Robust Image Filtering Using Joint Static and Dynamic Guidance Supplementary Material
نویسندگان
چکیده
1. Proof Proposition 1. Ψ y ν (x) is a surrogate function of ψ ν (x). Proof. A function g y (x) is said to a surrogate function of f (x) provided ⇢ f (x) ≤ g y (x), ∀x, y ∈ [0, ∞) f (y)=g y (y), ∀y ∈ [0, ∞). (1) Let us consider the following function: f ν (x)=(1− exp(−νx))/ν. (2) Since f 00 ν (x) < 0, this function is strictly concave, ∀x, y, f ν (x) ≤ f ν (y)+(x − y)f 0 ν (y), (3) with equality holding at x = y. Thus, the surrogate function g y ν (x) is g y ν (x)=f ν (y)+(x − y)f 0 ν (y) (4) = f ν (y)+(x − y)exp(−νy) = f ν (y)+(x − y)(1 − νf ν (y)). Let us substitute x and y with x 2 and y 2 , respectively, then ψ ν (x)=f ν (x 2) (5) ≤ g y 2 ν (x 2), and g y 2 ν (x 2)=f ν (y 2)+(x 2 − y 2)(1 − νf ν (y 2)) (6) = ψ ν (y)+(x 2 − y 2)(1 − νψ ν (y)) =Ψ y ν (x).
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