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Simplification of Jacobi Sets
"... Abstract. The Jacobi set of two Morse functions defined on a 2manifold 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 importan ..."
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Abstract. The Jacobi set of two Morse functions defined on a 2manifold 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 multifield visualization. The Jacobi set of two piecewise linear functions may contain several components indicative of noisy or a featurerich 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. 1
Segmentation and Cross . . .
, 2008
"... Scientific visualization provides useful insights and analysis for scientists through graphical representations of data. As numerical simulations become more sophisticated, interpretation of the underlying physical phenomena requires new tools to be developed that perform complex feature analysis an ..."
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Scientific visualization provides useful insights and analysis for scientists through graphical representations of data. As numerical simulations become more sophisticated, interpretation of the underlying physical phenomena requires new tools to be developed that perform complex feature analysis and cross correlation of data parameters. Towards this end, this thesis presents two new frameworks that contribute to more effective data understanding. The first is a segmentation framework that can be used to effectively identify regions of interest in data sets. A format of higherlevel data is defined from which meaningful derived features can be efficiently extracted and visualized. Furthermore, a higherorder interpolation scheme is presented that allows for extraction of smoother feature surfaces from scalar data. The second framework robustly computes cross parameterizations between two triangulated meshes of arbitrary and possibly unequal genus. Cross parameterizations can be used to describe the evolution of surfaces over time as well as to establish a shape deviation metric. Results are presented that demonstrate the effectiveness of both of the frameworks in a
Local, Smooth, and Consistent Jacobi Set Simplification
"... The relation between two Morse functions defined on a smooth, compact, and orientable 2manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of t ..."
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The relation between two Morse functions defined on a smooth, compact, and orientable 2manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two 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. In practice, unfortunately, functions often contain noise and discretization artifacts, causing their Jacobi set to become unmanageably large and complex. Although there exist techniques to simplify Jacobi sets, they are unsuitable for most applications as they lack finegrained control over the process, and heavily restrict the type of simplifications possible. This paper introduces the theoretical foundations of a new simplification framework for Jacobi sets. We present a new interpretation of Jacobi set simplification based on the perspective of domain segmentation. Generalizing the cancellation of critical points from scalar functions to Jacobi sets, we focus on simplifications that can be realized by smooth approximations of the corresponding functions, and show how these cancellations imply simultaneous simplification of contiguous subsets of the Jacobi set. Using these extended cancellations as atomic operations, we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications to some userdefined metric. We show that for simply connected domains, our algorithm reduces a given Jacobi set to its minimal
Local, Smooth, and Consistent Jacobi Set Simplification
"... 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 t ..."
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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 finegrained 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 userdefined 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. ar X iv