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Image Editing in the Gradient Domain

Shai AvidanTel Aviv University

Slide Credits (partial list)

• Rick Szeliski

• Steve Seitz• Alyosha Efros

• Yacov Hel-Or• Marc Levoy

• Bill Freeman• Fredo Durand

• Sylvain Paris

Image Composition

Target ImageSource Images

Basics

• Images as scalar fields

– R2 -> R

Vector Field

• A vector function G: R2 →R2

• Each point (x,y) is associated with a vector (u,v)

G(x,y)=[ u(x,y) , v(x,y) ]

Gradient Field• Partial derivatives of scalar field • Direction

– Maximum rate of change of scalar field

• Magnitude– Rate of change

• Not all vector fields are gradients of an image.– Only if they are curl-free (a.k.a conservative)– What’s the difference between 1D and 2D Gradient field?

,y

I

x

II

∂∂

∂∂

=∇

),( yxI

Continues v.s. Discrete

Image I(x,y)

Ix Iy

Continues case → derivative

Discrete case → Finite differences

[ ]

[ ] Iy

I

Ix

I

T ∗−→∂∂

∗−→∂∂

110

110

Interpolation

Ω

Ω

Ω∂Ω

over function scalar unknown :f

\over function scalar known :f

boundary with , ofsubset closed a :

ofsubset closed a :

*

2

S

S

RS

I

Intuition – hole filling

• 1D:

• 2D:

x1 x2

Membrane Interpolation2

min ∫∫Ω

∇ff

Solve the following minimization problem:

Subject to Dirichlet boundary conditions:Ω∂Ω∂

= *ff

Variational Methods to the Rescue!

Calculus:

When we want to minimize g(x) over the space of real values xWe derive and set g’(x)=0

What’s the derivative of a function?

Variational Methods:Express your problem as an energy minimization over a space of functions

Derivative Definition

1D Derivative: ( ) ( ) ( )ε

εε

xfxfxf

−+=′

→0lim

Multidimensional derivative for some direction vector w

( ) ( ) ( )ε

εε

xfwxfxfDw

rrrr

r−+

=→0

lim

We want to minimize ( ) ( ) ( )∫ ==′2

1

112 and with

x

x

bxfaxfdxxf

Assume we have a solution f and try to define some notion of 1D derivativewrt to a 1D parameter ε in a given direction of functional space:

For a perturbation function η(x) that also respects boundary conditions (i.e.η(x_1)=η(x_2) = 0) and a scalar ε the integral

( ) ( )( )∫ ′′+′2

1

alone n bigger tha be should 2x

x

fdxxxf ηε

And we are left with:

But since this must be true for every η, it holds that f’’(x) = 0 everywhere.

Calculus of VariationsLets open the parenthesis: ( ) ( ) ( ) ( ) dxxηεxfxηεxf

x

x∫ ′+′′+′

2

1

222 2

The third term is always positive and is negligible when ε is going to zero.

So derive the rest with respect to ε and set to zero: ( ) ( )∫ =′′2

1

02x

x

dxxfxη

Integrate by parts: ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ′′′−′=′′2

1

2

1

2

1

x

x

x

x

xx dxxfxη xfxηdxxfxη

Where: ( ) ( )[ ] ( ) ( ) ( ) ( )agafbgbfxgxf ba −=

And since η(x_1)=η(x_2) = 0 then the expression in the squared brackets is equal to zero

( ) ( ) 02

1

=′′′∫x

xdxxfxη

Intuitioninterval over the integrated slove theis ofmin The ∫ ′f

Locally, if the second derivative was not zero, this would mean that the First derivative is varying which is bad since we want to be minimized∫ ′f

Recap:

We start with the functional we need to minimizeIntroduce the perturbation functionUse Calculus of variationSet to zeroIntegrate by partsAnd obtain the solution.

