Designed especially for neurobiologists, FluoRender is an interactive tool for multi-channel fluorescence microscopy data visualization and analysis.
Large scale visualization on the Powerwall.
BrainStimulator is a set of networks that are used in SCIRun to perform simulations of brain stimulation such as transcranial direct current stimulation (tDCS) and magnetic transcranial stimulation (TMS).
Developing software tools for science has always been a central vision of the SCI Institute.

SCI Publications

2014


P. Muralidharan, J. Fishbaugh, H.J. Johnson, S. Durrleman, J.S. Paulsen, G. Gerig, P.T. Fletcher. “Diffeomorphic Shape Trajectories for Improved Longitudinal Segmentation and Statistics,” In Proceedings of Medical Image Computing and Computer Assisted Intervention (MICCAI), 2014.

ABSTRACT

Longitudinal imaging studies involve tracking changes in individuals by repeated image acquisition over time. The goal of these studies is to quantify biological shape variability within and across individuals, and also to distinguish between normal and disease populations. However, data variability is influenced by outside sources such as image acquisition, image calibration, human expert judgment, and limited robustness of segmentation and registration algorithms. In this paper, we propose a two-stage method for the statistical analysis of longitu- dinal shape. In the first stage, we estimate diffeomorphic shape trajectories for each individual that minimize inconsistencies in segmented shapes across time. This is followed by a longitudinal mixed-effects statistical model in the second stage for testing differences in shape trajectories between groups. We apply our method to a longitudinal database from PREDICT-HD and demonstrate our ap- proach reduces unwanted variability for both shape and derived measures, such as volume. This leads to greater statistical power to distinguish differences in shape trajectory between healthy subjects and subjects with a genetic biomarker for Huntington's disease (HD).



J.A. Nielsen, B.A. Zielinski, P.T. Fletcher, A.L. Alexander, N. Lange, E.D. Bigler, J.E. Lainhart, J.S. Anderson. “Abnormal lateralization of functional connectivity between language and default mode regions in autism,” In Molecular Autism, Vol. 5, No. 1, pp. 8. 2014.
DOI: 10.1186/2040-2392-5-8

ABSTRACT

Background: Lateralization of brain structure and function occurs in typical development, and abnormal lateralization is present in various neuropsychiatric disorders. Autism is characterized by a lack of left lateralization in structure and function of regions involved in language, such as Broca and Wernicke areas.
Methods: Using functional connectivity magnetic resonance imaging from a large publicly available sample (n = 964), we tested whether abnormal functional lateralization in autism exists preferentially in language regions or in a more diffuse pattern across networks of lateralized brain regions.
Results: The autism group exhibited significantly reduced left lateralization in a few connections involving language regions and regions from the default mode network, but results were not significant throughout left- and right-lateralized networks. There is a trend that suggests the lack of left lateralization in a connection involving Wernicke area and the posterior cingulate cortex associates with more severe autism.
Conclusions: Abnormal language lateralization in autism may be due to abnormal language development rather than to a deficit in hemispheric specialization of the entire brain.

Keywords: brain lateralization, brain asymmetry, autism, autism spectrum disorder, language, functional magnetic resonance imaging, functional connectivity



K.A. Nestor, J.D. Jones, C.R. Butson, T. Morishita, C.E. Jacobson, D.A. Peace, D. Chen, K.D. Foote, M.S. Okun. “Coordinate-based lead location does not predict Parkinson's disease deep brain stimulation outcome,” In PloS One, Vol. 9, No. 4, pp. e93524. January, 2014.
ISSN: 1932-6203
DOI: 10.1371/journal.pone.0093524
PubMed ID: 24691109

ABSTRACT

BACKGROUND: Effective target regions for deep brain stimulation (DBS) in Parkinson's disease (PD) have been well characterized. We sought to study whether the measured Cartesian coordinates of an implanted DBS lead are predictive of motor outcome(s). We tested the hypothesis that the position and trajectory of the DBS lead relative to the mid-commissural point (MCP) are significant predictors of clinical outcomes. We expected that due to neuroanatomical variation among individuals, a simple measure of the position of the DBS lead relative to MCP (commonly used in clinical practice) may not be a reliable predictor of clinical outcomes when utilized alone.

