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Gradient image processing.pdf
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《Digital and Computational Photography》课程 梯度图像处理讲义。
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1
6.098 Digital and Computational Photography
6.882 Advanced Computational Photography
Gradient image processing
Bill Freeman
Frédo Durand
MIT - EECS
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How was pset 2?
What have we learnt last time?
• Log is good
• Luminance is different from chrominance
• Separate components:
– Low and high frequencies
• Strong edges are important
OutputLarge-scale
Detail
Color
Homomorphic filtering
• Oppenhein, in the sixties
• Images are the product of illumination and albedo
– Similarly, many sounds are the product of an
envelope and a modulation
• Illumination is usually slow-varying
• Perform albedo-illumination using low-pass filtering
of the log image
• http://www.cs.sfu.ca/~stella/papers/blairthesis/main/node33.html
• See also Koenderink "Image processing done right"
http://www.springerlink.com/(l1bpumaapconcbjngteojwqv)/app/home/contribution
.asp?referrer=parent&backto=issue,11,53;journal,1538,3333;linkingpublicationres
ults,1:105633,1
What's great about the bilateral filter
• Separate image into two components
• Preserve strong edges
• Non-iterative
– More controllable, stable
• Can be accelerated
• Lots of other applications
Edit materials and lighting
• With Oh, Chen and Dorsey
2
A Simple Relighting Example
• With Oh, Chen and Dorsey
Flash Photography (Elmar Eisemann)
No-flash
Flash
Result
Bilateral filtering on meshes
• http://www.cs.tau.ac.il/~dcor/onli
ne_papers/papers/shachar03.pdf
• http://people.csail.mit.edu/thouis/
JDD03.pdf
Questions?
Questions? Today: Gradient manipulation
Idea:
• Human visual system is very sensitive to gradient
• Gradient encode edges and local contrast quite well
• Do your editing in the gradient domain
• Reconstruct image from gradient
• Various instances of this idea, I’ll mostly follow Perez et al. Siggraph 2003
http://research.microsoft.com/vision/cambridge/papers/perez_siggraph03.pdf
∇
3
Problems with direct cloning
From Perez et al. 2003
Solution: clone gradient
Gradients and grayscale images
• Grayscale image: n× n scalars
• Gradient:
• Overcomplete!
• What’s up with this?
• Not all vector fields are the gradient of an image!
• Only if they are curl-free (a.k.a. conservative)
– But it does not matter for us
n× n 2D vectors
Today message II
• Variational approach
– Express your problem as an energy minimization over
a space of functions
• And we are going to spend our time going back and
force between minimization and setting derivatives to
zero. Your head will spin.
Questions? Seamless Poisson cloning
• Given vector field v (pasted gradient), find the value
of f in unknown region that optimize:
Pasted gradient
Mask
Background
unknown
region
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Warning:
• What follows is not strictly necessary to implement
Poisson image editing
• But
– It helps understand the properties of the equation
– It helps to read the literature
– It's cool math
Membrane interpolation
• What if v is null?
• Laplace equation (a.k.a. membrane equation )
Membrane interpolation
• What if v is null?
• Laplace equation (a.k.a. membrane equation )
• Mathematicians will tell you there is an
Associated Euler-Lagrange equation:
• Kind of the idea that we want a minimum, so we kind
of derive and get a simpler equation
Calculus
Simplified version:
• Want to minimize g(x) over the space of real values x
• Derive and set g'(x)=0
• Now we have a more complex equation: we want to
minimize a variational equation over the space of
functions f
• It's a complex business to derive wrt functions
– In general, derivatives are well defined only for
functions over 1D domains
Derivative definition
• 1D derivative:
• multidimensional derivative:
– For a direction v, directional derivative is
• For functionals ?
– Do something similar, replace vector by function
Calculus of variation – 1D
• We want to minimize with f(x
1
)=a, f(x
2
)=b
• Assume we have a solution f
Try to define some notion of 1D derivative wrt to a 1D
parameter ε in a given direction of functional space:
• For a perturbation function η(x) that also respects
the boundary condition (i.e. η(x
1
)=η(x
2
)=0)
and scalar ε,
the integral ∫ (f'(x)+εη'(x))
2
dx should be bigger than
for f’ alone
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