The visualization of temporal data on urban buildings, such as shadows, noise, and solar potential, plays a critical role in the analysis of dynamic urban phenomena. However, in dense and geographically constrained 3D urban environments, visual representations of time-varying building data often suffer from occlusion and visual clutter. To address these two challenges, we introduce an immersive lens visualization that integrates i) a view-dependent cutaway de-occlusion technique and ii) a temporal display derived from a conformal mapping algorithm. The mapping process first partitions irregular building footprints into smaller, sufficiently regular subregions that serve as structural primitives. These subregions are then seamlessly recombined to form a conformal, layered layout for our temporal lens visualization. The view-responsive cutaway is inspired by traditional architectural illustrations, preserving the overall layout of the building and its surroundings to maintain users' sense of spatial orientation. This lens design enables the occlusion-free embedding of shape-adaptive temporal displays across building facades on demand, supporting rapid time-space association for the discovery, access and interpretation of spatiotemporal urban patterns. Guided by domain and design goals, we outline the rationale behind the lens visual and interaction design choices, such as the encoding of time progression and temporal values in the conforming lens image. A controlled user study compares our approach against conventional juxtaposition and x-ray spatiotemporal designs. Results validate the usage and utility of our lens, showing that it improves task accuracy and completion time, reduces navigation effort, and increases user confidence. From these findings, we distill design recommendations and promising directions for future research on spatially-embedded lenses in 3D visualization, urban analytics, and related domains.

Haptic feedback provides an essential sensory stimulus crucial for interaction and analyzing three-dimensional spatio-temporal phenomena on surface visualizations. Given its ability to provide enhanced spatial perception and scene maneuverability, virtual reality (VR) catalyzes haptic interactions on surface visualizations. Various interaction modes, encompassing both mid-air and on-surface interactions-with or without the application of assisting force stimuli-have been explored using haptic force feedback devices. In this paper, we evaluate the use of on-surface and assisted on-surface haptic modes of interaction compared to a no-haptic interaction mode. A force-based haptic stylus is used for all three modalities; the on-surface mode uses collision based forces, whereas the assisted on-surface mode is accompanied by an additional snapping force. We conducted a within-subjects user study involving fundamental interaction tasks performed on surface visualizations. Keeping a consistent visual design across all three modes, our study incorporates tasks that require the localization of the highest, lowest, and random points on surfaces; and tasks that focus on brushing curves on surfaces with varying complexity and occlusion levels. Our findings show that participants took almost the same time to brush curves using all the interaction modes. They could draw smoother curves using the on-surface interaction modes compared to the no-haptic mode. However, the assisted on-surface mode provided better accuracy than the on-surface mode. The on-surface mode was slower in point localization, but the accuracy depended on the visual cues and occlusions associated with the tasks. Finally, we discuss participant feedback on using haptic force feedback as a tangible input modality and share takeaways to aid the design of haptics-based tangible interactions for surface visualizations.

The use of non-Cartesian grids is a niche but important topic in sub-fields of the numerical sciences such as simulation and scientific visualization. However, non-Cartesian approaches are virtually unexplored in machine learning. This is likely due to the difficulties in the representation of data on non-Cartesian domains and the lack of support for standard machine learning operations on non-Cartesian data. This paper proposes a new data structure called the lattice tensor which generalizes traditional tensor spatio-temporal operations to lattice tensors, enabling the use of standard machine learning algorithms on non-Cartesian data. However, data need not reside on a non-Cartesian structure, we use non-Dyadic downsampling schemes to bring Cartesian data into a non-Cartesian space for further processing. We introduce a software library that implements the lattice tensor container (with some common machine learning operations), and demonstrate its effectiveness. Our method provides a general framework for machine learning on non-Cartesian domains, addressing the challenges mentioned above and filling a gap in the current literature.

Interpolation is a foundational concept in scientific computing and is at the heart of many scientific visualization techniques. There is usually a tradeoff between the approximation capabilities of an interpolation scheme and its evaluation efficiency. For many applications, it is important for a user to navigate their data in real time. In practice, evaluation efficiency outweighs any incremental improvements in reconstruction fidelity. We first analyze, from a general standpoint, the use of compact piece-wise polynomial basis functions to efficiently interpolate data that is sampled on a lattice. We then detail our automatic code-generation framework on both CPU and GPU architectures. Specifically, we propose a general framework that can produce a fast evaluation scheme by analyzing the algebro-geometric structure of the convolution sum for a given lattice and basis function combination. We demonstrate the utility and generality of our framework by providing fast implementations of various box splines on the Body Centered and Face Centered Cubic lattices, as well as some non-separable box splines on the Cartesian lattice. We also provide fast implementations for certain Voronoi-splines that have not yet appeared in the literature. Finally, we demonstrate that this framework may also be used for non-Cartesian lattices in 4D.

