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Poisson Image Editing
Ligang LiuGraphics&Geometric Computing Lab
USTChttp://staff.ustc.edu.cn/~lgliu
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Images in the Internet
Images everywhere!
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Images Everywhere
• Digital camera
• Personal digital album
• Photoshop
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Digital Image
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Math Model of Image
• Continuous– f(x,y)
• Discrete– I (i,j)
– Pixel
(R, G, B)
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Image Processing
• Image filtering
• Image coding
• Image transformation
• Image enhancement
• Image segmentation
• Image understanding
• Image recognition
• …
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Gradient Domain Image Editing
• Motivation:– Human visual system is very sensitive to gradient
– Gradient encode edges and local contrast quite well
• Approach:– Edit in the gradient domain
– Reconstruct image from gradient
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Gradient Domain: View of Image
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Membrane Interpolation
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Membrane Interpolation
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Poisson Equations
)(4 xρπG−=ΔΦ
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Siméon Denis Poisson
• His teachers: Laplace, Lagrange, …• Poisson’s terms:
– Poisson's equation
– Poisson's integral
– Poisson distribution
– Poisson brackets
– Poisson's ratio
– Poisson's constant
1781-1840, France
“Life is good for only two things: to study mathematics and to teach it.”
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Background:
02 =+⋅+⋅+⋅+⋅+⋅+⋅ GfFfEfDfCfBfA yxyyxyxx
• Partial Differential Equations (PDE)
• The PDE's which occur in physics are mostly second order and linear:
0),,,,,( =yyxyxxyx ffffffE
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– Hyberbolic• wave equation:
– Parobolic• heat equation:
– Elliptic• Laplace equation:
• Poisson equation:
:2BCA <⋅
:2BCA =⋅
2
2
21
tf
vf
∂∂
=Δ
fktf
Δ⋅=∂∂
:2BCA >⋅
0=Δf
ρ−=Δf
02 =+⋅+⋅+⋅+⋅+⋅+⋅ GfFfEfDfCfBfA yxyyxyxx
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Poisson Equation
ρ−=Δf
yx 2
2
2
2
∂∂
+∂∂
≡Δ
),( yxρρ =
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Boundary conditions
• Dirichlet boundary conditions:
• Neumann boundary conditions:
dsΩ∂Ω
Ω∂|f
Ω∂∂∂ |sf
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Existence of solution
dsΩ∂Ω
The solution of an Poisson Equation is uniquelydetermined in , if Dirichlet boundary conditions or Neumann boundary conditions are specified on
ΩΩ∂
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Discrete Poisson Equation
jijijijijijiji
hfff
hfff
,2,1,1,
2,,1,1 22
ρ=−+
+−+ −+−+
ρ−=Δf
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Matrix Nature of Poisson Equation
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−
−−
−−
−
33
23
13
32
22
12
31
21
11
33
23
13
32
22
12
31
21
11
41141
141141
141141
141141
14
bbbbbbbbb
fffffffff
Large sparse linear system
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Poisson Equation Solver
• Direct method
• Iterative methods– Jacobi, Gauss‐Seidel, SOR
• Multigrid method
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Variational interpretation
Ω∂Ω∂Ω=−∇= ∫∫ ||s.t minarg *2* ffff f v
Ω∂Ω∂ ==Δ ||s.t )( * ffdivf v
Euler Equation: 0=∂∂
−∂∂
−yx fff F
yF
xF
F
v is a guidance field, needs not to be a gradient field.
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Physical Origins of Poisson Equation
• Electrostatic potential
• Gravitational potential
)(4 xρπG−=ΔΦ
0
)(ερ x
−=ΔΦ
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Electrostatic potential
Charge Density
Electric Potential
Electric Field Φ
E
E
Φ
)(xρ
)(xρ
Φ−∇=E
30
21
4 rqq
πεrF =
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Derivations
∫∫ =⋅VS
dvd0
)(ερ xsE
0
)(ερ x
−=ΔΦ
Gauss’s Law: V
S
ds
∫∫ ⋅∇=⋅VS
dvd EsEGauss’stheorem:
0
)(ερ xE =⋅∇
Φ−∇=E Poisson Equation
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Φ
E
ρ
0
)(ερ x
−=ΔΦ
Φ−∇=E V
ddiv S
V
∫ ⋅≡=
→
sEEx
0
lim)()(
0ερ
Relationships
DensityPotential
Field
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Φ E),()( 00 yxδρ =x
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Gravitational potential
Mass Density
Gravitational Potential
Force Field
(acceleration)
g
Φ
)(xρ
Φ−∇=g
gF m=
m
)(xρ
Φ
3rmMGrF =
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V
ddivG S
V
∫ ⋅≡=
→
sggx
0
lim)()(4 ρπ
Φ
g
ρ
Φ−∇=g
Relationships
DensityPotential
Field
)(4 xρπG−=ΔΦ
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Analogy for Image
Image Density
Image (Potential)
Image Gradientg
I
)(xρ
I−∇=g
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V
ddiv S
V
∫ ⋅≡=
→
sggx
0
lim)()(ρ
I
g
ρ
I−∇=g
Relationships
DensityPotential
Field
)(xρ−=ΔI
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V
ddiv S
V
∫ ⋅≡=
→
sggx
0
lim)()(ρI−∇=g
)(xρ−=ΔI
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)(xρ−=ΔI
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Motivation for Image Editing
I
ρDensity
Potential
)(xρ−=ΔI
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Poisson Image Editing
P. Pérez, M. Gangnet, and A. BlakeSIGGRAPH 2003
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Seamless Cloning
Precise selection: tedious and unsatisfactory
Alpha-Matting: powerful but involved
Seamless cloning: loose selection but no seams?
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Seamless Poisson Cloning
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Ω∂Ω∂ =∇=Δ || s.t. )( BA IIIdivI
AI BI
Cloning by solving Poisson Equation
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Why we do analogy for image?
• Easier in the image gradient domain
– Local editing global effects
– Seamless ‐ cloning, editing, tilling
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Variational Interpretation
Ω∂Ω∂Ω=∇−∇= ∫∫ ||s.t minarg *2*
BAf IIIII
Euler Equation: 0=∂∂
−∂∂
−yx fff F
yF
xF
F
AI∇ is a gradient field to be cloned.
Ω∂Ω∂ =∇=Δ || s.t. )( BA IIIdivI
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ComposeDemo
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Change Features
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Change Texture
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Conceal
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Mix Lights
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Change Colors
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Change Colors
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Seamless TilingSingle image
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Seamless TilingMultipleimagestiled at random
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Image Stitching
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Alpha Blending / Feathering
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Affect of Window Size
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Affect of Window Size
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Good Window Size
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Gradient based Stitching
• Input images
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Recap: Gradient Domain Image Editing
• Motivation:– Human visual system is very sensitive to gradient
– Gradient encode edges and local contrast quite well
• Approach:– Edit in the gradient domain
– Reconstruct image from gradient
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Q&A