Warping Pictures Nicely

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David Swart: Warping Pictures Nicely. In: Bridges 2011. Pages 303–310



We present a new algorithm, using only simple geometry, to warp imagery from an arbitrarily shaped source region to an arbitrarily shaped target region. Mathematically speaking, the algorithm outputs harmonic maps (every point is at the average position of its neighbors) using new boundary conditions to curb excessive non-uniform stretching and shearing in order to appear more conformal. The algorithm has typical running times measured in seconds. We give some artistic examples to demonstrate how the results can be used in digital photography and other graphical work.

Extended Abstract


Used References

[1] Vladimir Bulatov. Conformal Models of the Hyperbolic Geometry. Presented at MAA-AMS Joint Mathematics Meeting, San Francisco 2010. http://bulatov.org/math/1001/. Accessed February 1, 2011.

[2] Chuck Collins and Kenneth Stephenson. A Circle Packing Algorithm. Computational Geometry: Theory and Applications, vol. 25, pages 233–256, 2003.

[3] Bart de Smit and Hendrik W. Lenstra Jr. The Mathematical Structure of Escher’s Print Gallery. In Notices of the AMS 50, no 4, pages 446–451, 2003.

[4] Tobin A. Driscoll and Lloyd N. Trefethen. Schwarz-Christoffel Mapping. Cambridge Monographs on Applied and Computational Mathematics, 2002.

[5] Daniel M. Germán, Lloyd Burchill, Alexandre Duret-Lutz, Sébastien Pérez-Duarte, Emmanuel Pérez-Duarte, Josh Sommers. Flattening the Viewable Sphere. Computational Aesthetics 2007, pages 23–28. 2007.

[6] Owen Jones. The Grammar of Ornament. Folio edition, Bernard Quaritch, 1910.

[7] Liliya Kharevych, Boris Springborn and Peter Schröder. Discrete Conformal Mappings via Circle Patterns. In ACM Transactions on Graphics 25(2), pages 412-438, 2006

[8] Boris Springborn, P. Schröder, and Ulrich Pinkall. Conformal Equivalence of Triangle Meshes. In ACM Transactions on Graphics, 27(3), article 77, 2008.

[9] Martin von Gagern and Jürgen Richter-Gebert. Hyperbolization of Euclidean Ornaments. The Electronic Journal of Combinatorics, 16(2), 2009.


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