DETECTION OF EDGES IN SPECTRAL DATA II. NONLINEAR ENHANCEMENT

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1 SIAM J. UMER. AAL. Vol. 38, o. 4, pp c 2 Society for Industrial and Applied Mathematics DETECTIO OF EDGES I SPECTRAL DATA II. OLIEAR EHACEMET AE GELB AD EITA TADMOR Abstract. We discuss a general framewor for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f]() := f(+) f( ). Our approach is based on two main aspects localization using appropriate concentration ernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration ernels, K ɛ( ), depending on the small scale ɛ. Itis shown thatodd ernels, properly scaled, and admissible (in the sense of having small W, - moments of order O(ɛ)) satisfy K ɛ f() =[f]() +O(ɛ), thus recovering both the location and amplitudes of all edges. As an eample we consider general concentration ernels of the form K σ (t) = σ(/) sin t to detect edges from the first /ɛ = spectral modes of piecewise smooth f s. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (999), pp. 35]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of eponential factors, σ ep ( ), with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where K ɛ f() [f](), and the smooth regions where K ɛ f = O(ɛ). umerical eamples demonstrate that by coupling concentration ernels with nonlinear enhancement one arrives at effective edge detectors. Key words. piecewise smoothness, concentration ernels, spectral epansions AMS subject classifications. 42A, 42A5, 65T PII. S Introduction. We discuss a general framewor for recovering edges from the spectral projections of piecewise smooth functions. Our approach for edge detection is based on two fundamental aspects localization to the neighborhood of the edges using appropriate concentration ernels and separation of scales by nonlinear enhancement. Both the locations and amplitudes of all edges are recovered. Let S f() denote the spectral projection of a piecewise smooth f. Given S f, one can accurately reconstruct f away from its discontinuous jumps, e.g., [], [4, sect. 2.], as well as up to the discontinuities []. In either case, an a priori nowledge of the location of the edges and their amplitudes is required. This issue was treated in recent literature; consult [], [5], [3], [5]. In [7], we unified several types of treatments as special cases of appropriate concentration ernels, specifically those discussed in [] and [3]. Here we improve on these results in both generality and simplicity. To this end, let [f]() :=f(+) f( ) denote the local jump function and let us consider a concentration ernel K ɛ ( ), depending on a small scale ɛ. It is shown that odd ernels, properly scaled, and admissible (in the sense of having small W, -moments of order O(ɛ); see eq. (2.5)), recover both the locations and the amplitudes of the jumps so that (.) K ɛ f() =[f]()+o(ɛ). Received by the editors July 9, 999; accepted for publication (in revised form) May 6, 2; published electronically October 3, 2. Department of Mathematics, Arizona State University, P.O. Bo 8784, Tempe, AZ (ag@math.la.asu.edu). This author s research was supported in part by the Sloan Foundation. Department of Mathematics, UCLA, Los Angeles, CA 995 (tadmor@math.ucla.edu). This author s research was supported by SF grant DMS and OR grant 4--J

2 39 AE GELB AD EITA TADMOR Thus, K ɛ tends to concentrate near the singular support of f. Differentiation of ɛ-supported mollifiers is one eample for local concentration ernels outlined in section In section 2 we also address the issue of detecting edges in global Fourier projections. Given the first /ɛ = Fourier modes, we see concentration ernels of the form K σ (t) = ( ) σ sin t. It is shown that if the concentration factors σ(ξ) ξµ(ξ) are normalized so that µ(ξ)dξ =, then K σ (t) is an admissible concentration ernel, K σ S f() [f](), and the following error estimate holds: π = µ ( ) ˆf ie i [f]() Const log. The nonperiodic case is studied in section 3. The analogous results for the Chebyshev case (consult Corollary 3.2) are written as π 2 ( ) µ ˆf T () [f]() Const log. The special cases of Fourier concentration factors σ α (ξ) sin αξ and σ p (ξ) =pξ p were considered earlier in [], [7], [9], [3], and [5]. Our general framewor motivates a new set of C -eponential concentration factors which yield superior localization properties away from the detected edges. While (.) refers to the asymptotic behavior of the concentration ernel as a function of the small parameter ɛ, it is essential to recover the eact locations of the edges of f for the accurate reconstruction of f. In section 4 we discuss another essential aspect of edge detection, namely, nonlinear enhancement. To this end, one introduces a critical threshold, J crit, for the amplitude of admissible edges and an enhancement eponent, p, to amplify the separation of scales in (.) between the edges, where K ɛ f() [f](), and the smooth regions where K ɛ f() =O(ɛ). Consider the enhanced ernel { Kɛ f if ɛ K ɛ,j [f]() = p/2 K ɛ f() p >J crit, otherwise. Clearly, with p large enough, one ends up with a sharp edge detector where K ɛ,j [f]() = at all but O(ɛ)-neighborhoods of the jump discontinuities. In this sense, the enhancement procedure actually pinpoints the location jump discontinuities, allowing an accurate reconstruction of f. The particular case p = 2 corresponds to the quadratic filter studied in [2], [22], in the special contet of concentration ernels based on localized mollifiers. 2. Edge detection by concentration ernels. 2.. Concentration ernels. We want to detect the edges in piecewise smooth functions. Assume that f( ) has jump discontinuities of the first ind with well-defined

