Blue Noise Sampling Research Articles

Non‐Euclidean Sliced Optimal Transport Sampling

AbstractIn machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well‐dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real‐world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space 𝕊d, the hyperbolic plane ℍd, and the real projective plane ℙd. Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

Attribute-Aware RBFs: Interactive Visualization of Time Series Particle Volumes Using RT Core Range Queries.

Smoothed-particle hydrodynamics (SPH) is a mesh-free method used to simulate volumetric media in fluids, astrophysics, and solid mechanics. Visualizing these simulations is problematic because these datasets often contain millions, if not billions of particles carrying physical attributes and moving over time. Radial basis functions (RBFs) are used to model particles, and overlapping particles are interpolated to reconstruct a high-quality volumetric field; however, this interpolation process is expensive and makes interactive visualization difficult. Existing RBF interpolation schemes do not account for color-mapped attributes and are instead constrained to visualizing just the density field. To address these challenges, we exploit ray tracing cores in modern GPU architectures to accelerate scalar field reconstruction. We use a novel RBF interpolation scheme to integrate per-particle colors and densities, and leverage GPU-parallel tree construction and refitting to quickly update the tree as the simulation animates over time or when the user manipulates particle radii. We also propose a Hilbert reordering scheme to cluster particles together at the leaves of the tree to reduce tree memory consumption. Finally, we reduce the noise of volumetric shadows by adopting a spatially temporal blue noise sampling scheme. Our method can provide a more detailed and interactive view of these large, volumetric, time-series particle datasets than traditional methods, leading to new insights into these physics simulations.

Parallel Graph Signal Processing: Sampling and Reconstruction

Graph signal processing (GSP) extends classical signal processing methods to analyzing signals supported over irregular grids represented by graphs. Within the scope of GSP, sampling and reconstruction represent fundamental tools that have received considerable attention. For very large graphs, however, many of the current methods struggle with the computational and memory requirements. Vertex domain and randomized sampling strategies somewhat ameliorate the computational requirements, but these algorithms perform poorly at preserving signal fidelity for graphs with hundreds of thousands of vertices. To address these shortcomings, this paper introduces a new and scalable approach that can be easily parallelized. This new approach uses existing graph partitioning algorithms in concert with vertex-domain blue-noise sampling and reconstruction, performed independently across partitions. In the reconstruction, some degree of overlapping is added to the partitions to induce trans-partition smoothness in the recovered signal. We also propose two sampling schemes based on the spatial characteristic of the graph that minimizes the recovery error. The first combines graph partitioning with the Void-and-Cluster algorithm, while the second approach uses Error Diffusion. We conclude this paper with experiments on synthetic and real data that show the effectiveness of these new approaches on very large graphs.

Visual abstraction of dynamic network via improved multi-class blue noise sampling

Visual abstraction of dynamic network via improved multi-class blue noise sampling

Multi-class inverted stippling

We introduce inverted stippling , a method to mimic an inversion technique used by artists when performing stippling. To this end, we extend Linde-Buzo-Gray (LBG) stippling to multi-class LBG (MLBG) stippling with multiple layers. MLBG stippling couples the layers stochastically to optimize for per-layer and overall blue-noise properties. We propose a stipple-based filling method to generate solid color backgrounds for inverting areas. Our experiments demonstrate the effectiveness of MLBG in terms of reducing overlapping and intensity accuracy. In addition, we showcase MLBG with color stippling and dynamic multi-class blue-noise sampling, which is possible due to its support for temporal coherence.

Fast spectral clustering method based on graph similarity matrix completion

Spectral clustering (SC) is a widely used technique to perform group unsupervised classification of graph signals. However, SC is sometimes computationally intensive due to the need to calculate the graph similarity matrices on large high-dimensional data sets. This paper proposes an efficient SC method that rapidly calculates the similarity matrix using a matrix completion algorithm. First, a portion of the elements in the similarity matrix are selected by a blue noise sampling mask, and their similarity values are calculated directly from the original dataset. After that, a split Bregman algorithm based on the Schatten capped p norm is developed to rapidly retrieve the rest of the matrix elements. Finally, spectral clustering is performed based on the completed similarity matrix. A set of simulations based on different data sets are used to assess the performance of the proposed method. It is shown that for a sufficiently large sampling rate, the proposed method can accurately calculate the completed similarity matrix, and attain good clustering results while improving on computational efficiency.

