Image Super-Resolution and Applications

Appendix C: Newton’s Method

According to the regularized solution:

Minimizes‖Cf^‖

(C.1)

subject‖H¯jf^−gi‖2=‖nj‖2=εi2fori=1,2,.............,N

(C.2)

Using the method of Lagrange multipliers, the solution f^ is obtained by minimizing the following cost function [130]:

Ψ(f^,λ1,λ2,.......,λN)=∑i=1Nλj‖H¯if^−gi‖2−‖nj‖2+‖Cf^‖2

(C.3)

The Lagrangian method is based on the possibility of solving a system of equations that constitute necessary conditions for optimality. For the problem described by (C.1) and (C.2), these conditions are given by [130]:

∇f^Ψ(f^,λ)=0

(C.4)

∇λΨ(f^,λ)=0

(C.5)

where the vector λ = (λ₁, λ₂,........, λ_N) and Δ_z denotes the gradient of a function with respect to vector z. The Lagrange ...