3 EHACED DETECTIO OF EDGES I SPECTRAL DATA 39 one-sided limits, f(±) = lim ± f(), and let [f]() :=f(+) f( ) denote the local jump function. By piecewise smoothness we mean f( + t) f( t) [f]() (2.) F (t) := BV [,δ]. t In practice one encounters functions f() with finitely many jump discontinuities, and (2.) requires the differential of f() on each side of the discontinuity to have bounded variation. For eample, if f (±) are well defined (for finitely many jumps), then (2.) holds. We will detect the edges in such piecewise smooth f s using smooth concentration ernels, K(t) = K ɛ (t), depending on a small parameter ɛ. Such ernels are characterized by (2.2) K ɛ f() [f]() as ɛ. Thus the support of K ɛ f() tends to concentrate near the edges of f(). One recovers both the location of the jump discontinuities as well as their amplitudes. To guarantee the concentration property of K ɛ, we see odd ernels, (2.3) K ɛ ( t) = K ɛ (t), which are normalized so that (2.4) K ɛ (t)dt = +O(ɛ) t and which satisfy the main admissibility requirement (2.5) tk ɛ (t)φ(t)dt Const ɛ φ BV. t Remars.. For eample, if K ɛ (t) concentrates near the origin so that its first moment does not eceed (2.6) tk ɛ (t) dt Const ɛ, t then it is clearly admissible in the sense that (2.5) holds. We note that our admissibility condition also allows for more general oscillatory ernels, K ɛ (t), where (2.6) might fail, yet (2.5) is satisfied due to the cancellation effect of the oscillations. Consult (2.7) below. 2. Observe that the admissibility requirement (2.5) generalizes both properties P 3 and P 4 in the definition of admissible ernel [7, Def. 2.]. We state our main result as follows. Theorem 2.. Consider an odd ernel K ɛ (t), (2.3), normalized so that (2.4) holds, and satisfying the admissibility requirement (2.5). Then the ernel K ɛ (t) satisfies the concentration property (2.2) for all piecewise smooth f s, and the following error estimate holds: (2.7) K ɛ f() [f]() Const ɛ. Here and below we use BV [a, b] to denote the space of functions with bounded variation, endowed with the usual seminorm φ BV [a,b] := b a φ d.

4 392 AE GELB AD EITA TADMOR Proof. Using the fact that K ɛ (t) is odd, we have K ɛ f() = K ɛ (t)(f( + t) f( t))dt t f( + t) f( t) [f]() = tk ɛ (t) dt [f]() K ɛ (t)dt. t t t Applying (2.4) yields (2.8) K ɛ f() [f]() = tk ɛ (t)f (t)dt + O(ɛ). t By our assumption in (2.), F (t) isbv and it is therefore bounded. Consequently, in the particular case that the moment bound (2.6) holds, the first term on the right of (2.8) is of order O(ɛ), yielding K ɛ f() [f]() Const tk ɛ (t) dt + O(ɛ) =O(ɛ). In the general case, F (t) has bounded variation, and the admissibility requirement (2.5) implies that the first term on the right of (2.8) is of order O(ɛ), and we conclude K ɛ f() [f]() Const F (t) BV ɛ + O(ɛ) =O(ɛ) Eamples of concentration ernels Compactly supported ernels. Our first eample consists of concentration ernels which concentrate near the origin, so that (2.6) holds. We consider a standard mollifier, φ ɛ (t) := ɛ φ( t ɛ ), based on an even, compactly supported bump function, φ C(, ) with φ() =. We then set (2.9) K ɛ (t) = ( ) t ɛ φ φ ɛ ɛ(t). Clearly, K ɛ is an odd ernel satisfying the required normalization (2.4) K ɛ (t)dt = ( ) t φ dt = φ() =. t ɛ t ɛ In addition, its first moment is of order tk ɛ (t) dt = ɛ s φ (s) ds = O(ɛ), t s and hence (2.6) holds. Theorem 2. then implies the following. Corollary 2.2. Consider the odd ernel K ɛ (t) =φ ɛ(t), based on even φ C(, ) with φ( )=. Then K ɛ (t) satisfies the concentration property (2.2), and the following error estimate holds: (2.) φ ɛ f() =[f]()+o(ɛ).