METHODS: 55 PD subjects implanted with subthalamic nucleus (STN) DBS and 41 subjects implanted with globus pallidus internus (GPi) DBS were included. Lead locations in AC-PC space (x, y, z coordinates of the active contact and sagittal and coronal entry angles) measured on high-resolution CT-MRI fused images, and motor outcomes (Unified Parkinson's Disease Rating Scale) were analyzed to confirm or refute a correlation between coordinate-based lead locations and DBS motor outcomes.

RESULTS: Coordinate-based lead locations were not a significant predictor of change in UPDRS III motor scores when comparing pre- versus post-operative values. The only potentially significant individual predictor of change in UPDRS motor scores was the antero-posterior coordinate of the GPi lead (more anterior lead locations resulted in a worse outcome), but this was only a statistical trend (p<.082).

CONCLUSION: The results of the study showed that a simple measure of the position of the DBS lead relative to the MCP is not significantly correlated with PD motor outcomes, presumably because this method fails to account for individual neuroanatomical variability. However, there is broad agreement that motor outcomes depend strongly on lead location. The results suggest the need for more detailed identification of stimulation location relative to anatomical targets.



I. Oguz, M. Farzinfar, J. Matsui, F. Budin, Z. Liu, G. Gerig, H.J. Johnson, M.A. Styner. “DTIPrep: Quality Control of Diffusion-Weighted Images,” In Frontiers in Neuroinformatics, Vol. 8, No. 4, 2014.
DOI: 10.3389/fninf.2014.00004

ABSTRACT

In the last decade, diffusion MRI (dMRI) studies of the human and animal brain have been used to investigate a multitude of pathologies and drug-related effects in neuroscience research. Study after study identifies white matter (WM) degeneration as a crucial biomarker for all these diseases. The tool of choice for studying WM is dMRI. However, dMRI has inherently low signal-to-noise ratio and its acquisition requires a relatively long scan time; in fact, the high loads required occasionally stress scanner hardware past the point of physical failure. As a result, many types of artifacts implicate the quality of diffusion imagery. Using these complex scans containing artifacts without quality control (QC) can result in considerable error and bias in the subsequent analysis, negatively affecting the results of research studies using them. However, dMRI QC remains an under-recognized issue in the dMRI community as there are no user-friendly tools commonly available to comprehensively address the issue of dMRI QC. As a result, current dMRI studies often perform a poor job at dMRI QC.

Thorough QC of diffusion MRI will reduce measurement noise and improve reproducibility, and sensitivity in neuroimaging studies; this will allow researchers to more fully exploit the power of the dMRI technique and will ultimately advance neuroscience. Therefore, in this manuscript, we present our open-source software, DTIPrep, as a unified, user friendly platform for thorough quality control of dMRI data. These include artifacts caused by eddy-currents, head motion, bed vibration and pulsation, venetian blind artifacts, as well as slice-wise and gradient-wise intensity inconsistencies. This paper summarizes a basic set of features of DTIPrep described earlier and focuses on newly added capabilities related to directional artifacts and bias analysis.

Keywords: diffusion MRI, Diffusion Tensor Imaging, Quality control, Software, open-source, preprocessing



D.C.B. de Oliveira, A. Humphrey, Q. Meng, Z. Rakamaric, M. Berzins, G. Gopalakrishnan. “Systematic Debugging of Concurrent Systems Using Coalesced Stack Trace Graphs,” In Proceedings of the 27th International Workshop on Languages and Compilers for Parallel Computing (LCPC), September, 2014.

ABSTRACT

A central need during software development of large-scale parallel systems is tools that help help to identify the root causes of bugs quickly. Given the massive scale of these systems, tools that highlight changes--say introduced across software versions or their operating conditions (e.g., inputs, schedules)--can prove to be highly effective in practice. Conventional debuggers, while good at presenting details at the problem-site (e.g., crash), often omit contextual information to identify the root causes of the bug. We present a new approach to collect and coalesce stack traces, leading to an efficient summary display of salient system control flow differences in a graphical form called Coalesced Stack Trace Graphs (CSTG). CSTGs have helped us understand and debug situations within a computational framework called Uintah that has been deployed at large scale, and undergoes frequent version updates. In this paper, we detail CSTGs through case studies in the context of Uintah where unexpected behaviors caused by different vesions of software or occurring across different time-steps of a system (e.g., due to non-determinism) are debugged. We show that CSTG also gives conventional debuggers a far more productive and guided role to play.



