2018

M. Hajij, B. Wang, C. Scheidegger, P. Rosen.
**“Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology,”** In *2018 IEEE Pacific Visualization Symposium (PacificVis)*, IEEE, April, 2018.

DOI: 10.1109/pacificvis.2018.00024

Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in time-varying graphs. Specifically, we transform each instance of the time-varying graph into a metric space, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in time-varying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on real-world datasets and show how our method can find cyclic patterns, deviations from those patterns, and one-time events in time-varying graphs. We also examine whether a persistence-based similarity measure satisfies a set of well-established, desirable properties for graph metrics.

M. Hajij, B. Wang, P. Rosen.
**“MOG: Mapper on Graphs for Relationship Preserving Clustering,”** In *CoRR*, 2018.

The interconnected nature of graphs often results in difficult to interpret clutter. Typically techniques focus on either decluttering by clustering nodes with similar properties or grouping edges with similar relationship. We propose using mapper, a powerful topological data analysis tool, to summarize the structure of a graph in a way that both clusters data with similar properties and preserves relationships. Typically, mapper operates on a given data by utilizing a scalar function defined on every point in the data and a cover for scalar function codomain. The output of mapper is a graph that summarize the shape of the space. In this paper, we outline how to use this mapper construction on an input graphs, outline three filter functions that capture important structures of the input graph, and provide an interface for interactively modifying the cover. To validate our approach, we conduct several case studies on synthetic and real world data sets and demonstrate how our method can give meaningful summaries for graphs with various complexities

2017

A. Suh, M. Hajij, B. Wang, C. Scheidegger, P. Rosen.
**“Driving Interactive Graph Exploration Using 0-Dimensional Persistent Homology Features,”** In *CoRR*, 2017.

Graphs are commonly used to encode relationships among entities, yet, their abstractness makes them incredibly difficult to analyze. Node-link diagrams are a popular method for drawing graphs. Classical techniques for the node-link diagrams include various layout methods that rely on derived information to position points, which often lack interactive exploration functionalities; and force-directed layouts, which ignore global structures of the graph. This paper addresses the graph drawing challenge by leveraging topological features of a graph as derived information for interactive graph drawing. We first discuss extracting topological features from a graph using persistent homology. We then introduce an interactive persistence barcodes to study the substructures of a force-directed graph layout; in particular, we add contracting and repulsing forces guided by the 0-dimensional persistent homology features. Finally, we demonstrate the utility of our approach across three datasets.

2016

P. Rosen, B. Burton, K. Potter, C.R. Johnson.
**“muView: A Visual Analysis System for Exploring Uncertainty in Myocardial Ischemia Simulations,”** In *Visualization in Medicine and Life Sciences III*, Springer Nature, pp. 49--69. 2016.

DOI: 10.1007/978-3-319-24523-2_3

In this paper we describe the Myocardial Uncertainty Viewer (muView or µView) system for exploring data stemming from the simulation of cardiac ischemia. The simulation uses a collection of conductivity values to understand how ischemic regions effect the undamaged anisotropic heart tissue. The data resulting from the simulation is multi-valued and volumetric, and thus, for every data point, we have a collection of samples describing cardiac electrical properties. µView combines a suite of visual analysis methods to explore the area surrounding the ischemic zone and identify how perturbations of variables change the propagation of their effects. In addition to presenting a collection of visualization techniques, which individually highlight different aspects of the data, the coordinated view system forms a cohesive environment for exploring the simulations.We also discuss the findings of our study, which are helping to steer further development of the simulation and strengthening our collaboration with the biomedical engineers attempting to understand the phenomenon.

2015

P. Skraba, Bei Wang, G. Chen, P. Rosen.
**“Robustness-Based Simplification of 2D Steady and Unsteady Vector Fields,”** In *IEEE Transactions on Visualization and Computer Graphics (to appear)*, 2015.

Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness which enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory and has minimal boundary restrictions. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets. We show local and complete hierarchical simplifications for steady as well as unsteady vector fields.

2014

Y. Joon Ahn, C. Hoffmann, P. Rosen.
**“Geometric constraints on quadratic Bézier curves using minimal length and energy,”** In *Journal of Computational and Applied Mathematics*, Vol. 255, pp. 887--897. 2014.

