Raymond Chan

Xiaohao Cai, Raymond Chan, Serena Morigi et al.

Tight-frame, a generalization of orthogonal wavelets, has been used successfully in various problems in image processing, including inpainting, impulse noise removal, super-resolution image restoration, etc. Segmentation is the process of identifying object outlines within images. There are quite a few efficient algorithms for segmentation that depend on the variational approach and the partial differential equation (PDE) modeling. In this paper, we propose to apply the tight-frame approach to automatically identify tube-like structures such as blood vessels in Magnetic Resonance Angiography (MRA) images. Our method iteratively refines a region that encloses the possible boundary or surface of the vessels. In each iteration, we apply the tight-frame algorithm to denoise and smooth the possible boundary and sharpen the region. We prove the convergence of our algorithm. Numerical experiments on real 2D/3D MRA images demonstrate that our method is very efficient with convergence usually within a few iterations, and it outperforms existing PDE and variational methods as it can extract more tubular objects and fine details in the images.

Hao Liu, Jun Liu, Raymond H. Chan et al.

In this study, our goal is to integrate classical mathematical models with deep neural networks by introducing two novel deep neural network models for image segmentation known as Double-well Nets. Drawing inspirations from the Potts model, our models leverage neural networks to represent a region force functional. We extend the well-know MBO (Merriman-Bence-Osher) scheme to solve the Potts model. The widely recognized Potts model is approximated using a double-well potential and then solved by an operator-splitting method, which turns out to be an extension of the well-known MBO scheme. Subsequently, we replace the region force functional in the Potts model with a UNet-type network, which is data-driven and is designed to capture multiscale features of images, and also introduce control variables to enhance effectiveness. The resulting algorithm is a neural network activated by a function that minimizes the double-well potential. What sets our proposed Double-well Nets apart from many existing deep learning methods for image segmentation is their strong mathematical foundation. They are derived from the network approximation theory and employ the MBO scheme to approximately solve the Potts model. By incorporating mathematical principles, Double-well Nets bridge the MBO scheme and neural networks, and offer an alternative perspective for designing networks with mathematical backgrounds. Through comprehensive experiments, we demonstrate the performance of Double-well Nets, showcasing their superior accuracy and robustness compared to state-of-the-art neural networks. Overall, our work represents a valuable contribution to the field of image segmentation by combining the strengths of classical variational models and deep neural networks. The Double-well Nets introduce an innovative approach that leverages mathematical foundations to enhance segmentation performance.