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...s: min {1 2 p∑ ∥RSiu − bi∥ 2 + τ∥∇u∥1 ∣ u ∈ Ω } , (5.9) i=1 Note that the model (5.9) differs from the super-resolution model in [9] in that the additional constraint Ω is required for u. In fact, as =-=[28]-=- shows, this additional constraint Ω often results in a reconstructed image with much higher quality. We now illustrate that the model (5.9) is a special case of (1.3). In fact, introducing the auxili...

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...d image restoration model (1.2). For constrained L2-L2 problems, i.e. the regularization term is the L2-norm of some derivatives of x, there are several methods that based on Newton-like methods, see =-=[10, 22]-=-. For constrained TV problems, the singularity of the TV functional prohibits the application of Newton-like methods. Recently, Beck and Teboulle [4] proposed a fast gradient-based algorithm for solvi...

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...g ill-conditioned inner linear system at each Newton iteration when the iterate approaches a solution on the boundary of the feasible set. Their work inspired immediately the reduced Newton method in =-=[23]-=-, which solves a subsystem of the inner linear system corresponding only to components of the iterate that are not close to the boundary. Since the resulting subsystem is smaller and less ill-conditio...

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...nkage and selection operator (Lasso) problem [76], the Dantzig Selector problem [13, 80], the robust principal component analysis models [10, 16, 62, 87], the constrained linear least-squares problem =-=[18, 58]-=-, the inverse eigenvalue problem [79], etc. 3. Applications of (1.4). The robust principal component analysis model with incomplete and noisy observations [74], the non-negative matrix factorization a...

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...where A ∈ Rm×d , b ∈ Rm×1 are given. To increase the numerical stability, we apply the regularization in NNLS of the form min x>0 ‖ Ax − b ‖2 +λ ‖ x ‖2 (λ ≥ 0) where λ is the regularization parameter =-=[3]-=-. Thus, Phoenix gives preference to solutions with smaller Euclidean norms, i.e., smaller ‖ x ‖. Similar regularization is used in DMF NC system [22] without enforcing x > 0. According to the test usi...

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...,where ...

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...our constrained image restoration method has fast convergence when compared with some of the existing solvers. Box constraints have been considered for image restoration by many authors, such as [3], =-=[4]-=- and [5], and the references therein. However, most of these methods have the following two major weaknesses: (i) they mostly consider the Gaussian noise (equivalently least-squares restoration), and ...