5 EHACED DETECTIO OF EDGES I SPECTRAL DATA The conjugate Dirichlet ernel. The conjugate Dirichlet ernel, K (t) = log D (t), D (t) := sin t, is an eample of an oscillatory concentration ernel. Clearly, K (t) is an odd ernel. Moreover, the normalization (2.4) holds with ɛ log, π t= Finally, summing K (t)dt = 2 log D (t) = odd s = +O(ɛ), ɛ log. sin t = cos t 2 cos ( + 2 )t 2 sin t, 2 we find that the first moment of K = D (t)/ log does not eceed π tk (t) dt Const ɛ, ɛ = t= log, so that the requirement (2.6) is fulfilled. Theorem 2. then yields the classical result regarding the concentration of conjugate partial sums, [2, sect. 42], [23, sect. II, Thm. 8.3], (2.) log D f() =[f]()+o ( log We note in passing that in the case of the Dirichlet conjugate ernel, K (t) does not concentrate near the origin, but instead (2.6) is fulfilled thans to its uniformly small amplitude of order O(/ log ). The error, however, is only of logarithmic order; consult [7, sect. 2] Oscillatory ernels: General concentration factors. To accelerate the unacceptable logarithmically slow rate of the Dirichlet conjugate ernel in (2.), we consider a general form of odd concentration ernels (2.2) K σ (t) := ( ) σ sin t, based on concentration factors σ( ) which are yet to be determined. Clearly Kσ (t) is odd. et, for the normalization (2.4) we note that π t= K σ (t)dt = 2 odd σ( ) ). σ() d. In fact, the above Riemann s sum amounts to the midpoint quadrature, so that for C 2 [, ], one has σ(ξ) ξ (2.3) π t= K σ (t)dt = 2 odd σ( ) = ( ) σ(ξ) ξ dξ + O 2,

6 394 AE GELB AD EITA TADMOR and thus (2.4) holds for normalized concentration factors σ(ξ), (2.4) σ(ξ) dξ =. ξ Consult [7] for further refinement concerning the assumed regularity of σ( ) (We note that σ( ) is rescaled here with an additional factor of π compared to [7].) Finally, we address the admissibility requirement (2.5) (and in particular (2.6)). To this end, we proceed along the lines of [7, Assertion 3.3], utilizing the identity (abbreviating ξ = ) 2 2 sin(t/2)k(t) σ sin ( +)t (σ(ξ ) 2σ(ξ + )+σ(ξ +2 )) 2 sin(t/2) sin t (σ() σ(ξ )) 2 sin(t/2) +(σ(ξ sin t 2) σ(ξ )) 2 sin(t/2) + σ(ξ ) cos t ( 2 σ() cos + ) t 2 (2.5) =: I (t)+i 2 (t)+i 3 (t)+i 4 (t) I 5 (t). This leads to the corresponding decomposition of K σ (t) K σ (t) =R σ (t) σ() 2 cos ( + 2 )t sin t. 2 Here, R σ (t) consists of the first four terms on the right-hand side of (2.5), 4 j= I j(t)/2 sin(t/2), and it is easily verified that each one of these terms has a small first moment satisfying (2.6) (and consequently, (2.5) holds); i.e., (2.6) π t= tr(t) dt σ log Const σ C 2 [,]. For eample, using the standard bound sin(t)/2 sin(t/2) min{, /t}, the contribution corresponding to the first term, I (t), does not eceed π [ t I (t) t= 2 sin(t/2) dt ma σ 2 / ] π ( ) sin(t) 2 + log t= t=/ 2 sin(t/2) dt = O. Similar estimates hold for the remaining contributions of I 2,I 3, and I 4. In particular, since σ(ξ)/ξ is bounded, σ(/ ) O(/ ), and hence ti 4 (t)/2 sin(t/2) dt = O(/ ). Finally, the admissibility of the fifth term on the right of (2.5) is due to standard cancellation which guarantees that (2.5) holds: (2.7) σ() 2 π t= t cos ( + 2 )t sin t 2 φ(t)dt Const σ() φ BV. It is in this contet of spectral concentration ernels that admissibility requires the more intricate property of cancellation of oscillations. Summarizing (2.3), (2.6), and (2.7), we obtain as a corollary an improved version of the main result in [7, Thm. 3.] regarding spectral edge detection using concentration ernels, K σ (t). In

7 EHACED DETECTIO OF EDGES I SPECTRAL DATA 395 particular, since K σ (t) are -degree trigonometric polynomials, one detects the edges of the piecewise smooth function f() directly from its spectral projection S (f) := ˆf = e i, K σ f K σ S (f) =iπ = sgn()σ ( ) ˆf e i. Corollary 2.3. Consider the odd concentration ernel (2.2) K σ (t) = ( ) σ sin t, Assume that σ( ) is normalized so that (2.4) holds: σ(ξ) dξ =. ξ σ(ξ) ξ C 2 [, ]. Then K σ (t) admits the concentration property (2.2), and the following estimate holds: (2.8) K σ S (f) [f]() = Const log. Remar. One can rela the regularity on the concentration factor σ( ) [7]. Corollary 2.3 is a generalization of [7, Thm. 3.]; 2 in particular, the error estimate (2.8) is valid throughout the interval, including at the location of the jump discontinuities. Let us introduce few prototypical eamples of concentration factors σ( ) for the detection of edges from spectral data. In this contet we note that other detection methods of discontinuities in periodic spectral data can be found in the wors of Echoff [5], [6] and of Mhasar and Prestin, e.g., [5] and the references therein. We note that our results apply to the nonperiodic epansions as discussed in section below.. Trigonometric factors. We consider concentration factors of the form σ(ξ) = σ α (ξ) :=sinαξ/si(α) with the proper normalization Si(α) := α (sin η/η)dη. The edge detector introduced originally by Banerjee and Geer [] corresponds to σ π (ξ); the general case is found in [7, sect. 3.2]. 2. Polynomial factors. As a first eample consider σ(ξ) = ξ. In this case, K f = π S (f), and Corollary 2.3 recovers Fejér s result, [23, sect. III, Thm. 9.3], with the following error estimate: π (2.9) S (f) [f]() Const log. This is the first member of a whole family of polynomial concentration factors, e.g., [7, sect. 3.4], (2.2) σ(ξ) =σ p (ξ) :=pξ p, which correlate to concentration ernels satisfying (2.3), (2.4), and (2.5). For odd p s, K σp f = ( )[p/2] p p S (p) (f); for even p s, Kσp f = 2 We note the different rescaling here of σ( ) by an additional factor of π, compared with the formulation in [7, Thm. 3.].

