Sparse Wasserstein Blue Noise Sampling

Objects in real space contain complex shape information,a new technology called three-dimensional scanning was used which can be reproduced in computer initially.However,the completeness of these information was restricted by the incompleteness of scanning equipment technology,resulting in many defects in the data obtained.To facilitate the follow-up work,the blue noise sampling was studied according to the actual needs.First,the relevant theories have been strictly improved and gradually developed into practical application.Based on the discrete optimal transport theory of multi-scale hierarchical partition,the sparse blue noise sampling algorithm was completed through probability measure and probability density model under multi-scale hierarchical partition model.Next,the sparse blue noise sampling framework is used to study the single-class and multi-class sampling problems,and extended to adaptive sampling and color-stippling sampling.This thesis discusses the sparse representation of Wasserstein blue noise sampling algorithm which can significantly reduce the memory requirement.The innovation lies in the following two aspects:1.Sparse blue noise sampling algorithm based on multi-scale hierarchical partition optimal transport theoryWasserstein blue noise sampling method provides a relaxed sampling method by constraining Wasserstein barycenter with a set of density distributions,but the global dense distance matrix used in solving the optimal transport problem consumes a lot of memory.To reduce the memory use to adapt to large-scale data,a sparse blue noise sampling algorithm was proposed by multi-scale hierarchical partition in this thesis.The global optimal sparse transport plan is obtained through solving a series of discrete sparse problems to deanalogize continuous problems.According to transport plan,the barycenter in Wasserstein space is obtained to approximate the sampling point position,and then get a group of sampling points with high quality blue noise.2.Multi-class sparse blue noise sampling algorithm under multi-scale hierarchical partition frameworkSingle-class blue noise sampling is difficult to describe these de facto changes since the phenomena produced by things in nature vary widely.Multi-class blue noise sampling can well expand the application of sampling technology,but the high cost is still an urgent problem to be solved.Memory requirement can be reduced by multi-class sparse blue noise sampling algorithm in hierarchical framework and Wasserstein barycenter redefines the way to update the position of sampling point under the constraint of multi-probability distribution.Good blue noise properties can also be maintained between categories.Experiments of constant density function sampling,halftone sampling and colorstippling are carried out according to the sparse blue noise sampling method proposed in this thesis.The results fully prove that high quality blue noise distribution points set can be obtained and memory consumption can be reduced by sparse method.