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Interactive Tensor Field Design and Visualization on Surfaces

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Eugene Zhang, James Hays, and Greg Turk
IEEE Transactions on Visualization and Computer Graphics, 2007, Vol 13(1), pp 94-107.

Paper (PDF, 3.21Mb).

Video (MOV, 86.1Mb)

Talk slides (PPT, 52.3Mb)

This material is based upon work supported by the National Science Foundation under Grant No. CCF-0546881.

Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).


Designing tensor fields in the plane and on surfaces is a necessary task in many graphics applications, such as painterly rendering, pen-and-ink sketch of smooth surfaces, and anisotropic remeshing. In this paper, we present an interactive design system that allows a user to create a wide variety of surface tensor fields with control over the number and location of degenerate points. Our system combines basis tensor fields to make an initial tensor field that satisfies a set of userspecifications. However, such a field often contains unwanted degenerate points that cannot always be eliminated due to topological constraints of the underlying surface. To reduce the artifacts caused by these degenerate points, our system allows the user to move a degenerate point or to cancel a pair of degenerate points that have opposite tensor indices.

We observe that a tensor field can be locally converted into a vector field such that there is a one-to-one correspondence between the set of degenerate points in the tensor field and the set of singularities in the vector field. This conversion allows us to effectively perform degenerate point pair cancellation and movement by using similar operations for vector fields. In addition, we adapt the image-based flow visualization technique to tensor fields, therefore allowing interactive display of tensor fields on surfaces.

We demonstrate the capabilities of our tensor field design system with painterly rendering, pen-and-ink sketch of surfaces, and anisotropic remeshing.


1. Our tensor field visualization technique. Given a tensor field (an example is given in d), we create two (possibly discontinuous) vector fields (a, b) whose regions of discontinuity only intersect at degenerate points. By applying the image-based flow visualization (IBFV) to both vector fields and combining the resulting images through a properly constructed weight map (shown in c), we are able to obtain a high-quality interactive visualization of the tensor field.


2. Conversions between a vector field and a tensor field. This conversion and its inverse provide a one-to-one correspondence between the degenerate points in a tensor field and the singularities in the related vector field. These conversions allow us to adapt singularity pair cancellation and movement operations from vector fields and apply them to tensor fields.


3. Anisotropic remeshing of a horse model. Numerical estimation of the curvature tensor (left) contains a wedge and trisector pair that cause difficulty in the remeshing. By removing them through our tensor field system (right), the quality of the mesh is greatly improved.