8 396 AE GELB AD EITA TADMOR ( ) p/2 p p H S (p) (f), where H() =i sgn()e i. These edge detectors were introduced in [9] and were recently analyzed by Kvernadze in [3]. Corollary 2.3 yields iπ p = sgn() p ˆf e i [f]() Const log. The last error estimate is (essentially) first order. It is sharp. It was noted in [7, sect. 3.4], however, that σ p s with higher p s lead to faster convergence rates at selected interior points, bounded away from the singularities of f. This leads us to the net eample. 3. Eponential factors. Polynomial concentration factors (of odd degree) correspond to differentiation in physical space; trigonometric factors correspond to divided differences in the physical space consult the original derivation in []. Our main result stated in Corollary 2.3 provides us with the framewor of general concentration ernels which are not necessarily limited to a realization in the physical space. In particular, we see concentration factors, σ( ), which vanish at ξ =, to any prescribed order, (2.2) d j dξ j σ(ξ) ξ= = dj dξ j σ(ξ) =, j =,, 2,...,p. ξ= The higher p is, the more localized the corresponding concentration ernel, K σ ( ), becomes. Here is why. Evaluating K σ (t) at the equidistant points t l =2πl/, K σ ( ) 2πl = ( ) σ sin 2πl, we observe that K σ (t l)/ coincide with the l-discrete Fourier coefficient of σ( ); since σ(ξ) and its first p-derivatives vanish at both ends, ξ =,, there is a rapid decay of its (discrete) Fourier coefficients, ˆσ l Const l p, t= π K(t σ l ) Const σ C p+ [,] (t l ) p. Thus, for t away from the origin, K σ (t) is rapidly decaying for large enough s. Moreover, we claim an increasing number of moments of K σ ( ) vanish. To this end we consider the odd moments of K σ ( ) (its even moments vanish, of course). With ξ = / we find ( π ( ) ) (2.22) t j σ sin t dt = 2 j = ±2 j π ξ= τ= σ(ξ) τ j σ(ξ ) sin(ξ τ)dτ d j π dξ j cos(ξτ)dτ dξ, τ= j odd.

9 EHACED DETECTIO OF EDGES I SPECTRAL DATA 397 Integrate by parts; respectively, sum by parts the summation on the right of (2.22). Thans to (2.2) the boundary terms vanish and we have π t= π t j K(t)dt σ j = j ξ= d j dξ sin(πξ) σ(ξ) dξ j ξ d p j dξ p j ( ξ d j ) sin(πξ) ξ= dξ j σ(ξ) (π) j < p, p odd. ( dξ = O p j As an eample, we consider the eponential concentration factors ( ) σ ep (2.23) (ξ) =Const ξe αξ(ξ ), Const = ep dη αη(η ) normalized so that ξ= σep (ξ)/ξdξ =. Here, the C [, ] concentration factor σ ep (ξ) vanishes eponentially at both ends; ξ =, so that (2.2) holds for all p s. Figures 3 and 4 confirm the improved localization of these eponential concentration factors. 4. Band pass filter. Bauer [3] has considered a family of what he termed as band pass filter, η( ), supported in the range of middle frequencies, say, suppη [/4, 3/4]. We note in passing that these are special cases of p-order admissible concentration factors (see (2.2)), although the normalization used in [3, eq. (.35)], η()/ sin(π)d =, prevented the recovery of the amplitude of the jumps. To demonstrate the detection of edges by the concentration factors outlined above, consider the following two eamples of discontinuous f s (defined on [ π, π]): ( ) f a () := sgn cos 2 (2+sgn), f b () := p ), cos( 2 sgn( π 2 )), <, cos( sgn( π 2 )), >. In both cases, f a () and f b () are recovered from their Fourier coefficients using the Fourier partial sums S [f](), and we wish to recover their jump discontinuities 2, =, ± 2, = ± π 2, [f a ]() = [f b ]() = else, else. Figures and 2 demonstrate the use of trigonometric and polynomial concentration factors for the detection of edges from Fourier spectral data. As noted in [, sect. 3.4], polynomial factors of higher degree yield improved results away from the jump discontinuities. Indeed, the corresponding concentration ernels, K σp ( ), have additional vanishing moments. In the limit, one arrives at eponential factors, K σep ( ); Figures 3 and 4 demonstrate the edge detection of these factors in Fourier epansions S f a () and S f b (). The improved localization is evident, due to the faster convergence rate in the smooth parts of f s. In particular, the superiority of the eponential factors is illustrated in the figures on the left, when compared with the first-order accurate polynomial concentration factor, σ p= (ξ) = ξ. At the same time, Gibbs oscillations can be noticed in the vicinity of the jump discontinuities.

