Source-linked AI summary
Bilevel parameter learning for higher-order total variation regularisation models
J. C. De los Reyes, C. -B. Schönlieb, T. Valkonen
TL;DR
The paper addresses how to choose regularisation parameters and assess reconstruction fitness for higher-order image denoising. It develops a function-space bilevel learning approach for TGV and ICTV, including a Huberised TV cost. Its comparison finds that average performance and the better choice depend on image structure and noise level.
Problem
Choosing the regulariser, fidelity term, and parameters affects reconstruction, while functional models can have limited adaptivity to real data.
Source: §1 · p. 2Method
The paper learns higher-order TV parameters in function space using a smoothed bilevel problem, considering both TGV and ICTV and a Huberised TV cost.
Source: §1 · pp. 2–5Results
ICTV performs slightly better than TGV on average, both outperform TV on average, and TGV performs better than ICTV for images with large smooth areas.
Source: Conclusion and Outlook · p. 27Takeaways & Limitations
ICTV appears better for arbitrary-structure images with moderate noise, while ICTV and TGV score comparably at high noise levels.
Source: §1 · p. 5, Conclusion and Outlook · p. 27The Fréchet differentiability proof uses TGV2's quasilinear structure, making extension to ICTV difficult without further regularisation.
Source: §3.1 · p. 11
Abstract
from arXiv · showhide
We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost, based on a Huber regularised TV-seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a quasi-Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV$^2$ and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.
1. Introduction
The introduction motivates learning regularisation parameters for higher-order image denoising and presents a function-space bilevel approach using TGV and ICTV. It also previews a comparison showing that the better-performing regulariser depends on image structure and noise level.
1. Introduction
- Choosing the regularisation weight, prior, fidelity term, and forward operator affects reconstruction, while functional models can lack adaptivity to real data.The introduction motivates combining model-based structure with data-based adaptation.Source: §1 · p. 2
- The bilevel approach learns first- and higher-order regularisation parameters in function space rather than optimising a discretised model.The parameters balance regularisation strength and the different orders of regularity; the paper considers denoising with K = Id and Gaussian noise.Source: §1 · pp. 2–3
- Higher-order TV addresses staircasing in images with structures such as piecewise-linear regions; the paper focuses on second-order TGV and ICTV.Standard TV is well suited to piecewise-constant images, while higher-order regularisation adds derivative information.Source: §1 · p. 3
- The analysis uses a smoothed constraint problem and box constraints on α and β for its existence proof, then derives an optimality system and BFGS learning algorithm.The paper also describes a semi-smooth Newton method for the denoising subproblems.Source: §1 · pp. 4–5
- Alongside the usual L2-squared fitness cost, the paper proposes a Huberised total-variation cost whose choice affects reconstructed-image quality.The L2-squared cost is closely related to PSNR; the Huberised cost is introduced as an alternative.Source: §1 · pp. 4–5
- ICTV performs slightly better than TGV2 on average, while TGV2 performs better on images with large smooth areas.For arbitrary-structure images with moderate noise, ICTV appears to perform better.Source: §1 · p. 5
2. Problem statement and existence analysis
The section formulates parameter learning for higher-order TV reconstruction from noisy and ground-truth image pairs, using either an L2^2 or Huberised TV cost. It establishes existence of optimal parameters for both TGV2 and ICTV under box constraints.
2. Problem statement and existence analysis
- The learning task selects α and β for a noisy image with fixed-variation normally distributed noise, using a ground-truth or sufficiently accurate target image.The lower-level reconstruction is a regularised version of the denoising problem; the authors work in infinite-dimensional function spaces.Source: §2 · p. 6
2.1. Formal statement.
- The Huber regularisation replaces the model's regularisation term, while an elliptic term is also added to the denoising functional.The Huberised TV cost is an alternative to the L2^2 cost in the learning objective.Source: §2.1 · pp. 6–7
2.2. Existence of an optimal solution.
- The existence proof uses weak-* lower semicontinuity of the Huber-regularised functional on measures to pass to a limit of minimising sequences.The bounded parameter box and convergence of the states are used to obtain an optimal limit point.Source: §2.2 · pp. 7–8
- For TGV2 and ICTV, an optimal parameter pair exists under finite box constraints with either the L2^2 or Huberised TV cost.The constraints are 0 ≤ α ≤ ᾱ and 0 ≤ β ≤ β̄, where the upper bounds are fixed and finite.Source: §2.2 · p. 8
3. Lagrange multipliers
The section establishes differentiability and adjoint results for the learning problem, then derives an optimality system and gradient formula for local minima. The Frechet differentiability result is for TGV2; ICTV has a weaker Gateaux result under the stated approach.
