The overdraw problem of scatterplots seriously interferes with the visual tasks. Existing methods, such as data sampling, node dispersion, subspace mapping, and visual abstraction, cannot guarantee the correspondence and consistency between the data points that reflect the intrinsic original data distribution and the corresponding visual units that reveal the presented data distribution, thus failing to obtain an overlap-free scatterplot with unbiased and lossless data distribution. A dual space coupling model is proposed in this paper to represent the complex bilateral relationship between data space and visual space theoretically and analytically. Under the guidance of the model, an overlap-free scatterplot method is developed through integration of the following: a geometry-based data transformation algorithm, namely DistributionTranscriptor; an efficient spatial mutual exclusion guided view transformation algorithm, namely PolarPacking; an overlap-free oriented visual encoding configuration model and a radius adjustment tool, namelyfrdraw. Our method can ensure complete and accurate information transfer between the two spaces, maintaining consistency between the newly created scatterplot and the original data distribution on global and local features. Quantitative evaluation proves our remarkable progress on computational efficiency compared with the state-of-the-art methods. Three applications involving pattern enhancement, interaction improvement, and overdraw mitigation of trajectory visualization demonstrate the broad prospects of our method.

For real-world optimization problems, a uniformly and widely distributed Pareto optimal set (PS) in the decision space can provide more choices for decision makers. However, most of multi-objective evolutionary algorithms (MOEAs) only consider convergence and diversity in the objective space, which rarely pay attention to diversity in the decision space. Especially for multimodal multi-objective optimization problems (MMOPs), there may exist multiple distinct PSs corresponding to the same Pareto front (PF). Thus, we propose a novel multimodal multi-objective evolutionary algorithm using a density-based one-by-one update strategy in this paper, which considers diversity in both the objective and decision spaces. In the proposed algorithm, once an offspring is generated during evolution, the most crowded subregion with the largest niche count in the objective space has to be identified again, helpful to maintain diversity in the objective space. Furthermore, the harmonic average distance approach is used to estimate the global density of solutions in the decision space, trying to maintain the population's diversity in the decision space. Our proposed algorithm is compared with several state-of-the-art algorithms on MMOPs. The experimental results demonstrate that our algorithm is capable of preserving promising solutions with even distribution in both of decision space and objective space and also shows the superiority on solving the adopted MMOPs.