Recent technological innovations have led to an increase in the availability of 3D urban data, such as shadow, noise, solar potential, and earthquake simulations. These spatiotemporal datasets create opportunities for new visualizations to engage experts from different domains to study the dynamic behavior of urban spaces in this under explored dimension. However, designing 3D spatiotemporal urban visualizations is challenging, as it requires visual strategies to support analysis of time-varying data referent to the city geometry. Although different visual strategies have been used in 3D urban visual analytics, the question of how effective these visual designs are at supporting spatiotemporal analysis on building surfaces remains open. To investigate this, in this paper we first contribute a series of analytical tasks elicited after interviews with practitioners from three urban domains. We also contribute a quantitative user study comparing the effectiveness of four representative visual designs used to visualize 3D spatiotemporal urban data: spatial juxtaposition, temporal juxtaposition, linked view, and embedded view. Participants performed a series of tasks that required them to identify extreme values on building surfaces over time. Tasks varied in granularity for both space and time dimensions. Our results demonstrate that participants were more accurate using plot-based visualizations (linked view, embedded view) but faster using color-coded visualizations (spatial juxtaposition, temporal juxtaposition). Our results also show that, with increasing task complexity, plot-based visualizations perform better in preserving efficiency (time, accuracy) compared to color-coded visualizations. Based on our findings, we present a set of takeaways with design recommendations for 3D spatiotemporal urban visualizations for researchers and practitioners. Lastly, we report on a series of interviews with four practitioners, and their feedback and suggestions for further work on the visualizations to support 3D spatiotemporal urban data analysis.

We present decal-lenses, a new interaction technique that extends the concept of magic lenses to augment and manage multivariate visualizations on arbitrary surfaces. Our object-space lenses follow the surface geometry and allow the user to change the point of view during data exploration while maintaining a spatial reference to positions where one or more lenses were placed. Each lens delimits specific regions of the surface where one or more attributes can be selected or combined. Similar to 2D lenses, the user interacts with our lenses in real-time, switching between different attributes within the lens context. The user can also visualize the surface data representations from the point of view of each lens by using local cameras. To place lenses on surfaces of intricate geometry, such as the human brain, we introduce the concept of support surfaces for designing interaction techniques. Support surfaces provide a way to place and interact with the lenses while avoiding holes and occluded regions during data exploration. We further extend decal-lenses to arbitrary regions using brushing and lassoing operations. We discuss the applicability of our technique and present several examples where our lenses can be useful to create a customized exploration of multivariate data on surfaces.

Abstract In this paper, we present a novel illustrative multivariate visualization for geological modelling to assist geologists and reservoir engineers in visualizing multivariate datasets in superimposed representations, in contrast to the single‐attribute visualizations supported by commercial software. Our approach extends the use of decals from a single surface to 3D irregular grids, using the layering concept to represent multiple attributes. We also build upon prior work to augment the design and implementation of different geological attributes (namely, rock type, porosity, and permeability). More specifically, we propose a new sampling strategy to generate decals for porosity on the geological grid, a hybrid visualization for permeability which combines 2D decals and 3D ellipsoid glyphs, and a perceptually‐based design that allows us to visualize additional attributes (e.g., oil saturation) while avoiding visual interference between layers. Furthermore, our visual design draws from traditional geological illustrations, facilitating the understanding and communication between interdisciplinary teams. An evaluation by domain experts highlights the potential of our approach for geological modelling and interpretation in this complex domain.

Abstract In this work, we present a family of compact, biorthogonal wavelet filter banks that are applicable to the Body Centered Cubic (BCC) lattice. While the BCC lattice has been shown to have superior approximation properties for volumetric data when compared to the Cartesian Cubic (CC) lattice, there has been little work in the way of designing wavelet filter banks that respect the geometry of the BCC lattice. Since wavelets have applications in signal de‐noising, compression, and sparse signal reconstruction, these filter banks are an important tool that addresses some of the scalability concerns presented by the BCC lattice. We use these filters in the context of volumetric data compression and reconstruction and qualitatively evaluate our results by rendering images of isosurfaces from compressed data.

We introduce the use of decals for multivariate visualization design. Decals are visual representations that are used for communication; for example, a pattern, a text, a glyph, or a symbol, transferred from a 2D-image to a surface upon contact. By creating what we define as decal-maps, we can design a set of images or patterns that represent one or more data attributes. We place decals on the surface considering the data pertaining to the locations we choose. We propose a (texture mapping) local parametrization that allows placing decals on arbitrary surfaces interactively, even when dealing with a high number of decals. Moreover, we extend the concept of layering to allow the co-visualization of an increased number of attributes on arbitrary surfaces. By combining decal-maps, color-maps and a layered visualization, we aim to facilitate and encourage the creative process of designing multivariate visualizations. Finally, we demonstrate the general applicability of our technique by providing examples of its use in a variety of contexts.

