tailieunhanh - OPTICAL IMAGING AND SPECTROSCOPY Phần 3

Tham khảo tài liệu 'optical imaging and spectroscopy phần 3', ngoại ngữ, anh ngữ phổ thông phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | B-SPLINES 91 A function ff x E V f may be represented as ff x X Cnfn x nEZ In contrast with the sampling theorem and with the Haar wavelet expansion the expansion coefficients are not samples of ff or inner products between ff and the basis vectors. For the B-splines it turns out that we can derive complementary functions fn x for each fn x bm x n such that fn fn 8nn0. The complementary functions can be used to produce a continuous estimate for f x that is completely consistent with the discrete measurements. This interpolated function is fest x X hfn ffn x nEZ Given the orthogonality relationship between the sampling functions and the complementary functions fest is by design consistent with the measurements. We can further state that fest ff if the complementary functions are such that f E V f in which case there exist discrete coefficients p k such that f x X p k f x k ÉÍ Using the convolution theorem the Fourier transform of f x is f u f u x .e p kEZ The orthogonality between the dual bases may be expressed as hfn fn dnn0 X p k a n - k - n kEZ where a n f x f x n . Without loss of generality we set n 0 and sum both sides of Eqn. 128 against the discrete kernel e i2pn u to obtain EC i2pn u I Ỡ0n e 1 n EZ p k a n k e i2pn u kEZ n EZ Xp k e i2pku kEZ X a n e-i2pn n EZ where we use the substitution of variables n00 n0 k. 92 ANALYSIS Poisson s summation formula is helpful in analyzing the sums in Eqn. . The summation formula states that for g x e L1 R .n. g u k nez kez where g u is the Fourier transform of g x . To prove the summation formula note that h u 52 g u k kez is periodic in u with period 1. The Fourier series coefficients for h u are 1 hn h u e2pinudu 0 1 X g u k e2pinudu kez j k 1 X g u e2pinudu k 1 g ù e2pinudu 1 g n Since a x is the autocorrelation of f its Fourier transform is I f u 2. Thus by the Poisson summation formula a n e-i2pnu Yf u k 2 nez kez Reconsidering Eqn. we find X p k