B.R. Parmar, T.R. Jarrett, N.S. Burgon, E.G. Kholmovski, N.W. Akoum, N. Hu, R.S. Macleod, N.F. Marrouche, R. Ranjan. “Comparison of Left Atrial Area Marked Ablated in Electroanatomical Maps with Scar in MRI,” In Journal of Cardiovascular Electrophysiology, 2014.
DOI: 10.1111/jce.12357

ABSTRACT

Background

Three-dimensional electroanatomic mapping (EAM) is routinely used to mark ablated areas during radiofrequency ablation. We hypothesized that, in atrial fibrillation (AF) ablation, EAM overestimates scar formation in the left atrium (LA) when compared to the scar seen on late-gadolinium enhancement magnetic resonance imaging (LGE-MRI).

Methods and Results

Of the 235 patients who underwent initial ablation for AF at our institution between August 2011 and December 2012, we retrospectively identified 70 patients who had preprocedural magnetic resonance angiography merged with LA anatomy in EAM software and had a 3-month postablation LGE-MRI for assessment of scar. Ablated area was marked intraprocedurally using EAM software and quantified retrospectively. Scarred area was quantified in 3-month postablation LGE-MRI. The mean ablated area in EAM was 30.5 ± 7.5% of the LA endocardial surface and the mean scarred area in LGE-MRI was 13.9 ± 5.9% (P < 0.001). This significant difference in the ablated area marked in the EAM and scar area in the LGE-MRI was present for each of the 3 independent operators. Complete pulmonary vein (PV) encirclement representing electrical isolation was observed in 87.8% of the PVs in EAM as compared to only 37.4% in LGE-MRI (P < 0.001).

Conclusions

In AF ablation, EAM significantly overestimates the resultant scar as assessed with a follow-up LGE-MRI.

Keywords: atrial fibrillation, magnetic resonance imaging, radiofrequency ablation



Christian Partl, Alexander Lex, Marc Streit, Hendrik Strobelt, Anne-Mai Wasserman, Hanspeter Pfister,, Dieter Schmalstieg. “ConTour: Data-Driven Exploration of Multi-Relational Datasets for Drug Discovery,” In IEEE Transactions on Visualization and Computer Graphics (VAST '14), Vol. 20, No. 12, pp. 1883--1892. 2014.
ISSN: 1077-2626
DOI: 10.1109/TVCG.2014.2346752

ABSTRACT

Large scale data analysis is nowadays a crucial part of drug discovery. Biologists and chemists need to quickly explore and evaluate potentially effective yet safe compounds based on many datasets that are in relationship with each other. However, there is a lack of tools that support them in these processes. To remedy this, we developed ConTour, an interactive visual analytics technique that enables the exploration of these complex, multi-relational datasets. At its core ConTour lists all items of each dataset in a column. Relationships between the columns are revealed through interaction: selecting one or multiple items in one column highlights and re-sorts the items in other columns. Filters based on relationships enable drilling down into the large data space. To identify interesting items in the first place, ConTour employs advanced sorting strategies, including strategies based on connectivity strength and uniqueness, as well as sorting based on item attributes. ConTour also introduces interactive nesting of columns, a powerful method to show the related items of a child column for each item in the parent column. Within the columns, ConTour shows rich attribute data about the items as well as information about the connection strengths to other datasets. Finally, ConTour provides a number of detail views, which can show items from multiple datasets and their associated data at the same time. We demonstrate the utility of our system in case studies conducted with a team of chemical biologists, who investigate the effects of chemical compounds on cells and need to understand the underlying mechanisms.



A. J. Perez, M. Seyedhosseini, T. J. Deerinck, E. A. Bushong, S. Panda, T. Tasdizen, M. H. Ellisman. “A workflow for the automatic segmentation of organelles in electron microscopy image stacks,” In Frontiers in Neuroanatomy, Vol. 8, No. 126, 2014.
DOI: 10.3389/fnana.2014.00126

