Axioms Arjan Kuijper [email protected]. Axioms; PhD course on Scale Space, Cph 1-5 Dec 2003 2 27 Axioms of measurements and scale I1I2I3O KYBL1F1APN L2F2 Convolution

AxiomsAxioms

Arjan KuijperArjan Kuijper

[email protected]@itu.dk

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

227

Axioms of measurements and Axioms of measurements and scalescale

I1 I2 I3 O K Y B L1 F1 A P N L2 F2

Convolution kernel x x x x x x x x x x x

Semigroup property x x x x x x x x x

Locality x

Regularity x x x x x x x x

Infinitesimal generator x

Max. loss principle x

Causality x x x x x

Nonnegativity x x x x x x

Tikhonov regularization x

Average grey level invar. x x x x x x

Flat kernel for t to infinity x

Isometry invariance x x x x x x x x x x x

Homogeneity & isotropy x

Separability x x

Scale invariance x x x x x x x x

Valid for dimension 1 2 2 2 1,2 1,2 1 1 >1 N 1,2 N N N


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Backgrounds for axiomsBackgrounds for axioms

• Device-based Physics: Device-based Physics: dimension analysis, “we know nothing”dimension analysis, “we know nothing”

• Vision-based Physics: Vision-based Physics: causalitycausality

• MathematicsMathematics Distribution theoryDistribution theory

• StatisticsStatistics Entropy maximizationEntropy maximization

•RegularizationRegularization TomorrowTomorrow


427

Physics: device-basedPhysics: device-based

Physical properties:Physical properties:

[candela/meter[candela/meter22] ”<>” [meters]] ”<>” [meters]

Uncommitted assumptions:Uncommitted assumptions:• linearitylinearity (no memory or model) (no memory or model)

• spatial shift invariancespatial shift invariance (no preferred (no preferred location)location)

• isotropyisotropy (no preferred orientation) (no preferred orientation)

• scale invariancescale invariance (no preferred scale) (no preferred scale)

• separabilityseparability (for the sake of (for the sake of computationally ease) computationally ease)


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Physical properties:Physical properties:

L [candela/meterL [candela/meter22] ”<>” ] ”<>” xx [meters] [meters]

Intensity <> spatialityIntensity <> spatiality

Pi-teorem:Pi-teorem:

Physical laws must be independent of the Physical laws must be independent of the choice of the fundamental parameterschoice of the fundamental parameters

So So L/ LL/ L00 = G(s = G(s xx))


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•Linear shift invarianceLinear shift invariance• Convolution : Convolution :

L = LL = L00 * G((s * G((s xx))pp))

• In Fourier domain equal to multiplication: In Fourier domain equal to multiplication: LL = = LL00 G ((s G ((s xx) ) pp))

•IsotropyIsotropy• (s (s xx) ) is a scalaris a scalar

•Scale invarianceScale invariance• G G ((((ss11 xx) ) pp)) G G (((( s s2 2 xx ) ) pp)) ==G G (((( s s11 xx ++ s s22 xx ) ) pp))

Which implies Which implies G G ((((s s xx) ) pp) = Exp ((a ) = Exp ((a s s xx) ) pp))


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•SeparabilitySeparability• G G ((((s s xx) ) pp) = Exp ((a ) = Exp ((a s s xx) ) 22))

•InnerInner scale (image is not scaled) scale (image is not scaled)• Exp ((a Exp ((a s s xx) ) 22)) = 1 for = 1 for s s xx limiting to 0 limiting to 0

•OuterOuter scale (image averages) scale (image averages)• Exp ((a Exp ((a s s xx) ) 22)) = 0 for = 0 for s s xx limiting to infinity limiting to infinity

•So aSo a22 < 0, say -1/2 for later convenience. < 0, say -1/2 for later convenience.


