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An } F ( p , q ) : P × Q → reals outlined and {B1 , . . . , Bm } to be the next functionality f nm i i as n m F ( pi , q j ) = f nm Fkl Ak ( pi )Bl (q j ) (6. 2) k=1 l=1 the next approximation theorem holds [32]: Theorem 6. 1 enable f : P × Q → reals be an assigned functionality, P × Q ⊆ [a, b] × [c, d], being P = { p1 , . . . , p N } and Q = {q1 , . . . , q M }. Then for each ε > zero, there exist integers n(ε), m(ε) and similar fuzzy walls {A1 , A2 , . . . , An(ε) } of [a, b] and {B1 , B2 , . . . , Bm(ε) } of [c, d] such that the units of issues P and Q are sufficiently dense with admire to such walls and the inequality | f ( pi , q j ) − F ( pi , q j )| < ε holds for each i ∈ {1, . . . , N } and j ∈ {1, . . . , M}. f n(ε)m(ε) Now we exhibit how the F-transforms paintings for coding and deciphering grey pictures. permit R be a grey picture divided in N × M pixels interpreted as a fuzzy relation R : (i, j) ∈ {1, . . . , N } × {1, . . . , M} → [0, 1], R(i, j) being the normalized worth of the pixel P(i, j), that's, R(i, j) = P(i, j)/255 if the size of the grey scale, for example, has 256 degrees. In [6] the picture R is compressed through the use of an F-transform in variables [Fkl ] outlined for every okay = 1, . . . , n and l = 1, . . . , m, as Fkl = M j=1 N i=1 M j=1 R(i, j)Ak (i)Bl ( j) N i=1 Ak (i)Bl ( j) (6. three) the place we think pi = i, q j = j, a = c = 1, b = N , d = M and A1 , . . . , An (resp. , B1 , . . . , Bm ) with n N (resp. , m M), shape a fuzzy partition of [1, N ] (resp. , [1, M]). by way of interpreting with the inverse F-transform, we have now the next fuzzy relation outlined as n m F (i, j) = Rnm Fkl Ak (i)Bl ( j) k=1 l=1 (6. four) 112 F. Di Martino and S. Sessa desk 6. 1 a number of compression charges (L = 2) F-transforms L F-transforms Compression premiums Rows Columns Coded Coded Rows Columns Coded Coded ρ Lρ Lρtot rows columns rows columns four 12 6 eight sixteen four 12 6 eight sixteen three 6 2 2 three three 6 2 2 three four eight eight 12 sixteen four eight eight 12 sixteen 2 three 2 2 2 2 three 2 2 2 zero. 562 zero. 250 zero. 111 zero. 062 zero. 035 zero. 250 zero. a hundred and forty zero. 062 zero. 028 zero. 015 zero. 500 zero. 281 zero. a hundred twenty five zero. 056 zero. 031 for each (i, j) ∈ {1, . . . , N } × {1, . . . , M}. now we have subdivided the picture R of N × M pixels in submatrices R B of sizes N (B) × M(B), referred to as blocks (cf. , e. g. , [1, 2]), each one compressed to a block facebook of sizes n(B) × m(B)(3 ≤ n(B) < N (B), three ≤ m(B) < M(B)) through the direct F-transform [FklB ] outlined for every ok = 1, . . . , n(B) and l = 1, . . . , m(B) as FklB = M(B) N (B) j=1 i=1 R B (i, j)Ak (i)Bl ( j) M(B) N (B) j=1 i=1 Ak (i)Bl ( j) (6. five) the next easy capabilities A1 , . . . , An(B) (resp. , B1 , . . . , Bm(B) ) shape a uniform fuzzy partition of [1, N (B)] (resp. , [1, M(B)]): zero. 5(1 + cos πh (x − x1 )) zero zero. 5(1 + cos πh (x − xk )) Al (x) = zero zero. 5(1 + cos πh (x − xn )) An (x) = zero A1 (x) = if x ∈ [x1 , x2 ] another way if x ∈ [xk−1 , xk+1 ] in a different way if x ∈ [xn−1 , xn ] another way (6. 6) the place n = n(B), okay = 2, . . . , n, h = (N (B) − 1)/(n − 1), xk = 1 + h · (k − 1) and zero. 5(1 + cos πs (y − y1 )) if y ∈ [y1 , y2 ] zero differently zero. 5(1 + cos πs (y − yt )) if y ∈ [yt−1 , yt+1 ] Bt (y) = zero another way zero.