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Solving Poisson Equations Using Least Square Technique in Image Editing

Solving Poisson Equations Using Least Square Technique in Image Editing. Colin Zheng Yi Li. Roadmap. Poisson Image Editing Poisson Blending Poisson Matting Least Square Techniques Conjugate Gradient With Pre-conditioning Multi-grid. Intro to Blending. source. target. paste. blend.

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Solving Poisson Equations Using Least Square Technique in Image Editing

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  1. Solving Poisson Equations Using Least Square Technique in Image Editing Colin Zheng Yi Li

  2. Roadmap • Poisson Image Editing • Poisson Blending • Poisson Matting • Least Square Techniques • Conjugate Gradient • With Pre-conditioning • Multi-grid

  3. Intro to Blending source target paste blend

  4. Gradient Transfer

  5. Gradient Transfer

  6. Gradient Transfer

  7. Gradient Transfer

  8. Results

  9. Results

  10. Into to Matting I = α F + (1 – α) B ∇I = (F −B)∇α+ α∇F +(1− α)∇B ∇I ≈ (F −B)∇α

  11. Poisson Matting with

  12. Poisson Matting with with

  13. Results

  14. Conjugate Gradient Method • Problem to solve: Ax=b • Sequences of iterates: x(i)=x(i-1)+(i)d(i) • The search directions are the residuals. • The update scalars are chosen to make the sequence conjugate (A-orthogonal) • Only a small number of vectors needs to be kept in memory: good for large problems

  15. Conjugate Gradient +

  16. Initialized as the source image (50 iterations) Initialized as the target image (50 iterations) Conjugate Gradient: Starting

  17. Precondition • We can solve Ax=b indirectly by solving M-1Ax= M-1b • If (M-1A) <<(A), we can solve the latter equation more quickly than the original problem. * If max and min are the largest and smallest eigenvalues of a symmetric positive definite matrix B, then the spectral condition number of B is

  18. Symmetric Successive Over Relaxation (SSOR) Barrett, R.; Berry, M.; Chan, T. F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; and Van der Vorst, H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. Philadelphia, PA: SIAM, 1994.

  19. Precondition

  20. Precondition (Cont) Step=0 Step=5 Step=10 Step=20 Step=40 Without Precondition Without Precondition

  21. Precondition Demo (20 iterations)

  22. Multigrid Use coarse grids to computer an improved initial guess for the fine-grid.

  23. Multigrid: Different Starting Initialized as Target (bad starting)

  24. Multigrid (Cont) Looser threshold for the coarse grids:

  25. Multigrid + Precondition Combine Multigrid with Precondition

  26. Multigrid Demo

  27. Conclusion • Applications • Poisson Blending • Poisson Matting • Least Square Techniques • Conjugate Gradient • With Pre-conditioning • Multi-grid • Performance Analysis • Sensitivity • Convergence

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