Some Aspects of Band-limited Extrapolations

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1 October 3, 2009

2 I. Introduction Definition of Fourier transform: F [f ](ω) := ˆf (ω) := + f (t)e iωt dt, ω R (1)

3 I. Introduction Definition of Fourier transform: F [f ](ω) := ˆf (ω) := + f (t)e iωt dt, ω R (1) Definition of band-limited function: A function f L 2 (R) is said to be Ω-band-limited if ˆf (ω) = 0, ω / [ Ω, Ω]. Here ˆf (ω) is the Fourier transform of f.

4 The inversion formula: f (t) = 1 Ω ˆf (ω)e iωt dω, a.e. t R (2) 2π Ω

5 The inversion formula: f (t) = 1 Ω ˆf (ω)e iωt dω, a.e. t R (2) 2π Ω Remark 1: This integral is meaningful for all t C and furnishes an analytic extension of f to C.

6 The inversion formula: f (t) = 1 Ω ˆf (ω)e iωt dω, a.e. t R (2) 2π Ω Remark 1: This integral is meaningful for all t C and furnishes an analytic extension of f to C. Remark 2: By the uniqueness theorem of analytical functions, f (t) is uniquely determined by its values on the interval [ T, T ] where T = const. > 0.

7 Uniqueness Theorem of Analytical Functions Suppose two single-valued analytic functions f (z) and g(z), defined on a domain G, coincide on a set E G with a limit point z 0 in G. Then f (z) and g(z) coincide on the whole domain G.

8 The Extrapolation Model and Its Applications Band-limited extrapolation problem: Assume that f : R R is a band-limited function and T is a positive constant: given f (t) t [ T, T ] (3) find f (t) t / [ T, T ].

9 The Extrapolation Model and Its Applications Band-limited extrapolation problem: Assume that f : R R is a band-limited function and T is a positive constant: given f (t) t [ T, T ] (3) find f (t) t / [ T, T ]. Applications in engineering and medical/healthcare: This is widely used in image restoration [11], spectral estimation and denoising. The denoising can be used in the case of Sensor Deviations due to the noise.

10 A brief review of some previous results in this problem I 1. Papoulis and Gerchberg algorithm in 1975 and 1974: For k = 0, 1, 2,..., where and f 0 (t) = P T f (t). f k+1 (t) = P T f (t) + (I P T )F 1 P Ω Ff k (t), P T f (t) = { f (t), t [ T, T ] 0, t / [ T, T ] P Ωˆf (ω) = { ˆf (ω), ω [ Ω, Ω] 0, ω / [ Ω, Ω]. The convergence f k f L 2 0 is proven by Papoulis.

11 A Fast Convergence Algorithm By the sampling theorem the kth iteration of the Papoulis and Gerchberg algorithm is + n= ˆf k+1 (ω) = T T [ g k (nh) he inhω 1 [ Ω,Ω] (ω) f (t)e iωt dt t T ] sin Ω(t nh) Ω(t nh) eiωt dt. (4)

12 In the formula above, for g k (nh), if nh [ T, T ] we can replace them by f (nh) and obtain another function: + nh T + nh >T ˆf C k+1 (ω) := T T [ f (nh) he inhω 1 [ Ω,Ω] (ω) [ g k (nh) he inhω 1 [ Ω,Ω] (ω) f (t)e iωt dt t T t T ] sin Ω(t nh) Ω(t nh) eiωt dt ] sin Ω(t nh) Ω(t nh) eiωt dt. (5)

13 The iterative fast convergence algorithm Let For k = 0, 1, 2,..., f C 0 := P T f. f C k+1 := P T f +(I P T )F 1 P Ωˆf C k. (6)

14 II. Computation of The Inverse Fourier Transform In many cases we need the computation of the inverse Fourier transform given by (2) for extrapolation: f (t) = F 1 (ˆf )(t) 1 2π M 1 j= M where ω j = j ω and ω = Ω/M. ˆf (ω j )e iω j t ω, a.e. t R (7)

15 II. Computation of The Inverse Fourier Transform In many cases we need the computation of the inverse Fourier transform given by (2) for extrapolation: f (t) = F 1 (ˆf )(t) 1 2π M 1 j= M where ω j = j ω and ω = Ω/M. ˆf (ω j )e iω j t ω, a.e. t R (7) Remark: This is a periodic function with the period 2Mπ/Ω.

16 Proposition. If there is a peak of the function f (t) in [ Mπ/Ω, Mπ/Ω] then there is a peak of the approximate sum in (7) in each of the interval [ Mπ/Ω + 2kMπ/Ω, Mπ/Ω + 2kMπ/Ω] for any integer k.

17 Suppose Then ˆf = (1 ω )1 [ 1,1]. f (t) := 1 cos t πt 2. Example 1. We compute the inverse Fourier transform by ˆf = (1 ω )1 [ 1,1] with M = 10. The numerical results of ˆf and the exact inverse Fourier transform are in fig.1.

18 Remark. By the proposition and the example above, we can see that if we choose the sum (7) to approximate the extrapolation, the result out of the interval [ Mπ/Ω, Mπ/Ω] is not reliable at all.

19 III. The Accuracy of The Fast Convergence Algorithm Theorem 1. In each step of the iteration, if the signal used for extrapolation are same, we have ĝk C ˆf 2 L = ĝ 2 k ˆf 2 L 2Ω h 2 g 2 k (nh) f (nh) 2. nh T

20 III. The Accuracy of The Fast Convergence Algorithm Theorem 1. In each step of the iteration, if the signal used for extrapolation are same, we have ĝk C ˆf 2 L = ĝ 2 k ˆf 2 L 2Ω h 2 g 2 k (nh) f (nh) 2. nh T Remark: By this theorem, we can see that if 2Ω h 2 g k (nh) f (nh) 2 nh T is larger, the error of the fast convergence algorithm is smaller. So this is a criterion for us to choose Ω and N such that ĝk C is a good approximation of ˆf.

