Te Qi

We propose a new approach to perform the boosted difference of convex functions algorithm (BDCA) on non-smooth and non-convex problems involving the difference of convex (DC) functions. The recently proposed BDCA uses an extrapolation step from the point computed by the classical DC algorithm (DCA) via a line search procedure in a descent direction to get an additional decrease of the objective function and accelerate the convergence of DCA. However, when the first function in DC decomposition is non-smooth, the direction computed by BDCA can be ascent and a monotone line search cannot be performed. In this work, we proposed a monotone improved boosted difference of convex functions algorithm (IBDCA) for certain types of non-smooth DC programs, namely those that can be formulated as the difference of a possibly non-smooth function and a smooth one. We show that any cluster point of the sequence generated by IBDCA is a critical point of the problem under consideration and that the corresponding objective value is monotonically decreasing and convergent. We also present the global convergence and the convergent rate under the Kurdyka-Lojasiewicz property. The applications of IBDCA in image recovery show the effectiveness of our proposed method. The corresponding numerical experiments demonstrate that our IBDCA outperforms DCA and other state-of-the-art DC methods in both computational time and number of iterations.

Image deblurring is a challenging inverse problem due to its ill-posed nature, leading to various strategies for enhancing stability. The Rudin-Osher-Fatemi (ROF) model is a seminal framework recognized for its effectiveness in image denoising and deblurring, inspiring numerous variational techniques. Despite existing approaches focusing on solving analysis-based, synthesis-based, and balanced models, achieving a more accurate solution to the ROF model remains an open question. The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is a widely applied numerical algorithm known for its efficiency in large-scale optimization. However, FISTA’s effectiveness in solving the ROF model is constrained because it is primarily designed for synthesis models when the sparsifying transform is orthonormal. To overcome this limitation, we propose a novel Kernel FISTA (K-FISTA), integrated with a structured Pseudoinverse Image Formulation (PIF) model. In practice, our approach outperforms classical methods like ADMM and sPADMM by achieving lower final energy values, offering a more accurate numerical approximation to the common minimizer while maintaining fast convergence. Extensive experiments validate the effectiveness of the K-FISTA method in image deblurring, and we provide theoretical guarantees for its convergence.

Image dehazing is regarded as a fundamental image processing task with a major impact on higher-level imaging tasks. Many existing haze removal methods are designed for homogeneous haze, but in real-world cases, the haze is normally non-homogeneous. Superpixels, which segment an image into a set of closely spaced regions, can be employed in real-world scenarios to deal with non-homogeneous haze. In our paper, an adaptive non-homogeneous image dehazing approach that utilizes the superpixel-guided algorithm is designed to segment different hazy regions. Considering that both ambient light and transmission map estimation have a significant impact on the results, our research focuses on the development of a variational dehazing model that takes into account non-uniform ambient light and non-uniform transmission maps to address varying levels of haze. A series of numerical results illustrate the superiority and efficacy of our method.

Blind image deblurring is a critical and challenging task in the field of imaging science due to its severe ill-posedness. Appropriate prior information and regularizations are normally introduced to alleviate this problem. Inspired by the fact that the matrix representing a natural image is intrinsically low-rank or approximately low-rank, we employ the low-rank matrix approximation (LRMA) approach for tackling blind image deblurring problems with unknown kernels. When applied to color image restoration tasks, making use of the quaternion representation in the hypercomplex domain enables us to better illustrate the inner relationships among color channels and thus more accurately characterize color image structure. Following this idea, we develop a novel model for color image blind deblurring by implementing the quaternion representation to the LRMA method. This proposed model facilitates better results for blur kernel estimation through preserving the sharper color intermediate latent image, which is first implemented for addressing the blind color image deblurring problem. Extensive numerical experiments demonstrate that our proposed quaternion-aware low-rank prior model greatly improves the performance when compared with the conventional low-rank based scheme and outperforms some of the state-of-the-art methods in terms of some criteria and visual quality.