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# An Iterative Image Registration Technique with an Application to

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```An Iterative Image Registration Technique
with an Application to Stereo Vision
Bruce D. Lucas & Takeo Kanade
&
Determining Optical Flow
B. K. P. Horn & B. G. Schunck
Andrew Cosand
ECE CVRR
CSE 291 11-1-01
Image Registration
Basic Problem
Image Registration
вЂў Align two images to achieve the best match.
вЂў Determine motion between sequence
images
вЂў There are a number of choices to make:
вЂ“ What error metric to use.
вЂ“ What type of search to perform.
вЂў How to control a search.
Optical Flow
вЂў Flow of brightness through image
вЂ“ Analogous to fluid flow
вЂ“ Optic flow field resembles projection of motion field
вЂў Problem is underconstrained:
вЂ“ For a single pixel, we only have information on the
velocity normal to the difference contour
вЂ“ Need 2 velocity vectors, only have one equation
вЂ“ Need another constraint
Aperture Problem
Aperture Problem
Aperture Problem
Aperture Problem
вЂў Assume images are roughly aligned
вЂ“ On the order of ВЅ feature size
вЂў Newton-Raphson type iteration
вЂ“ Assume linearity and move in that direction
вЂў Constant velocity constraint
One Dimensional Registration
Allowable Pixel Shift
вЂў Algorithm only works for small (<1) pixel
shifts
вЂў Larger motion can be dealt with in
subsampled images where it is sub pixel
Error Metrics
Error Metric
вЂў Use a linear approximation
F(x+h) п‚» F(x) + h FвЂ™(x)
вЂў L2 norm error
E = пЃ“x[F(x+h)-G(x)]2
вЂў Becomes
E = пЃ“x[F(x) + h FвЂ™(x) -G(x)]2
вЂў Set derivative wrt h = 0 to minimize error
Estimating h
п‚¶E = 0 п‚» п‚¶ пЃ“x[F(x) + h FвЂ™(x) -G(x)]2
п‚¶h
п‚¶h
= пЃ“x 2 FвЂ™(x)[F(x) + h FвЂ™(x) -G(x)]2
Solving for h
h п‚» пЃ“x FвЂ™(x)[G(x) -F(x)]
пЃ“xFвЂ™(x) 2
Weighting
вЂў Approximation works well in linear areas
(low FвЂќ(x)) and not so well in areas with
large FвЂќ(x)
вЂў Add a weighting factor to account for this.
вЂў FвЂќ п‚» (FвЂ™-GвЂ™)/h
1D Algorithm
First Iteration
More Dimensions
вЂў Images are two dimensional signals.
вЂў Goal is to figure out how each pixel moves
from one image to the next.
вЂў Conservation of image brightness
( пѓ‘E)Tv+Et=0
Exv + Eyu + Et = 0
Constant Velocity Constraint
вЂў Single pixel gives one equation
( пѓ‘E)Tv+Et=0
вЂў But this wonвЂ™t solve 2 components of v
вЂў Force pixel to be similar to neighbors in
order to get many constraining equations
вЂ“ 5x5 block of neighbors is common
вЂў Find a good simultaneous solution for entire
block of solutions
Aperature Problem
Constant Velocity Solution
вЂў For a 5x5 block, we get a vector of 25
constraints
вЂў Find least squares solution
вЂў AT (Av=b) , Av=b пѓ ( пѓ‘E)Tv+Et=0
вЂ“ A is gradients, v is velocities, b is time
вЂў ATAv = Atb
вЂў ATA= пЃ“(Ex)2 пЃ“ExEy
пЃ“ExEy пЃ“(Ey)2
[
]
пЃ¬1, пЃ¬2
Corner Features
вЂў C= пЃ“(Ex)2 пЃ“ExEy
=
пЃ“ExEy пЃ“(Ey)2
вЂў Rank 0
пЃ¬1= пЃ¬2=0
[
пЃ¬1 0
0 пЃ¬2
] [ ]
вЂў Rank 1
пЃ¬1> пЃ¬2=0
вЂў Rank 2
пЃ¬1> пЃ¬2>0
Multiple Pixel Smoothness
вЂў Single Pixels, rank
deficient,
Underconstrained
вЂў Too Similar, rank
deficient,
Underconstrained
вЂў Non-parallel
contours, overcomes
aperature problem,
overconstrained
(Solvable!)
More Dimensions
Generalizing
вЂў Linear transformations with a matrix A
G( x) = F( xA + h)
вЂў Brightness and contrast scalars a and b
F( x) = aG( x) + b
вЂў Error measure to minmize
Horn & Schunck
( пѓ‘E)Tv+Et=0
вЂў Sum ( пѓ‘E)Tv+Et over the entire image,
minimize the sum
H(u,v)= пЃ“[Ex(i)u(i) + Ey(i)v(i) + Et(i)]2
i
вЂў Simply minimizing this can get ugly
Regularization
вЂў Use regularization to impose a smoothness
constraint on the solution
вЂў Try to reduce higher derivative terms
пЃ¬в€«в€«[(п‚¶2u/ п‚¶x2)2 + (п‚¶2u/ п‚¶y2)2 + (п‚¶2v/ п‚¶x2)2 +
(п‚¶2v/ п‚¶y2)2 ]dxdy
Iterative Solution
H(u,v)= пЃ“[Ex(i)u(i) + Ey(i)v(i) + Et(i)]2 +
пЃ¬в€«в€«[(п‚¶2u/ п‚¶x2)2 + (п‚¶2u/ п‚¶y2)2 + (п‚¶2v/ п‚¶x2)2 +
(п‚¶2v/ п‚¶y2)2 ]dxdy
вЂў Simultaneously minimize both to get a
smooth solution
вЂ“ пЃ¬ determines how smooth to make it
вЂў An iterative version propagates information
to pixels without enough local info
Iterative Propagation
Results
Results
Issues
вЂў When does optic flow work?
вЂў When does it fail?
вЂ“ Edges, large movement, even sphere, barber
pole
вЂў Recent improvements
вЂ“ Multi-resolution
вЂ“ Multi-body for independently moving obejcts
вЂ“ Robust methods
h
```
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