Image Decomposition with G-norm Weighted by Total Symmetric Variation

TL;DR

The paper addresses image decomposition into cartoon and texture components by leveraging a non-local Total Symmetric Variation (TSV) to guide a boundary-aware weighting. It combines a TV-based cartoon term with a weighted Meyer's $G$-norm on the texture, where the weight $\eta$ is constructed from TSV-derived information, and proves the existence of minimizers for BV inputs with bounded TSV. An efficient operator-splitting algorithm (Lie/Marchuk–Yanenko scheme) with barrier and inverse Sobolev approximations solves the resulting non-convex problem, achieving scalable per-iteration cost. Numerical experiments on mosaics and real images demonstrate effective structure-texture separation, robustness to TSV discretization, and a re-starting strategy that accumulates texture components to improve results. The approach offers boundary-aware texture segmentation with potential extensions to color data and multiscale processing, enhancing practical deployment in image analysis and synthesis.

Abstract

In this paper, we propose a novel variational model for decomposing images into their respective cartoon and texture parts. Our model characterizes certain non-local features of any Bounded Variation (BV) image by its Total Symmetric Variation (TSV). We demonstrate that TSV is effective in identifying regional boundaries. Based on this property, we introduce a weighted Meyer's $G$-norm to identify texture interiors without including contour edges. For BV images with bounded TSV, we show that the proposed model admits a solution. Additionally, we design a fast algorithm based on operator-splitting to tackle the associated non-convex optimization problem. The performance of our method is validated by a series of numerical experiments.

Figures (4)

• Figure 1: Image gradient (b) and proposed Total Symmetric Variation (TSV) \ref{['eq_TSV']} (c), computed for (a). TSV is high at regional boundaries while remains low in homogeneous and textural interiors. We exploit this property for image decomposition.
• Figure 2: Re-starting technique. (a) Input image $u^{(0)}$. (b) Normalized weight $\eta(u^{(0)})$ weight function. (c) $u^{(1)}$ component. (d) Zoom-in of (c). (e) $v^{(1)}$ component. Restart the algorithm to decompose (f) $u^{(1)}$ which is identical to (c). (g) $\eta(u^{(1)})$ weight function. (h) $u^{(2)}$ component. (i) Zoom-in of (h). (j) The accumulated texture part $v^{(1)}+v^{(2)}$. Here we set $\alpha_1=0.03, \alpha_2 = 0.3, \theta = 1\times10^{-6}$, $\sigma_1=2.75$, $\sigma_2=0.75$, and $\kappa = 0.1$.
• Figure 3: Results by proposed method for a mosaic picture (a) and its zoom-ins in red boxes (a$_1$) and (a$_2$); normalized scale $\eta$ (d), (d$_1$), (d$_2$); resulting component $u$ (b), (b$_1$), (b$_2$), and the texture component (c), (c$_1$), (c$_2$) when using constant $\eta$ values reproducing the model in osher2003image. The corresponding component $u$ (e), (e$_1$), (e$_2$) and texture (f), (f$_1$), (f$_2$) are obtained using $\alpha_1=0.09, \alpha_2 = 0.01, \theta = 1\times10^{-6}$, $\sigma_1=1.5$, $\sigma_1=0.1$, and $\kappa = 0.01$. These parameters vary depending on the image sizes and texture scales.
• Figure 4: Effects of the TSV parameter $\sigma$. Input image (a); resulting component $u$ (left half) and normalized $\eta$ (right half) in (b) and texture component in (d) when using $\sigma_1=1.5$ and $\sigma_2=0.1$; the corresponding results when using $\sigma_1=1.5$ and $\sigma_2=2$ are in (c) and (e). The second row shows the zoom-ins of the first row in the red box. The other parameters are $\alpha_1=0.15, \alpha_2 = 0.01, \theta = 1\times10^{-6}$, and $\kappa = 0.01$.

Theorems & Definitions (3)

• Remark 1
• Theorem 1
• proof