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Simple Formulas for Quasiconformal Plane Deformations

ACM Transactions on Graphics, August 2012

Yaron Lipman, Vladimir G. Kim, Thomas A. Funkhouser
Image deformations computed with quasiconformal maps

We introduce a simple formula for 4-point planar warping that produces provably good 2D deformations. In contrast to previous work, the new deformations minimizes the maximum conformal distortion and spreads the distortion equally across the domain. We derive closed-form formulas for computing the 4-point interpolant and analyze its properties.We further explore applications to 2D shape deformations by building local deformation operators that use Thin-Plate Splines to further deform the 4-point interpolant to satisfy certain boundary conditions. Although this modification no longer has any theoretical guarantees, we demonstrate that, practically, these local operators can be used to create compound deformations with fewer control points and smaller worst-case distortions in comparisons to the state-of-the-art.

Yaron Lipman, Vladimir G. Kim, and Thomas A. Funkhouser.
"Simple Formulas for Quasiconformal Plane Deformations."
ACM Transactions on Graphics 31(5), August 2012.


   author = "Yaron Lipman and Vladimir G. Kim and Thomas A. Funkhouser",
   title = "Simple Formulas for Quasiconformal Plane Deformations",
   journal = "ACM Transactions on Graphics",
   year = "2012",
   month = aug,
   volume = "31",
   number = "5"