[Paper Review] Inverse problems with second-order Total Generalized Variation constraints

Kristian Bredies, Tuomo Valkonen|arXiv (Cornell University)|May 19, 2020

Image and Signal Denoising Methods10 references30 citations

TL;DR

The paper analyzes second-order Total Generalized Variation (TGV^2) as a regularizer for ill-posed linear inverse problems, proves well-posedness and BV equivalence, and demonstrates deblurring with numerical evidence.

ABSTRACT

Total Generalized Variation (TGV) has recently been introduced as penalty functional for modelling images with edges as well as smooth variations. It can be interpreted as a "sparse" penalization of optimal balancing from the first up to the $k$-th distributional derivative and leads to desirable results when applied to image denoising, i.e., $L^2$-fitting with TGV penalty. The present paper studies TGV of second order in the context of solving ill-posed linear inverse problems. Existence and stability for solutions of Tikhonov-functional minimization with respect to the data is shown and applied to the problem of recovering an image from blurred and noisy data.

Motivation & Objective

• Motivate and formulate TGV^2 as a regularizer for ill-posed linear inverse problems.
• Establish that TGV^2 is a semi-norm and topologically equivalent to BV in appropriate spaces.
• Prove existence and stability of Tikhonov-type minimizers with TGV^2 regularization.
• Relate TGV^2 to BV via a minimization over BD vector fields and Sobolev-Korn inequality.
• Apply the theory to a deconvolution (deblurring) problem and illustrate with numerical results.

Proposed method

• Define Total Generalized Variation of second order (TGV^2) and its dual/saddle-point representation.
• Show TGV^2(u)=min_{w in BD(Ω)} α1||Du − w||_M + α0||Ew||_M (via Fenchel–Rockafellar duality).
• Demonstrate topological equivalence: c||u||_BV ≤ ||u||_1 + TGV^2_α(u) ≤ C||u||_BV.
• Use Poincaré–Wirtinger inequality and BD/Sobolev–Korn arguments to obtain coercivity and BV-equivalence.
• Prove existence of minimizers for the Tikhonov problem and stability under data perturbations; discuss relative weak compactness.
• Provide a saddle-point formulation suitable for primal–dual algorithms and apply to a deconvolution example (u ∗ k).
• Reference a primal–dual algorithm as a practical solver for the convex problem.

Experimental results

Research questions

• RQ1 Does TGV^2 provide a well-posed regularization for linear inverse problems?
• RQ2 Is TGV^2 equivalent to BV in a topological sense, enabling BV-based existence/stability analysis?
• RQ3 Can TGV^2 regularization achieve stable deblurring with noisy data?
• RQ4 How does TGV^2 compare to TV in handling edges and smooth regions in inverse problems?

Key findings

• TGV^2 is a semi-norm on the space BGV^2(Ω).
• TGV^2(u)=0 iff u is a polynomial of degree < 2.
• TGV^2 and TGV^2 for scaled α are equivalent; TGV^2 is rotationally invariant.
• TGV^2 has a scaling property under resampling: TGV^2∘ρ_r = r^{-d} TGV^2 with α̃=(α0 r^2, α1 r).
• TGV^2 is proper, convex, and lower semi-continuous on L^p(Ω).
• BGV^2(Ω) is topologically equivalent to BV(Ω): c||u||_BV ≤ ||u||_1 + TGV^2_α(u) ≤ C||u||_BV."
• Existence of minimizers for the Tikhonov problem with TGV^2 and stability to data perturbations are established.
• The deblurring problem with TGV^2 admits a solution and the method is stable to data; numerically, TGV^2 improves over TV in deconvolution tasks.