Variable-Growth TV Regularization

• Variable-Growth TV Regularization is a framework that adapts spatial penalization, allowing variable exponent penalties and dual formulations to better preserve edges and reduce artifacts.
• It generalizes classical TV by incorporating models like variable exponent modulars, double-phase functions, and bilevel-optimized parameters to tailor regularization to local image features.
• Advanced numerical schemes such as primal-dual methods and augmented Lagrangian frameworks enable efficient solution of the associated non-smooth optimization problems in inverse imaging.

Variable-growth total variation regularization (VGTV) refers to a broad class of variational models in which the local penalization of gradient magnitudes is allowed to change spatially and/or functionally, in contrast to classical total variation (TV) regularization that enforces homogeneous first-order sparsity. VGTV encompasses models with variable exponent modulars, double-phase functionals, spatially adaptive Lipschitz constraints, infimal convolution forms, and bilevel-optimized parameter fields. This framework systematically generalizes the concept of edge-preserving regularization to adaptively accommodate spatially-varying smoothness and feature preservation, thereby reducing artifacts such as staircasing and over-smoothing in inverse and imaging problems.

1. Mathematical Formalisms and Classes of Variable-Growth TV

VGTV regularization realizes spatial adaptivity by modifying the TV integrand. A prototypical formulation involves an integral functional of Musielak-Orlicz type: $$TV_\phi(u) = \int_\Omega \phi(x, |\nabla u(x)|) \, dx$$ where $\phi(x, t)$ embodies the spatially-varying growth rate. Common choices include $p(x)$-power type functions ($\phi(x, t) = |t|^{p(x)}$ with $p(x) \geq 1$) and double-phase forms (e.g., switch between linear and quadratic growth depending on $x$ or $t$) (Górny et al., 18 Apr 2025). Composite regularizers of infimal convolution form (e.g., TVLp) and piecewise-Lipschitz constraints further extend this paradigm. In addition, convex modulars such as

$\rho_p(f) = \int_\Omega |f(x)|^{p(x)} dx$

where $p(x)\in[1,2]$ and variable Huberized TV terms (with spatially variable thresholds) provide further flexibility (Kohr, 2017, Pagliari et al., 2021).

The associated variational problem is generally posed as the minimization of a fidelity-regularization sum: $$\min_{u\in \mathcal{K}} \left\{ TV_\phi(u) + \lambda\, \text{Fidelity}(Su, g^\delta) \right\}$$ where $S$ is the forward operator dictated by the imaging or inverse setup (e.g., fractional diffusion (Jia et al., 2016), tomography (Kohr, 2017)).

2. Subdifferential Characterization and Euler–Lagrange Equations

A central aspect of VGTV theory is the derivation of Euler–Lagrange equations governing optimality. For integral functionals with variable growth and spatially-dependent convex integrands,

$$TV_\phi(v) = \sup_{\xi \in C_c^1(\Omega)^n} \left\{ \int_{\Omega} v \, \mathrm{Div}\, \xi \, dx - \int_{\Omega} \phi^*(x, |\xi(x)|)\, dx \right\}$$

where $\phi^*$ is the Legendre–Fenchel dual (Górny et al., 18 Apr 2025). The $L^2$-subdifferential $\partial_\phi(u)$ can be locally characterized using an Anzellotti-type pairing $(\xi, Dv)$ defined via integration by parts and bounded using Young's inequality: $$\langle (\xi, Dv), \psi \rangle = -\int_{\Omega} \psi\,v\,\mathrm{Div}\,\xi\,dx - \int_{\Omega} v\,\xi\cdot\nabla\psi \,dx$$ with the measure-valued identity

$|(\xi, Dv)| \leq \phi(|Dv|) + \phi^*(x,|\xi|)\,dx$

and the optimality condition expressed as: $$\int_{\Omega} v\,w\,dx = \int_{\Omega} \phi(|Dv|) + \int_{\Omega} \phi^*(x,|\xi|)\,dx$$ for $w \in \partial_\phi(v)$ and $-\mathrm{Div}\,\xi = w$, subject to zero-flux (Neumann) boundary conditions (Górny et al., 18 Apr 2025).

These results generalize the known dual characterizations for TV and enable the rigorous specification of optimality and algorithmic design for variable-growth regularizers, including for double-phase ($p$–$q$) and $p(x)$ exponent cases.

3. Spatially Adaptive and Bilevel Regularization Parameter Selection

VGTV models commonly employ spatially dependent regularization parameters, either as part of the integrand (e.g., $\phi(x, t) = a(x)|t|$) or as weight fields and thresholds in Huberized or piecewise-Lipschitz-type functionals (Burger et al., 2019, Burger et al., 2019, Pagliari et al., 2021). Bilevel optimization frameworks have been developed for automatic selection of spatially distributed parameters, with upper-level objectives matching local residual statistics (e.g., noise variance) and lower-level constraints enforcing regularization via TV or TGV-type terms (Hintermüller et al., 2020, Pagliari et al., 2021).