ABSTRACT

Electron microscopy (EM) facilitates analysis of the form, distribution, and functional status of key organelle systems in various pathological processes, including those associated with neurodegenerative disease. Such EM data often provide important new insights into the underlying disease mechanisms. The development of more accurate and efficient methods to quantify changes in subcellular microanatomy has already proven key to understanding the pathogenesis of Parkinson's and Alzheimer's diseases, as well as glaucoma. While our ability to acquire large volumes of 3D EM data is progressing rapidly, more advanced analysis tools are needed to assist in measuring precise three-dimensional morphologies of organelles within data sets that can include hundreds to thousands of whole cells. Although new imaging instrument throughputs can exceed teravoxels of data per day, image segmentation and analysis remain significant bottlenecks to achieving quantitative descriptions of whole cell structural organellomes. Here, we present a novel method for the automatic segmentation of organelles in 3D EM image stacks. Segmentations are generated using only 2D image information, making the method suitable for anisotropic imaging techniques such as serial block-face scanning electron microscopy (SBEM). Additionally, no assumptions about 3D organelle morphology are made, ensuring the method can be easily expanded to any number of structurally and functionally diverse organelles. Following the presentation of our algorithm, we validate its performance by assessing the segmentation accuracy of different organelle targets in an example SBEM dataset and demonstrate that it can be efficiently parallelized on supercomputing resources, resulting in a dramatic reduction in runtime.



A. Perez, M. Seyedhosseini, T. Tasdizen, M. Ellisman. “Automated workflows for the morphological characterization of organelles in electron microscopy image stacks (LB72),” In The FASEB Journal, Vol. 28, No. 1 Supplement LB72, April, 2014.

ABSTRACT

Advances in three-dimensional electron microscopy (EM) have facilitated the collection of image stacks with a field-of-view that is large enough to cover a significant percentage of anatomical subdivisions at nano-resolution. When coupled with enhanced staining protocols, such techniques produce data that can be mined to establish the morphologies of all organelles across hundreds of whole cells in their in situ environments. Although instrument throughputs are approaching terabytes of data per day, image segmentation and analysis remain significant bottlenecks in achieving quantitative descriptions of whole cell organellomes. Here we describe computational workflows that achieve the automatic segmentation of organelles from regions of the central nervous system by applying supervised machine learning algorithms to slices of serial block-face scanning EM (SBEM) datasets. We also demonstrate that our workflows can be parallelized on supercomputing resources, resulting in a dramatic reduction of their run times. These methods significantly expedite the development of anatomical models at the subcellular scale and facilitate the study of how these models may be perturbed following pathological insults.



F. Rousset, C. Vachet, C. Conlin, M. Heilbrun, J.L. Zhang, V.S. Lee, G. Gerig. “Semi-automated application for kidney motion correction and filtration analysis in MR renography,” In Proceeding of the 2014 Joint Annual Meeting ISMRM-ESMRMB, pp. (accepted). 2014.

ABSTRACT

Altered renal function commonly affects patients with cirrhosis, a consequence of chronic liver disease. From lowdose contrast material-enhanced magnetic resonance (MR) renography, we can estimate the Glomerular Filtration Rate (GFR), an important parameter to assess renal function. Two-dimensional MR images are acquired every 2 seconds for approximately 5 minutes during free breathing, which results in a dynamic series of 140 images representing kidney filtration over time. This specific acquisition presents dynamic contrast changes but is also challenged by organ motion due to breathing. Rather than use conventional image registration techniques, we opted for an alternative method based on object detection. We developed a novel analysis framework available under a stand-alone toolkit to efficiently register dynamic kidney series, manually select regions of interest, visualize the concentration curves for these ROIs, and fit them into a model to obtain GFR values. This open-source cross-platform application is written in C++, using the Insight Segmentation and Registration Toolkit (ITK) library, and QT4 as a graphical user interface.



N. Sadeghi, J.H. Gilmore, W. Lin, G. Gerig. “Normative Modeling of Early Brain Maturation from Longitudinal DTI Reveals Twin-Singleton Differences,” In Proceeding of the 2014 Joint Annual Meeting ISMRM-ESMRMB, pp. (accepted). 2014.

ABSTRACT

Early brain development of white matter is characterized by rapid organization and structuring. Magnetic Resonance diffusion tensor imaging (MR-DTI) provides the possibility of capturing these changes non-invasively by following individuals longitudinally in order to better understand departures from normal brain development in subjects at risk for mental illness [1]. Longitudinal imaging of individuals suggests the use of 4D (3D, time) image analysis and longitudinal statistical modeling [3].



