Nonconvex TV$^q$-Models in Image Restoration: Analysis and a Trust-Region Regularization--Based Superlinearly Convergent Solver

Abstract

A nonconvex variational model is introduced which contains the q-"norm", q ∈ (0, 1), of the gradient of the image to be reconstructed as the regularization term together with a leastsquares type data fidelity term which may depend on a possibly spatially dependent weighting parameter. Hence, the regularization term in this functional is a nonconvex compromise between the minimization of the support of the reconstruction and the classical convex total variation model. In the discrete setting, existence of a minimizer is proven, a Newton-type solution algorithm is introduced and its global as well as locally superlinear convergence is established. The potential indefiniteness (or negative definiteness) of the Hessian of the objective during the iteration is handled by a trust-region based regularization scheme. The performance of the new algorithm is studied by means of a series of numerical tests. For the associated infinite dimensional model an existence result based on the weakly lower semicontinuous envelope is established and its relation to the original problem is discussed. * This research was supported by the Austrian Science Fund (FWF) through START project Y305 "Interfaces and Free Boundaries" and through SFB project F3204 "Mathematical Optimization and Applications in Biomedical Sciences".

Limiting Aspects of Nonconvex ${TV}^{\phi}$ Models

“…Note that ψ is locally Lipschitz and monotonically increasing, and in the following we shall denote by ∂ψ a subdifferential of ψ. We remark that the consistency of the Huberized stationary points induced by (9.3) towards the stationary points of the original model (9.1) was investigated in the previous work [17,18]. Moreover, the system (9.3) is not differentiable in the classical sense.…”

“…In fact, in [17] Huber regularisation was used with ϕ(t) = t q for q ∈ (0, 1) for algorithmic reasons. For small γ > 0, this is defined as…”

“…Given the current iterate (u i , p i ), our solver relies on the following regularized linear system arising from differentiating (9.3) (and further straightforward manipulations; see [17,18]):…”

“…In particular, (1.1) also serves as a definition of TV ϕ c for continuous piecewise affine discretisations of u ∈ C 1 (Ω). We observe that if u ∈ C 1 (Ω) on a bounded domain Ω, and we set u h (k) = u(k) for k ∈ Ω h , then This approximate model TV ϕ h with h = 1 is essentially what is considered in [18,17,25]. On an abstract level, it is also covered by [23].…”

Semi-calibrated Near-Light Photometric Stereo

“…In this work, our choices for H (k,k) θ and H (k+1,k) z (see formulas (18) and (21) below) will be structured Hessian approximations motivated from the iteratively reweighted least-squares (IRLS) method [13,14]. The solution of the resulting subproblem can be interpreted as a regularized Newton step; see [15,16,17].…”