Abstract Compressive rendering refers to the process of reconstructing a full image from a small subset of the rendered pixels, thereby expediting the rendering task. In this paper, we empirically investigate three image order techniques for compressive rendering that are suitable for direct volume rendering. The first technique is based on the theory of compressed sensing and leverages the sparsity of the image gradient in the Fourier domain. The latter techniques exploit smoothness properties of the rendered image; the second technique recovers the missing pixels via a total variation minimization procedure while the third technique incorporates a smoothness prior in a variational reconstruction framework employing interpolating cubic B‐splines. We compare and contrast the three techniques in terms of quality, efficiency and sensitivity to the distribution of pixels. Our results show that smoothness‐based techniques significantly outperform techniques that are based on compressed sensing and are also robust in the presence of highly incomplete information. We achieve high quality recovery with as little as 20% of the pixels distributed uniformly in screen space.

In this paper, we present two methods for accurate gradient estimation from scalar field data sampled on regular lattices. The first method is based on the multidimensional Taylor series expansion of the convolution sum and allows us to specify design criteria such as compactness and approximation power. The second method is based on a Hilbert space framework and provides a minimum error solution in the form of an orthogonal projection operating between two approximation spaces. Both methods lead to discrete filters, which can be combined with continuous reconstruction kernels to yield highly accurate estimators as compared to the current state of the art. We demonstrate the advantages of our methods in the context of volume rendering of data sampled on Cartesian and Body-Centered Cubic lattices. Our results show significant qualitative and quantitative improvements for both synthetic and real data, while incurring a moderate preprocessing and storage overhead.

We investigate the use of a Fourier-domain derivative error kernel to quantify the error incurred while estimating the gradient of a function from scalar point samples on a regular lattice. We use the error kernel to show that gradient reconstruction quality is significantly enhanced merely by shifting the reconstruction kernel to the centers of the principal lattice directions. Additionally, we exploit the algebraic similarities between the scalar and derivative error kernels to design asymptotically optimal gradient estimation filters that can be factored into an infinite impulse response interpolation prefilter and a finite impulse response directional derivative filter. This leads to a significant performance gain both in terms of accuracy and computational efficiency. The interpolation prefilter provides an accurate scalar approximation and can be re-used to cheaply compute directional derivatives on-the-fly without the need to store gradients. We demonstrate the impact of our filters in the context of volume rendering of scalar data sampled on the Cartesian and Body-Centered Cubic lattices. Our results rival those obtained from other competitive gradient estimation methods while incurring no additional computational or storage overhead.

Abstract Within the context of emission tomography, we study volumetric reconstruction methods based on the Expectation Maximization (EM) algorithm. We show, for the first time, the equivalence of the standard implementation of the EM‐based reconstruction with an implementation based on hardware‐accelerated volume rendering for nearest‐neighbor (NN) interpolation. This equivalence suggests that higher‐order kernels should be used with caution and do not necessarily lead to better performance. We also show that the EM algorithm can easily be adapted for different lattices, the body‐centered cubic (BCC) one in particular. For validation purposes, we use the 3D version of the Shepp‐Logan synthetic phantom, for which we derive closed‐form analytical expressions of the projection data. The experimental results show the theoretically‐predicted optimality of NN interpolation in combination with the EM algorithm, for both the noiseless and the noisy case. Moreover, reconstruction on the BCC lattice leads to superior accuracy, more compact data representation, and better noise reduction compared to the Cartesian one. Finally, we show the usefulness of the proposed method for optical projection tomography of a mouse embryo.

In this paper, we extend the single relaxation time Lattice-Boltzmann Method (LBM) to the 3D body-centered cubic (BCC) lattice. We show that the D3bQ15 lattice defined by a 15 neighborhood connectivity of the BCC lattice is not only capable of more accurately discretizing the velocity space of the continuous Boltzmann equation as compared to the D3Q15 Cartesian lattice, it also achieves a comparable spatial discretization with 30 percent less samples. We validate the accuracy of our proposed lattice by investigating its performance on the 3D lid-driven cavity flow problem and show that the D3bQ15 lattice offers significant cost savings while maintaining a comparable accuracy. We demonstrate the efficiency of our method and the impact on graphics and visualization techniques via the application of line-integral convolution on 2D slices as well as the extraction of streamlines of the 3D flow. We further study the benefits of our proposed lattice by applying it to the problem of simulating smoke and show that the D3bQ15 lattice yields more detail and turbulence at a reduced computational cost.