This paper derives expressions for the arc length and the bending energy of quadratic Bézier curves. The formulas are in terms of the control point coordinates. For fixed start and end points of the Bézier curve, the locus of the middle control point is analyzed for curves of fixed arc length or bending energy. In the case of arc length this locus is convex. For bending energy it is not. Given a line or a circle and fixed end points, the locus of the middle control point is determined for those curves that are tangent to a given line or circle. For line tangency, this locus is a parallel line. In the case of the circle, the locus can be classified into one of six major types. In some of these cases, the locus contains circular arcs. These results are then used to implement fast algorithms that construct quadratic Bézier curves tangent to a given line or circle, with given end points, that minimize bending energy or arc length.

P. Skraba, Bei Wang, G. Chen, P. Rosen.
**“2D Vector Field Simplification Based on Robustness,”** In *Proceedings of the 2014 IEEE Pacific Visualization Symposium*, PacificVis, Note: *Awarded Best Paper!*, 2014.

Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. These geometric metrics do not consider the flow magnitude, an important physical property of the flow. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness, which provides a complementary view on flow structure compared to the traditional topological-skeleton-based approaches. Robustness enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory, has fewer boundary restrictions, and so can handle more general cases. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets.

**Keywords:** vector field, topology-based techniques, flow visualization

2013

B. Burton, B. Erem, K. Potter, P. Rosen, C.R. Johnson, D. Brooks, R.S. Macleod.
**“Uncertainty Visualization in Forward and Inverse Cardiac Models,”** In *Computing in Cardiology CinC*, pp. 57--60. 2013.

ISSN: 2325-8861

Quantification and visualization of uncertainty in cardiac forward and inverse problems with complex geometries is subject to various challenges. Specific to visualization is the observation that occlusion and clutter obscure important regions of interest, making visual assessment difficult. In order to overcome these limitations in uncertainty visualization, we have developed and implemented a collection of novel approaches. To highlight the utility of these techniques, we evaluated the uncertainty associated with two examples of modeling myocardial activity. In one case we studied cardiac potentials during the repolarization phase as a function of variability in tissue conductivities of the ischemic heart (forward case). In a second case, we evaluated uncertainty in reconstructed activation times on the epicardium resulting from variation in the control parameter of Tikhonov regularization (inverse case). To overcome difficulties associated with uncertainty visualization, we implemented linked-view windows and interactive animation to the two respective cases. Through dimensionality reduction and superimposed mean and standard deviation measures over time, we were able to display key features in large ensembles of data and highlight regions of interest where larger uncertainties exist.

P. Rosen, B. Burton, K. Potter, C.R. Johnson.
**“Visualization for understanding uncertainty in the simulation of myocardial ischemia,”** In *Proceedings of the 2013 Workshop on Visualization in Medicine and Life Sciences*, 2013.

We have created the Myocardial Uncertainty Viewer (muView) tool for exploring data stemming from the forward simulation of cardiac ischemia. The simulation uses a collection of conductivity values to understand how ischemic regions effect the undamaged anisotropic heart tissue. The data resulting from the simulation is multivalued and volumetric and thus, for every data point, we have a collection of samples describing cardiac electrical properties. muView combines a suite of visual analysis methods to explore the area surrounding the ischemic zone and identify how perturbations of variables changes the propagation of their effects.

P. Rosen.
**“A Visual Approach to Investigating Shared and Global Memory Behavior of CUDA Kernels,”** In *Computer Graphics Forum*, Vol. 32, No. 3, Wiley-Blackwell, pp. 161--170. June, 2013.

DOI: 10.1111/cgf.12103

We present an approach to investigate the memory behavior of a parallel kernel executing on thousands of threads simultaneously within the CUDA architecture. Our top-down approach allows for quickly identifying any significant differences between the execution of the many blocks and warps. As interesting warps are identified, we allow further investigation of memory behavior by visualizing the shared memory bank conflicts and global memory coalescence, first with an overview of a single warp with many operations and, subsequently, with a detailed view of a single warp and a single operation. We demonstrate the strength of our approach in the context of a parallel matrix transpose kernel and a parallel 1D Haar Wavelet transform kernel.

P. Skraba, Bei Wang, G. Chen, P. Rosen.
**“2D Vector Field Simplification Based on Robustness,”** SCI Technical Report, No. UUSCI-2013-004, *SCI Institute, University of Utah*, 2013.

Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. Further, the distance and area-based metrics are used to determine the cancellation ordering of features from a geometric point of view. Specifically, these metrics do not consider the flow magnitude, which is an important physical property of the flow. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness, which provides a complementary flow structure hierarchy to the traditional topological skeleton-based approach. Robustness enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them within a local neighborhood. This leads to a natural hierarchical simplification scheme with more physical consideration than purely topological-skeleton-based methods. Such a simplification does not depend on the topological skeleton of the vector field and therefore can handle more general situations (e.g. centers and pairs not connected by separatrices). We also provide a novel simplification algorithm based on degree theory with fewer restrictions and so can handle more general boundary conditions. We provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets.

Bei Wang, P. Rosen, P. Skraba, H. Bhatia, V. Pascucci.
**“Visualizing Robustness of Critical Points for 2D Time-Varying Vector Fields,”** In *Computer Graphics Forum*, Vol. 32, No. 3, Wiley-Blackwell, pp. 221--230. jun, 2013.

DOI: 10.1111/cgf.12109

Analyzing critical points and their temporal evolutions plays a crucial role in understanding the behavior of vector fields. A key challenge is to quantify the stability of critical points: more stable points may represent more important phenomena or vice versa. The topological notion of robustness is a tool which allows us to quantify rigorously the stability of each critical point. Intuitively, the robustness of a critical point is the minimum amount of perturbation necessary to cancel it within a local neighborhood, measured under an appropriate metric. In this paper, we introduce a new analysis and visualization framework which enables interactive exploration of robustness of critical points for both stationary and time-varying 2D vector fields. This framework allows the end-users, for the first time, to investigate how the stability of a critical point evolves over time. We show that this depends heavily on the global properties of the vector field and that structural changes can correspond to interesting behavior. We demonstrate the practicality of our theories and techniques on several datasets involving combustion and oceanic eddy simulations and obtain some key insights regarding their stable and unstable features.

W. Widanagamaachchi, P. Rosen, V. Pascucci.
**“A Flexible Framework for Fusing Image Collections into Panoramas,”** In *Proceedings of the 2013 SIBGRAPI Conference on Graphics, Patterns, and Images*, Note: *Awarded Best Paper.*, pp. 195-202. 2013.

DOI: 10.1109/SIBGRAPI.2013.35

Panoramas create summary views of multiple images, which make them a valuable means of analyzing huge quantities of image and video data. This paper introduces the Ray Graph - a general framework for panorama construction. With rays as its vertices, the Ray Graph uses its edges to specify a set of coherency relationships among all input rays. Consequently, by using a set of simple graph traversal rules, a diverse set of panorama structures can be enumerated, which can be used to efficiently and robustly generate panoramic images from image collections. To demonstrate this framework, we first use it to recreate both 360° and street panoramas. We further introduce two new panorama models, the centipede panorama - a hybrid of the 360° and street panoramas, and the storytelling panorama - a time encoding panorama. Finally, we demonstrate the flexibility of this framework by enabling interactive brushing of panoramic regions for removal of undesired features such as occlusions and moving objects.

2012

Y.-J. Ahn, C. Hoffmann, P. Rosen.
**“A Note on Circle Packing,”** In *Journal of Zhejiang University SCIENCE C*, Vol. 13, No. 8, pp. 559--564. 2012.

The problem of packing circles into a domain of prescribed topology is considered. The circles need not have equal radii. The Collins-Stephenson algorithm computes such a circle packing. This algorithm is parallelized in two different ways and its performance is reported for a triangular, planar domain test case. The implementation uses the highly parallel graphics processing unit (GPU) on commodity hardware. The speedups so achieved are discussed based on a number of experiments.

**Keywords:** Circle packing, Algorithm performance, Parallel computation, Graphics processing unit (GPU)

J. Beezley, M. Martin, P. Rosen, J. Mandel, A. Kochanski.
**“Data management and analysis with WRF and SFIRE,”** In *Proceedings of the IEEE International Geoscience and Remote Sensing Symposium*, Note: *UCD CCM Report 312*, 2012.

We introduce several useful utilities in development for the creation and analysis of real wildland fire simulations using WRF and SFIRE. These utilities exist as standalone programs and scripts as well as extensions to other well known software. Python web scrapers automate the process of downloading and preprocessing atmospheric and surface data from common sources. Other scripts simplify the domain setup by creating parameter files automatically. Integration with Google Earth allows users to explore the simulation in a 3D environment along with real surface imagery. Postprocessing scripts provide the user with a number of output data formats compatible with many commonly used visualization suites allowing for the creation of high quality 3D renderings. As a whole, these improvements build toward a unified web application that brings a sophisticated wildland fire modeling environment to scientists and users alike.

