[论文解读] Inverse problems with second-order Total Generalized Variation constraints

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

Image and Signal Denoising Methods参考文献 10被引用 30

一句话总结

本文将二阶 Total Generalized Variation (TGV^2) 作为病态线性逆问题的正则化项进行分析，证明了良态性和 BV 等价性，并给出带有数值证据的去模糊结果。

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.

研究动机与目标

• 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.

提出的方法

• 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.

实验结果

研究问题

• 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?

主要发现

• 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.