Euler's Elastica Based Cartoon-Smooth-Texture Image Decomposition

TL;DR

This work tackles the cartoon–smooth–texture image decomposition by introducing a three-component variational model $f=v+w+n$ that leverages an $L^0$-gradient energy together with a curvature regularization on level lines (Euler's elastica) for the structural part, a squared Laplacian penalty for isotropic smoothness of the shading component, and an inverse Sobolev seminorm to capture oscillations. The authors reformulate the problem with auxiliary fields and develop a novel operator-splitting scheme, enabling four subproblems that are either solvable in closed form or efficiently with FFT-based linear solves, achieving per-iteration complexity $O(MN(\log M+\log N))$. Through extensive numerical experiments, including ablation studies and comparisons with CEP and HKLM, the method demonstrates superior decomposition quality, reduced staircase artifacts, and substantial computational speedups. The approach is robust to parameter choices, demonstrates clear visual improvements on real images, and opens avenues for extensions to color data and separate modeling of texture/noise components.

Abstract

We propose a novel model for decomposing grayscale images into three distinct components: the structural part, representing sharp boundaries and regions with strong light-to-dark transitions; the smooth part, capturing soft shadows and shades; and the oscillatory part, characterizing textures and noise. To capture the homogeneous structures, we introduce a combination of $L^0$-gradient and curvature regularization on level lines. This new regularization term enforces strong sparsity on the image gradient while reducing the undesirable staircase effects as well as preserving the geometry of contours. For the smoothly varying component, we utilize the $L^2$-norm of the Laplacian that favors isotropic smoothness. To capture the oscillation, we use the inverse Sobolev seminorm. To solve the associated minimization problem, we design an efficient operator-splitting algorithm. Our algorithm effectively addresses the challenging non-convex non-smooth problem by separating it into sub-problems. Each sub-problem can be solved either directly using closed-form solutions or efficiently using the Fast Fourier Transform (FFT). We provide systematic experiments, including ablation and comparison studies, to analyze our model's behaviors and demonstrate its effectiveness as well as efficiency.

Figures (16)

• Figure 1: Motivation for curvature regularuzation on level lines. (a) Clean image. (b) Noisy image ($\sigma=120/255$). (c) Denoised image by $L^0$-gradient. (d) Denoised image by the MC penalty function zhang2010nearly. (e) Denoised image by TV. (f) Denoised image by Euler's elastica mumford1994elastica. The noise is captured by $H^{-1}$-norm approximating the Meyer's $G$-norm. Observe that $L^0$-gradient and MC penalty function produce irregular boundaries, and TV-norm creates staircase effects. These artifacts are reduced in (f) when incorporating the curvature regularization on the level lines.
• Figure 2: Three-component image decomposition by the proposed method. A grayscale image $f$ is decomposed into a structural part $v^*$, a smooth part $w^*$, and an oscillatory part $n^*$. The sum of $v^*$ and $w^*$, denoted by $u^*$, can be regarded as the denoised version of $f$ if the clean image does not have rich textures. Our effectively reduces the staircase effects.
• Figure 3: Decomposition of synthetic images with noise ($\sigma=20/255$). The first column ($f_{\text{clean}}$) shows the clean images; the second column ($f_{\text{noise}}$) shows the noisy inputs; the third column ($u^*$) shows summations of the identified $v^*$ and $w^*$ components; the subsequent columns show respectively $v^*$, $w^*$ and $n^*$, and for visualization, they are linearly scaled between 0 and 1. Model parameters for these examples are: $\alpha_0=2\times10^{-2}$, $\alpha_\text{curv}=0.1$, $\alpha_w = 80$, and $\alpha_n=1\times10^{-5}$. PSNR values for $f_{\text{noise}}$ and $u^*$ are computed, and the standard deviations (STD) of respective $n^*$ are reported.
• Figure 4: Illustration of the proposed method without scaling the components. (a) Input noisy image ($\sigma=20/255$). (b) Identified structure component $v^*$. (c) Identified smooth component $w^*$. (d) Identified noise component $n^*$. (f) The reconstructed $u^*$ component (PSNR$=36.05$). (g)-(i) are the scaled versions of (b)-(d), respectively. The proposed method successfully separates visually distinctive components. Noticeably, $n^*$ in (d) captures only unstructured oscillations (STD=$19.77/255$).
• Figure 5: Decomposition of photographic images with noise ($\sigma=20/255$). The first column ($f_{\text{clean}}$) shows the clean images; the second column ($f_{\text{noise}}$) shows the noisy inputs; the third column ($u^*$) shows summations of the identified $v^*$ and $w^*$ components; the subsequent columns show respectively $v^*$, $w^*$ and $n^*$, and for visualization, they are linearly scaled between 0 and 1. Model parameters for these examples are: $\alpha_0=2\times10^{-3}$, $\alpha_\text{curv}=0.5$, $\alpha_w = 50$, and $\alpha_n=0.1$. PSNR values for $f_{\text{noise}}$ and $u^*$ are computed, and the standard deviations (STD) of respective $n^*$ are reported.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Remark 3
• Proposition 2.1
• Remark 4
• Remark 5
• Remark 6
• proof