10 398 AE GELB AD EITA TADMOR.3 [f]() 2. [f]() Fig.. Trigonometric concentration factor σ(ξ) = sin ξ for (left) fa(), where the eact jump Si() value is [f]() = 2 and (right) f b (), where the eact jump values are [f](± π 2 )=± 2..3 [f]() 2. [f]() Fig. 2. Jump value obtained by the polynomial concentration factor σ p= (ξ) =ξ for (left) f a() and (right) f b () Fig. 3. Edge detection using the eponential concentration factor σ(ξ) = 3ep( 6ξ(ξ ) ) (left) σ ep vs. σ p= for S 4 f a() and (right) σ ep for S f a() with =2, 4, 8 modes.

11 EHACED DETECTIO OF EDGES I SPECTRAL DATA σ p σ e Fig. 4. Edge detection using the eponential concentration factor σ(ξ) = 3ep( 6ξ(ξ ) ) (left) σ ep vs. σ p= for S 4 f b () and (right) σ ep for S f b () with =2, 4, 8 modes. 3. Edge detection in nonperiodic projections. Consider a piecewise smooth f( ). To simplify our presentation, we assume f eperiences a single jump discontinuity at = c. The localization property of the appropriate concentration ernel in the presence of a single jump applies to the case with finitely many jump discontinuities. We begin with an alternative derivation of our results for the periodic case. 3.. Revisiting the periodic case. Ifa2π-periodic f( ) eperiences a single jump, [f](c), then it dictates the Fourier coefficients decay [4],[23], ( 2 ˆf =[f](c) e ic 2πi + O To etract information about the location of the jump from the phase of the leading term, we eamine the special concentration ernel, K σ with σ(ξ) =ξ, where Kξ S (f) = π S (f), (3.) π S (f) = π = i {[f](c) ei( c) 2πi ( + O =[f](c) { ( )} ( ) log (3.2) +O e i( c) =[f]()+o. 2 Here we used the concentration property of the Dirichlet ernel localized at = c: ( ) O, c, e i( c) c = 2 =, = c. The same property applies to the class of concentration factors, σ(ξ) := ξµ(ξ), such that (2.4) holds, µ(ξ)dξ =, ( ) O, c, 2 = µ ( ) e i( c) = 2 ). )} c, = c.

12 4 AE GELB AD EITA TADMOR It then follows that the corresponding K σ ernel, so that K σ f() [f]() onperiodic epansions. in (2.2) is an admissible concentration General Jacobi epansions. We begin with the Jacobi epansion of a piecewise smooth f( ), (3.3) S (f) = ˆf P (), ˆf = = f()ω()p ()d. Here P () are the Jacobi polynomials the eigenfunctions of the singular Sturm Liouville problem (3.4) (( 2 )ω()p ()) = λ ω()p (), with corresponding eigenvalues λ = λ (α) = ( + 2α + ). Different families of Jacobi polynomials are associated with different weight functions ω() ω α () := ( 2 ) α. To simplify the computations, we assume that the P s are normalized so that P () ω =. As in the periodic case, integration by parts (against (3.4)) shows that a single jump discontinuity, [f](c), dictates the decay of the Jacobi coefficients, (3.5) ˆf = f()(( 2 )ω()p λ ()) d =[f](c) ( ) ( c 2 )ω(c)p λ (c)+o λ 2. To etract information about the location of the jump, we consider the conjugate sum of the form π 2 ( ) (3.6) µ ˆf P () =[f](c) π 2 ( c 2 )ω(c) ( ){ ( )} µ + O λ λ 2 P (c)p (), corresponding to concentration factors σ(ξ) = ξµ(ξ). We shall focus our attention on the particular case µ(ξ), (3.7) π 2 S (f) () = π 2 ˆf P () =[f](c) π 2 ( c 2 )ω(c) { ( )} + O λ λ 2 P (c)p (). This is the nonperiodic analogue of the Fourier concentration ernel K ξ f with the additional prefactor weight of 2. We want to quantify the localization property of the last summation. To this end we note that if {P (α) ()} are the Jacobi polynomials w.r.t. the weight function

