By Yasumichi Hasegawa
This monograph bargains with approximation and noise cancellation of dynamical structures which come with linear and nonlinear input/output relationships. It additionally care for approximation and noise cancellation of 2 dimensional arrays. it is going to be of distinct curiosity to researchers, engineers and graduate scholars who've really expert in filtering conception and procedure thought and electronic photos. This monograph consists of 2 components. half I and half II will take care of approximation and noise cancellation of dynamical structures or electronic photographs respectively. From noiseless or noisy info, aid could be made. a mode which reduces version info or noise used to be proposed within the reference vol. 376 in LNCIS [Hasegawa, 2008]. utilizing this technique will let version description to be taken care of as noise aid or version relief with no need to trouble, for instance, with fixing many partial differential equations. This monograph will suggest a brand new and simple technique which produces an identical effects because the procedure handled within the reference. As evidence of its effective impact, this monograph offers a brand new legislation within the experience of numerical experiments. the recent and simple strategy is performed utilizing the algebraic calculations with out fixing partial differential equations. For our function, many real examples of version details and noise aid can be provided.
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Additional resources for Algebraically Approximate and Noisy Realization of Discrete-Time Systems and Digital Images
25. 24) is called an algebraic Constrained Least Square method, abbreviated, the algebraic CLS method. We show examples for the algebraic CLS method. We can show that the analytic approximate realization algorithm given in the reference [Hasegawa, 2008] produces the same systems as the ones obtained by the algebraic CLS method in the sense of the numerical calculation. In addition, we compare the method with the common method for noise processing which is called AIC. 26. 3 ⎦ , h = [10, 5, −5]. 41 Let an added noise be given in Fig.
3 Let F ∈ Rn×n be given as below. Let g0 be g0 = e1 , where e1 = [1, 0, · · · , 0]T ∈ Rn . 4 Let hs be hs = [Ia (1 )− I¯a (1 ), Ia (1)− I¯a (1),· · ·, Ia (0n1 −2 |1)− I¯a (0n1 −2 |1)]. ⎡ ⎤ 0 · · · 0 α1 ⎢ .. ⎥ ⎢ 1 . α2 ⎥ ⎢ ⎥ F0 = ⎢ . ⎥. ⎣ .. . 0 .. ⎦ 0 1 αn [proof] By 1), the approximate part of the data can be excluded in the sense of the norm of Hankel matrix Ha (p,p) ¯ . 14) corresponds to the matrix composed of eigenvectors of Ha (p,p) ¯ . 17). 17) in the sense of a linear combination. ˆ a (n +1,p) Therefore, we obtain the approximate Hankel matrices H ¯ (n1 + 1 ˆ 1, 0).
Ia (i + ¯i) ⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠ where i ≤ p and ¯i ≤ p¯. i ˆ a (p,p) Note that the column vectors of H ¯ is represented by S l Ia . When we actually treat the approximate and noisy realization problem, we will use a notation Ha (n1 ,N−n1 ) expressed as follows: Ha (n1 ,N−n1 ) = [Ia , · · · , Sln1 −1 Ia ]. 14. Let the rank of a ﬁnite-sized Hankel matrix H ¯ be n. 9) for S nl Ia = 2) Let e1 be e1 = [1, 0, · · · , 0]T . 3) Let hs be hs = [Ia (1), Ia (2), · · · , Ia (n)]. n i=1 αi S i−1 Ia . l [proof] It is obvious from the deﬁnition of behavior of the system.