21 IV. Experimental Results Let and Err criterion := 2Ω h 2 g k (nh) f (nh) 2 nh T Err energy := ĝ C k ˆf 2 L 2. Example 2. The signal is same as the signal in Example 1. Suppose the signal is known on [ T, T ] = [ π/5, π/5].

22 We choose Ω = 1 and N = 10. The numerical results by the fast convergence algorithm for the iterations 1,2,3,4 are in fig.2 and fig.3. The Err criterion and the Err energy are in the next table. Iteration Err criterion e e e-006 Err energy We choose Ω = 10 and N = 10. The numerical results by the fast convergence algorithm for the iterations 1,2,3,4 are in fig.4 and fig.5. The Err criterion and the Err energy are in the next table. Iteration Err criterion Err energy

23 IV. The Ill-posedness and Regularization Methods 1. In 1983, Sanz and Huang presented the following theorem: Theorem: Given any positive numbers ɛ < θ and given f as above, let P be any real number with P > T. Then there exists a Ω-band-limited function Ψ ɛ,p that satisfies but Ψ ɛ,p f L [ T,T ] ɛ, Ψ ɛ,p (P) f (P) θ.

24 IV. The Ill-posedness and Regularization Methods 1. In 1983, Sanz and Huang presented the following theorem: Theorem: Given any positive numbers ɛ < θ and given f as above, let P be any real number with P > T. Then there exists a Ω-band-limited function Ψ ɛ,p that satisfies but Ψ ɛ,p f L [ T,T ] ɛ, Ψ ɛ,p (P) f (P) θ. 2. This theorem shows that small noise in the interval [ T, T ] of the signal may produce large errors outside the interval. Worse, the solution of the extrapolation no longer exists if η δ is not the restriction of a band-limited function on the interval [ T, T ].

25 An Example of The Ill-posedness: Let f n (t) := n[1 cos Ω(t n)] (t n) 2 L 1 (R) L (R). Then ˆf n (ω) = e inω nπ 1 [ Ω,Ω] (ω) (Ω ω ) and f n 0 uniformly in [ T, T ] as n. But f n (n) = Ω 2 n/2 as n. Therefore, given 0 < ɛ < θ, we can pick n so that Ω 2 n/2 > θ and so that f n (t) < ɛ for all t [ T, T ], and we can consider η := f n, P := n. Then η L (R) L 1 (R) is Ω-bandlimited and satisfies η(t) < ɛ for all t [ T, T ] and η(p) > θ.

26 The regularized fast convergence algorithm: For k = 0, 1, 2,..., f0 C P T f (t) (t) := 1 + 2πα + 2παt 2. (10) where + nh T + nh >T k+1(t) := P T f + (I P T )F 1 P Ωˆf k C 1 + 2πα + 2παt 2. (11) f C T ˆf k+1(ω) C := f (nh) 1 + 2πα + 2πα(nh) 2 f (t)e iωt T 1 + 2πα + 2παt 2 dt [ he inhω 1 [ Ω,Ω] (ω) t T [ gk C (nh) 1 + 2πα + 2πα(nh) 2 he inhω 1 [ Ω,Ω] (ω) t T ] sin Ω(t nh) Ω(t nh) eiωt dt ] sin Ω(t nh) Ω(t nh) eiωt dt.

27 Example 3. The signal is same as the signal in Example 1. Suppose the signal is known on [ T, T ] = [ 10, 10]. We add white noise that is uniformly distributed in [ 0.2, 0.2] to the signal on [ T, T ]. The numerical results by the fast convergence algorithm for the iterations 1,2,3,4 are in fig.6 and fig.7. The numerical results by the regularized fast convergence extrapolation algorithm with α = 0.01 for the iterations 1,2,3,4 are in fig.8 and fig.9.

28 References I A. Papoulis, A new algorithm in spectral analysis and band-limited extrapolation. IEEE Trans. Circuits Syst., vol. CAS-22, pp , Sept R. Gerchberg, Supperresolution through error energy reduction. Opt. Acta, vol. 12, no. 9, C. Chamzas and W. Xu, An improved version of Papoulis-Gerchberg algorithm on band-limited extrapolation. IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, no.2, Apr

29 References II Xia, X. G. and Kuo, C. C. J. and Zhang, Z., Signal extrapolation in wavelet subspaces. SIAM J. Sci. Comput., vol.16, no.1 pp.50-73, T. Strohmer, On discrete band-limited signal extrapolation. AMS Contemp. Math., pp , vol.190, B. G. Salomon and H. Ur, Accelerated iterative band-limited extrapolation algorithms. IEEE Signal Processing Letters, vol. 11, Nov

30 References III W. Chen, A new extrapolation algorithm for band-limited signals using regularization method. IEEE Trans on Signal Processing vol. SP-41, pp , Mar W. Chen, An Efficient Method for An Ill-posed Problem Band-limited Extrapolation by Regularization. IEEE Trans on Signal Processing vol. SP-54, pp , Dec Drouiche K, Kateb D and Noiret C, Regularization of the ill-posed problem of extrapolation with the Malvar-Wilson wavelets. Inverse Problems, 17(2001),

31 References IV P. J. S. G. Ferreira, Noniterative and faster iterative methods for interpolation and extrapolation. IEEE Transactions Signal Processing, vol. 42, n. 11, p , Nov Priyam Chatterjee, Sujata Mukherjee, Subhasis Chaudhuri and Guna Seetharaman, Application of Papoulis-Gerchberg Method in Image Super-resolution and Inpainting. The Computer Journal, vol. 00, no. 0, 2007.