13 EHACED DETECTIO OF EDGES I SPECTRAL DATA 4 ω α (), then {P ()} are the Jacobi polynomials w.r.t. the modified weight function ω β () =( 2 )ω α () with β := α + : indeed, their ω β -orthogonality follows from integration by parts of (3.4) against P (α). Thus, P (α) () =C,β P (β), β = α +. The coefficients C,β are determined by normalization, where by using (3.4) once more we find = P (α) 2 ω α = λ (( 2 )ω α ()P (α) ) P (α) ()d = C2,β P (β) λ 2 ω β, and hence we set C,β = λ, so that {P (β) } is the orthonormal family w.r.t. ω β weight. Inserted into the leading term of (3.7), we end up with a Jacobi ernel associated with weight function ω β () =( 2 ) β, π 2 S (f) () [f](c) π 2 ω β (c) =[f](c) π 2 ω β (c) C 2,β λ P (β) (β) (c)p () P (β) (β) (c)p (). We rewrite this as π 2 S (f) () [f](c) π 2 ω β (c) (3.8) K (c, ). By virtue of the Christoffel Darbou formula, e.g., [9, Thm ], the ernel K (c, ) is given by K (c, ) = P (β) (β) (β) (β) (3.9) ()P (c) P (c)p (), c 2, and it remains to quantify the concentration property of K (c, ). To this end we use the asymptotic behavior of P (β) which is stated as3 ( ) (3.) P (β) 2 β+/2 2 () min, Const = πω β+/2 () πω β+/2 ( ), where := sgn() min{, Const/} denotes the separation between the interior and boundary regions. Using this to upper bound K (c, ) in (3.9), we find (3.) K (c, ) π ω β+/2 (c )ω β+/2 ( ) c, c. 3 The first term on the right of (3.) follows from the classical asymptotic formula, e.g., [9, Thm. 2..4], which tells us the behavior of the L 2 (β) ω-normalized P () at the interior = cos θ ω β+/2 ()P (β) () 2/π cos{θ + γ(θ)}, = cos θ. The second term on the right of (3.) reflects the fact that as approaches the ±-boundaries, the L 2 (β) ω-normalized P () approaches its maimal value, e.g., [9, eqs. (4.7.3), (4.7.5)] P (β) () = ( +2λ ) Γ( + )( + λ) π2 2λ Γ( +2λ) Γ(λ) Γ( +2λ)( + λ) λ, λ = β +/2.!

14 42 AE GELB AD EITA TADMOR The upper bound on the right is in fact the leading term in the asymptotics of K (c, ) for large s as long as < c<, e.g., [9, sect. 3.4]. Similarly, the behavior at = c yields (3.2) K (c, c) + Const πω β+/2 (c), <c<. The desired concentration property now follows, similar to the localization of the periodic Dirichlet ernel D ( c)/ in (3.3). We restrict our attention to interior jumps, <c<, so that for large enough, c = c, and (3.), (3.2), and (3.8) yield (3.3) ( ) π ωα+/2 (c) 2 ω β (c) O K (c, ) ωα+/2 ( ) α+/2 c, c,, = c. We summarize by stating the following. Corollary 3.. Let S (f) denote the truncated Jacobi epansion (3.3) of a piecewise smooth f, associated with a weight function ω α =( 2 ) α, <α. Then π 2 S (f) ()/ admits the concentration property π 2 (3.4) S (f) () [f]() Const log ( 2 ), α/2+/4 +Const 2 << Const 2. It is instructive to eamine the above discussion for the special case of Chebyshev epansion corresponding to α = 2, S (f)() = ˆf T (), ˆf = 2 π = = f()t () d. 2 (Observe that ecept for Chebyshev epansion, the concentration bound (3.3) deteriorates as we approach the boundaries, depending on whether for α> 2 or c for α< 2.) The conjugate sum corresponding to (3.8) reads π 2 S (f) () [f](c) π 2 π 2 c2 T (c) 2 T (). In this case, we can sum the corresponding Chebyshev ernel: setting = cos θ and <c= cos η<wefind (3.5) π 2 S (f) () [f](c) 2 =[f](c) sin(η) sin(θ) = [ ] D (θ η) D (θ + η) = O( ), c, [f](c), = c.

