Roy Y. He

Roy Y. He, Hao Liu

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.

Jin Guo, Roy Y. He, Jean-Michel Morel

In 2020 Domingos introduced an interpolation formula valid for "every model trained by gradient descent". He concluded that such models behave approximately as kernel machines. In this work, we extend the Domingos formula to stochastic training. We introduce a stochastic gradient kernel that extends the deterministic version via a continuous-time diffusion approximation. We prove stochastic Domingos theorems and show that the expected network output admits a kernel-machine representation with optimizer-specific weighting. It reveals that training samples contribute through loss-dependent weights and gradient alignment along the training trajectory. We then link the generalization error to the null space of the integral operator induced by the stochastic gradient kernel. The same path-kernel viewpoint provides a unified interpretation of diffusion models and GANs: diffusion induces stage-wise, noise-localized corrections, whereas GANs induce distribution-guided corrections shaped by discriminator geometry. We visualize the evolution of implicit kernels during optimization and quantify out-of-distribution behaviors through a series of numerical experiments. Our results support a feature-space memory view of learning: training stores data-dependent information in an evolving tangent feature geometry, and predictions at test time arise from kernel-weighted retrieval and aggregation of these stored features, with generalization governed by alignment between test points and the learned feature memory.

Jianbo Cui, Roy Y. He

In this paper, we propose Stoch-IDENT, a novel framework for identifying stochastic partial differential equations (SPDEs) from observational data. Our method can handle linear and nonlinear high-order SPDEs driven by time-dependent Wiener processes, accommodating both additive and multiplicative noise structures. To investigate the identifiability of SPDEs from trajectory data, we analyze the spectral properties of the solution's mean and covariance for linear SPDEs with constant coefficients, as well as the dimension of the solution space for parabolic and hyperbolic types, generalizing the identifiability theory for deterministic PDEs. Algorithmically, the drift term is identified via a sample-mean generalization of existing methods for PDE identification. For the diffusion term, we formulate a sparse regression problem with quadratic measurements induced from drift residuals and feature covariances. To address this challenging non-convex and non-smooth optimization, we develop a new greedy algorithm, Quadratic Subspace Pursuit (QSP), and prove that QSP enjoys stable support recovery under certain conditions. We validate Stoch-IDENT on various SPDEs, demonstrating its effectiveness through quantitative and qualitative evaluations.

Roy Y. He, Sung Ha Kang, Jean-Michel Morel

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.

Roy Y. He, Martin Huska, Hao Liu

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.

Chaozhi Zhang, Wenxiang Ding, Roy Y. He et al.

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.