poisson image editing - computer sciencelazebnik/research/fall08/jia_pan.pdfmicrosoft powerpoint -...
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Poisson Image Editing
SIGGRAPH 2003Patric Perez Michel GangnetAndrew Black
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Most Frequently Used PDE
Wave Equation
Heat Equation
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Most Frequently Used PDE
Poisson Equation, Steady State of Wave Equation and Heat Equation
Laplace’s Equation
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Boundary Conditions
Dirichlet Boundary ConditionsSpecify the value of the function on a surface
Neumann Boundary ConditionSpecify the normal derivative of the function on a surface
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Guided Interpolation
f: to be solved, f*: known regionv: guided field, g: v is prob. gradient of g
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Simple Interpolation
Maximize the Smoothness
Solution: Laplace Equation with DirichletBoundary Conditions
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Guided Interpolation
Interpolation-> minimization
Solution: Poisson Equation with DirichletBoundary Conditions
Relationship with Laplace case?
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Or from vector field decomposition
Helmholtz-Hodge decomp.
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Discrete Poisson SolverDiscretize the Minimization Directly
Partial Derivative
Partial Derivative for Interior Points
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Discrete Poisson Solver
Linear System of EquationsGauss-Seidel Method with Successive OverrelaxationV-cycle MultigridDiscretize Laplacian with Discrete Laplacian of Gaussianhttp://www.tau.ac.il/~stoledo/taucs/ Taucs
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Seamless Cloning :Importing Gradients
Importing Gradients from a Source Image
Discretize
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Seamless Cloning Results
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Seamless Cloning Results
Texture
Alignment
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Transfer intensity only
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Seamless Cloning: Mixing Gradients
Two ProposalsDefine v as Linear Combination of Source and Destination GradientsSelect Stronger one from Source and Destination Gradients (not conservative!)
Discretization
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Mixing Gradients Results
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Mixing Gradients Results
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Mixing Gradients Results
Just reduce terrible things ! Another solution
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Texture Flattening
Remain Only Salient Gradients
Discretization
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Texture Flattening
Edge mask
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Local Illumination Changes
Fattal Transformation
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Local Color Changes
Mix two different colored version of original image
One provide f* outsideOne provide g inside
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Local Color Changes
Monochrome of backgroundWhite in flower
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Seamless Tiling
Select original image as gBoundary condition:
f*north=f*
south=0.5(gnorth+gsouth)Similarly for the east and west
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Seamless Tiling
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Discussion
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Discussion
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Discussion
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Discussion
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Drag-and-Drop Pasting
Leo Jiaya Jia Leo Jiaya Jia Jian Sun Jian Sun ChiChi--Keung TangKeung TangHeungHeung--Yeung ShumYeung Shum
The Chinese University of Hong KongMicrosoft Research AsiaThe H.K. University of Sci. & Tech.Microsoft Research Asia
SIGGRAPH 2006
Slides by the authors
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Introduction to our method
Our method improves the Poisson image editing with
a new boundary optimization algorithm,an easier user interface,and an integration of alpha values.
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Poisson equations in images
A case study
+
sftf
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Poisson equations in imagesA case study
+
sftf
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Poisson equations in imagesThe optimization problem in image blending [Perez et al. 2003]
Taking , we have
What does it imply?
0 00
2
pmin | | with | |f s tf f dp f f∂Ω ∂Ω∈Ω
∇ −∇ =∫ 0 00
2
pmin | | with | |tsf dp ff ff ∂Ω ∂Ω∈Ω
=∇ −∇∫ 0 00
2
pmin | | with | | f s tf f fdp f∂Ω Ω ∂Ω∈
∇ −∇ =∫sf∇
tf'
sf f f= −
0 00
2' p
min | ' | with ' | |f t sf dp f f f∂Ω ∂Ω∈Ω∇ = −∫ 0
t sf f−
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Poisson equations in images
The minimization problem equals to solving the Laplace equation:
Image blending should take both the source and the target images into consideration.Property of solving the Laplace equation:
The variational energy will approach zero if and only if all boundary pixels satisfy , where k is a constant value.
0 0' 0 with ' | |t sf f f f∂Ω ∂ΩΔ = = −
0
2| ' |fΩ
∇∫0
( ) |t sf f k∂Ω− =
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Poisson equations in images
Where is the optimal boundary ?Inside the user drawn regionOutside the object of interest
How is the object extracted?Lazy snapping or Grabcut [Rother et al. 2004] (September 23)
How to optimize it?Minimum color variance
∂Ω
2min (( ( ) ( )) ) , s.t. bluet sp
f p f p k∈∂Ω
− − ∂Ω∈∑
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Boundary optimization
∂Ω and k are all unknownsAn iterative optimization
Initialize ∂Ω as the user drawn boundary.Given new ∂Ω, the optimal k is computed:
Given new k, optimize the boundary ∂Ω.Repeat the previous two steps until convergence.
2( , ) (( ( ) ( )) ) , s.t. bluet sp
E k f p f p k∈∂Ω
∂Ω = − − ∂Ω∈∑
( , ) 0E kk
∂ ∂Ω=
∂Shortest path problem
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Boundary optimizationIn 2D graph, computing the shortest path between any two points: Dynamic ProgrammingOur problem is to compute a closed path
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Boundary optimizationA shortest closed-path algorithm
Breaking closed boundary
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Boundary optimizationA shortest closed-path algorithm
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Boundary optimizationA shortest closed-path algorithm
Computation complexity O(N)
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Boundary optimizationA shortest closed-path algorithm
Total computation complexity O(NM)
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Boundary optimization discussion
OptimalityAvoiding that the path twists around the cut by selecting the initial cut position.
