Guided Variational Network for Image Decomposition

TL;DR

The paper tackles cartoon–texture image decomposition by replacing globally weighted variational terms with spatially adaptive quadratic norms, enabling efficient, interpretable, and robust separation. It introduces Guided Variational Decomposition (GVD) and its neural variant NGVD, which learn pixel-wise weights and global regularization via a bilevel framework; the inner problem remains a quadratic solve, solved by a stable linear system, while the outer network-guided updates promote adaptive structure discovery. The authors provide a rigorous fixed-point convergence analysis with explicit constants, Lipschitz stability, and contraction conditions, along with extensive experiments showing superior decomposition quality and edge preservation versus state-of-the-art methods. The work offers automatic parameter selection and self-tuning through data-driven and model-based weight estimation, with practical implications for preprocessing in denoising, recognition, and medical imaging.

Abstract

Cartoon-texture image decomposition is a critical preprocessing problem bottlenecked by the numerical intractability of classical variational or optimization models and the tedious manual tuning of global regularization parameters.We propose a Guided Variational Decomposition (GVD) model which introduces spatially adaptive quadratic norms whose pixel-wise weights are learned either through local probabilistic statistics or via a lightweight neural network within a bilevel framework.This leads to a unified, interpretable, and computationally efficient model that bridges classical variational ideas with modern adaptive and data-driven methodologies. Numerical experiments on this framework, which inherently includes automatic parameter selection, delivers GVD as a robust, self-tuning, and superior solution for reliable image decomposition.

Figures (5)

• Figure 1: Overview of the proposed NGVD framework: (a) the guided variational network for image decomposition; (b): the scalar MLP predicting global regularization scalars $\lambda_1,\lambda_2$; (c): the detailed structure of weight predictor $\mathcal{W}_{\Theta_2}$.
• Figure 2: Example 1 -- Iterative image decomposition for a synthetic image and probabilistic baseline. First column: observed image, ground-truth cartoon, and ground-truth texture (each spanning two rows). Columns two to four: results at iterations $k = 1,4,8$ of the proposed method, showing from top to bottom $W_1$, $W_2$, reconstructed cartoon (with PSNR), cartoon residual (with RMSE), reconstructed texture (with PSNR), and texture residual (with RMSE). The last column reports the same quantities for the probabilistic baseline.
• Figure 3: Example 1- Comparison of isotropic vs. anisotropic weights. Left: observed and weight maps (isotropic $W_1/W_2$; anisotropic split into $x/y$). Right: reconstructed cartoon/texture with overlaid SSIM.
• Figure 4: Example2 - Image decomposition results. Left column: observed input (spanning two rows). Top row per sample: reconstructed cartoon; bottom row: reconstructed texture. PSNR values overlaid at higher-left of each method result (ground truth column intentionally has no PSNR).
• Figure 5: Example2 - Real-world image decomposition results. Left column: observed input (spanning two rows, with zoomed regions integrated below). Top row per sample: cartoon component; bottom row: texture component.

Theorems & Definitions (15)

• proposition thmcounterproposition
• proof
• proposition thmcounterproposition
• proposition thmcounterproposition: Computable Lipschitz bound for $\mathcal{W}_{\Theta}$
• lemma thmcounterlemma
• theorem 1: Lipschitz bound
• theorem 2: Existence and contractive convergence
• proposition thmcounterproposition: Stability to data perturbation
• proof
• proof