Euler-Lagrange Equation

( )∫=2

1

,,x

xx

dxffxFJ

A fundamental equation of calculus of variations, that states that if J is defined by an integral of the form

Then J has stationary value if the following differential equation is satisfied

0=∂∂

−∂∂

xf

F

dx

d

f

FEquation (2)

Equation (1)

2min ∫∫

Ω

∇ff

Recall, we want to solve the following minimization problem:

Subject to Dirichlet boundary conditions:Ω∂Ω∂

= *ff

Membrane Interpolation( )222

:caseour In yx fffF +=∇=

0 :becomes (2)equation Then =∂∂

−∂∂

−∂∂

yx f

F

dy

d

f

F

dx

d

f

F

( )

( )

( )2

222

2

222

22

22

22

0

y

ff

dy

d

f

ff

dx

d

f

F

dy

d

x

ff

dx

d

f

ff

dx

d

f

F

dx

d

f

ff

f

F

yy

yx

y

xx

yx

x

yx

∂∂

==∂

+∂=

∂∂

∂∂

==∂

+∂=

∂∂

=∂

+∂=

∂∂

0 :Laplacian get the weand2

2

2

2

=∆=∂∂

+∂∂

fy

f

x

f

Smooth image completion

Ω∂Ω∂Ω=∇∫∫ *2

.. minarg :Lagrange-Euler fftsff

The minimum is achieved when:

Ω∂Ω∂=Ω=∆ * .. 0 fftsoverf

Discrete Approximation (Membrane interpolation)

Ω∂Ω∂=Ω=∆ * .. over 0 fftsf

2

2

2

2

y

f

x

ff

∂∂

+∂∂

=∆

yxyxyxyxyx fffx

fff

x

f,1,,12

2

,,1 2 −++ +−≅∂∂

−≅∂∂

( )04

22,

,1,1,,1,1

1,,1,,1,,1

=−+++=

+−++−≅∆

−+−+

−+−+

yxyxyxyxyx

yxyxyxyxyxyx

fffff

ffffffyxf

Discrete Approximation

=

−

−

−

b

x

x

0

0

0

1

11411

1411

11411

2

1

Each fx,y is an unknown variable xi, there are N unknown (the pixel values)

This reduces to a sparse linear system of equations:

We have A_x * I = 0 A_y * I = 0 A_boundary * I = boundary so We can combine all and get Ax = b

Gradient constraints

Boundary conditions

What’s in the picture?

What’s in the picture?

What’s in the picture?

Editing in Gradient Domain

• Given vector field G=(u(x,y),v(x,y)) (pasted gradient) in a bounded region Ω. Find the values of f in Ω that optimize:

Ω∂∗

Ω∂Ω=−∇∫∫ ffwithGf

f

2min

G=(u,v)

f*

fI

f*

Ω

Intuition - What if G is null?

• 1D:

• 2D:

x1 x2

Ω∂∗

Ω∂Ω=∇∫∫ ffwithf

f

2min

What if G is not null?

• 1D case

Seamlessly paste onto

- Add a linear function so that the boundary condition is respected

- Gradient error is equally distributed all over Ω in order to respect the boundary condition

2D case

From Perez et al. 2003

2D case

From Perez et al. 2003

2D case

Poisson Equation( ) ( )222

:caseour In yyxx GfGfGfF −+−=−∇=

0=∂∂

−∂∂

−∂∂

yx f

F

dx

d

f

F

dx

d

f

F

∂

∂−

∂

∂=

−

∂

∂=

∂∂

∂∂

−∂∂

=

−∂∂

=∂∂

y

G

y

fG

y

f

dy

d

f

F

dy

d

x

G

x

fG

x

f

dx

d

f

F

dx

d

yyy

y

y

xxx

x

x

2

2

2

2

22

22

divGf

y

G

x

Gf

y

G

x

G

y

f

x

f

yx

yx

=∆

∂

∂+

∂∂

=∆

=∂

∂−

∂∂

−∂∂

+∂∂

0 :get weand2

2

2

2

Discrete Approximation (Poisson Cloning)

Ω∂Ω∂=Ω=∆ * .. fftsoverdivGf

2

2

2

2

y

f

x

ff

∂∂

+∂∂

=∆

yxyxyxyxyx fffx

fff

x

f,1,,12

2

,,1 2 −++ +−≅∂∂

−≅∂∂

( )04

22,

,1,1,,1,1

1,,1,,1,,1

=−+++=

+−++−≅∆

−+−+

−+−+

yxyxyxyxyx

yxyxyxyxyxyx

fffff

ffffffyxf

( ) ( ) ( ) ( )1,,,1, −−+−−≅∂

∂+

∂∂

= yxGyxGyxGyxGy

G

x

GdivG yyxx

yx

Alternative Derivation (discrete notation)