Context-Aware Visual Abstraction of Crowded Parallel Coordinates

As the size of dataset increases, it has become a difficult task to explore structures of interest from crowded parallel coordinates due to visual clutter. Numerous methods have been proposed to simplify the visualization of crowded parallel coordinates, such as filtering, bundling and sampling. However, contextual structures are hardly preserved in the course of simplification, which make significant features easily lost in the simplified parallel coordinates. In this paper, we propose a context-aware visual sampling method for the exploration of crowded parallel coordinates. A Doc2Vec model, widely used in the field of Natural Language Processing (NLP), is utilized to represent the contextual structures across a series of attribute axes with quantifiable vectors. Then, an adaptive blue noise sampling model is employed to reduce the size of original dataset in the vectorized space, guarantying that data items with different contextual structures would be retained in the simplified parallel coordinates. A set of meaningful visual interfaces are designed, enabling users to easily capture the contextual features and evaluate the sampled parallel coordinates. Case studies based on real-world datasets and quantitative comparison have demonstrated the effectiveness of our method in the simplification of crowded parallel coordinates and the exploration of large scale multi-dimensional datasets.

Visual selection of standard wells for large scale logging data via discrete choice model

It is a time-consuming and laborious task to conduct accurate geological interpretation based on large-scale logging data. A subset of drilling wells are always selected undergoing detailed stratigraphic correlation, aiming to enhance the efficiency of geological interpretation and retain the matching accuracy of stratigraphic structures. However, standard wells are selected by hand in the course of traditional standard well selection, which will easily bring many uncertainties for subsequent interpretation of geological structures. In this paper, we propose a visual analytics system to support supervised standard well selection via a discrete choice model. Firstly, an adaptive blue noise sampling model is applied to determine spatial distribution of standard wells, and a stratigraphic correlation model is designed to measure the similarity between drilling wells from different attribute perspectives. Then, we utilize a discrete choice model to select standard wells with the spatial distribution and multiple attributes taken into consideration. Furthermore, several meaningful visualizations are designed allowing users to get deeper insights into the selection of standard wells, and a rich set of interactions are also provided enabling users to further optimize the discrete choice model. At last, a visualization framework is implemented to integrate the models and visual designs, by means of which the geological interpreters are able to visually select, evaluate and optimize the standard wells. The effectiveness of our work and its application values for geological interpretation are further demonstrated through case studies based on real-world datasets and interviews with domain experts.

Context-aware Sampling of Large Networks via Graph Representation Learning.

Numerous sampling strategies have been proposed to simplify large-scale networks for highly readable visualizations. It is of great challenge to preserve contextual structures formed by nodes and edges with tight relationships in a sampled graph, because they are easily overlooked during the process of sampling due to their irregular distribution and immunity to scale. In this paper, a new graph sampling method is proposed oriented to the preservation of contextual structures. We first utilize a graph representation learning (GRL) model to transform nodes into vectors so that the contextual structures in a network can be effectively extracted and organized. Then, we propose a multi-objective blue noise sampling model to select a subset of nodes in the vectorized space to preserve contextual structures with the retention of relative data and cluster densities in addition to those features of significance, such as bridging nodes and graph connections. We also design a set of visual interfaces enabling users to interactively conduct context-aware sampling, visually compare results with various sampling strategies, and deeply explore large networks. Case studies and quantitative comparisons based on real-world datasets have demonstrated the effectiveness of our method in the abstraction and exploration of large networks.

Visual exploration of urban functional zones based on augmented nonnegative tensor factorization

A city contains a variety of different urban functions with referring to the purpose of land use to support the diverse needs of urban residents, such as residence, working and recreation. Exploring urban functional zones is a critical task, which provides valuable applications for business site selection, transportation management and urban planning. It is well known that location information and human mobility semantics both are significant factors for identifying urban functional zones, just considering one factor is not effective. However, most of existing techniques capture the transformation of functional zones and interpret the results based solely on location or mobility semantics, and lack the capacity to deal with the multifaceted features of urban data. To tackle these problems, in this paper, we propose an interactive visual analytics system for effectively exploring urban functional zones based on spatio-temporal OD data and Points of Interest data. We first adaptively partition the territory into region units based on adaptive blue noise sampling method, extract POI feature matrix (location information) as a prior knowledge, model multidimensional spatio-temporal OD data as a tensor for addressing multifaceted features and propose an augment tensor-based algorithm that enables users to simultaneously combine mobility semantics and inherent location information for identifying functional zones. In addition, we design a set of visual encodings to better understand and interpret the results in a visual and intuitive manner. This system has been demonstrated using two case studies with a real-world dataset of HangZhou city and domain-expert interviews.