N. Sadeghi, P.T. Fletcher, M. Prastawa, J.H. Gilmore, G. Gerig. “Subject-specific prediction using nonlinear population modeling: Application to early brain maturation from DTI,” In Proceedings of Medical Image Computing and Computer-Assisted Intervention (MICCAI 2014), 2014.

ABSTRACT

The term prediction implies expected outcome in the future, often based on a model and statistical inference. Longitudinal imaging studies offer the possibility to model temporal change trajectories of anatomy across populations of subjects. In the spirit of subject-specific analysis, such normative models can then be used to compare data from new subjects to the norm and to study progression of disease or to predict outcome. This paper follows a statistical inference approach and presents a framework for prediction of future observations based on past measurements and population statistics. We describe prediction in the context of nonlinear mixed effects modeling (NLME) where the full reference population's statistics (estimated fixed effects, variance-covariance of random effects, variance of noise) is used along with the individual's available observations to predict its trajectory. The proposed methodology is generic in regard to application domains. Here, we demonstrate analysis of early infant brain maturation from longitudinal DTI with up to three time points. Growth as observed in DTI-derived scalar invariants is modeled with a parametric function, its parameters being input to NLME population modeling. Trajectories of new subject's data are estimated when using no observation, only the rst or the first two time points. Leave-one-out experiments result in statistics on differences between actual and predicted observations. We also simulate a clinical scenario of prediction on multiple categories, where trajectories predicted from multiple models are classified based on maximum likelihood criteria.



A.R. Sanderson. “An Alternative Formulation of Lyapunov Exponents for Computing Lagrangian Coherent Structures,” In Proceedings of the 2014 IEEE Pacific Visualization Symposium (PacificVis), Yokahama Japan, 2014.

ABSTRACT

Lagrangian coherent structures are time-evolving surfaces that highlight areas in flow fields where neighboring advected particles diverge or converge. The detection and understanding of such structures is an important part of many applications such as in oceanography where there is a need to predict the dispersion of oil and other materials in the ocean. One of the most widely used tools for revealing Lagrangian coherent structures has been to calculate the finite-time Lyapunov exponents, whose maximal values appear as ridgelines to reveal Lagrangian coherent structures. In this paper we explore an alternative formulation of Lyapunov exponents for computing Lagrangian coherent structures.



M. Seyedhosseini, T. Tasdizen. “Disjunctive normal random forests,” In Pattern Recognition, September, 2014.
DOI: 10.1016/j.patcog.2014.08.023

ABSTRACT

We develop a novel supervised learning/classification method, called disjunctive normal random forest (DNRF). A DNRF is an ensemble of randomly trained disjunctive normal decision trees (DNDT). To construct a DNDT, we formulate each decision tree in the random forest as a disjunction of rules, which are conjunctions of Boolean functions. We then approximate this disjunction of conjunctions with a differentiable function and approach the learning process as a risk minimization problem that incorporates the classification error into a single global objective function. The minimization problem is solved using gradient descent. DNRFs are able to learn complex decision boundaries and achieve low generalization error. We present experimental results demonstrating the improved performance of DNDTs and DNRFs over conventional decision trees and random forests. We also show the superior performance of DNRFs over state-of-the-art classification methods on benchmark datasets.

Keywords: Random forest, Decision tree, Classifier, Supervised learning, Disjunctive normal form



M. Seyedhosseini, T. Tasdizen. “Scene Labeling with Contextual Hierarchical Models,” In CoRR, Vol. abs/1402.0595, 2014.

ABSTRACT

Scene labeling is the problem of assigning an object label to each pixel. It unifies the image segmentation and object recognition problems. The importance of using contextual information in scene labeling frameworks has been widely realized in the field. We propose a contextual framework, called contextual hierarchical model (CHM), which learns contextual information in a hierarchical framework for scene labeling. At each level of the hierarchy, a classifier is trained based on downsampled input images and outputs of previous levels. Our model then incorporates the resulting multi-resolution contextual information into a classifier to segment the input image at original resolution. This training strategy allows for optimization of a joint posterior probability at multiple resolutions through the hierarchy. Contextual hierarchical model is purely based on the input image patches and does not make use of any fragments or shape examples. Hence, it is applicable to a variety of problems such as object segmentation and edge detection. We demonstrate that CHM outperforms state-of-the-art on Stanford background and Weizmann horse datasets. It also outperforms state-of-the-art edge detection methods on NYU depth dataset and achieves state-of-the-art on Berkeley segmentation dataset (BSDS 500).