3.1. Differentiability of the solution operator.
- For TGV2, the denoising solution operator is Frechet differentiable, with its derivative characterized by a unique solution of a linearized equation.The proof uses the TGV2 variational form; ICTV admits a Gateaux differentiability result by the same proof technique, while extending the Frechet proof requires further regularisation.Source: §3.1 · pp. 9, 11
3.2. The adjoint equation.
- The adjoint equations are derived by a Lagrangian approach, and their solutions exist uniquely because the linearized equation is well-posed.The passage describes the derivation for both TGV2 and ICTV learning problems.Source: §3.2 · p. 11
3.3. Optimality condition.
- For local minima, the optimality system includes Lagrange multipliers and yields a gradient formula for the reduced cost functional.The gradient formula is identified as useful for designing solution algorithms.Source: §3.3 · p. 12
4. Numerical algorithms
The numerical approach uses a BFGS outer method with adjoint gradients and a semismooth Newton method for the lower-level problem. TGV2 learning requires warm initialisation in the reported experiments.
4. Numerical algorithms
- The proposed scalar-parameter learning method combines BFGS with line-search feasibility safeguards and a semismooth Newton solver for the lower-level problem.The BFGS matrix update is skipped when curvature is negative; the lower-level Newton method uses a modified Jacobi matrix.Source: §4 · p. 13, §4.1 · p. 14, §4.2 · p. 16
4.1. BFGS algorithm.
- The BFGS iteration solves the adjoint equation to calculate the gradient, updates its matrix cyclically, and uses Armijo line search while keeping parameters away from the constraint boundary.The step-length limit preserves positivity of the iterates without a projection step.Source: §4.1 · p. 14
4.2. An infeasible semi-smooth Newton method.
- The semismooth Newton method may temporarily take infeasible steps, and the cited convergence result requires nonsingular operator subdifferentials near the solution.A reduced system eliminates the state increment to improve performance with TGV2.Source: §4.2 · p. 16
4.3. Warm initialisation.
- TGV2 parameter learning needs warm initialisation in the authors' experiments, using a TV solution to choose the starting parameters and BFGS matrix.The authors report that the algorithm generally performed well for TV parameter learning, whereas TGV2 required this initialisation to obtain convergence.Source: §4.3 · p. 16
5. Experiments
The experiments assess parameter learning and compare regularisers and cost functionals across denoising settings. Across the statistical study, ICTV generally ranks ahead of TGV2 and TV, with results varying by image structure and noise level.
5.1. Gaussian denoising.
- For TGV2 learning, L1 η∇ yields better visual quality and SSIM than L2², while L2² yields better PSNR; TV shows no significant visual-quality difference between costs.The authors note that L2² is equivalent to PSNR.Source: §5.1 · p. 17
- Optimal TGV2 parameters generally satisfy β*/α* ∈ (0.75, 1.5)/ℓ and avoid the stair-casing observed with TV in the reported results.For ℓ≈128 or 256, the authors note the heuristic β ∈ (1, 1.5)α.Source: §5.1 · p. 17
5.2. Statistical testing.
- ICTV > TGV2 > TV overall; only at high noise does TGV2 appear to come on par with ICTV using the L1 η∇ cost.The ordering is based on the statistical tests and other reported data.Source: §5.2 · p. 21
- TGV2 performs better on images with large smooth areas, while ICTV generally performs better on more chaotic images.For the first dataset image ICTV does better, while for the second TGV2 does; where ICTV wins, little visual difference is apparent, while TGV2 has less stair-casing in smooth areas, especially at σ=20.Source: §5.2 · pp. 21, 24
- Batch-learning results are roughly similar to individual learning; TV frequently wins on individual images but statistically performs worse than ICTV and TGV2.Individual learning chooses image-specific parameters, while batch learning uses one parameter set for images at the same noise level.Source: §5.2 · pp. 22–23
5.3. The choice of cost functional.
- L1 η∇ generally achieves higher SSIM at low noise and in batch learning, whereas L2² yields better PSNR; the authors recommend L1 η∇ for visual quality.Direct SSIM optimisation is non-convex, while the proposed total-variation cost is convex.Source: §5.3 · p. 25
Conclusion and Outlook
- Across 200 images, ICTV performs slightly better than TGV on average, both outperform TV on average, and TGV performs better on images with large smooth areas.For high-noise denoising, ICTV and TGV score comparably well.Source: Conclusion and Outlook · pp. 26–27
- The proposed cost is presented as a compromise between image-quality measurement and analytical tractability, with better behavior than the previously used PSNR-related L2 cost.The authors note that using SSIM directly as the cost could pose analytical and numerical difficulties because it is non-convex.Source: Conclusion and Outlook · p. 27