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• G G ((((s s xx) ) pp) = Exp (-1/2 () = Exp (-1/2 (s s xx) ) 22))

SoSo

• G(sx) = Exp (- G(sx) = Exp (- xx22/(2s/(2s22)) / Sqrt[ (2 / Sqrt[ (2 s s22) ) DD ] ]

4 2 2 4

0.1

0.2

0.3

0.4


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Gaussian scale spaceGaussian scale space

Therefore, Therefore,

L(L(xx;s) = L;s) = L00((xx) *) * Exp (- Exp (- xx22/(2s/(2s22)) / Sqrt[ (2 / Sqrt[ (2 s s22) ) DD ]]

L(L(xx;s) is called the ;s) is called the Gaussian scale space Gaussian scale space imageimage..


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How does it look likeHow does it look like


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How to calculateHow to calculate

Sample a GaussianSample a Gaussian

•In the spatial domainIn the spatial domain• Large scalesLarge scales

•In Fourier domainIn Fourier domain• Small scalesSmall scales

Boundary effectsBoundary effects

•Repeated imageRepeated image

•Mirrored imageMirrored image

•Padding with zerosPadding with zeros


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Physics: Vision-basedPhysics: Vision-based

Whatever you Whatever you do on this do on this image, you image, you don’t want the don’t want the introduction of introduction of white regions in white regions in the black ones.the black ones.

No new level No new level lines are to be lines are to be created:created:

CausalityCausality


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CausalityCausality: non-enhancement of local extrema.: non-enhancement of local extrema.

Let Let L = LL = Lxxxx +L +Lyyyy

L equals the sum of the eigenvalues of the L equals the sum of the eigenvalues of the Hessian.Hessian.

Then at a maximum Then at a maximum L < 0 and LL < 0 and Ltt < 0 < 0

and at a minimum and at a minimum L > 0 and LL > 0 and Ltt > 0 > 0

So So L LL Ltt > 0 and L > 0 and Ltt = a = a L, a > 0L, a > 0

Say a = 1 and LSay a = 1 and Ltt = = LL


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LLtt (x,y;t) = (x,y;t) = L (x,y;t) L (x,y;t)

Obviously, for t to 0, LObviously, for t to 0, L (x,y;t) = L(x,y;t) = L00

The general solution (Greens function) for The general solution (Greens function) for this this diffusion equationdiffusion equation is convolution of the is convolution of the original image with an Gaussian:original image with an Gaussian:

•G(sx) = Exp (-(xG(sx) = Exp (-(x22+y+y2 2 )/(4 t))/(4 t) / √ (4 / √ (4 t) t)22

Note: one uses rater 4t than 2sNote: one uses rater 4t than 2s22


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Mathematics: Distribution theoryMathematics: Distribution theory

Differentiation is not Differentiation is not well-definedwell-defined..

-3 -2 -1 1 2 3-2

2

4

6

8

10fx,e x2e Cosxe2

-3 -2 -1 1 2 3

-6

-4

-2

2

4

6

fxx,e 2x1e Sinxe2-3 -2 -1 1 2 3-2

2

4

6

8

10fx,e x2

-3 -2 -1 1 2 3

-6

-4

-2

2

4

6

fxx,e 2x


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•A solution is A solution is well-defined well-defined in the sense of in the sense of Hadamard if the solutionHadamard if the solution• ExistsExists

• Is uniquely definedIs uniquely defined

• Depends continuously on the initial or boundary Depends continuously on the initial or boundary datadata

•So there is one stable solution.So there is one stable solution.

•The The operationoperation is the problem, not the is the problem, not the functionfunction..

•And what about discrete data?And what about discrete data?


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How should the derivative of this thing look like?How should the derivative of this thing look like?

-0.3

-0.2

-0.1

0.1

0.2

0.3


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•Laurent Schwartz: use my Laurent Schwartz: use my Schwartz spaceSchwartz space with with Smooth test functionsSmooth test functions• Infinitely differentiable Infinitely differentiable

• decrease fast to zero at the boundariesdecrease fast to zero at the boundaries

•Construct a Construct a regular tempered distributionregular tempered distribution• i.e. the correlation of a test function and i.e. the correlation of a test function and

“something”“something”

•The regular tempered distribution now has the The regular tempered distribution now has the nice properties of the test function.nice properties of the test function.

