Poisson Image Editing

f
Ω
〈 p,q〉∩Ω≠0
p+q w here vpq is the projection of v( ) on the oriented edge [ p, q] 2 p +q i.e., vpq = v( ) ⋅ pq 2 Its solution satisfies the following simultaneous linear equation:
Guided Interpolation
A guided field is a vector field v used in an extended version of minimization problem (1):
m ∫∫ ∇f − v in
f Ω
2
with f
= f* ∂Ω
∂Ω
(3)
This solution is the unique solution of following Poisson equation with Dirichlet boundary condition:
It computes the function whose gradient is the closet to some prescribed vector field-the guidance vector field-under given boundary conditions
Partial differential equation (PDE)
∂Ω
∂2. ∂2. w ere ∆. = 2 + 2 is th laplacian operator h e ∂x ∂y
(2)
Guided Interpolation
Equation 2 is a Laplace equation with Dirichlet boundary conditions.
Introduction
The Poisson equation has been used extensively in computer vision. The mathematical tool at the heart of this approach is the Poisson partial differential equation with Dirichlet boundary conditions which specifies the Laplacian of an unknown function over the domain of interest, along with the unknown function values over the boundary of domain. Solving the Poisson equation also has an alternative interpretation as a minimization problem:
i.e., 0.4s. per system on a Pentium 4 for a disk-shaped region of 60,000 pixels.
Seamless cloning
Importing gradients The basic choice for the guidance field v is a gradient field taken directly from a source image. Denoting by g this source image, the interpolation is performed under the guidance of
Importing gradients
Importing gradients
Seamless cloning
Mixing gradients
There are situations where it is desirable to combine properties of f* with those of g, for example to add objects with holes ,or partially transparent ones, on top of a textured or cluttered background. An example is shown in Fig.6
Guided Interpolation
The simplest interpolant f of f* over Ω is the membrane interpolant defined as the solution of the minimization 2 problem: *
m ∫∫ ∇f in
T = f (t) , at x = 0 , t > 0
T0
T = T1 ,
t>0
T= T , 1
T = T0 ,
at x =1, t > 0
at x > 0 , t = 0
0
1
x
圖 平版Dirichlet Condition 示意圖
Neumann condition
Neumann condition係指依變數之變化率之邊 界條件為定值，抑或獨立變數之函數之情況。 例如 ∂T = f (t) , at t = 0 , 0 ≤ x ≤1 或 ∂x ∂T = 0 , at x = 1, t ≥0 Neumann型邊界條件，亦稱為natural ∂x boundary condition。
Discrete Poisson solver
The finite difference discretization of (3) yields the following discrete, quadratic optimization problem: * m in ∑( f p − fq −vpq )2, w f p = f p , for all p∈∂Ω ith (6)
Poisson Image Editing
Patrick Perez Michel Gangnet Andrew Blake Microsoft Research UK
Outline
Introduction Poisson solution to guided interpolation
Guided Interpolation Discrete Poisson solver
方程式類別 橢圓型 拋物線型 雙曲線型 判斷式
b − 4ac < 0
2
代表性範例 Laplace方程式 Poisson方程式 波動方程式
∂2u ∂2u + 2 =0 2 ∂x ∂y
b2 − 4ac = 0
b − 4ac > 0
2
∂2u ∂2u + 2 = f (x, y) 2 ∂x ∂y
∂2u ∂2u = 2 α 2 ∂x ∂t
Seamless cloning
Importing gradients Mixing gradients
Selection editing
Texture flattening Local illumination change Local color change Seamless tiling
Conclusion
for all p∈Ω, Np f p −
q q∈Np ∩Ω
∑f
=
q* q∈Np ∩∂Ω
∑f
+ ∑vpq
q∈Np
(7)
Discrete Poisson solver
Equation (7) form a classical, sparse (banded), symmetric, positive-definite system. Results shown in this paper have been computed using either Gauss-Seidel iteration with successive overrelaxation or V-cycle multigrid. Both methods are fast enough
2
起始條件和邊界條件的分類
為了能獲得偏微分方程式之解答，其起始條 件和邊界條件可依其特性區分為三類。
第一類：Dirichlet Condition Dirichlet 第二類：Neumann condition 第三類：Robbins condition
Dirichlet Condiction
若依變數(T)本身，在某個獨立變數值時， 被指定，則此條件稱為Dirichlet Condition，亦稱為essential邊界條件。 下圖為一典型的Dirichlet條件示意圖 由圖中很清楚的顯示，該平板之邊界條 = f (t ), T 件為 t >0
for all 〈 p, q〉, vpq = gp − gq
(11)
Importing gradients
The seamless cloning tool thus obtained ensures the compliance of source and destination boundaries. It can be used to conceal undesirable image features or to insert new elements in an image, but with much more flexibility
∂u ∂v + is the divergence of v = (u, v) ∂x ∂y
Discrete Poisson solver
S Ωnow become finite point sets defined on an infinite discrete grid. Note that S can include all the pixels of an image or only a subset of them. For each pixel p in S, let Np be the set of its 4connected neighbors which are in S, and let <p,q> denote a pixel pair such that q∈Np The boundary of Ω is now ∂Ω ={p∈S \ Ω: Np ∩Ω ≠ 0} Let fp be the value of f at p and f Ω ={ f p , p∈Ω }