Blue-Noise Sampling of Graph and Multigraph Signals: Dithering on Non-Euclidean Domains

With the surge in the volumes and dimensions of data defined in non-Euclidean spaces, graph signal processing (GSP) techniques are emerging as important tools in our understanding of these domains [1]. A fundamental problem for GSP is to determine which nodes play the most important role; so, graph signal sampling and recovery thus become essential [2]. In general, most of the current sampling methods are based on graph spectral decompositions where the graph Fourier transform (GFT) plays a central role [2]. Although adequate in many cases, they are not applicable when the graphs are large and where spectral decompositions are computationally difficult [3]. After years of beautiful and useful theoretical insights developed in this problem, the interest has now centered on finding more efficient methods for the computation of good sampling sets. Looking to the spatial domain for inspiration, substantial research has been performed that looks at the use of spatial point processes to define stochastic sampling grids with a particular interest at point processes that generate blue noise.

Evaluation of Sampling Methods for Scatterplots.

Given a scatterplot with tens of thousands of points or even more, a natural question is which sampling method should be used to create a small but "good" scatterplot for a better abstraction. We present the results of a user study that investigates the influence of different sampling strategies on multi-class scatterplots. The main goal of this study is to understand the capability of sampling methods in preserving the density, outliers, and overall shape of a scatterplot. To this end, we comprehensively review the literature and select seven typical sampling strategies as well as eight representative datasets. We then design four experiments to understand the performance of different strategies in maintaining: 1) region density; 2) class density; 3) outliers; and 4) overall shape in the sampling results. The results show that: 1) random sampling is preferred for preserving region density; 2) blue noise sampling and random sampling have comparable performance with the three multi-class sampling strategies in preserving class density; 3) outlier biased density based sampling, recursive subdivision based sampling, and blue noise sampling perform the best in keeping outliers; and 4) blue noise sampling outperforms the others in maintaining the overall shape of a scatterplot.

Blue-noise sampling for human retinal cone spatial distribution modeling

This paper proposes a novel method for modeling retinal cone distribution in humans. It is based on Blue-noise sampling algorithms being strongly related with the mosaic sampling performed by cone photoreceptors in the human retina. Here we present the method together with a series of examples of various real retinal patches. The same samples have also been created with alternative algorithms and compared with plots of the center of the inner segments of cone photoreceptors from imaged retinas. Results are evaluated with different distance measure used in the field, like nearest-neighbor analysis and pair correlation function. The proposed method can effectively describe features of a human retinal cone distribution by allowing to create samples similar to the available data. For this reason, we believe that the proposed algorithm may be a promising solution when modeling local patches of retina.

Blue-Noise Sampling on Graphs

In the area of graph signal processing, a graph is a set of nodes arbitrarily connected by weighted links; a graph signal is a set of scalar values associated with each node; and sampling is the problem of selecting an optimal subset of nodes from which a graph signal can be reconstructed. This paper proposes the use of spatial dithering on the vertex domain of the graph, as a way to conveniently find statistically good sampling sets. This is done establishing that there is a family of good sampling sets characterized on the vertex domain by a maximization of the distance between sampling nodes; in the Fourier domain, these are characterized by spectrums that are dominated by high frequencies referred to as blue-noise. The theoretical connection between blue-noise sampling on graphs and previous results in graph signal processing is also established, explaining the advantages of the proposed approach. Restricting our analysis to undirected and connected graphs, numerical tests are performed in order to compare the effectiveness of blue-noise sampling against other approaches.

GenerativeMap: Visualization and Exploration of Dynamic Density Maps via Generative Learning Model.

The density map is widely used for data sampling, time-varying detection, ensemble representation, etc. The visualization of dynamic evolution is a challenging task when exploring spatiotemporal data. Many approaches have been provided to explore the variation of data patterns over time, which commonly need multiple parameters and preprocessing works. Image generation is a well-known topic in deep learning, and a variety of generating models have been promoted in recent years. In this paper, we introduce a general pipeline called GenerativeMap to extract dynamics of density maps by generating interpolation information. First, a trained generative model comprises an important part of our approach, which can generate nonlinear and natural results by implementing a few parameters. Second, a visual presentation is proposed to show the density change, which is combined with the level of detail and blue noise sampling for a better visual effect. Third, for dynamic visualization of large-scale density maps, we extend this approach to show the evolution in regions of interest, which costs less to overcome the drawback of the learning-based generative model. We demonstrate our method on different types of cases, and we evaluate and compare the approach from multiple aspects. The results help identify the effectiveness of our approach and confirm its applicability in different scenarios.