For example, in the weighted Huber TV model, the regularizer is

$$TV_{a,\gamma}(u) = \int_\Omega a(x)\,f_\gamma(x, \nabla u(x))\,dx + \int_\Omega a(x)d|D^su|$$

where $f_\gamma(x, z)$ is the spatially-adaptive Huber function, with parameters $a(x)$ and $\gamma(x)$ (Huber threshold) derived from a bilevel training procedure. The parameter fields may be optimized for improved local edge preservation, artifact reduction, and detail sensitivity.

4. Algorithmic Approaches and Numerical Schemes

VGTV models typically require non-smooth, possibly non-convex minimization procedures with spatially-variable integrands. Prominent algorithmic choices include primal-dual hybrid gradient (PDHG), augmented Lagrangian methods (ALM), split Bregman, and conditional gradient strategies.

• In the variable-exponent modular setting, the proximal operator and convex conjugate are evaluated pointwise, exploiting closed-form or efficiently computable solutions for $p(x)\in[1,2]$ (Kohr, 2017).
• For piecewise-Lipschitz regularization, the functional $$\operatorname{pwL}^\gamma(u) = \min_{|g| \leq \gamma} \|Du-g\|_M$$ admits primal-dual optimization and equivalently $(|Du|-\gamma)_+$ representation, with closed-form proximal mappings for the constraint (Burger et al., 2019, Burger et al., 2019).
• Bregman iterative algorithms have been developed for variable $p(x)$-TV regularization, with the exponent updated at every iteration and the regularizing term recomputed from smoothed/edge-detected images (Jia et al., 2016).
• Bilevel frameworks embed Newton-type methods for the lower-level problem (primal or dual TGV reconstruction) within projected gradient search at the upper level for spatial parameter tuning, relying on dualization and smoothness properties (Hintermüller et al., 2020, Pagliari et al., 2021).
• For dynamic settings, fully-corrective generalized conditional gradient algorithms have been adapted to solve for sparse BV curves in Wasserstein-$1$ space (Carioni et al., 12 May 2025).

5. Functional Properties, Topological Equivalence, and Analytical Results

VGTV functionals are designed to be convex (when integrands are convex) and lower semicontinuous with respect to weak-* BV convergence. Topological equivalence results situate new regularizers—such as piecewise-Lipschitz TV (pwL$^\gamma$)—within the broader TV framework, establishing that boundedness in one implies boundedness in another up to additive constants (Burger et al., 2019, Burger et al., 2019). For instance,

$$TV(u) - \gamma(\Omega) \leq \operatorname{pwL}^\gamma(u) \leq TV(u)$$

and kernel characterizations determine the set of functions unpenalized by the regularizer (e.g., functions with pointwise gradient bounded by $\gamma$).

Dual formulations (infimal convolution, supremal, or via Fenchel duality) enable the exploitation of convex geometry and measure-theoretic analysis, yielding explicit representation formulas and supporting convergence and optimality proofs for the associated numerical methods.

6. Applications in Inverse Problems and Imaging

VGTV regularization has found utility in a variety of imaging and inverse problem contexts:

• Fractional diffusion: variable $p(x)$ TV regularization enables stable inversion from noisy data while preserving edges and reducing staircase artifacts (Jia et al., 2016).
• Tomography and denoising: variable-exponent, modular-based TV supports edge sensitivity in dual-channel imaging, with competitive or superior performance relative to TGV in artifact suppression and smooth region reconstruction (Kohr, 2017, Hugonnet et al., 2022).
• Image restoration: models coupling TV (or its variable-growth modification) with optimal transport-based dual norms balance cartoon and oscillatory components, leveraging algorithmic advances in PDHG and ALM for improved contrast and artifact reduction (Huang et al., 19 Mar 2025).
• Dynamic inverse problems: extension to BV curves with essential temporal variation and sparse Dirac trajectory representations generalizes Lasso and TV regularization to time-evolving sources, supporting abrupt change modeling (Carioni et al., 12 May 2025).
• Machine learning: TV-based regularization efficiently controls complexity in generalized additive models (GAMs), with tight generalization error bounds and favorable statistical learnability (Matsushima, 2018).

7. Impact on Artifact Reduction and Feature Adaptation

VGTV models systematically address limitations of classical TV regularization, particularly staircasing, over-smoothing, and insufficient detail preservation. By enabling locally adaptive penalization—through either variable exponents, spatially-varying thresholds, or piecewise constraints—VGTV enhances reconstruction quality for inhomogeneous signals and images, particularly those combining sharp edges with smooth transitions or textures. The mathematical and algorithmic underpinnings facilitate provable control on regularizer-induced bias, allow for computational efficiency even in high-dimensional or temporally dynamic settings, and provide actionable frameworks for automated parameter selection via bilevel optimization.

VGTV therefore represents a unifying and flexible approach for regularization in modern variational imaging, inverse problems, and statistical modeling, with detailed theoretical guarantees and broad empirical validation.