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

2. Roadmap Poisson Image Editing Least Square Techniques 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. ∇I = (F −B)∇α+ α∇F +(1− α)∇B
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. Conjugate Gradient: Starting
Initialized as the source image
(50 iterations)
Initialized as the target image
(50 iterations)

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

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 Least Square Techniques Performance Analysis
Poisson Blending
Poisson Matting
Least Square Techniques
Conjugate Gradient
With Pre-conditioning
Multi-grid
Performance Analysis
Sensitivity
Convergence