There exist many multi-objective optimization problems (MOPs) containing several inequality and equality constraints in practical applications, which are known as CMOPs. CMOPs pose great challenges for existing multi-objective evolutionary algorithms (MOEAs) since the difficulty in balancing the objective minimization and constraint satisfaction. Without loss of generality, the distribution of the Pareto set for a continuous m-objective CMOP can be regarded as a piecewise continuous manifold of dimension (m − 1). According to this property, a self-organizing map (SOM) approach for constrained multi-objective optimization problems is proposed in this article. In the proposed approach, we adopt the strategy of two population evolution, in which one population is evolved by considering all the constraints and the other population is used to assist in exploring the areas. In the evolutionary stage, each population is assigned a self-organizing map for discovering the population distribution structure in the decision space. After the topological mapping, we utilize the extracted neighborhood relationship information to generate promising offspring solutions. Afterwards, the neuron weight vectors of SOM are updated by the objective vectors of the surviving offsprings. Through the proposed approach, we can make the population efficiently converge to the feasible region with suitable levels of diversity. In the experiments, we compare the proposed method with several state-of-the-art approaches by using 48 benchmark problems. The evaluation results indicate that the overwhelmingly superior performance of the proposed method over the other peer algorithms on most of the tested problems. The source code is available at https://github.com/hccccc92918/CMOSMA.

Though the accuracy and robustness of optical flow has been dramatically enhanced over the past few years, the issue of edge-blurring near the image and motion boundaries has remained a challenge in flow field estimation. In this paper, we propose a refined total variation with L1 norm (TV-L1) optical flow estimation approach using joint filtering, named JOF. First, we divide the image into three categorized regions: mutual-structure regions, inconsistent structure regions and smooth regions. The mutual-structure guided filter for optical flow estimation is constructed by extracting the mutual- structure regions of the flow field. Second, the refined TV-L1 optical flow model is proposed by incorporating the non-local term and mutual-structure guided filter objective function into the classical TV-L1 energy function. Furthermore, the novel TV-L1 optical flow objective function is minimized using a joint filtering program composed of a weighted median filter and a mutual-structure guided filter to optimize the estimated flow field during the coarse-to-fine optical flow computation scheme. Finally, we compare the proposed JOF method with several state-of-the-art approaches including variational and deep learning based optical flow models using the Middlebury, MPI-Sintel and UCF101 test databases. The evaluation results indicate that the proposed method has high accuracy and good robustness for flow field computation, especially owns the significant benefit of edge-preserving.

Even though optical flow approaches based on convolutional neural networks have achieved remarkable performance with respect to both accuracy and efficiency, large displacements and motion occlusions remain challenges for most of the existing learning-based models. To address the abovementioned issues, we propose in this paper a self-attention-based multiscale feature learning optical flow computation method with occlusion feature map prediction. First, we exploit a self-attention mechanism-based multiscale feature learning module to compensate for large displacement optical flows, and the presented module is able to capture long-range dependencies from the input frames. Second, we design a simple but effective self-learning module to acquire an occlusion feature map, in which the predicted occlusion map is utilized to correct the optical flow estimation in occluded areas. Third, we explore a hybrid loss function that integrates the photometric and smoothness losses into the classical endpoint error (EPE) based loss to ensure the accuracy and robustness of the presented network. Finally, we compare the proposed method with some state-of-the-art approaches using the MPI-Sintel and KITTI test databases. The experimental results demonstrate that the proposed method achieved a competitive performance with respect to both accuracy and robustness, and it produced the best results than other methods did under large displacements and occlusions.