C.-S. Chiang, C. Hoffmann, P. Rosen.
**“A Generalized Malfatti Problem,”** In *Computational Geometry: Theory and Applications*, Vol. 45, No. 8, pp. 425--435. 2012.

Malfatti's problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle's sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly.

**Keywords:** Malfatti's problem, circle packing, geometric constraint solving, GPU programming

A.N.M. Imroz Choudhury, Bei Wang, P. Rosen, V. Pascucci.
**“Topological Analysis and Visualization of Cyclical Behavior in Memory Reference Traces,”** In *Proceedings of the IEEE Pacific Visualization Symposium (PacificVis 2012)*, pp. 9--16. 2012.

DOI: 10.1109/PacificVis.2012.6183557

We demonstrate the application of topological analysis techniques to the rather unexpected domain of software visualization. We collect a memory reference trace from a running program, recasting the linear flow of trace records as a high-dimensional point cloud in a metric space. We use topological persistence to automatically detect significant circular structures in the point cloud, which represent recurrent or cyclical runtime program behaviors. We visualize such recurrences using radial plots to display their time evolution, offering multi-scale visual insights, and detecting potential candidates for memory performance optimization. We then present several case studies to demonstrate some key insights obtained using our techniques.

**Keywords:** scidac

K. Potter, P. Rosen, C.R. Johnson.
**“From Quantification to Visualization: A Taxonomy of Uncertainty Visualization Approaches,”** In *Uncertainty Quantification in Scientific Computing*, IFIP Advances in Information and Communication Technology Series, Vol. 377, Edited by Andrew Dienstfrey and Ronald Boisvert, Springer, pp. 226--249. 2012.

DOI: 10.1007/978-3-642-32677-6_15

Quantifying uncertainty is an increasingly important topic across many domains. The uncertainties present in data come with many diverse representations having originated from a wide variety of domains. Communicating these uncertainties is a task often left to visualization without clear connection between the quantification and visualization. In this paper, we first identify frequently occurring types of uncertainty. Second, we connect those uncertainty representations to ones commonly used in visualization. We then look at various approaches to visualizing this uncertainty by partitioning the work based on the dimensionality of the data and the dimensionality of the uncertainty. We also discuss noteworthy exceptions to our taxonomy along with future research directions for the uncertainty visualization community.

**Keywords:** scidac, netl, uncertainty visualization

P. Rosen, V. Popescu.
**“Simplification of Node Position Data for Interactive Visualization of Dynamic Datasets,”** In *IEEE Transactions on Visualization and Computer Graphics (IEEE Visweek 2012 TVCG Track)*, pp. 1537--1548. 2012.

PubMed ID: 22025753

PubMed Central ID: PMC3411892

We propose to aid the interactive visualization of time-varying spatial datasets by simplifying node position data over the entire simulation as opposed to over individual states. Our approach is based on two observations. The first observation is that the trajectory of some nodes can be approximated well without recording the position of the node for every state. The second observation is that there are groups of nodes whose motion from one state to the next can be approximated well with a single transformation. We present dataset simplification techniques that take advantage of this node data redundancy. Our techniques are general, supporting many types of simulations, they achieve good compression factors, and they allow rigorous control of the maximum node position approximation error. We demonstrate our approach in the context of finite element analysis data, of liquid flow simulation data, and of fusion simulation data.

P. Rosen.
**“Rectilinear Texture Warping for Fast Adaptive Shadow Mapping,”** In *Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games (I3D '12)*, pp. 151--158. 2012.

Conventional shadow mapping relies on uniform sampling for producing hard shadow in an efficient manner. This approach trades image quality in favor of efficiency. A number of approaches improve upon shadow mapping by combining multiple shadow maps or using complex data structures to produce shadow maps with multiple resolutions. By sacrificing some performance, these adaptive methods produce shadows that closely match ground truth.

This paper introduces Rectilinear Texture Warping (RTW) for efficiently generating adaptive shadow maps. RTW images combine the advantages of conventional shadow mapping - a single shadow map, quick construction, and constant time pixel shadow tests, with the primary advantage of adaptive techniques - shadow map resolutions which more closely match those requested by output images. RTW images consist of a conventional texture paired with two 1-D warping maps that form a rectilinear grid defining the variation in sampling rate. The quality of shadows produced with RTW shadow maps of standard resolutions, i.e. 2,048×2,048 texture for 1080p output images, approaches that of raytraced results while low overhead permits rendering at hundreds of frames per second.

**Keywords:** Rendering, Shadow Algorithms, Adaptive Sampling