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Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. They are among the most widely used topological tools in visualization. In this paper, we are interested in sketching a set of merge trees using techniques from matrix sketching. That is, given a large set T of merge trees, we would like to find a much smaller set of basis trees S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be approximated via matrix sketching techniques such as principal component analysis and column subset selection. However, until now, there has not been any work on sketching a set of merge trees. We develop a framework for sketching a set of merge trees that combines matrix sketching with tools from optimal transport. In particular, we vectorize a set of merge trees into high-dimensional vectors while preserving their structures and structural relations. We demonstrate the applications of our framework in sketching merge trees that arise from time-varying scientific simulations. Specifically, our framework obtains a set of basis trees as representatives that capture the “modes” of physical phenomena for downstream analysis and visualization. 

Morse complexes and Morse-Smale complexes are topological descriptors popular in topology-based visualization. Comparing these complexes plays an important role in their applications in feature correspondences, feature tracking, symmetry detection, and uncertainty visualization. Leveraging recent advances in optimal transport, we apply a class of optimal transport distances to the comparative analysis of Morse complexes. Contrasting with existing comparative measures, such distances are easy and efficient to compute, and naturally provide structural matching between Morse complexes. We perform an experimental study involving scientific simulation datasets and discuss the effectiveness of these distances as comparative measures for Morse complexes. We also provide an initial guideline for choosing the optimal transport distances under various data assumptions. 

Abstract Fabricating polymeric composites with desirable characteristics for electronic applications is a complex and costly process. Digital light processing (DLP) 3D printing emerges as a promising technique for manufacturing intricate structures. In this study, polymeric samples are fabricated with a conductivity difference exceeding three orders of magnitude in various portions of a part by employing grayscale DLP (g‐DLP) single‐vat single‐cure 3D printing deliberate resin design. This is realized through the manipulation of light intensity during the curing process. Specifically, the rational resin design with added lithium ions results in the polymer cured under the maximum UV‐light intensity exhibiting higher electrical resistance. Conversely, sections that are only partially cured retains uncured monomers, serving as a medium that facilitates ion mobility, consequently leading to higher conductivity. The versatility of g‐DLP allows precise control of light intensity in different regions during the printing process. This characteristic opens up possibilities for applications, notably the low‐cost, facile, and rapid production of complex electrical circuits and sensors. The utilization of this technique makes it feasible to fabricate materials with tailored conductivity and functionality, providing an innovative pathway to advance the accelerated and facile creation of sophisticated electronic devices.