On the Solvability in Sobolev Spaces and Related Regularity Results for a Variant of the TV-Image Recovery Model: The Vector-Valued Case

Abstract

Keywords: variational problems of linear growth, TV-regularization, denoising and inpainting of multicolor images, existence of solutions in Sobolev spaces. AbstractWe study classes of variational problems with energy densities of linear growth acting on vector-valued functions. Our energies are strictly convex variants of the TV-regularization model introduced by Rudin, Osher and Fatemi [15] as a powerful tool in the field of image recovery. In contrast to our previous work we here try to figure out conditions under which we can solve these variational problems in classical spaces, e.g. in the Sobolev class W 1,1 .

Coupled variational problems of linear growth related to the denoising and inpainting of images

“…ii) The results from [8] indicate, that the bound µ < 3 /2 is not optimal, whereas in [16] Fuchs, Tietz and the author have shown, that µ, ν < 2 is indeed necessary for the existence of a solution in the Sobolev class, if µ-elliptic densities are considered.…”

“…In order to obtain more regular minimizers, we need to refine our assumptions on the functions F and G. In fact, previous work in this regard (see e.g. [5], [6], [7] or [16] and particularly [8] for more recent results) indicate, that the correct framework for establishing "classical" solvability (i.e. in a Sobolev space) of our primal problem (P) is the concept of "µ-ellipticity".…”

The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

“…-(H 3 ) is satisfied with σ 0 = σ 1 = σ 2 = σ ≥ 2, for the following class of nonlinearities a(z, ξ) := |ξ | σ −1 sgn(ξ ) (ϕ(z) + |ξ | σ ) 1−1/σ or (1.12) a(z, ξ) := ξ (ϕ(z) + |ξ | σ ) 1/σ (1. 13) with ϕ ∈ C 1 (R) ∩ W 1,∞ (R), ϕ(z) ≥ c > 0 for all z ∈ R (see Remark 3.2 below). -In the case F ≡ 0, the condition σ i ≤ σ 0 is not necessary for the results in Theorem 3.6 to hold.…”

“…We point out that the condition of μ-ellipticity is similar to condition (H 4 ) with σ = μ + 1. This study has been then generalized to the vector valued case [12] or to Dirichlet problems for radially symmetric data [13]. Concerning the Dirichlet problem we also mention the results in [9] about global Lipschitz minimizers or [15] about boundary regularity.…”

On the local boundedness of generalized minimizers of variational problems with linear growth

“…REMARK 1.6. In the paper [16] the reader will find some intermediate regularity results for solutions u of (1.14) saying that even without the assumption f ∈ L ∞ (Ω − D) the solution u belongs to some Sobolev class. With respect to these results we can even replace the "data term" Ω−D |u − f | 2 dx by more general expressions (with appropriate variants of (1.12)), however, in any case µ-ellipticity (1.9) together with an upper bound on µ is required.…”