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

Roadmap Poisson Image Editing Least Square Techniques 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

∇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

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

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

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

Precondition Demo (20 iterations)

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

Multigrid: Different Starting Initialized as Target (bad starting)

Multigrid (Cont) Looser threshold for the coarse grids:

Multigrid + Precondition Combine Multigrid with Precondition

Multigrid Demo

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