Functions, ICASE Report 95-7, ASA Langley Research Center, Bary, Treatise of Trigonometric Series, Macmillan, ew York, H. S. Carslaw, Introduction to the Theory of Fourier s Series and Integrals, Dover, ew York, I. Daubechies, Ten Lectures on Wavelets, CBMS-SF, R. DeVore and B. Lucier, Wavelets, Acta umer. 9 (99), K. S. Eckhoff, Accurate reconstructions of functions of finite regularity from truncated series expansions, Math. Comp. 64 (995), K. S. Eckhoff and C. E. Wasberg, On the numerical approximation of derivatives by a modified Fourier collocation method, Report 99, Department of Mathematics, University of Bergen, orway, B. I. Golubov, Determination of the jump of a function of bounded p-variation by its Fourier series, Math. otes (97), D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in Progress and Supercomputing in Computational Fluid Dynamics (Proceedings of a 984 U.S. Israel Workshop, Progress in Scientific Computing; (E. M. Murman and S. S. Abarbanel, Eds.), Vol. 6, pp , Birkhauser, Boston, 985.

35 DETECTIO OF EDGES I SPECTRAL DATA 35. D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (998), D. Gottlieb, C.-W. Shu, A. Solomonoff, and H. Vandeven, On the Gibbs phenomenon. I. Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43 (99), G. Kvernadze, Determination of the jump of a bounded function by its Fourier series, J. Approx. Theory 9 (998), Y. Maday, S. M. Ould Kaber, and E. Tadmor, Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. umer. Anal. 30 (993), A. Majda, J. McDonough, and S. Osher, The Fourier method for nonsmooth initial data, Math. Comp. 30 (978), S. Mallat, Multiresolution approximations and wavelets orthonormal bases of L (R), Trans. Amer. Math. Soc. 35 (989), Y. Meyers, Wavelets, Algorithms and Applications, SIAM, Philadelphia, E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. umer. Anal. 6 (989), H. Vandeven, Family of spectral filters for discontinuous problems, SIAM J. Sci. Comput. 48 (99), A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, UK, 959.