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

Input Images source image target image

Editing Results Simple Cloning Result Poisson Seamless Cloning

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

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.

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.

Related work– poisson equation Fattal et al. 2002 Entire images Lewis 2001 harmonic interpolation Our method Local images Our method Guided Interpolation

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

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. 2000 ) a guided interpolation framework , with the guidance being selected by the user ,and this framework is not limited in seamless cloning

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

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

Solution :Euler-Lagrange Equation

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

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

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

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

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

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)

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

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

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

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

Seamless Cloning Results Insertion

Seamless Cloning Results Feature exchange

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

Mixing Gradients Variant of V: Discretization :

Mixing Gradients Results Inserting objects with holes

Mixing Gradients Results Inserting one object close to another

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

Texture Flattening results input output

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

Local illumination Changes results input output

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

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

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

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

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

Thank you!

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 .

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.