Chapter XXI : MeasurementsChapter XXI : Measurementscmm.ensmp.fr/~serra/cours/pdf/en/ch21en.pdf ·...

J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXI. 1

Chapter XXI : MeasurementsChapter XXI : Measurements

Euler-Poincaré Characteristic Minkowski Functionals Steiner Formulaindividual Analysis and StereologyPassage to Numerical FunctionsOther Measurements

• Convexity Number• Fractal Dimension


Set FunctionalsSet Functionals

• A set functional is a number W measured on a set X∈ P(Rn), for describing it. It is also called measurement or parameter. For both physical and logical reasons, it often fulfills the following requirements :– to lend itself to sampling, what allows the condition of c-additivity:

W(X∪∪∪∪Y) + W(X∩∩∩∩Y) = W(X) + W(Y) X,Y ∈ ∈ ∈ ∈ PPPP(Rn) ;– to be homogenous, i.e. autonomous under magnification :

W(kX) = k p .W(X) k positive integer ; 0 ≤ p ≤ n ;– to be invariant under translation :

W(Xb) = W(X) b vector or, more severely, under translation and rotation ;

– to satisfy a certain robustness, e.g. that W be continuous, or be increasing , on the class of the compact convex sets.

• A set functional is a number W measured on a set X∈ P(Rn), for describing it. It is also called measurement or parameter. For both physical and logical reasons, it often fulfills the following requirements :– to lend itself to sampling, what allows the condition of c-additivity:

W(X∪∪∪∪Y) + W(X∩∩∩∩Y) = W(X) + W(Y) X,Y ∈ ∈ ∈ ∈ PPPP(Rn) ;– to be homogenous, i.e. autonomous under magnification :

W(kX) = k p .W(X) k positive integer ; 0 ≤ p ≤ n ;– to be invariant under translation :

W(Xb) = W(X) b vector or, more severely, under translation and rotation ;

– to satisfy a certain robustness, e.g. that W be continuous, or be increasing , on the class of the compact convex sets.


Model and NotationModel and Notation

• Convex Ring : The functionals which are studied here hold on the convex ring, i.e.

on the class of the finite unions of compact convex sets in !n .

• Symbols for displacements :− α stands for the angles in the plane, and ω for the solid angles in the space ;

− {x} represents the point x ∈ E considered as an element of P(E);

− ∆(x, α), resp. ∆(x, ω), represents the straight line of !2 , resp. !3 , passing through point x and with a direction α, resp. ω ;

− Π(x, ω) stands for the plane of R3 going through x, with normal ω.

• Convex Ring : The functionals which are studied here hold on the convex ring, i.e.

on the class of the finite unions of compact convex sets in !n .

• Symbols for displacements :− α stands for the angles in the plane, and ω for the solid angles in the space ;

− {x} represents the point x ∈ E considered as an element of P(E);

− ∆(x, α), resp. ∆(x, ω), represents the straight line of !2 , resp. !3 , passing through point x and with a direction α, resp. ω ;

− Π(x, ω) stands for the plane of R3 going through x, with normal ω.


Euler-Poincaré Characteristic in !!!!1Euler-Poincaré Characteristic in !!!!1

• Euler-Poincaré characteristic ν (also called connectivity number or again EPC) is defined by means of an induction on the dimensions of the space.

• For n = 0, the space is conventionally reduced to a unique point, and ν0(X) = 1 or 0 according as set X is this point or the empty set .

• Pour n = 1, puth1(x) = ν0( X ∩{x} ) - ν0( X ∩{x + 0} ) ( x + 0, right limit at point x )The term h1(x) is ≠ 0 only at the right ends xi of the segments that form

X, where it is equal to 1. The sum

νννν1111(X) = Σ Σ Σ Σ h1(xi) then defines the C.E.P. in !1.

• Euler-Poincaré characteristic ν (also called connectivity number or again EPC) is defined by means of an induction on the dimensions of the space.

• For n = 0, the space is conventionally reduced to a unique point, and ν0(X) = 1 or 0 according as set X is this point or the empty set .

