By Prof. Yasumichi Hasegawa, Dr. Tatsuo Suzuki (auth.)

Thismonographisconcernedwithdescriptionanddesignfortwo-dimensional and third-dimensional photographs; will probably be of distinct curiosity to researchers and graduate scholars who really good in photo processing and process conception. From the information in electronic photos, mathematical versions can be built. Then new platforms which describe faithfully any two-dimensional or thr- dimensional electronic photographs should be proposed. utilizing the platforms therefore permits description to be handled as consciousness challenge and layout. by means of advantage of this method, this monograph presents new effects and their extensions that are designing of two-dimensional and three-d pictures. a few genuine layout examples may be additionally proven. In traditional snapshot processing this present day, two-dimensional pictures are reworked into one-dimensional indications, then that are analyzed via numerous confirmed equipment in sign processing conception. Likewise, 3-dimensional photographs are reworked into two-dimensional indications and those signs are analyzedbyestablishedmethodsintwo-dimensionalsignalprocessingtheory. one other universal processing strategy employs tree constructions akin to qu- bushes for two-dimensional pictures and oct-trees for third-dimensional ones.

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Also see Matsuo and Hasegawa [1981] for the case of two variables. 20). 21. 19). Then the Quasi-reachable Standard System σ = ((K n , Fαs , Fβs ), e1 , hs ) which realizes the two-dimensional image a is obtained by the following procedure: 1) 2) k) Find an integer ν1 and coeﬃcients {c11l ; 1 ≤ l ≤ ν1 } such that the vectors {Sαi a; 1 ≤ i ≤ ν1 − 1} of the set {Sαi a; i ≤ n − 1, i ∈ N } are ν1 linearly independent and Sαν1 a = l=1 c11l Sαl−1 a. Find an integer ν2 and coeﬃcients {c2ml ; 1 ≤ l ≤ νm , 1 ≤ m ≤ 2} such that the vectors {Sβj−1 Sαi−1 a; 1 ≤ i ≤ νj − 1, 1 ≤ j ≤ 2} of the set {Sβj Sαi a; i ≤ n − 1, j ≤ n − 2 ∈ N } are linearly independent and 2 νm 2 cml Sβm−1 Sαl−1 a.

8. 5). Proof. Clearly, K[zα , zβ ] is an algebra. 6), an algebra mor˜ phism is given by S(λ) = λ · e for λ = i,j λ(i, j)zαi zβj ∈ K[zα , zβ ], where the map e : N × N → L(K[zα , zβ ]) is given as follows: e : N × N → L(K[zα , zβ ]); (i, j) → zαi zβj . 9. 3), the corresponding K[zα , zβ ]-module (F (N × N, Y ), S) i j i j ˜ setting S(λ) := i,j λ(i, j)Sα Sβ for λ = i,j λ(i, j)zα zβ ∈ K[zα , zβ ]. 10. Let (X1 , Fα1 , Fβ1 ) [ (X1 , φ1 ), (X1 , φ˜1 )] and (X2 , Fα2 , Fβ2 ) [(X2 , φ2 ) , (X2 , φ˜2 )] be {α, β}-actions [N ×N -module , K[zα , zβ ]-module].

We also remember that homogeneous bilinear systems and K − U automaton are a sort of Linear Representation Systems. See Tarn and Nonoyama [1976] and Fliess [1978] for discrete-time homogeneous bilinear systems. See also Brockett [1976] for continuous-time systems and Paz [1966] for probabilistic automaton. Schuzenberger [1961] considered generalized automata, which are called the K − U automata by Eilenberg [1974]. See also Matsuo and Hasegawa [2003]. The set A(Ω) in Linear Representation Systems may be equivalent to the algebra of polynomial in non-commutative variable introduced by Fliess [1970] and [1974].