Image Decomposition with G-norm Weighted by Total Symmetric Variation

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.

Dynamic PET Image Reconstruction via Non-negative INR Factorization

The reconstruction of dynamic positron emission tomography (PET) images from noisy projection data is a significant but challenging problem. In this paper, we introduce an unsupervised learning approach, Non-negative Implicit Neural Representation Factorization (\texttt{NINRF}), based on low rank matrix factorization of unknown images and employing neural networks to represent both coefficients and bases. Mathematically, we demonstrate that if a sequence of dynamic PET images satisfies a generalized non-negative low-rank property, it can be decomposed into a set of non-negative continuous functions varying in the temporal-spatial domain. This bridges the well-established non-negative matrix factorization (NMF) with continuous functions and we propose using implicit neural representations (INRs) to connect matrix with continuous functions. The neural network parameters are obtained by minimizing the KL divergence, with additional sparsity regularization on coefficients and bases. Extensive experiments on dynamic PET reconstruction with Poisson noise demonstrate the effectiveness of the proposed method compared to other methods, while giving continuous representations for object's detailed geometric features and regional concentration variation.

A Formalization of Image Vectorization by Region Merging

Image vectorization converts raster images into vector graphics composed of regions separated by curves. Typical vectorization methods first define the regions by grouping similar colored regions via color quantization, then approximate their boundaries by Bezier curves. In that way, the raster input is converted into an SVG format parameterizing the regions' colors and the Bezier control points. This compact representation has many graphical applications thanks to its universality and resolution-independence. In this paper, we remark that image vectorization is nothing but an image segmentation, and that it can be built by fine to coarse region merging. Our analysis of the problem leads us to propose a vectorization method alternating region merging and curve smoothing. We formalize the method by alternate operations on the dual and primal graph induced from any domain partition. In that way, we address a limitation of current vectorization methods, which separate the update of regional information from curve approximation. We formalize region merging methods by associating them with various gain functionals, including the classic Beaulieu-Goldberg and Mumford-Shah functionals. More generally, we introduce and compare region merging criteria involving region number, scale, area, and internal standard deviation. We also show that the curve smoothing, implicit in all vectorization methods, can be performed by the shape-preserving affine scale space. We extend this flow to a network of curves and give a sufficient condition for the topological preservation of the segmentation. The general vectorization method that follows from this analysis shows explainable behaviors, explicitly controlled by a few intuitive parameters. It is experimentally compared to state-of-the-art software and proved to have comparable or superior fidelity and cost efficiency.

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

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.