Poisson Noise Removal Scheme Based on Fourth-Order PDE by Alternating Minimization Algorithm

and Applied Analysis 3 2. Existence and Uniqueness In this section, the existence and uniqueness of the minimizer for the model 1.4 are proved. Some basic notations and properties concerning with the BV 2 space can be seen in 8, 13 . Motivated by 5 , we have the following existence and uniqueness result for the optimization problem 1.4 . Theorem 2.1. Assume Ω is an open, bounded Lipschitz domain in R and f is a positive bounded function. Let u ∈ BV 2 Ω such that logu ∈ L1 Ω . Then there exists a unique minimizer for model 1.4 in BV 2 Ω . Proof. Obviously, the objective functional E u is lower bounded. Then there exists a minimizing sequence {ui}i 1. Let F u ∫ Ω u − f logu dx. Then, using Jensen inequality, we conclude that ‖ui‖1 − ∥f∥∞ log ‖ui‖1 ≤ F ui , 2.1 which indicates that ‖ui‖1 is bounded. This, together with the boundedness of { ∫ Ω |Dui|}, yields that {ui}i 1 is bounded in BV 2 Ω . Then, following the compactness theorem in BV 2 Ω space, we deduce that there exists a function u∗ ∈ BV 2 Ω such that a subsequence of {ui} denoted also by {ui} converges to u∗ a.e. in L1 Ω . In addition, by the lower semicontinuity of the BV 2 Ω space, we have lim infi→∞‖Dui‖1 ≥ ‖Du‖1. Combining this with Fatou’s Lemma, we obtain ∫