15 EHACED DETECTIO OF EDGES I SPECTRAL DATA Chebyshev epansion. Our discussion above on edge detection in the nonperiodic epansions is based on epansion of the Jacobi coefficients to their leading order in (3.5). More precise information is obtained using the general framewor introduced in the main theorem, Theorem 2.. Corollary 3.2. Let f( ) be a piecewise smooth function with Chebyshev epansion S f() ˆf()T (). Consider the concentration factors, σ(ξ) :=ξµ(ξ), with µ( ) normalized so that µ(ξ)dξ =, µ(ξ) C 2 [, ]. Then K σ (t) f(cosθ) admits the concentration property (2.2), and the following estimate holds: (3.6) π 2 ( ) µ ˆf()T () [f]() = Const log. Proof. With a piecewise smooth f() defined over the interval [, ] we utilize the usual Chebyshev transformation = cos θ, θ π. We consider the even etension f(cos θ), π θ π. Using Theorem 2. along the lines of Corollary 2.3, we find that the odd concentration ernel, K σ (t), recovers the jumps of f(cos θ), i.e., ( ) µ log sin(t) f(cos θ) [f](cos θ) Const. With T () = sin θ/ 2, a straightforward computation shows the sum on the left equals ( ) σ sin(t) f(cos θ) = π ( ) µ ˆf() sin(θ) = π 2 ( ) µ ˆf()T () and the result follows. We turn to numerical eamples. The following tables summarize our results for the edge detection in Legendre epansion, corresponding to α =, and in Chebyshev epansion, corresponding to α = /2. Scaled to the unit interval [, ], we consider f a ( π ) and f b( π ). The results confirm the linear convergence rate stated in Corollary 3., both away from the jumps consult Tables and 2 as well as at the jump itself (Table 3). We note that the critical threshold must be very high for =2, 4 to eliminate the artificial jumps. This indicates that 4 nodes are not enough to resolve the jumps of f b () in either the Chebyshev or Legendre case. It is clear from Tables and 2 that convergence is nonuniform at the boundaries. We have observed in our numerical eperiments that the edge detector, π 2 S [f] (), eperiences larger oscillations near the boundaries which do affect the linear convergence rate there. In this contet we note the dependence of the error bounds on the smoothness of f( 2 ). The first-order convergence is reconfirmed, in Table 4 below, when measuring the L -error away from the jumps discontinuities (and up to the boundaries).

16 44 AE GELB AD EITA TADMOR Table Pointwise error estimate π 2 / S (f a) (/π) [f a](/π) away from the jump discontinuity at =. Legendre epansion Chebyshev epansion =.998 =.985 =.5 =.75 =.998 =.985 =.5 = E-2 7.E E-2 3.8E-2.6.3E-2 2.E-2 5.9E E E-3 4.E-3 3.E-2 5.6E-3 5.3E-3 8.7E-3 3.3E-2.6E-2 Table 2 Pointwise error estimate π 2 / S (f b ) (/π) [f b ](/π) away from the jump discontinuities at = ± 2. Legendre epansion Chebyshev epansion =.998 =.985 =.5 =.75 =.998 =.985 =.5 = E-2 7.4E-2 5.4E E-2.5E-2 9.8E E-2 7.E-4.8.E-2.3E-2 4.5E E-3 7.2E-3 3.9E-4.2.8E-3 4.6E-3 2.3E-2.2 Table 3 Pointwise error estimate π 2 / S (f) (c) [f](c) at the point(s) of discontinuity, = c. Legendre Chebyshev f a() f b ( π/2) f b (π/2) f a() f b ( π/2) f b (π/2) 4.9E-2 8.5E-2.87 E-2 5.3E E-2 8.4E-3 3.9E-2. E-2 4.7E-2 2.5E-2 6.3E-2.7E-2 5.5E-2 8.7E-3.E-2 2.2E-2 Table 4 L [, ] {c} error estimate π 2 / S (f) () [f]() away from discontinuities. Legendre Chebyshev f = f a f = f b f = f a f = f b E E E-2 6.8E-2 4.6E-2 6.9E-2 4. onlinear enhancement. The detection of edges in Theorem 2. is based on separation of scales. Thus, consider for eample a piecewise smooth f with finitely many jump discontinuities at = c,c 2,...,c n.ifk ɛ is an admissible concentration ernel, then K ɛ f() for away from these jumps, whereas at = c j, K ɛ f() =cj O(), (4.) K ɛ f() = { O(ɛ), c,c 2,..., [f](c j ), = c j. The last statement refers to the asymptotic behavior of the concentration ernel as a function of the small parameter ɛ. In this section we outline a new, nonlinear enhancement procedure, which is easily implemented to pinpoint finitely many edges in piecewise smooth f s. To this end we enhance the separated scales in (4.) by considering (4.2) ɛ p 2 (Kɛ f()) p = { p Const ɛ 2, c,c 2,..., ([f](c j )) p ɛ p 2, = c,c 2,...

