We introduce a versatile framework for characterizing and extracting salient structures

We introduce a versatile framework for characterizing and extracting salient structures in three-dimensional symmetric second-order tensor fields. stress tensor field used in engineering research. exactly characterizes the singular behavior of tensor fields. Hence, if topology is the focus of the data analysis problem, the ridge and valley lines of mode can be extracted from tensor fields using crease line extraction schemes to yield the desired degenerate lines. More importantly, considering tensor topology from this perspective suggests that the crease lines of other tensor invariants can be substituted to mode in the same versatile framework to yield an insightful picture of the important structural properties present in the data. One significant example that we consider in the following concerns the fractional anisotropy (FA) commonly used in the analysis of DTI data. Our results show that the ridge lines of FA capture certain important white matter tracts. Another important contribution of this paper, which provides the algorithmic foundation of our crease-based approach, is a robust and accurate method for the extraction of these feature lines from nonlinear quantities. Indeed, measures like mode and FA are nonlinear invariants whose computation from the tensor coefficients requires caution. Since the definition of ridge and valley lines involves the first and second-order derivatives of the considered scalar measure, our implementation uses smooth reconstruction kernels in the computation of tensor invariants. Additionally, we combine these kernels with an adaptive scheme that automatically adjusts the resolution of the crease line extraction to the spatial variations of the invariant. As a result our method enables the application of existing crease collection extraction schemes to the structural analysis of tensor fields. Our fresh platform is definitely algorithmically simpler and also theoretically more general, since it allows for the definition of structural saliency in terms of several invariants that can be naturally adapted to the focus of a particular application. The remainder of this paper is definitely organized as follows. Related work, with emphasis on topological methods for tensor fields and crease manifolds GHRP-6 Acetate in image data is definitely discussed in Section 2. Our demonstration proceeds by critiquing fundamental theoretical notions relevant to this work in Section 3. buy PF-04449913 Implementation considerations, centered around the specific challenges posed from the nonlinearity of tensor invariants and by their clean reconstruction, are detailed in Section 4. Results are proposed in Section 5 for any synthetic dataset of a stress tensor buy PF-04449913 field on one hand and for a DTI dataset of the brain white matter on the buy PF-04449913 other hand. We conclude by discussing our findings and mentioning interesting avenues to extend this work in Section 6. 2 RELATED WORK The research offered with this paper is definitely closely related to previous work on tensor field topology visualization and crease detection in image data. 2.1 Topological Methods The topological framework was first applied to the visualization of second-order tensor field by Delmarcelle and Hesselink [6]. Leveraging suggestions launched previously for the topology-based visualization of vector fields [16, 13], these authors proposed to display a planar tensor field through the topological structure of its two orthogonal eigenvector fields. As discussed in their work, the lack of buy PF-04449913 orientation of eigenvector fields prospects to singularities that are not seen in regular vector fields. Those correspond namely to locations where the tensor field becomes isotropic, i.e. where both eigenvalues are equivalent.