How to select the initial cut?Making it short to reduce O(MN)Passing smooth region
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One example
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Integrating fractional boundary
Fractional boundary is important in image composting: (transparency)
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MattingComposition Image I is generated by foreground and background with alpha matteI = alpha*F + (1-alpha)*BMatting is a problem to get alpha,F,B from a given image I.User have to devide the image into three region: Foreground, Background and Unknown area. In foreground area, F = I, alpha = 1, B = 0; In background area, F = 0, alpha = 0, B = I. Our task is to get F,B,alphain unknown area.
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Integrating fractional boundary
Where to use the fractional values?only the pixels where the optimized boundary is near the blue ribbon
∂Ω
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Integrating fractional boundary
Where to use the fractional values?only the pixels where the optimized boundary is near the blue ribbon
fractional integration: the green region
otherwise: the yellow region
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'
( , ) ( , ), ( 1, )( , ) ( (1 ) ) ( , ), ( 1, )
0 otherwise
x s
x x s t
f x y x y x y yellowv x y f f x y x y greenα α
∇ + ∈⎧⎪= ∇ + − + ∈⎨⎪⎩
Integrating fractional boundary
How to integrate the fractional values in Poisson blending?
A blended guidance field
'
( , ) ( , ), ( 1, )( , ) ( (1 ) ) ( , ), ( 1, )
0 otherwise
s
s
x
x x t
f x y x y x y yellowv x y f f x y x y greenα α
∇ + ∈⎧⎪= ∇ + − + ∈⎨⎪⎩
'
( , ) ( , ), (( , ) ( (1 ) ) ( , ), ( 1, )
0 otherwi
1, )
sex x s
x s
tv x y f f x y x y greef x y x y x y
nyellow
α α⎧⎪= ∇ + −
∇ ∈+ ∈⎨
+
⎪⎩
( , ) ( 1, ) ( , )x f x y f x y f x y∇ = + −
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' ( (1 ) ) ( , ), ( 1, )( , ) ( , ), ( 1, )
( , )0 otherwise
x s t
x s
x
f x y x yf f x y x y gr
x y yellowv y nx eeα α
∇ +∇ + − +
∈= ∈⎧⎪⎨⎪⎩
Integrating fractional boundaryHow to integrate the fractional values in Poisson blending?
A blended guidance field
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'
0 o
( , ) ( , ), (
therw
1, )( , ) ( (1 ) ) ( , ), ( 1,
s)i e
x s
x x s t
f x y x y x y yellowv x y f f x y x y greenα α
∇ + ∈⎧⎪= ∇ + − + ∈⎨⎪⎩
Integrating fractional boundaryHow to integrate the fractional values in Poisson blending?
A blended guidance field
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Integrating fractional boundary
Final minimization:
New boundary: Solving the corresponding Poisson equation.
2* *p *
min | ' | with | |f tf v dp f f∂Ω ∂Ω∈Ω∇ − =∫
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Results and comparison
Alpha blending
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Results and comparison
Poisson blendingOur method
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Results and comparison
Alpha blendingOur method
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Results and comparison
Poisson blendingOur method
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Poisson Image Editing Extended
SIGGRAPH 2006 sketch
Daniel Leventhal Brown Univ.Bernard Gordon Brown Univ.Peter G. Sibley Brown Univ.
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Problem caused by texture
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Alpha control
Alpha in [0, 1], background 0, and foreground 1.Obtain unselected area’s Alpha by blurring.
Implemented in YUV rather than RGB
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Result
Luminance rescaling
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Far from solved…
Only pure texture, how about mixture of texture and non-textureFar from perfect…
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Parallel Method on Mesh
Poisson based mesh editingOptical Boundaries for mesh mergingExtend from image space to mesh manifold
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Mesh Editing with Poisson-Based Gradient Field Manipulation
SIGGRAPH 2004Yizhou Yu UIUCKun Zhou MSRADong Xu Zhejiang Univ, MSRAXiaohan Shi Zhejiang Univ, MSRAHujun Bao Zhejiang Univ.Baining Guo MSRAHeung-Yueng Shum MSRA
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Mesh merging
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Deformation
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Basic idea
Quite similar to the case of imageVector field decomposition can be extended to manifold
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Discretization
NOT real gradient and divergence, but it works
Sparse linear system
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Mesh Merging
Some ‘ugly’ detailsBoundary interactionBoundary correspondenceRe-parameterization
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Mesh deformation
Interactively change guided vector fieldChange normal on a curvePropagate to other areas
Smooth normal field – smooth
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Optimal Boundaries for Poisson Mesh Merging
SPM 2007Xiaohuang Huang Zhejiang Univ. & HUSTHongbu Fu HUSTOscar Kin-Chung Au HUSTChiew-Lan Tai HUST
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Basic idea
Best boundary for Poisson MergingSimilar to ‘Drag-and-Drop Pasting’.
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Optimal boundary
2( , ) (( ( ) ( )) ) , s.t. bluet sp
E k f p f p k∈∂Ω
∂Ω = − − ∂Ω∈∑
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Mesh ‘ugly’ limitation
Correspondence between source and target images are trivialMeaningful correspondence between source and target meshes are VERY difficult.
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Results
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Failure?