Scale Space Geometry Arjan Kuijper [email protected]. Scale Space Geometry; PhD course on Scale Space,...

Scale Space GeometryScale Space Geometry

Arjan KuijperArjan Kuijper

[email protected]@itu.dk

Scale Space Geometry; PhD course on Scale Space, Cph 1-5 Dec 2003

222

Deep structureDeep structure

The challenge is to understand the imageThe challenge is to understand the imagereally on all the levels simultaneously,really on all the levels simultaneously,and not as an unrelated set of derived and not as an unrelated set of derived

imagesimagesat different levels of blurring.at different levels of blurring.

Jan Koenderink (1984)Jan Koenderink (1984)


322

What to look forWhat to look for

• Gaussian scale space is Gaussian scale space is intensity-basedintensity-based..

• Consider an n - dimensional image, i.e. a (n+1) Consider an n - dimensional image, i.e. a (n+1) dimensional Gaussian scale space (dimensional Gaussian scale space (GssGss) image. ) image.

• Investigated intensity-related items.Investigated intensity-related items.

• ““Things” with specialties w.r.t. intensity.Things” with specialties w.r.t. intensity.• Equal intensities – isophotes, iso-intensity manifolds: L=cEqual intensities – isophotes, iso-intensity manifolds: L=c

n - dimensional n - dimensional iso-manifoldsiso-manifolds in the Gss image in the Gss image (n-1) - dimensional manifolds in the image.(n-1) - dimensional manifolds in the image.

• Critical intensities – maxima, minima, saddle points: Critical intensities – maxima, minima, saddle points: L=0L=0 0 – dimensional points in the Gss image.0 – dimensional points in the Gss image.

• Critical intensities – maxima, minima, saddle points, .....:Critical intensities – maxima, minima, saddle points, .....: 0 – dimensional critical points in the blurred image,0 – dimensional critical points in the blurred image, 1 – dimensional 1 – dimensional critical curvescritical curves in the Gss image. in the Gss image.


422

Example imageExample image

Consider a simple 2D image.Consider a simple 2D image.

In this image, and its blurred In this image, and its blurred versions we have versions we have

Critical points Critical points L=0:L=0:• Extrema (green)Extrema (green)

• MinimumMinimum• MaximaMaxima

• Saddles (Red)Saddles (Red)

Isophotes L=0:Isophotes L=0:• 1-d curves, only intersecting 1-d curves, only intersecting

in saddle pointsin saddle points


522

What happens with these What happens with these structures?structures?Causality: no creation of new level linesCausality: no creation of new level lines

Outer scale: flat kernelOuter scale: flat kernel

•All level lines disappearAll level lines disappear

•All but one extrema disappearAll but one extrema disappear

ExampleExample

View critical points in scale space: the View critical points in scale space: the criticalcritical curvescurves..


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Critical curvesCritical curves

x

yt


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Critical pointsCritical points

Let L(x,y) describe the image landscape.Let L(x,y) describe the image landscape.

At At criticalcritical points, points, TTL = (∂L = (∂xxL,∂L,∂yyL) = (LL) = (Lxx,L,Lyy) = (0,0).) = (0,0).

To determine the type, consider de To determine the type, consider de HessianHessian matrixmatrix

H = H = TTL(x,y) = ((LL(x,y) = ((Lxxxx , L , Lxyxy), (L), (Lxyxy , L , Lyyyy)). )).

•MaximumMaximum: H has two negative eigenvalues: H has two negative eigenvalues

•MinimumMinimum: H has two positive eigenvalues: H has two positive eigenvalues

•SaddleSaddle: H has a positive and a negative : H has a positive and a negative eigenvalue. eigenvalue.


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When things disappearWhen things disappear

Generically, det [H] = LGenerically, det [H] = Lxx xx LLyyyy - L - Lxyxy L Lxyxy <> = 0, <> = 0, there is no eigenvalue equal to 0.there is no eigenvalue equal to 0.This yields an This yields an over-determinedover-determined system. system.

In scale space there is an extra parameter, In scale space there is an extra parameter, so an extra possibility: det [H] = 0.so an extra possibility: det [H] = 0.

So, what happens if det [H] = 0? So, what happens if det [H] = 0? -> Consider the scale space image-> Consider the scale space image


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Diffusion equationDiffusion equation

We know that We know that LLt t = L= Lxx xx + L+ Lyyyy So we can construct So we can construct polynomialspolynomials in scale in scale space. space.

Let’s make a Let’s make a HessianHessian with zero determinant: with zero determinant:

H=((6x,0),(0,2))H=((6x,0),(0,2))

Thus Thus LLxx xx = 6x, L= 6x, Lyyyy = 2, L = 2, Lxyxy = 0 = 0And And LLtt = 6x +2 = 6x +2

Thus L = xThus L = x33 + 6xt + y + 6xt + y22 + 2t + 2t

Consider the Consider the critical curvescritical curves


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Critical CurvesCritical Curves

L = xL = x33 + 6xt + y + 6xt + y22 + 2t + 2t

LLx x = 3x= 3x2 2 + 6t, L+ 6t, Lyy = 2y = 2y

For (x,y;t) we have For (x,y;t) we have

•A A minimumminimum at (x,0;-x at (x,0;-x22/2), or (√-2t,0;t)/2), or (√-2t,0;t)

•A A saddlesaddle at (-x,0;- x at (-x,0;- x22/2), or (-√-2t,0;t)/2), or (-√-2t,0;t)

•A A catastrophe pointcatastrophe point at (0,0;0), an annihilation. at (0,0;0), an annihilation.

