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Curvature Driven Flows
Allen Tannenbaum
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Basic curve evolution: Invariant Flows
Planar curve:
General flow:
General geometric flow:
C p R : ( ) [ , ]0 1 2
C
tT N
C
tN
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Smoothing by classical heat flow
Linear (curve parameter p is independent of t)
Equivalent to Gaussian filteringUnique linear scale-space
Non geometricShrinks the shapeImplementation problems
pp
pp
t
tyx
yx
ppCtC = =
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Invariant differential geometryFor every Lie group we will consider,
exists and invariant parametrization s, the group arc-length
For every such a group exists an invariant signature, the group curvature, k
High curvature
Low curvature
Negative curvature
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What and why invariant
Camera motion Deformation
Camera/object movement in the space
Transformations description (for “flat” objects):Euclidean
Motion parallel to the camera and planar projectionAffine
Planar projectionProjective
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Euclidean geometric heat flow Use the Euclidean arc-length: The deformation:
Smoothly deforms to a circle (Gage-Hamilton, Grayson)
Geometric smoothingReduces length as fast as possible
Cs = 1
=
=
C
t
C
sN
2
2
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Affine geometric heat flowUse the affine arc-length: The flow:
=
non - inflection
inflection
C
t
Css0
C N f Tss = 1/3
( , )
C Nt = 1/3
det [Cs;Css] = 1
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Affine geometric heat flow-(cont.)
Theorem (Angenent-Sapiro-Tannenbaum):
Let be a maximal classical solution of the
affine heat flow.
Then shrinks to an elliptically shaped point
as .
Equation also introduced by Alvarez, Guichard, Lions,and Morelin a viscosity framework.
f Ct : 0 ô t < Tg
Ct
t " T
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Affine geometric heat flow (cont.)
Nonconvex curve becomes convex and then deforms into an ellipse.
Decreases area as fast as possible (in an affine form)
Applications:Curvature computation for shape recognition:
reduce noiseSimplify curvature computation (Faugeras ‘95)Object recognition for robot manipulation
(Cipolla ‘95)
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General invariant flows Theorem: For every sub-group of the projective
group the most general invariant curve deformation has the form
Theorem: In general dimensions, the most general invariant flow is given by
u: graph locally representing the hypersurfaceg: invariant metricE(g): variational derivative of g I differential invariant
=
C
t
C
sf s ss
2
2 ( , , ,...)
IgE
ut )(
g =
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From Curves to Smoothing Filters
Ðt = ì jjr Ðjj
Ðt = (ÐxxÐ2y à 2ÐxÐyÐxy + ÐyyÐ
2x)
1=3
C0 = f Ð0(x;y) = 0g
C(t) = f Ð(x;y; t) = 0g; @t@C = ì N~ )
Embed initial curve as zero level set of surface:
Want evolution of surface to track motion of curve as zero level set:
For affine geometric heat equation this leads to filter:
Here is interpreted as a gray-level image.Ð0 : R 2 ! R
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Smoothing with Linear Heat Equation
256 by 256 MRI brain image smoothed by linear heat equation:t=2, 6, 32, 128
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Smoothing with Geometric Heat Equation
Smoothing with kappa filter: t=0, 4, 16, 64, 256, 1024
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Smoothing with Affine Heat Equation-I
Smoothing withkappa-shleesh:t=0, 16, 128, 1024
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Smoothing with Affine Heat Equation-II
Magnification of original image and image after 256 iterations of kappa-shleesh filter.