1.5 Contributions of This D issertation
1.5.3 Development of Combinatorial Representations
As discussed in Section 1 .2.1 , computational models for experiments, simulations, and data analysis are typically based upon mathematical theories that assume smooth fields defined over smooth domains. However, practical adaptations o f these theories are numerical in nature, and are fundamentally limited in how the data is represented and computations are performed. This analysis model typically incurs large amounts of errors, which, more importantly, are nontrivial to track.
To address the issues caused by the lack of numerical robustness in vector field analysis, Part III of this dissertation develops a new discrete theory o f representation and analysis of vector fields. As compared to smooth theory, the primary advantage of the discrete world is that its data and concepts can be adapted for computers exactly using combinatorial representations. The primary goal of this work is to guarantee consistency in feature extraction for vector fields, and at the same time, track the uncertainties in representation and analysis. New representations and algorithms developed in Part III achieve this goal, and answer Question 2 asked above to lend credibility to analysis.
C h a p te r 6 introduces Edge Maps, which define a new theory o f representation for 2D piecewise linear (PL) vector fields. They are an explicit representation o f flow as they encode the most fundamental property— streamlines— needed by analysis, and eliminate the need for numerical integration of streamlines. Using Edge Maps, which can also encode the discretization errors incurred during their construction, errors in streamline tracing can be tracked to provide candid uncertainty visualizations. This chapter discusses the construction and properties of Edge Maps in detail, and provides an exhaustive list of equivalence classes of Edge Maps for PL vector fields.
C h a p te r 7 introduces a quantized version of Edge Maps, which eliminates the need for floating-point arithmetic by using integer arithmetic to compute streamlines. Thus, these
quantized Edge Maps define a new discrete theory based upon the PL theory developed
by Edge Maps. Using the quantized Edge Maps, all possible streamlines in the flow can be encoded into a combinatorial datastructure through an integer-based representation. This combinatorial representation of vector fields allows guaranteeing consistency during
streamline tracing. This chapter also introduces a new combinatorial algorithm to perform consistent critical point detection in both 2D and 3D.
The research corresponding to Chapters 6 and 7 has been published as the following journal articles, conference proceedings, and book chapters.
[5] B h a t i a , H., J a d h a v S., B r e m e r P .-T ., C h e n G ., L e v i n e J. A ., N o n a t o L. G.,
a n d P a s c u c c i V . Edge maps: Representing flow with bounded error. In Proceedings
o f the 4th IEEE Pacific Visualization Symposium (2011) pp. 75-82, Hong Kong,
China. h ttp ://d x .d o i.o r g /1 0 .1 1 0 9 /P A C IF IC V IS .2 0 1 1 .5 7 4 2 3 7 5 .
[6] J a d h a v S., B h a t i a , H., B r e m e r P .-T ., L e v i n e J. A ., N o n a t o L. G ., a n d
P a s c u c c i V . Consistent approximation of local flow behavior for 2D vector fields.
In Topological Methods in Data Analysis and Visualization II - Theory, Algorithms,
and Applications, R. Peikert, H. Hauser, H. Carr, and R. Fuchs, Eds., Mathematics
and Visualization. Springer Berlin Heidelberg, 2012, pp. 141-159. h t t p ://d x .d o i .o r g /1 0 .1 0 0 7 /9 7 8 - 3 -6 4 2 -2 3 1 7 5 - 9 _ 1 0 .
[7] B h a t i a , H., J a d h a v S., B r e m e r P .-T ., C h e n G ., L e v i n e J. A ., N o n a t o
L. G ., a n d P a s c u c c i V . Flow visualization with quantified spatial and temporal errors using edge maps. IEEE Transactions on Visualization and Computer Graphics
(T V C G ) 18 (2012) 1386-1396. h t t p ://d x .d o i.o r g /1 0 .1 1 0 9 /T V C G .2 0 1 1 .2 6 5 .
[8] L e v i n e J. A ., J a d h a v S., B h a t i a , H., P a s c u c c i V ., a n d B r e m e r P .-T . A quantized boundary representation of 2D flow. Computer Graphics Forum 31 (2012), 945-954. h t t p ://d x .d o i .o r g /1 0 .1 1 1 1 /j.1 4 6 7 - 8 6 5 9 .2 0 1 2 .0 3 0 8 7 .x .
[9] B h a t i a , H., G y u l a s s y A ., W a n g H., B r e m e r P .-T ., a n d P a s c u c c i V . Robust detection o f singularities in vector fields. In Topological Methods in Data Analysis
and Visualization III - Theory, Algorithms, and Applications, P.-T. Bremer, I. Hotz,
V. Pascucci, R. Peikert, Eds., Mathematics and Visualization. Springer Berlin Hei delberg, 2014, pp. 3-18.