What about the What about the speedspeed at such a at such a catastrophe? catastrophe?


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Speed of critical pointsSpeed of critical pointsHigher order derivatives: -Higher order derivatives: -L = H L = H xx + + L tL t

xx = -H = -H-1-1((L + L + L t) L t)

Obviously goes wrong at catastrophe points, since then det(H)=0. Obviously goes wrong at catastrophe points, since then det(H)=0.

The velocity becomes infinite: ∂The velocity becomes infinite: ∂t t (√-2t,0;t)= (-1/√-2t,0;1)(√-2t,0;t)= (-1/√-2t,0;1)

-4 -2 2 4

-2

-1.5

-1

-0.5

0.5

1


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Speed of critical pointsSpeed of critical points

Reparametrize t = det(H) Reparametrize t = det(H) xx = -H = -H-1-1((L + L + L det(H) L det(H) ))

Perfectly defined at catastrophe points Perfectly defined at catastrophe points

The velocity becomes 0: The velocity becomes 0: -H-H-1-1((L det(H) L det(H) vv

-3 -2 -1 1 2

-4

-3

-2

-1


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To detect catastrophesTo detect catastrophes

Do the same trick for the determinant:Do the same trick for the determinant:

--L L = H x + = H x + L tL t-det(H) -det(H) = = det(H) x + det(H) x + det(H) tdet(H) t

Set M = ((H, Set M = ((H, L), (L), (det(H), det(H), det(H)) det(H))

Then if at catastrophesThen if at catastrophes

•det[M] < 0 : det[M] < 0 : annihilationsannihilations

•det[M] > 0 : det[M] > 0 : creationscreations


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CreationsCreations

Obviously, critical points Obviously, critical points can also be created.can also be created.

This does This does notnot violate violate the causality principle. the causality principle.

That only excluded new That only excluded new level lines to be created.level lines to be created.

At creations level lines At creations level lines split, think of a camel split, think of a camel with two humps. with two humps.


1522

To create a creationTo create a creation

Let’s again make a Hessian with zero Let’s again make a Hessian with zero determinant:determinant:

H=((6x,0),(0,2+f(x)))H=((6x,0),(0,2+f(x)))

With f(0)=0.With f(0)=0.

Thus Thus LLxx xx = 6x, L= 6x, Lyyyy = 2 + f(x), L = 2 + f(x), Lxyxy = 0 = 0

To obtain a path (√2t,0;t) require LTo obtain a path (√2t,0;t) require Ltt = -6x = -6x +2, so f(x) = -6x.+2, so f(x) = -6x.

Thus L = xThus L = x33 - 6xt + y - 6xt + y22 + 2t -6 x y + 2t -6 x y2 2


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How does it look like?How does it look like?

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1

0.2


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On creationsOn creations

For creations the y-direction is needed:For creations the y-direction is needed:

Creations only occur if D>1.Creations only occur if D>1.

Creations can be understood when they are Creations can be understood when they are regarded as perturbations of regarded as perturbations of non-generic non-generic catastrophes.catastrophes.

At non-generic catastrophes the Hessian is At non-generic catastrophes the Hessian is “more” “more” degenerateddegenerated: there are more zero : there are more zero eigenvalues and/or they are “more” zero.eigenvalues and/or they are “more” zero.


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Critical points in scale spaceCritical points in scale space

L = 0L = 0L = 0L = 0

•Scale space critical points are Scale space critical points are alwaysalways spatial spatial saddle points.saddle points.

•Scale space critical points are Scale space critical points are alwaysalways saddle saddle points.points.

•CausalityCausality: no new level lines implies no : no new level lines implies no extrema in scale space.extrema in scale space.


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Scale space saddlesScale space saddles

At a scale space saddle two manifolds intersectAt a scale space saddle two manifolds intersect


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Manifolds in scale spaceManifolds in scale space

Investigate structure through saddles.Investigate structure through saddles.


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Void scale space saddlesVoid scale space saddles

-1.5

-1

-0.5

0

-1.5

-1

-0.5

0

00.20.4

0.6

0.8

-1.5

-1

-0.5

0


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SourcesSources

• Local Morse theory for solutions to the heat equation Local Morse theory for solutions to the heat equation and Gaussian blurringand Gaussian blurringJ. DamonJ. DamonJournal of differential equationsJournal of differential equations 115 (2): 386-401, 1995 115 (2): 386-401, 1995

• The topological structure of scale-space images The topological structure of scale-space images L. M. J. Florack, A. KuijperL. M. J. Florack, A. KuijperJournal of Mathematical Imaging and VisionJournal of Mathematical Imaging and Vision 12 (1):65- 12 (1):65-79, 2000. 79, 2000.

• The deep structure of Gaussian scale space images The deep structure of Gaussian scale space images Arjan KuijperArjan Kuijper

• Superficial and deep structure in linear diffusion scale Superficial and deep structure in linear diffusion scale space:space:Isophotes, critical points and separatricesIsophotes, critical points and separatricesLewis Griffin and A. Colchester.Lewis Griffin and A. Colchester.Image and Vision ComputingImage and Vision Computing 13 (7): 543-557, 1995 13 (7): 543-557, 1995

Scale Space Geometry Arjan Kuijper [email protected]. Scale Space Geometry; PhD course on Scale Space,...

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