Bounded distortion harmonic shape interpolation

Planar shape interpolation is a classic problem in computer graphics. We present a novel shape interpolation method that blends C∞ planar harmonic mappings represented in closed-form. The intermediate mappings in the blending are guaranteed to be locally injective C∞ harmonic mappings, with conformal and isometric distortion bounded by that of the input mappings. The key to the success of our method is the fact that the blended differentials of our interpolated mapping have a simple closed-form expression, so they can be evaluated with unprecedented efficiency and accuracy. Moreover, in contrast to previous approaches, these differentials are integrable, and result in an actual mapping without further modification. Our algorithm is embarrassingly parallel and is orders of magnitude faster than state-of-the-art methods due to its simplicity, yet it still produces mappings that are superior to those of existing techniques due to its guaranteed bounds on geometric distortion.

[1]  Steven R. Bell,et al.  The Cauchy Transform, Potential Theory and Conformal Mapping , 2015 .

[2]  Denis Zorin,et al.  Strict minimizers for geometric optimization , 2014, ACM Trans. Graph..

[3]  J. Geelen.  ON HOW TO DRAW A GRAPH , 2012 .

[4]  Michael Garland,et al.  Free-form motion processing , 2008, TOGS.

[5]  Jaeil Choi,et al.  On Coherent Rotation Angles for As-Rigid-As-Possible Shape Interpolation , 2003, CCCG.

[6]  L. Ahlfors.  Complex Analysis , 1979 .

[7]  George Wolberg,et al.  Image morphing: a survey , 1998, The Visual Computer.

[8]  Olga Sorkine-Hornung,et al.  Context‐Aware Skeletal Shape Deformation , 2007, Comput. Graph. Forum.

[9]  Mirela Ben-Chen,et al.  Complex Barycentric Coordinates with Applications to Planar Shape Deformation , 2009, Comput. Graph. Forum.

[10]  Ronen Basri,et al.  Controlling singular values with semidefinite programming , 2014, ACM Trans. Graph..

[11]  Marc Alexa,et al.  As-rigid-as-possible shape interpolation , 2000, SIGGRAPH.

[12]  Peter Duren,et al.  Harmonic Mappings in the Plane , 2004 .

[13]  S. Lang.  Complex Analysis , 1977 .

[14]  Mirela Ben-Chen,et al.  Planar shape interpolation with bounded distortion , 2013, ACM Trans. Graph..

[15]  Hujun Bao,et al.  Poisson shape interpolation , 2005, SPM '05.

[16]  Denis Zorin,et al.  Locally injective parametrization with arbitrary fixed boundaries , 2014, ACM Trans. Graph..

[17]  Yaron Lipman,et al.  Bounded distortion mapping spaces for triangular meshes , 2012, ACM Trans. Graph..

[18]  Olga Sorkine-Hornung,et al.  Locally Injective Mappings , 2013 .

[19]  Keenan Crane,et al.  Globally optimal direction fields , 2013, ACM Trans. Graph..

[20]  Zohar Levi,et al.  On the convexity and feasibility of the bounded distortion harmonic mapping problem , 2016, ACM Trans. Graph..

[21]  Craig Gotsman,et al.  Controllable morphing of compatible planar triangulations , 2001, TOGS.

[22]  Craig Gotsman,et al.  Intrinsic Morphing of Compatible Triangulations , 2003, Int. J. Shape Model..

[23]  Denis Zorin,et al.  Computing Extremal Quasiconformal Maps , 2012, Comput. Graph. Forum.

[24]  Roi Poranne,et al.  Provably good planar mappings , 2014, ACM Trans. Graph..

[25]  Ofir Weber,et al.  Bounded distortion harmonic mappings in the plane , 2015, ACM Trans. Graph..

[26]  Ronen Basri,et al.  Large-scale bounded distortion mappings , 2015, ACM Trans. Graph..

[27]  Marc Alexa,et al.  Recent Advances in Mesh Morphing , 2002, Comput. Graph. Forum.

[28]  Ofir Weber,et al.  Controllable conformal maps for shape deformation and interpolation , 2010, ACM Trans. Graph..

[29]  William V. Baxter,et al.  Rigid shape interpolation using normal equations , 2008, NPAR.

[30]  Yaron Lipman,et al.  Injective and bounded distortion mappings in 3D , 2013, ACM Trans. Graph..