“Biharmonic diffusion curve images from boundary elements”
Conference:
Type(s):
Title:
- Biharmonic diffusion curve images from boundary elements
Session/Category Title: Rendering and Thinking Inside the Box
Presenter(s)/Author(s):
Abstract:
There is currently significant interest in freeform, curve-based authoring of graphic images. In particular, “diffusion curves” facilitate graphic image creation by allowing an image designer to specify naturalistic images by drawing curves and setting colour values along either side of those curves. Recently, extensions to diffusion curves based on the biharmonic equation have been proposed which provide smooth interpolation through specified colour values and allow image designers to specify colour gradient constraints at curves. We present a Boundary Element Method (BEM) for rendering diffusion curve images with smooth interpolation and gradient constraints, which generates a solved boundary element image representation. The diffusion curve image can be evaluated from the solved representation using a novel and efficient line-by-line approach. We also describe “curve-aware” upsampling, in which a full resolution diffusion curve image can be upsampled from a lower resolution image using formula evaluated orrections near curves. The BEM solved image representation is compact. It therefore offers advantages in scenarios where solved image representations are transmitted to devices for rendering and where PDE solving at the device is undesirable due to time or processing constraints.
References:
1. Andrews, J., Joshi, P., and Carr, N. 2011. A linear variational system for modeling from curves. Computer Graphics Forum 30, 6, 1850–1861.
2. Bezerra, H., Eisemann, E., DeCarlo, D., and Thollot, J. 2010. Diffusion constraints for vector graphics. NPAR 2010: Proceedings of the 8th International Symposium on Non-Photorealistic Animation and Rendering.
3. Boyé, S., Barla, P., and Guennebaud, G. 2012. A vectorial solver for free-form vector gradients. Transaction on Graphics (Nov.).
4. Boyé, S. 2012. Représentation hybride pour la modélisation géométrique interactive. PhD thesis, Université Sciences et Technologies-Bordeaux 1.
5. Brown, J., and Churchill, R. 1996. Complex variables and applications. McGraw-Hill Inc.
6. Finch, M., Snyder, J., and Hoppe, H. 2011. Freeform vector graphics with controlled thin-plate splines. ACM Trans. Graph. 30, 6 (Dec.), 166:1–166:10.
7. Ilbery, P. 2008. Scanline calculation of radial influence for image processing. PhD thesis, University of NSW.
8. Jaswon, M. A., and Symm, G. 1977. Integral equation methods in potential theory and elastostatics. Academic Press Inc.
9. Jeschke, S., Cline, D., and Wonka, P. 2009. A GPU Laplacian solver for diffusion curves and Poisson image editing. Transaction on Graphics (Siggraph Asia 2009) 28, 5 (Dec.), 1–8.
10. Joshi, P., and Carr, N. A. 2008. Repoussé: automatic inflation of 2D artwork. In Proceedings of the Fifth Eurographics conference on Sketch-Based Interfaces and Modeling, Eurographics Association, SBM’08, 49–55.
11. Kreyszig, E. 1999. Advanced engineering mathematics (8th ed.). John Wiley.
12. Lerma, M., 2002. The Bernoulli periodic functions. http://www.math.northwestern.edu/~mlerma/papers/bern_period_func.pdf.
13. Mai-Duy, N., and Tanner, R. 2005. An effective high order interpolation scheme in BIEM for biharmonic boundary value problems. Engineering Analysis with Boundary Elements 29, 3, 210–223.
14. Nealen, A., Igarashi, T., Sorkine, O., and Alexa, M. 2007. Fibermesh: designing freeform surfaces with 3D curves. ACM Trans. Graph. 26, 3 (July).
15. Orzan, A., Bousseau, A., Winnemöller, H., Barla, P., Thollot, J., and Salesin, D. 2008. Diffusion curves: a vector representation for smooth-shaded images. ACM Trans. Graph. 27, 3 (Aug.), 92:1–92:8.
16. Orzan, A., Bousseau, A., Winnemöller, H., Barla, P., Thollot, J., and Salesin, D. 2013. Diffusion curves: a vector representation for smooth-shaded images. Commun. ACM 56, 7 (Jul.), 101–108.
17. Poritsky, H. 1946. Application of analytic functions to two-dimensional biharmonic analysis. Transactions of the American Mathematical Society 59, 2 (Mar.), 248–279.
18. Sun, X., Xie, G., Dong, Y., Lin, S., Xu, W., Wang, W., Tong, X., and Guo, B. 2012. Diffusion curve textures for resolution independent texture mapping. ACM Trans. Graph. 31, 4 (July), 74:1–74:9.
19. van de Gronde, J. 2010. A High Quality Solver for Diffusion Curves. Master’s thesis, University of Groningen.
20. Weber, O., Poranne, R., and Gotsman, C. 2012. Biharmonic coordinates. In Computer Graphics Forum, vol. 31, Wiley Online Library, 2409–2422.
