Lingfeng Li

Raymond Chan, Lingfeng Li

Convolution-type integral equations arise from various fields, \textit{e.g.}, finite impulse response filters in signal processing and deblurring problems in image processing. When solving these equations, conventional numerical methods, like the multigrid method, can only efficiently solve the low-frequency components in the error, but not the high-frequency components. In this paper, we apply neural operators to address this issue. By adopting a preconditioning approach, we propose a novel training strategy that trains neural operators to solve the high-frequency components efficiently. Then, we combine the neural operators with some classical iterative solvers, like the weighted Jacobi method, to obtain an efficient hybrid iterative algorithm for the integral equations. We analyze the generalization error of our training strategy and the convergence of the hybrid iterative algorithm. We test our algorithms on large-scale and ill-conditioned linear systems discretized from one- and two-dimensional convolution-type integral equations. Our proposed algorithm significantly outperforms the multigrid method and the preconditioned conjugate gradient method in both iteration numbers and computational time.

Kehui Zhang, Lingfeng Li, Hao Liu et al.

Shape compactness is a key geometrical property to describe interesting regions in many image segmentation tasks. In this paper, we propose two novel algorithms to solve the introduced image segmentation problem that incorporates a shape-compactness prior. Existing algorithms for such a problem often suffer from computational inefficiency, difficulty in reaching a local minimum, and the need to fine-tune the hyperparameters. To address these issues, we propose a novel optimization model along with its equivalent primal-dual model and introduce a new optimization algorithm based on primal-dual threshold dynamics (PD-TD). Additionally, we relax the solution constraint and propose another novel primal-dual soft threshold-dynamics algorithm (PD-STD) to achieve superior performance. Based on the variational explanation of the sigmoid layer, the proposed PD-STD algorithm can be integrated into Deep Neural Networks (DNNs) to enforce compact regions as image segmentation results. Compared to existing deep learning methods, extensive experiments demonstrated that the proposed algorithms outperformed state-of-the-art algorithms in numerical efficiency and effectiveness, especially while applying to the popular networks of DeepLabV3 and IrisParseNet with higher IoU, dice, and compactness metrics on noisy Iris datasets. In particular, the proposed algorithms significantly improve IoU by 20% training on a highly noisy image dataset.