=

v

uI

y

x

D

D

−

−

−

−

−

−

−

=

11

11

11

11

11

11

11

xD[ ]*110* −=∂∂x

IxDIx

=∗∂∂

• Let Dx - Toeplitz matrix

2

min x

Iy

D

D

−

uI

v

( ) ( )

=

v

uI T

yTx

y

xTy

Tx DD

D

DDD

• Normal equation:

( ) vuI Ty

Txy

Tyx

Tx DDDDDD +=+

[ ] [ ]*011*110, −=−⇒ flipDNote Tx

Numerical Solution

• Discretize Laplacian

=+≡∇ yTyx

Tx DDDD2 [ ] ∗

−=∗

−+−

010

141

010

1

2

1

121

−

−

−

−

−

−

−

1411

1411

11411

11411

11411

1141

1141

Sparse Toeplitz Matrix

Comments:– A is a sparse.– A is symmetric and can be inverted.

– If Ω is rectangular A is a Toeplitz matrix.– Size of A is ∼NxN.– Impractical to form or store A.– Impractical to invert A

( ) vuI Ty

Txy

Tyx

Tx DDDDDD +=+

bI =A

Iterative Solution: Conjugate Gradient

• Solves a linear system Ax=b (in our case x=I)• A is square, symmetric, positive semi-definite.

• Advantages:– Fast!– No need to store A but calculating Ax – In our case Ax can be calculated using a single

convolution.– Can deal with constraints.

Conjugate Gradient as a minimization problem

• Minimizes

And since A is symmetric

Steepest Descent Method

• Pick gradient direction r(i)

• Find optimum along this direction x(i)+αr(i)

Gradient direction

Energy along the gradient

Behavior of gradient descent

• Zigzag or goes straight depending if we’re lucky– Ends up doing multiple steps in the same direction

Conjugate gradient

• For each step i: – Take the residual d(i)=b-Ax(i) (= -gradient)

– Make it A-orthogonal to the previous ones

– Find minimum along this direction

• Needs at most N iterations.

• Matlab command:

x=cgs(A,b)

A can be a function handle afun

such that afun(x) returns A*x

Solving Poisson equation with boundary conditions

Ω

*ΩS T

( ) ( )ΩΩ

+=+ SDDDDDDDD yTyx

Txy

Tyx

Tx I s.t. ∗∗ ΩΩ

= TI

Ω∇S

• Define a circumscribing square Π=Ω∪Ω*– Let Ω⊂ Π denotes the edited image area.– Let Ω*= Π-Ω denotes the surrounding area.

yTyx

Txk ∂∗∂+∂∗∂=

( ) ∗ΩΩ∗= IIkAI U

Ω

*ΩS T( ) *ΩΩ∗= TSkb U

x=cgs(A,b)

• The above requirements can be expressed as a linear set of equations:

Ω∇S

[ ] [ ]bAI

T

SDDDD

I

IDDDD yyT

xxT

yyT

xxT

=⇒

+=

+⇒

∗∗ Ω

Ω

Ω

Ω

Image stitching

Gradient Domain CompositionGradient Domain Composition

Cut & Paste Cut & Paste Paste in Gradient Domain Paste in Gradient Domain

Another example

Transparent CloningTransparent Cloning

I SΩ Ω∇ = ∇2

S TI Ω Ω

Ω

∇ + ∇∇ = ( )max ,I S TΩ Ω Ω∇ = ∇ ∇

Transparent Cloning

Another example

Another example

Changing local illumination

Defect concealment

High Dynamic Range Compression

Small exposure: Dark inside

High Dynamic Range Compression

Large exposure: Outside Saturated

Manipulate gradients

α is set to 0.1 of average gradient magnitudeβ is set between 0.8 and 0.9

Where the gradient is given by:

High Dynamic Range Compression

Desired Image

Software Tone Mapping

Short Exposure

Long Exposure

High Dynamic Range Compression

Shadow Removal

Color2Grey Algorithm

Optimization:

min Σ Σ ( (gi - gj) - δi,j )2

j=i-µ

If δij == ∆L then ideal image is gOtherwise, selectively modulated by ∆Cij

i

i+µ

Original Color2GreyPhotoshop

Grey

Results

Color2Grey+ Color

Color2Grey+ColorOriginal PhotoshopGrey Color2Grey

Original PhotoshopGrey Color2Grey

Original PhotoshopGrey Color2Grey