• It can be regarded as a probing of “something”It can be regarded as a probing of “something”with a mathematically nice filter.with a mathematically nice filter.


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•Smooth test functionsSmooth test functions• Infinitely differentiable Infinitely differentiable • decrease fast to zero at the boundariesdecrease fast to zero at the boundaries

•For example a Gaussian.For example a Gaussian.

The regular tempered distribution The regular tempered distribution = =

The filtered imageThe filtered image

•Choose the “best” test function based on Choose the “best” test function based on image-related axioms.image-related axioms.


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•Now everything is well-defined, since Now everything is well-defined, since integrating is well-defined.integrating is well-defined.

•Do everything under the integralDo everything under the integral

L(L(xx;s) = ∫ L(;s) = ∫ L(yy) G () G (xx--yy;s) d;s) dyy

∂∂xx11 L( L(xx;s) = ∂;s) = ∂xx11 ∫ L G = ∫ ∂∫ L G = ∫ ∂xx11

L G = ∫ L ∂ L G = ∫ L ∂xx11 G G


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Statistics: Entropy maximizationStatistics: Entropy maximization

A A statistical measurestatistical measure for the for the disorderdisorder of the of the filter is given by the entropy:filter is given by the entropy:

H(g) = ∫ g(x) Log[g(x)] dx H(g) = ∫ g(x) Log[g(x)] dx

1D for simplicity1D for simplicity

If it is maximized it states something like If it is maximized it states something like “there is nothing ordered” (we know “there is nothing ordered” (we know nothing).nothing).

Obviously, there are some constraints.Obviously, there are some constraints.


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ConstraintsConstraints

The function must be The function must be normalizednormalized; no global ; no global enhancement: enhancement:

∫ g(x) dx = 1∫ g(x) dx = 1

The The meanmean of the measurement is zero; al points are of the measurement is zero; al points are the same: the same:

∫ x g(x) dx = 0∫ x g(x) dx = 0

There is a There is a standard deviationstandard deviation, say s:, say s:∫ x∫ x22 g(x) dx = s g(x) dx = s22

The function is The function is positivepositive; it’s a real object: ; it’s a real object: g(x)>0g(x)>0


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So maximize the So maximize the Euler LagrangeEuler Lagrange equation equation

E(g) = ∫ g(x)(Log[g(x)] + E(g) = ∫ g(x)(Log[g(x)] + λλ00 + + λλ11 x + x + λλ22 xx22) dx) dx

Set the Set the variational derivativevariational derivative w.r.t. g equal to w.r.t. g equal to zero:zero:

1+ Log[g(x)] + 1+ Log[g(x)] + λλ00 + + λλ11 x + x + λλ22 xx22 = 0 = 0

So So

g(x) = Exp[-(1+ g(x) = Exp[-(1+ λλ00 + + λλ11 x + x + λλ22 xx22)].)].


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g(x) = Exp[-(1+ g(x) = Exp[-(1+ λλ00 + + λλ11 x + x + λλ22 xx22 )] )]

∫ ∫ x g(x) dx = 0 -> x g(x) dx = 0 -> λλ11 = 0 = 0

∫ ∫ xx22 g(x) dx = s g(x) dx = s2 2 -> -> λλ22 = 1/2 = 1/2 s s22

g(x)>0 -> OKg(x)>0 -> OK

∫ ∫ g(x) dx = 1 -> g(x) dx = 1 -> λλ00 = -1 + Log[4 = -1 + Log[422ss44]/4]/4

=> g(x) = Exp[-=> g(x) = Exp[-Log[4Log[422ss44] ] - x- x22 /2/2 s s22]]

