Genus Oblivious Cross Parameterization: Robust Topological Management of Inter-Surface Maps

House, Donald H.; Singh, Mayank Dev

doi:10.1109/pg.2007.42

Cited by 10 publications

(2 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…the functions f and g are Morse functions with the minimal number of critical points, the Jacobi set J(f, g) has the minimal number of loops and contains no BD points. Previously, Bennett et al [2] suggested that given a domain with genus γ the minimal Jacobi set has γ + 1 loops. We disprove this claim by showing that there exist functions f and g on M that give rise to at least one and at most two Jacobi loops.…”

Section: Minimal Jacobi Setsmentioning

confidence: 99%

Local, smooth, and consistent Jacobi set simplification

Bhatia

Wang

Norgard

et al. 2015

Computational Geometry

Self Cite

View full text Add to dashboard Cite

The relation between two Morse functions defined on a common domain can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces have shown to be useful in various applications. Unfortunately, in practice functions often contain noise and discretization artifacts causing their Jacobi set to become unmanageably large and complex. While there exist techniques to simplify Jacobi sets, these are unsuitable for most applications as they lack fine-grained control over the process and heavily restrict the type of simplifications possible.In this paper, we introduce a new framework that generalizes critical point cancellations in scalar functions to Jacobi sets in two dimensions. We focus on simplifications that can be realized by smooth approximations of the corresponding functions and show how this implies simultaneously simplifying contiguous subsets of the Jacobi set. These extended cancellations form the atomic operations in our framework, and we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications according to some user-defined metric. We prove that the algorithm is correct and terminates only once no more local, smooth and consistent simplifications are possible. We disprove a previous claim on the minimal Jacobi set for manifolds with arbitrary genus and show that for simply connected domains, our algorithm reduces a given Jacobi set to its simplest configuration. arXiv:1307.7752v1 [cs.CG] 29 Jul 2013In scientific modeling and simulation, one often defines multiple functions, e.g. temperature, pressure, species distributions etc. on a common domain. Understanding the relation between such functions is crucial in data exploration and analysis. The Jacobi set [4] of two scalar functions provides an important tool for such analysis by describing points in the domain where the two gradients are aligned, and thus partitioning the domain into regions based on relative gradient orientation. A variety of interesting physical phenomena such as the interplay between salinity and temperature of water in oceanography [1] and the critical paths of gravitational potentials of celestial bodies [18] (similar to the Lagrange points in astrophysics) can be modeled using Jacobi sets. In data analysis and image processing, Jacobi sets have been used to compare multiple scalar functions [6], as well as to express the paths of critical points overtime [3,4], silhouettes of objects [8], and ridges in image data [16].However, the Jacobi sets can be extremely detailed to the point at which their complexity impedes or even prevents a meaningful analysis. Often, one is not interested in the fine-scale details, e.g. minor silhouette components due to surface roughness, but rather in more prevalent features such as significant protrusions. The Jacobi sets are also highly sensitive to noise which further leads to undesired artifacts. Finally...

show abstract

Section: Minimal Jacobi Setsmentioning

confidence: 99%

Local, smooth, and consistent Jacobi set simplification

Bhatia

Wang

Norgard

et al. 2015

Computational Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

“…Edelsbrunner et al [1] have shown that the Jacobi sets can be used to compute a comparison measure between two scalar functions. Bennett et al [2] have used the Jacobi set to represent tunnels and the silhouette of a mesh, both of which are subsequently used to compute a cross parameterization. Jacobi sets have also been used to track features of time-varying events such as molecular interactions, combustion simulation, etc.…”

Section: Introductionmentioning

confidence: 99%

Simplification of Jacobi Sets

N¹,

Natarajan

2010

Mathematics and Visualization

View full text Add to dashboard Cite

Abstract. The Jacobi set of two Morse functions defined on a 2-manifold is the collection of points where the gradients of the functions align with each other or where one of the gradients vanish. It describes the relationship between functions defined on the same domain, and hence plays an important role in multi-field visualization. The Jacobi set of two piecewise linear functions may contain several components indicative of noisy or a feature-rich dataset. We pose the problem of simplification as the extraction of level sets and offset contours and describe an algorithm to compute and simplify Jacobi sets in a robust manner.

show abstract