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Fourier Analysis of the 2D Screened Poisson Equation for Gradient Domain Problems

  • Pravin Bhat
  • Brian Curless
  • Michael Cohen
  • C. Lawrence Zitnick
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5303)

Abstract

We analyze the problem of reconstructing a 2D function that approximates a set of desired gradients and a data term. The combined data and gradient terms enable operations like modifying the gradients of an image while staying close to the original image. Starting with a variational formulation, we arrive at the “screened Poisson equation” known in physics. Analysis of this equation in the Fourier domain leads to a direct, exact, and efficient solution to the problem. Further analysis reveals the structure of the spatial filters that solve the 2D screened Poisson equation and shows gradient scaling to be a well-defined sharpen filter that generalizes Laplacian sharpening, which itself can be mapped to gradient domain filtering. Results using a DCT-based screened Poisson solver are demonstrated on several applications including image blending for panoramas, image sharpening, and de-blocking of compressed images.

Keywords

Discrete Cosine Transform Discrete Fourier Transform Data Term Fourier Domain Photometric Stereo 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, London (2000)zbMATHGoogle Scholar
  2. 2.
    Ho, J., Lim, J., Yang, M., Kriegman, D.: Integrating surface normal vectors using fast marching method. In: European Conference on Computer Vision, vol. III, pp. 239–250 (2006)Google Scholar
  3. 3.
    Strang, G.: Introduction to Applied Mathematics. Wellesley-Cambridge Press (1986)Google Scholar
  4. 4.
    Fetter, A.L., Walecka, J.D.: Theoretical Mechanics of Particles and Continua. Courier Dover (2003)Google Scholar
  5. 5.
    Agarwala, A.: Efficient gradient-domain compositing using quadtrees. ACM Trans. Graph. 26, 94:1–94:5 (2007) CrossRefGoogle Scholar
  6. 6.
    Kazhdan, M., Hoppe, H.: Streaming multigrid for gradient-domain operations on large images. In: ACM Transactions on Graphics (Proc. of ACM SIGGRAPH 2008) (to appear, 2008)Google Scholar
  7. 7.
    Simchony, T., Chellappa, R., Shao, M.: Direct analytical methods for solving poisson equations in computer vision problems. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 435–446 (1990)CrossRefzbMATHGoogle Scholar
  8. 8.
    Horn, B.: Robot Vision. MIT Press, Cambridge (1986)Google Scholar
  9. 9.
    Zickler, T.E., Belhumeur, P.N., Kriegman, D.J.: Helmholtz stereopsis: Exploiting reciprocity for surface reconstruction. Int. J. Comput. Vision 49, 215–227 (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fattal, R., Lischinski, D., Werman, M.: Gradient domain high dynamic range compression. In: SIGGRAPH 2002: Proceedings of the 29th annual conference on Computer graphics and interactive techniques, pp. 249–256. ACM Press, New York (2002)CrossRefGoogle Scholar
  11. 11.
    Pérez, P., Gangnet, M., Blake, A.: Poisson image editing. In: SIGGRAPH 2003: ACM SIGGRAPH 2003 Papers, pp. 313–318. ACM Press, New York (2003)Google Scholar
  12. 12.
    Agarwala, A., Dontcheva, M., Agrawala, M., Drucker, S., Colburn, A., Curless, B., Salesin, D., Cohen, M.: Interactive digital photomontage. ACM Trans. Graph. 23, 294–302 (2004)CrossRefGoogle Scholar
  13. 13.
    Horn, B.K.P.: Height and gradient from shading. Int. J. Comput. Vision 5, 37–75 (1990)CrossRefGoogle Scholar
  14. 14.
    Horovitz, I., Kiryati, N.: Depth from gradient fields and control points: bias correction in photometric stereo. Image Vision Comput. 22, 681–694 (2004)CrossRefGoogle Scholar
  15. 15.
    Nehab, D., Rusinkiewicz, S., Davis, J., Ramamoorthi, R.: Efficiently combining positions and normals for precise 3d geometry. ACM Trans. Graph. 24, 536–543 (2005)CrossRefGoogle Scholar
  16. 16.
    Ng, H., Wu, T., Tang, C.: Surface-from-gradients with incomplete data for single view modeling. In: International Conference on Computer Vision, pp. 1–8 (2007)Google Scholar
  17. 17.
    Lischinski, D., Farbman, Z., Uyttendaele, M., Szeliski, R.: Interactive local adjustment of tonal values. In: SIGGRAPH 2006: ACM SIGGRAPH 2006 Papers, pp. 646–653. ACM Press, New York (2006)Google Scholar
  18. 18.
    Bhat, P., Zitnick, L., Cohen, M., Curless, B.: A perceptually-motivated optimization-framework for image and video processing. Technical Report UW-CSE-08-06-02, University of Washington (2008)Google Scholar
  19. 19.
    Agrawal, A.K., Raskar, R., Chellappa, R.: What is the range of surface reconstructions from a gradient field? In: ECCV, vol. (1), pp. 578–591 (2006)Google Scholar
  20. 20.
    Agrawal, A.: Scene Analysis under Variable Illumination using Gradient Domain Methods. Ph.D thesis, University of Maryland (2006)Google Scholar
  21. 21.
    Szeliski, R.: Fast surface interpolation using hierarchical basis functions. IEEE Trans. Pattern Anal. Mach. Intell. 12, 513–528 (1990)CrossRefGoogle Scholar
  22. 22.
    Frankot, R.T., Chellappa, R.: A method for enforcing integrability in shape from shading algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 10, 439–451 (1988)CrossRefzbMATHGoogle Scholar
  23. 23.
    Georghiades, A.S., Belhumeur, P.N., Kriegman, D.J.: From few to many: Illumination cone models for face recognition under variable lighting and pose. IEEE Transactions on Pattern Analysis and Machine Intelligence 23, 643–660 (2001)CrossRefGoogle Scholar
  24. 24.
    Weiss, Y.: Deriving intrinsic images from image sequences. In: International Conference on Computer Vision, vol. II, pp. 68–75 (2001)Google Scholar
  25. 25.
    Bracewell, R.N.: The Fourier Transform and Its Applications, 2nd edn. McGraw-Hill, New York (1986)zbMATHGoogle Scholar
  26. 26.
    Frigo, M., Johnson, S.G.: FFTW for version 3.0 (2003)Google Scholar
  27. 27.
    Szeliski, R.: Locally adapted hierarchical basis preconditioning. In: SIGGRAPH 2006: ACM SIGGRAPH 2006 Papers, pp. 1135–1143. ACM Press, New York (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Pravin Bhat
    • 1
  • Brian Curless
    • 1
  • Michael Cohen
    • 1
    • 2
  • C. Lawrence Zitnick
    • 1
  1. 1.University of WashingtonUSA
  2. 2.Microsoft ResearchUSA

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