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