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
Gradient Transfer
Gradient Transfer
Gradient Transfer
Gradient Transfer
Results
Results
Into to Matting
I = α F + (1 – α) B
∇I = (F −B) α+ α F +(1− α) B∇ ∇ ∇
∇I ≈ (F −B) α∇
Poisson Matting
with
Poisson Matting
with
with
Results
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
Conjugate Gradient
+
Conjugate Gradient: Starting•Initialized as the source image
(50 iterations)
•Initialized as the target image
(50 iterations)
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
min
max
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.
Precondition
0.01
0.1
1
10
100
1000
10000
0 25 50 75 100
Iteration
Res
idu
al (
log
)
source target source, SSOR
Precondition (Cont)
WithoutPrecondition
WithoutPrecondition
Step=0 Step=5 Step=10 Step=20 Step=40
Precondition Demo(20 iterations)
Multigrid
Use coarse grids to computer an improved initial guess for the fine-grid.
0.01
1
100
10000
0 50 100 150 200 250
iteration
|r|
Multigrid Precondition C.G.
Multigrid: Different Starting
Initialized as Target (bad starting)
0.01
1
100
10000
0 50 100 150 200 250
iteration
|r|
Multigrid CG Precondition
Multigrid (Cont)
Looser threshold for the coarse grids:
0.01
1
100
10000
0 50 100 150 200 250
iteration
|r|
Multigrid Precondition Multigrid (loose T)
Multigrid + Precondition
Combine Multigrid with Precondition
0.01
1
100
10000
0 10 20 30 40 50 60 70
iteration
|r|
Precondition Multigrid+Precondition
Multigrid Demo
Conclusion
• Applications– Poisson Blending– Poisson Matting
• Least Square Techniques– Conjugate Gradient – With Pre-conditioning– Multi-grid
• Performance Analysis– Sensitivity– Convergence