=> g(x) = Exp[-x=> g(x) = Exp[-x22/2/2 s s22] / ] / √(2√(2ss22))


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Axioms of measurements and Axioms of measurements and scalescale

I1 I2 I3 O K Y B L1 F1 A P N L2 F2

Convolution kernel x x x x x x x x x x x

Semigroup property x x x x x x x x x

Locality x

Regularity x x x x x x x x

Infinitesimal generator x

Max. loss principle x

Causality x x x x x

Nonnegativity x x x x x x

Tikhonov regularization x

Average grey level invar. x x x x x x

Flat kernel for t to infinity x

Isometry invariance x x x x x x x x x x x

Homogeneity & isotropy x

Separability x x

Scale invariance x x x x x x x x

Valid for dimension 1 2 2 2 1,2 1,2 1 1 >1 N 1,2 N N N


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Kinds of scale spacesKinds of scale spaces

LinearLinear• -scale spaces: -scale spaces: G G ((((s s xx) ) ) = Exp (-a() = Exp (-a(s s xx) ) ))

• Gaussian Gaussian =2, e.g. 2D Heat equation L=2, e.g. 2D Heat equation Ltt (x,y;t) = (x,y;t) = L (x,y;t) L (x,y;t) • Poisson Poisson =1, e.g. 2D Poisson equation L=1, e.g. 2D Poisson equation Ltttt (x,y;t) = - (x,y;t) = -L L

(x,y;t) (x,y;t)

• Anisotropic scale spaces (oval vs. circular)Anisotropic scale spaces (oval vs. circular)

Non-linearNon-linear• Based on PDE’s: LBased on PDE’s: Ltt (x;t) = (x;t) = g g L), with g depended on L), with g depended on

the image the image • Perona Malik (preserve edges while smoothing)Perona Malik (preserve edges while smoothing)• Reaction-diffusion (minimizing some energy)Reaction-diffusion (minimizing some energy)• Total Variation (keeping flat regions)Total Variation (keeping flat regions)• Curvature based (snakes)Curvature based (snakes)• Morphology (discrete approach)Morphology (discrete approach)


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SourcesSources• Linear Scale-Space has First been Proposed in Japan, Linear Scale-Space has First been Proposed in Japan,

Joachim Weickert, Seiji Ishikawa, Atsushi Imiya Joachim Weickert, Seiji Ishikawa, Atsushi Imiya Journal of Mathematical Imaging and Vision (10), 237-252, 1999. Journal of Mathematical Imaging and Vision (10), 237-252, 1999.

• The structure of images, The structure of images, Jan Koenderink,Jan Koenderink,Biological Cybernetics (50), 363-370, 1984.Biological Cybernetics (50), 363-370, 1984.

• Basic theory of pattern normalization, Basic theory of pattern normalization, Taizo IijimaTaizo IijimaBulletin of the Electrotechnical Laboratory (26), 368-388, 1962.Bulletin of the Electrotechnical Laboratory (26), 368-388, 1962.

• On the Gaussian Scale-Space On the Gaussian Scale-Space Taizo IijimaTaizo IijimaIEICE Transactions D (E86-D), 1162-1164, 2003.IEICE Transactions D (E86-D), 1162-1164, 2003.

• Anisotropic diffusion in image processing, Joachim WeickertAnisotropic diffusion in image processing, Joachim Weickert• Image Structure, Luc Florack Image Structure, Luc Florack • Front-End Vision and Multi-Scale Image Analysis, Bart ter Haar Front-End Vision and Multi-Scale Image Analysis, Bart ter Haar

Romeny Romeny • From Paradigm to Algorithms in Computer Vision, Mads Nielsen From Paradigm to Algorithms in Computer Vision, Mads Nielsen

Documents

Axioms Arjan Kuijper [email protected]. Axioms; PhD course on Scale Space, Cph 1-5 Dec 2003 2 27 Axioms of measurements and scale I1I2I3O KYBL1F1APN L2F2 Convolution