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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., Comment: Accepted at IEEE VIS 2022. Video: https://youtu.be/3TBx-njP_sU

In the prequel to this paper, we presented a systematic framework for processing spline spaces. In this paper, we take the results of that framework and provide a code generation pipeline that automatically generates efficient implementations of spline spaces. We decompose the final algorithm from Part I and translate the resulting components into LLVM-IR (a low level language that can be compiled to various targets/architectures). Our design provides a handful of parameters for a practitioner to tune - this is one of the avenues that provides us with the flexibility to target many different computational architectures and tune performance on those architectures. We also provide an evaluation of the effect of the different parameters on performance.

Interpolation is a fundamental technique in scientific computing and is at the heart of many scientific visualization techniques. There is usually a trade-off between the approximation capabilities of an interpolation scheme and its evaluation efficiency. For many applications, it is important for a user to be able to navigate their data in real time. In practice, the evaluation efficiency (or speed) outweighs any incremental improvements in reconstruction fidelity. In this two-part work, 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. In the sequel, we detail how we generate efficient implementations via automatic code generation on both CPU and GPU architectures. Specifically, in this paper, 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.