• Pour n = 1, puth1(x) = ν0( X ∩{x} ) - ν0( X ∩{x + 0} ) ( x + 0, right limit at point x )The term h1(x) is ≠ 0 only at the right ends xi of the segments that form

X, where it is equal to 1. The sum

νννν1111(X) = Σ Σ Σ Σ h1(xi) then defines the C.E.P. in !1.

h1(x) 1 1 1x



• In !2 , introduce similarly h2(x) = ν1( X ∩ ∆(x) ) −−−− ν1( X ∩ ∆(x + 0) )

Again, h2(x) differs from zero uniquely at the convex outputs xg of the grains, where it is equal to +1, and at the convex inputs xp of the pores, where it is equal to -1. In the convex ring, there is a limited number of such locations, so that the quantity

νννν2222(X) = Σ Σ Σ Σ h2(xg) −−−− Σ Σ Σ Σ h2(xp)is finite, and defines the EPC in !2 . As we can see, ν2(X) is equal to the number of the grains of set X minus that of its pores :

• In !2 , introduce similarly h2(x) = ν1( X ∩ ∆(x) ) −−−− ν1( X ∩ ∆(x + 0) )

Again, h2(x) differs from zero uniquely at the convex outputs xg of the grains, where it is equal to +1, and at the convex inputs xp of the pores, where it is equal to -1. In the convex ring, there is a limited number of such locations, so that the quantity

νννν2222(X) = Σ Σ Σ Σ h2(xg) −−−− Σ Σ Σ Σ h2(xp)is finite, and defines the EPC in !2 . As we can see, ν2(X) is equal to the number of the grains of set X minus that of its pores :

xh2(x) -1 1 1 1 -1 1



• In !3, the principle is the same, and relationh3(x) = ν2( X ∩ Π(x) ) - ν2( X ∩ Π(x + 0) )

induces, by summing in x, νννν3333(X) = Σ Σ Σ Σ h3(xi) .

• The genus of a connected boundary ∂Y is defined as the maximum number of the loops one can trace on ∂Y without disconnect it. It is equal to 0 for a sphere, to 1 for a torus, to 2 for a tore provided with an handle, etc... It can prove that the E.P. C. ν3(X) is equal to :

νννν3333(X) = ΣΣΣΣ [ 1 - G (∂∂∂∂Xi ) ](∂Xi : internal and external connected component, of the boundary ∂X) .

• In !3, the principle is the same, and relationh3(x) = ν2( X ∩ Π(x) ) - ν2( X ∩ Π(x + 0) )

induces, by summing in x, νννν3333(X) = Σ Σ Σ Σ h3(xi) .

• The genus of a connected boundary ∂Y is defined as the maximum number of the loops one can trace on ∂Y without disconnect it. It is equal to 0 for a sphere, to 1 for a torus, to 2 for a tore provided with an handle, etc... It can prove that the E.P. C. ν3(X) is equal to :

νννν3333(X) = ΣΣΣΣ [ 1 - G (∂∂∂∂Xi ) ](∂Xi : internal and external connected component, of the boundary ∂X) .

(ball) (torus) (sphericalcrown)

h3(x) x1111 - 1


Properties of the E.P. C.Properties of the E.P. C.

• Theorem : On the convex ring, Euler-Poincaré characteristic is the onlyfunctional to be c-additive and constant for the convex sets .

This isotropic and translation invariant EPC does not depend on the direction by which it has been constructed. It has dimension 0, therefore is invariant under magnification.

• Minkowski Functionals : When X is a set in !n, one can wonder about the ECP of the sections X∩Πκ of X by hyper-planes of dimension k (0 ≤ k ≤ n ) and also about their averages under displacements as Πκ varies. These sums result in n+1 functionals which are, by construction – invariant under displacement ; − homogenous of degree n - k ;– c-additives; − increasing and continuous for the compact convex sets.

They are the Minkowski functionals

• Theorem : On the convex ring, Euler-Poincaré characteristic is the onlyfunctional to be c-additive and constant for the convex sets .

This isotropic and translation invariant EPC does not depend on the direction by which it has been constructed. It has dimension 0, therefore is invariant under magnification.

• Minkowski Functionals : When X is a set in !n, one can wonder about the ECP of the sections X∩Πκ of X by hyper-planes of dimension k (0 ≤ k ≤ n ) and also about their averages under displacements as Πκ varies. These sums result in n+1 functionals which are, by construction – invariant under displacement ; − homogenous of degree n - k ;– c-additives; − increasing and continuous for the compact convex sets.

