You-Wei Wen

In this work, we propose a novel two-stage image segmentation method addressing the issue of intensity inhomogeneities in images. In the first stage, a non-convex variational decomposition model based on a tight wavelet frame is proposed. The proposed non-convex variational decomposition model is a weighted sum of three terms: the data fitting term, the non-convex regularization term based on the tight wavelet frame for the structural component (piecewise constant), and the second-order gradient regularization term for the intensity component (smooth). The existence of a solution for the proposed decomposition model is established, and the uniqueness is guaranteed under certain conditions. In the second stage, a thresholding technique is applied to the piecewise constant component obtained in the first stage to achieve the final image segmentation result. This two-stage strategy focuses on the structural features of the image during image segmentation, thereby mitigating the impact of intensity inhomogeneities. In addition, this paper develops an effective algorithm based on variational Bayesian inference that simultaneously segments images and selects model parameters, reducing the burden of manual parameter adjustment. Numerical experimental results demonstrate that the proposed method not only effectively reduces illumination and shadow interference but also exhibits significant competitiveness in processing images with intensity inhomogeneities.

Tensor robust principal component analysis (TRPCA) holds a crucial position in machine learning and computer vision. It aims to recover underlying low-rank structures and to characterize the sparse structures of noise. Current approaches often encounter difficulties in accurately capturing the low-rank properties of tensors and balancing the trade-off between low-rank and sparse components, especially in a mixed-noise scenario. To address these challenges, we introduce a Bayesian framework for TRPCA, which integrates a low-rank tensor nuclear norm prior and a generalized sparsity-inducing prior. By embedding the priors within the Bayesian framework, our method can automatically determine the optimal tensor nuclear norm and achieve a balance between the nuclear norm and sparse components. Furthermore, our method can be efficiently extended to the weighted tensor nuclear norm model. Experiments conducted on synthetic and real-world datasets demonstrate the effectiveness and superiority of our method compared to state-of-the-art approaches.

Sparse signal recovery from underdetermined systems presents significant challenges when using conventional [Formula: see text] and [Formula: see text] penalties, primarily due to computational complexity and estimation bias. This paper introduces a truncated Huber penalty, a nonconvex metric that effectively bridges the gap between unbiased sparse recovery and differentiable optimization. The proposed penalty applies quadratic regularization to small entries while truncating large magnitudes, avoiding nondifferentiable points at optimal solutions. Theoretical analysis demonstrates that, for an appropriately chosen threshold, any [Formula: see text]-sparse solution recoverable via conventional penalties remains a local optimum under the truncated Huber function. This property allows the exact and robust recovery theories developed for other penalty regularization functions to be directly extended to the truncated Huber function. To solve the optimization problem, we develop a block coordinate descent (BCD) algorithm with finite-step convergence guarantees under spark conditions. Numerical experiments validate the effectiveness and robustness of the proposed method in both synthetic and real scenarios. Furthermore, we demonstrate the flexibility of the truncated Huber framework through two extensions: one to an adaptively weighted variant inspired by sorted penalties, and the other to the gradient domain for applications such as signal denoising and image smoothing.

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.

Tensor Robust Principal Component Analysis (TRPCA) is a fundamental technique for decomposing multi-dimensional data into a low-rank tensor and an outlier tensor, yet existing methods relying on sparse outlier assumptions often fail under structured corruptions. In this paper, we propose a self-guided data augmentation approach that employs adaptive weighting to suppress outlier influence, reformulating the original TRPCA problem into a standard Tensor Principal Component Analysis (TPCA) problem. The proposed model involves an optimization-driven weighting scheme that dynamically identifies and downweights outlier contributions during tensor augmentation. We develop an efficient proximal block coordinate descent algorithm with closed-form updates to solve the resulting optimization problem, ensuring computational efficiency. Theoretical convergence is guaranteed through a framework combining block coordinate descent with majorization-minimization principles. Numerical experiments on synthetic and real-world datasets, including face recovery, background subtraction, and hyperspectral denoising, demonstrate that our method effectively handles various corruption patterns. The results show the improvements in both accuracy and computational efficiency compared to state-of-the-art methods.

Sparse signal recovery from under-determined systems presents significant challenges when using conventional L_0 and L_1 penalties, primarily due to computational complexity and estimation bias. This paper introduces a truncated Huber penalty, a non-convex metric that effectively bridges the gap between unbiased sparse recovery and differentiable optimization. The proposed penalty applies quadratic regularization to small entries while truncating large magnitudes, avoiding non-differentiable points at optima. Theoretical analysis demonstrates that, for an appropriately chosen threshold, any s-sparse solution recoverable via conventional penalties remains a local optimum under the truncated Huber function. This property allows the exact and robust recovery theories developed for other penalty regularization functions to be directly extended to the truncated Huber function. To solve the optimization problem, we develop a block coordinate descent (BCD) algorithm with finite-step convergence guarantees under spark conditions. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed approach. Furthermore, we extend the truncated Huber-penalized model to the gradient domain, illustrating its applicability in signal denoising and image smoothing.