Semantic-Aware Visual Abstraction of Large-Scale Social Media Data With Geo-Tags

With the rapid growth of geo-tagged social media data, it has become feasible to explore topics across different areas through text mining and geographical visualization. However, the visual elements of social media data always overlap with each other in the map view, which largely disturbs visual perception of semantic features and their geographical distribution. Thus, it is of great significance to reduce the visual clutter of large-scale social media data, and enhance the visibility of semantic features across local areas. In this paper, we utilize a doc2vec model to transform geo-tagged social media data into high-dimensional vectors, and the semantic correlation can be easily characterized in the dimensionality reduction space. Aiming at the reduction of visual clutter of geographical visualization, a dual-objective blue noise sampling model is proposed to select a subset of social media data, by means of which both the semantic correlation and spatial distribution of large scale social media data are well retained. A rich set of visual designs are implemented enabling users to evaluate the sampled results from multiple perspectives and explore the changes of semantic features across areas, such as heatmap, word cloud and text stream. The effectiveness and validity of the proposed visualization system are further demonstrated through case studies and expert reviews.

Spoke-Darts for High-Dimensional Blue-Noise Sampling

Blue noise sampling has proved useful for many graphics applications, but remains underexplored in high-dimensional spaces due to the difficulty of generating distributions and proving properties about them. We present a blue noise sampling method with good quality and performance across different dimensions. The method, spoke-dart sampling, shoots rays from prior samples and selects samples from these rays. It combines the advantages of two major high-dimensional sampling methods: the locality of advancing front with the dimensionality-reduction of hyperplanes, specifically line sampling. We prove that the output sampling is saturated with high probability, with bounds on distances between pairs of samples and between any domain point and its nearest sample. We demonstrate spoke-dart applications for approximate Delaunay graph construction, global optimization, and robotic motion planning. Both the blue-noise quality of the output distribution and the adaptability of the intermediate processes of our method are useful in these applications.

Wasserstein Blue Noise Sampling

In this article, we present a multi-class blue noise sampling algorithm by throwing samples as the constrained Wasserstein barycenter of multiple density distributions. Using an entropic regularization term, a constrained transport plan in the optimal transport problem is provided to break the partition required by the previous Capacity-Constrained Voronoi Tessellation method. The entropic regularization term cannot only control spatial regularity of blue noise sampling, but it also reduces conflicts between the desired centroids of Vornoi cells for multi-class sampling. Moreover, the adaptive blue noise property is guaranteed for each individual class, as well as their combined class. Our method can be easily extended to multi-class sampling on a point set surface. We also demonstrate applications in object distribution and color stippling.

Wasserstein Blue Noise Sampling

In this article, we present a multi-class blue noise sampling algorithm by throwing samples as the constrained Wasserstein barycenter of multiple density distributions. Using an entropic regularization term, a constrained transport plan in the optimal transport problem is provided to break the partition required by the previous Capacity-Constrained Voronoi Tessellation method. The entropic regularization term cannot only control spatial regularity of blue noise sampling, but it also reduces conflicts between the desired centroids of Vornoi cells for multi-class sampling. Moreover, the adaptive blue noise property is guaranteed for each individual class, as well as their combined class. Our method can be easily extended to multi-class sampling on a point set surface. We also demonstrate applications in object distribution and color stippling.

Wasserstein blue noise sampling

In this article, we present a multi-class blue noise sampling algorithm by throwing samples as the constrained Wasserstein barycenter of multiple density distributions. Using an entropic regularization term, a constrained transport plan in the optimal transport problem is provided to break the partition required by the previous Capacity-Constrained Voronoi Tessellation method. The entropic regularization term cannot only control spatial regularity of blue noise sampling, but it also reduces conflicts between the desired centroids of Vornoi cells for multi-class sampling. Moreover, the adaptive blue noise property is guaranteed for each individual class, as well as their combined class. Our method can be easily extended to multi-class sampling on a point set surface. We also demonstrate applications in object distribution and color stippling.