17 EHACED DETECTIO OF EDGES I SPECTRAL DATA 45 By increasing the eponent p>, we enhance the separation between the vanishing scale at the points of smoothness (of order O(ɛ p 2 )) and the growing scale at the jumps (of order O(ɛ p 2 )). et, one must introduce a critical threshold which will eliminate all the unacceptable jumps. Only those edges with amplitudes larger than the critical threshold, [f]() >Jcrit /p ɛ, will be detected. Thus J = Jcrit is a measure which defines the small scale in our computation of edge detection. We note that J = J crit is data dependent and is typically related to the variation of the smooth part of f. Given this critical threshold, we form our enhanced concentration ernel K ɛ,j [f] (4.3) { Kɛ f() if ɛ K ɛ,j [f]() = p 2 K ɛ f() p >J crit, otherwise. Clearly, with p large enough, one ends up with a sharp edge detector where K ɛ,j [f]() = at all but O(ɛ) neighborhoods of the jumps = c,c 2,... In practical applications, a moderate enhancement eponent, p 5, will suffice. We consider two eamples.. The quadratic filter. Consider the peaed concentration ernel (2.9) K ɛ (t) = φ ɛ(t). Then, with p = 2, one finds the so-called quadratic filter [2], [22], where (4.4) (K ɛ f()) 2 =(φ ɛ f()) 2 [f] 2 (). 2. Enhanced spectral concentration ernels. We apply the procedure of nonlinear enhancement in conjunction with spectral concentration ernels K ɛ = K σ (t) := σ( ) sin t by considering the corresponding enhanced spectral concentration ernel { K,J σ K σ = S f() if ɛ p 2 K σ S () p >J crit, otherwise. The enhanced spectral concentration ernel depends on four ingredients which are at our disposal: The number of modes, ; The enhancement eponent, p; The critical threshold, J; The concentration factor, σ(ξ). Figures 5 and 6 demonstrate the enhancement procedure to the spectral detection of edges depicted earlier in the corresponding Figures 2 and. We conclude with nonperiodic eamples. In Figure 7 we show the detection of a single edge in f a (/π) from its Legendre epansion, ( ˆf a ) P (). The detection in Chebyshev epansion is shown in Figure 8 for f b (/π). In both cases we used an enhancement factor p = 2 and a critical threshold J =5. 5. Concluding remars. Accurate reconstruction of piecewise smooth functions from their spectral projections is plausible only when the location (and amplitude) of the underlying jump discontinuities are nown; consult [], [7], [5], [6], [], [6], [2], [], and references therein. Theorem 2., and its Corollaries 2.3 and 3., provide the general framewor for the detection of edges from spectral data, in both periodic and nonperiodic cases. The detection is based on admissible concentration ernels which include as particular cases classical eamples of Fejér as well as additional eamples in recent literature

18 46 AE GELB AD EITA TADMOR.3 [f]() J=5 2. [f]() -.2. J = 7,9 -.7 J = 7,9. J= Fig. 5. Jump value obtained by applying the polynomial concentration factor σ(ξ) =ξ with =4with enhancement eponent p =2for (a) f a(), where the eact jump value is [f]() = 2 and (b) f b (), where the eact jump values are [f](± π 2 )=± 2..3 [f]() J=5 2. [f]() -.2. J=7, J = 7,9. J= Fig. 6. Jump value obtained by applying the trigonometric concentration factor σ (ξ) = sin ξ Si() with =4modes and enhancement eponent p =2for (left) f a(), where the eact jump value is [f]() = 2 and (right) f b (), where the eact jump values are [f](± π 2 )=± 2. [f]() [f]() Fig. 7. Detection of edges in Legendre epansion of f a(/π) with eact jump value is [f a]() = 2 (left) before and (right) after enhancement with p =2and J =5.

19 EHACED DETECTIO OF EDGES I SPECTRAL DATA 47.5 [f]().5 [f]() Fig. 8. Detection of edges in Chebyshev epansion of f b (/π) with eact jump value is [f b ](± 2 )=± 2 (left) before and (right) after enhancement with p =2and J =5. [], [9], [3]. In particular, we introduce here a new family of eponential concentration ernels, (2.23), with a superior convergence rate away from the edges. A linear convergence rate is observed near the detected edges. We also introduce a nonlinear enhancement (4.3) procedure which enables one to pinpoint edges with amplitude larger than a critical threshold. Recently, the edge detection and enhancement method was applied to nonlinear conservation laws [2], [8] as a postprocessing tool to improve the overall convergence rate of the spectral viscosity solution. Since the edge detection occurs only at the postprocessing stage, very little cost is added to the procedure yet the results are dramatically improved. Future applications, in both one- and several-space dimensions, will also include image processing, where edge detection is needed to denoise the contamination by the O()-Gibbs oscillations in the neighborhoods of the undetected edges. REFERECES [].S. Banerjee and J. Geer, Eponential Approimations Using Fourier Series Partial Sums, ICASE Report 97-56, ASA Langley Research Center, Hampton, VA, 997. [2]. Bary, Treatise of Trigonometric Series, The Macmillan Company, ew Yor, 964. [3] R. Bauer, Band Filters for Determining Shoc Locations, Ph.D. thesis, Department of Applied Mathematics, Brown University, Providence, RI, 995. [4] H.S. Carslaw, Introduction to the Theory of Fourier s Series and Integrals, Dover, ew Yor, 95. [5] K.S. Echoff, Accurate reconstructions of functions of finite regularity from truncated series epansions, Math. Comp., 64 (995), pp [6] K.S. Echoff, On a high order numerical method for functions with singularities, Math. Comp., 67 (998), pp [7] A. Gelb and E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal., 7 (999), pp. 35. [8] A. Gelb and E. Tadmor, Enhanced spectral viscosity method for nonlinear conservation laws, Appl. umer. Math., 33 (2), pp [9] B.I. Golubov, Determination of the jump of a function of bounded p-variation by its Fourier series, Math. otes, 2 (972), pp [] D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of Progress in Scientific Computing, Vol. 6, E. M. Murman and S. S. Abarbanel, eds., Birhäuser, Boston, 985, pp [] D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution, SIAM Rev., 39 (998), pp

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