1. Patric Perez, Michel Gangnet, and Andrew Black
Poisson Image Editing
Patric Perez, Michel Gangnet, and Andrew Black
(SIGGRAPH 2003)
Presentation by : Lingli He
6 , jun 2017

2. Input Images
source image
target image

3. Editing Results
Simple Cloning Result
Poisson Seamless Cloning

4. Motivation
Problems :how to remove the seams between the mixed images

5. Goals
importing (cloning) transparent and opaque source image regions into a destination image in a seamless and effortless manner.
Seamless modification of appearance of the image within a selected region.

6. Background
A famous psychologists show that the second-order variations extracted by the Laplacian operator are the most significant perceptually (Land and McCann 1971)
A scalar function on a bounded domain is uniquely defined by its values on the boundary and its Laplacian in the interior.
The Poisson equation has a unique solution.

7. Related work– poisson equation
Fattal et al Entire images
Lewis harmonic interpolation
Our method Local images
Our method Guided Interpolation

8. Related work—multiresolution images blending
Burt and Adelson1983 long range(larger gradient) mixing
Poisson seamless cloning small gradient

9. Related work– seamless cloning
Example-based interpolation methods ; handle large holes and textured boundaries ( Barret and Cheney 2002)
inpainting techniques-PDE . more complex than poisson equation ( Bertalmio et al )
a guided interpolation framework , with the guidance being selected by the user ,and this framework is not limited in seamless cloning

10. Interpolation Problem
S: a closed subset of R2
Ω: a closed subset of S, with boundary ∂Ω
f*: known scalar function over S\Ω
f: unknown scalar function over Ω

11. Membrane Interpolation
To find the value of f ,Solve the following minimization problem:
subject to Dirichlet boundary conditions:
the gradient operator

12. Solution :Euler-Lagrange Equation

13. Laplace Equation
Solution: Laplace Equation with Dirichlet boundary conditions
this method produces an unsatisfactory due to over-blurring
Laplacian operator

14. Guided Interpolation
v: guided field
v may be gradient of a function g

15. Guided Interpolation Solve the following minimization problem:
subject to Dirichlet boundary conditions:

16. Poisson Equation
Solution :This time the Euler-Lagrange equation reduces to the Poisson equation:
written more concisely as:
G denotes v

17. if v is conservative If v is the gradient of an image g
Correction function so that
performs membrane interpolation over Ω:

18. Discrete Poisson Solver
Discretize directly by :
for neighbors p and q with
Discretized gradient
Discretized v: g(p)-g(q)
all pairs that are in Ω)

19. Discrete Poisson Solver
Partial Derivative for neighborhood overlaps boundary(Big yet sparse linear system):
Partial Derivative for interior points:

20. Discrete Poisson Solver:solution
Linear system of equations
sparse (banded)
Symmetric
positive-definite
Irregular shape of boundary requires general solver, such as
Gauss-Seidel iteration with successive overrelaxation
V-cycle Multi-grid
System can be solved at interactive rates

21. Seamless Cloning Import Gradients from a Source Image g
Discretization ：
Solving Poisson equation:

22. Seamless Cloning Results
Concealment
By importing seamlessly a piece of the background
Multiple strokes
input
output

23. Seamless Cloning Results
Insertion

24. Seamless Cloning Results
Feature exchange

25. Mixing Gradients
To combine properties of background f* with those of selected resource g
Two Variants of V
v averaged from source and destination gradients (insert transparent images)
Select stronger one from source and destination gradients

26. Mixing Gradients
Variant of V:
Discretization :

27. Mixing Gradients Results
Inserting objects with holes

28. Mixing Gradients Results
Inserting one object close to another

29. × Texture Flattening A sparse sieve ：Preserve only salient gradients
Discretization : with masking function so that: ×

30. Texture Flattening results
input
output

31. Local Illumination Changes
Approximate tone mapping transformation after Fattal et al. 2002:
Attenuating large gradients ×

32. Local illumination Changes results
input
output

33. Local Color Changes
Mix two differently colored version of original image
– One provides f * outside
– One provides g inside

34. Seamless Tiling Select original image as g Boundary condition:
– f *north = f *south = 0.5 (gnorth + gsouth)
– Similarly for the east and west
Boundary condition changed
input
output

35. Conclusions
Using the generic framework of guided interpolation to develop a variety of tools to edit the contents of an images selection in a seamless and effortless manner .
Seamlessly edit images via poisson solution to guided interpolation under Dirichlet boundary conditions

36. Limitations Cloning requires either of the images to be smooth
Minimization only adapts low-frequency Content
The backgrounds in resource and destination should be similar

37. Future works To combine the cloning facilities and the editing ones
Extend the editing facilities to deal with the sharpness of objects of interest
Extend the cloning facilities the editing facilities to 3D images

38. Thank you!

39. Questions How to do the Concealment in the 22 page ?
By importing seamlessly a piece of the background, complete objects, parts of objects, and undesirable artifacts can easily be hidden . We need multiple strokes.
How to determine the number of neighbors of p in the 18 page ?
It is a experience point. The number of neighbors of p is two in the 1D,and four in the 2D .

40. Questions
Why can the decolorization of background be recolor when changing the local Color of a image ?
Poisson solver can produce three color channel independently by solving three independent poisson equation.