A. Sharma, P.T. Fletcher, J.H. Gilmore, M.L. Escolar, A. Gupta, M. Styner, G. Gerig. “Parametric Regression Scheme for Distributions: Analysis of DTI Fiber Tract Diffusion Changes in Early Brain Development,” In Proceedings of the 2014 IEEE International Symposium on Biomedical Imaging (ISBI), pp. (accepted). 2014.

ABSTRACT

Temporal modeling frameworks often operate on scalar variables by summarizing data at initial stages as statistical summaries of the underlying distributions. For instance, DTI analysis often employs summary statistics, like mean, for regions of interest and properties along fiber tracts for population studies and hypothesis testing. This reduction via discarding of variability information may introduce significant errors which propagate through the procedures. We propose a novel framework which uses distribution-valued variables to retain and utilize the local variability information. Classic linear regression is adapted to employ these variables for model estimation. The increased stability and reliability of our proposed method when compared with regression using single-valued statistical summaries, is demonstrated in a validation experiment with synthetic data. Our driving application is the modeling of age-related changes along DTI white matter tracts. Results are shown for the spatiotemporal population trajectory of genu tract estimated from 45 healthy infants and compared with a Krabbe's patient.

Keywords: linear regression, distribution-valued data, spatiotemporal growth trajectory, DTI, early neurodevelopment



N.P. Singh, J. Hinkle, S. Joshi, P.T. Fletcher. “An Efficient Parallel Algorithm for Hierarchical Geodesic Models in Diffeomorphisms,” In Proceedings of the 2014 IEEE International Symposium on Biomedical Imaging (ISBI), pp. (accepted). 2014.

ABSTRACT

We present a novel algorithm for computing hierarchical geodesic models (HGMs) for diffeomorphic longitudinal shape analysis. The proposed algorithm exploits the inherent parallelism arising out of the independence in the contributions of individual geodesics to the group geodesic. The previous serial implementation severely limits the use of HGMs to very small population sizes due to computation time and massive memory requirements. The conventional method makes it impossible to estimate the parameters of HGMs on large datasets due to limited memory available onboard current GPU computing devices. The proposed parallel algorithm easily scales to solve HGMs on a large collection of 3D images of several individuals. We demonstrate its effectiveness on longitudinal datasets of synthetically generated shapes and 3D magnetic resonance brain images (MRI).

Keywords: LDDMM, HGM, Vector Momentum, Diffeomorphisms, Longitudinal Analysis



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.

ABSTRACT

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



P. Skraba, Bei Wang. “Interpreting Feature Tracking Through the Lens of Robustness,” In Mathematics and Visualization, Springer, pp. 19-37. 2014.
DOI: 10.1007/978-3-319-04099-8_2

ABSTRACT

A key challenge in the study of a time-varying vector fields is to resolve the correspondences between features in successive time steps and to analyze the dynamic behaviors of such features, so-called feature tracking. Commonly tracked features, such as volumes, areas, contours, boundaries, vortices, shock waves and critical points, represent interesting properties or structures of the data. Recently, the topological notion of robustness, a relative of persistent homology, has been introduced to quantify the stability of critical points. Intuitively, the robustness of a critical point is the minimum amount of perturbation necessary to cancel it. In this chapter, we offer a fresh interpretation of the notion of feature tracking, in particular, critical point tracking, through the lens of robustness.We infer correspondences between critical points based on their closeness in stability, measured by robustness, instead of just distance proximities within the domain. We prove formally that robustness helps us understand the sampling conditions under which we can resolve the correspondence problem based on region overlap techniques, and the uniqueness and uncertainty associated with such techniques. These conditions also give a theoretical basis for visualizing the piecewise linear realizations of critical point trajectories over time.



P. Skraba, Bei Wang. “Approximating Local Homology from Samples,” In Proceedings 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 174-192. 2014.

ABSTRACT

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become diffcult to construct in higher dimensions. We show that the persistence diagrams used for estimating local homology can be approximated using families of Vietoris-Rips complexes, whose simpler construction are robust in any dimension. To the best of our knowledge, our results, for the first time make applications based on local homology, such as stratification learning, feasible in high dimensions.