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Interactive Visualization of Rotational Symmetry Fields on Surfaces

Jonathan Palacios and Eugene Zhang
IEEE Transactions on Visualization and Computer Graphics, Vol. 17(7), 2011, pp. 947-955.

Paper (PDF, 6.9Mb).

This material is based upon work supported by the National Science Foundation under Grant No. CCF-0546881 and CCF-0830808.

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).


Rotational symmetries have found uses in several computer graphics applications, such as global surface parameterization, geometry remeshing, texture and geometry synthesis, and non-photorealistic visualization of surfaces. The visualization of N-way rotational symmetry (N-RoSy) fields is a challenging problem due to the ambiguities in the N directions represented by an N-way symmetry. We provide an algorithm that allows faithful and interactive representation of N-RoSy fields in the plane and on surfaces, by adapting the well-known Line Integral Convolution (LIC) technique from vector and second-order tensor fields. Our algorithm captures the N directions associated with each point in a given field by decomposing the field into multiple different vector fields, generating LIC images of these fields, and then blending the results. To address the loss of contrast caused by the blending of images, we observe that the pixel values in LIC images closely approximate normally distributed random variables. This allows us to use concepts from probability theory to correct the loss of contrast without the need to perform any image analysis at each frame.


1. Visualization of N-RoSy fields have applications in surface hatching, quadrangular remeshing, surface tiling, architectual rendering, and semi-regular triangulation.


2. Showing individual directional fields will not convey the structures in an N-RoSy field (compare a-c with j). We address this by composing several images to capture all directions in the rotational symmetries. To overcome visual artifacts caused when rendering an N-RoSy field as a vector field, we include more images to remove visual artifacts (f-h).


3. Image composition can lead to loss of contrast. In our case, we can predict the amount of loss and correct this authomatically without having to perform image analysis or histogram equalization.