Advances in image processing and understanding: A - download pdf or read online

By Alan C. Bovik, Chang Wen Chen, Dmitry B. Goldgof, Thomas S. Huang

ISBN-10: 9812380914

ISBN-13: 9789812380913

ISBN-10: 9812776958

ISBN-13: 9789812776952

Directed in the direction of snapshot processing researchers, together with educational school, graduate scholars and researchers, in addition to towards pros operating in program components.

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Extra info for Advances in image processing and understanding: A festschrift for T.S. Huang

Sample text

Define tdX (f ) as the transcendental degree of {gq }q∈Q . The main step in Kalorkoti’s proof is the following lemma (he considers the case when division gates are allowed, which makes his proof somewhat more elaborate). 6. Let f ∈ F[X] and X1 , . . , Xt be a partition of X. Then every arithmetic formula for f must be of size at least Ω( i∈[t] tdXi (f )). Proof. [Sketch] The basic property we shall use is that for every X ⊆ X and two polynomials f1 and f2 , both tdX (f1 + f2 ) and tdX (f1 · f2 ) are at most tdX (f1 ) + tdX (f2 ).

This makes the task of proving lower bounds for noncommutative circuits easier than for commutative circuits. Indeed, in terms of proving lower bounds, more is known about the noncommutative world than the commutative one. We start by discussing Nisan’s noncommutative formula-size lower bound [98]. 8 ([98]). There exists an explicit noncommutative polynomial f of polynomial degree that can be computed by a noncommutative circuit of linear size, but every noncommutative formula for f has exponential size.

We call such a circuit Ψ a skeleton, as it contains no field elements. The circuit Φ is thus a projection of Ψ: the polynomial computed by Φ can be obtained from the polynomial computed by Ψ by an appropriate substitution of field elements to the variables Z. We now explain how the polynomial computed by Ψ can be viewed as a polynomial map F = FΨ from Fs to FN , with N = n+r r . The coordinates of F are labeled by monomials M of degree at most r in the variables X, and we denote F = (FM )M . There are N such monomials, and so there are N entries in F .

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Advances in image processing and understanding: A festschrift for T.S. Huang by Alan C. Bovik, Chang Wen Chen, Dmitry B. Goldgof, Thomas S. Huang


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