Total Generalized Variation for Inverse Problems Involving Piecewise Linear Finite Elements

1 Introduction

• Higher-order regularizers capture structural information beyond first-order variations, including higher derivatives, directionality, and curvature; TGV unifies edge preservation with higher-order smoothness.The paper frames refined structural recovery as important for high-quality inverse-problem reconstructions.Source: §1 · p. 1
• Prior convergence results as h tends to zero covered regularizers using first-order gradient information, including TV, but not higher-order regularization.Piecewise linear TV approximations were known to converge at rate h^1/4.Source: §1 · p. 2
• Spatiotemporal regularization incorporates temporal dependencies, and ECGI reconstructs electrical potential inside the heart from sparse torso-electrode measurements.The choice of regularization is critical because ECGI is highly ill-posed.Source: §1 · p. 2
• Piecewise linear discretizations of both u and the auxiliary variable w converge to continuous minimizers at rate h^1/4 for TGV2, with the same rate in the spatiotemporal extension.Numerical experiments confirm the theoretical rates and report improved reconstruction results against established methods in image reconstruction and ECGI.Source: §1 · p. 2
• The paper develops spatial and spatiotemporal convergence theory, then evaluates denoising, inpainting, and ECGI alongside suitable comparison methods.Its numerical section reports qualitative and quantitative results across these settings.Source: §1 · p. 2

2 Notation and Preliminaries

• The analysis assumes bounded, star-shaped polygonal domains and uses conforming triangulations with piecewise linear or piecewise constant finite-element spaces.The mesh parameter h is the largest triangle diameter, and the auxiliary spaces are chosen to support the later approximation arguments.Source: §2 · pp. 3–4
• TGV is introduced using distributional derivatives and Radon measures, with an equivalent minimization form involving an auxiliary variable and its gradient.The paper focuses mainly on order k = 2.Source: §2 · p. 3
• The auxiliary variable requires bounded-deformation regularity, while the TGV seminorms induce bounded-generalized-variation spaces.The paper notes that measure-valued minimizers can have unbounded distributional gradients, making the functional infinite for the lower-regularity choice.Source: §2 · p. 3
• The convergence analysis uses interpolation operators with approximation estimates that also extend to vector-valued functions and affine elements.These properties support the spatial and later prismatic spatiotemporal discretizations.Source: §2 · p. 4

3.1 Convergence of Semidiscrete TGV

• The semidiscrete proof approximates the minimizer by smooth functions and controls the interpolation error using Clément interpolation and TGV bounds.The auxiliary variable is initially treated as continuous before the fully discretized analysis.Source: §3.1 · pp. 5–6
• The semidiscrete TGV2 minimizer converges in L2 at rate h^1/4 under the stated star-shaped-domain assumption.The estimate depends on the TGV2 regularization and the L1 norm of the continuous minimizer.Source: §3.1 · pp. 5–6

3.2 Convergence of Fully Discrete, Non-symmetric TGV

• For the fully discrete problem, the authors use non-symmetric TGV because known convergence results for discretizing the auxiliary variable in BD are limited to specialized finite-element subspaces.The chosen regularizer requires convergence in the bounded-variation norm.Source: §3.2 · p. 6
• The fully discrete non-symmetric TGV minimizers are shown to converge in L2 on star-shaped domains.The proof extends the semidiscrete argument and accounts for discretizing the auxiliary variable.Source: §3.2 · p. 7

4 Spatiotemporal Total Generalized Variation on Finite Elements

• The spatiotemporal framework extends spatial TGV to product domains motivated by inverse problems whose unknowns evolve over time while their forward models are spatial PDEs.ECGI is cited as an example of this structure.Source: §4 · p. 8

4.1 Spatiotemporal Function Space

• The spatiotemporal bounded-generalized-variation space is Banach, and the regularizer vanishes exactly for functions polynomial below the specified degrees in space and time.The definition uses separate spatial and temporal orders and weights.Source: §4.1 · pp. 8–9

4.2 Discretization

• The product-domain discretization combines P1 finite elements in space and time to form prismatic elements with mixed spatial and temporal derivative spaces.Spatial derivatives are piecewise constant in space and affine in time; the temporal derivative uses the converse arrangement.Source: §4.2 · p. 10

4.3 Spatiotemporal Fully Discrete Convergence

• For star-shaped spatial and temporal domains, the fully discrete spatiotemporal denoising minimizer converges in L2 to the continuous minimizer.The proof applies spatial convergence results to the spatiotemporal objective, adding only a higher-order term.Source: §4.3 · p. 10

5.1 Numerical Implementation of Finite Element TGV

• A convexity-based bound avoids quadrature rules for the regularizer integrals and still yields an observed convergence rate of magnitude h^1/4.The bound uses γ = 1 or 2 to determine the norm for spatial gradients.Source: §5.1 · p. 12

