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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.