5.2 Primal-Dual Optimization Algorithm

• The authors use a first-order primal-dual algorithm for the convex regularized minimization, with convergence guaranteed when the stepsizes satisfy τσλmax(K*K) ≤ 1.Diagonal preconditioning is applied across gradient and divergence operators to speed practical convergence.Source: §5.2 · pp. 12–13

5.3 Image Reconstruction Results

• TGVP0 and TGV1 P1 outperform TVP1 visually, whose reconstructions show staircasing; the two second-order methods look similar, though TGVP0 uses twice as many degrees of freedom.On the inverted gradient image, TGV2 P1 has marginally lower SSIM than the other second-order methods; inpainting also shows visible piecewise-constant filling with TVP1, while TGV methods recover diagonal beams in occluded regions.Source: §5.3 · p. 14, Figure 1 · p. 15, Figure 2 · p. 16
• TGV2 P1 attains the highest PSNR on BSD68 for both denoising and inpainting, while denoising PSNR values are similar across methods.The inpainting result also reports TGV2 P1 as best on the full dataset.Source: §5.3 · p. 14, Table 2 · p. 15

5.4 Convergence results

• Experiments observe convergence rates of h^1/2 in both spatial tests and h in both spatiotemporal tests.The tests use regular and unstructured meshes and compare discrete solutions with a finest-grid solution used as a surrogate for the continuous optimum.Source: §5.4 · pp. 15, 17, Figure 4 · p. 17

5.5 Results in the Inverse Problem in Electrocardiographic Imaging

• ECGI is a severely ill-posed PDE-constrained inverse problem for reconstructing myocardial transmembrane potential from sparse body-surface measurements, motivating regularization.The forward model assumes static domains and space-dependent conductivity tensors; the inverse operator is generally unbounded.Source: §5.5 · pp. 17–19
• The ECGI forward PDE has solutions with regularity, unique up to an additive constant, and the Tikhonov-regularized variational problem has a unique minimizer.The existence result also holds for the regularizer symTGVε,(α,β).Source: §5.5 · pp. 18–19
• TGVP0 is unsuitable for this ECGI setting because computing the forward problem requires derivatives of v.Source: §5.5 · p. 20
• Three ECGI ground truths—linear and piecewise constant in time, rotating quadratic, and rotating dampened tanh—were reconstructed on 572-point meshes over 1001 timesteps with 50dB noise.Solutions are unique only up to an additive constant; computations used the Fredholm alternative rather than a Tikhonov term.Source: §5.5 · pp. 20–21
• Across three examples, mean runtimes were 0.16 seconds (T0), 1.26 seconds (T1), 19.29 seconds (TV1P1), 39.63 seconds (TGV1P1), 22.55 seconds (TV2P1), and 31.12 seconds (TGV2P1).Per iteration, TGVP1 methods take approximately 50% more time than TVP1 methods because they evaluate three proximal mappings instead of two.Source: §5.5 · p. 21
• TV2P1 and TGV2P1 both remove noise, while TGV2P1 recovers linear and quadratic regions where TV2P1 suffers from staircasing artifacts.The visual comparison is at timestep 250 across three test cases.Source: §5.5 · p. 21, Figure 6 · p. 21
• For the tanh case, TGV2P1 favors substantially larger α1 than α0, behaving closer to total variation, which appears sensible for tanh.Regularization parameters are similar across methods for the linear and quadratic cases but differ for the tanh example.Source: §5.5 · p. 21
• At 50dB, TGVP1 outperformed all comparison methods on every test case, gaining 12.5–43.1% in RE and 4.7–31.9% in CC over T1, with the greatest improvement for quadratic data.Full reconstruction results are reported in Table 3.Source: §5.5 · p. 21, Table 3 · p. 22

• The work extends TV’s best-known convergence rates to piecewise linear TGV2 and transfers the framework to the spatiotemporal setting.Source: §6 · p. 22
• In transmembrane ECGI, where piecewise constant discretization is unsuitable, the method outperformed state-of-the-art Tikhonov regularization in three model functions.Source: §6 · p. 22
• On a sample image, the method matched established piecewise constant discretization performance with half the degrees of freedom, and slightly improved quantitative results in image inpainting.Source: §6 · p. 22
• TGV2 is more computationally demanding because of spatiotemporal gradient computation and additional proximal maps, but the authors say its accuracy gains justify the cost.Source: §6 · p. 22
• Future work will extend the results to simulated cardiac activation sequences and arrhythmias such as reentry waves, where shift invariance in both forward and inverse problems is a challenge.Source: §6 · p. 22