They are the Minkowski functionals


Minkowski Functionals Minkowski Functionals

The importance of Minkowski functionals derives from the following result :• Theorem (Hadwiger, 1957) : Every functional defined on the convex ring,

and which is invariant under displacement, c-additive and continuous (or equivalently increasing) on the compact convex sets is a linear combination of the Minkowski functionals.

• Geometrical Interpretation. In !n, up to a multiplying constant:– the first functional (degree n) is the Lebesgue measure of X, it is

increasing and upper-semi continuous ,– the second one ( degree n-1) is the superficial measure of the

boundary ∂X,– the last but one ( degree 1) is the so called norm or mean width ; on

the convex class it satisfies the characteristic relationshipM ( λλλλX ⊕⊕⊕⊕ µµµµY) = λ λ λ λ M(X) + µ µ µ µ M(Y)

The importance of Minkowski functionals derives from the following result :• Theorem (Hadwiger, 1957) : Every functional defined on the convex ring,

and which is invariant under displacement, c-additive and continuous (or equivalently increasing) on the compact convex sets is a linear combination of the Minkowski functionals.

• Geometrical Interpretation. In !n, up to a multiplying constant:– the first functional (degree n) is the Lebesgue measure of X, it is

increasing and upper-semi continuous ,– the second one ( degree n-1) is the superficial measure of the

boundary ∂X,– the last but one ( degree 1) is the so called norm or mean width ; on

the convex class it satisfies the characteristic relationshipM ( λλλλX ⊕⊕⊕⊕ µµµµY) = λ λ λ λ M(X) + µ µ µ µ M(Y)


Minkowski Functionals in R1 and R2Minkowski Functionals in R1 and R2

• For example, in !1, the functionals reduce to– the length L(X) of set X,– and the number νννν0000(X) of the segments which constitute X.

• In !2, The three values are the area A, the perimeter U and the EPC νννν2222

with U(X) = ∫∫∫∫ππππ dα α α α ∫∫∫∫R νννν1111[ X ∩∩∩∩ ∆ ∆ ∆ ∆ (x , αααα) ] dx(the perimeter is equal to the sum of the intercepts taken in all directions).

Moreover, when set X is compact convex, the perimeter is related to the projections of X on the straight lines ∆α by the relation

U(X) = ∫∫∫∫2π2π2π2π L ( X ∆∆∆∆αααα ) dαααα ....

• For example, in !1, the functionals reduce to– the length L(X) of set X,– and the number νννν0000(X) of the segments which constitute X.

• In !2, The three values are the area A, the perimeter U and the EPC νννν2222

with U(X) = ∫∫∫∫ππππ dα α α α ∫∫∫∫R νννν1111[ X ∩∩∩∩ ∆ ∆ ∆ ∆ (x , αααα) ] dx(the perimeter is equal to the sum of the intercepts taken in all directions).

Moreover, when set X is compact convex, the perimeter is related to the projections of X on the straight lines ∆α by the relation

U(X) = ∫∫∫∫2π2π2π2π L ( X ∆∆∆∆αααα ) dαααα ....


StereologyStereology

• A measurement on a set family in !n is said to be stereological when it can be written as a function of measurements performed on sets of !k, k < n.

• By construction, all Minkowski functionals are stereological except the last one (the EPC). More precisely, in !3 (for example), for computing

a volume, it suffices to sample by points,an area « « lines,a width « « planes.

• The more often, stereology intervenes between specific measures, hence in a stationary framework. Denote by VV(X) the volume of X by unit volume of the space, by SV(X) the specific surface, by NA and NL the specific EPC in !2 and !1 (or, if so, their rotation averages). The above relationships become :

VV = AA = LL SV = 4 NL = (4/ππππ) UA MV = 2ππππNA

• A measurement on a set family in !n is said to be stereological when it can be written as a function of measurements performed on sets of !k, k < n.

• By construction, all Minkowski functionals are stereological except the last one (the EPC). More precisely, in !3 (for example), for computing

a volume, it suffices to sample by points,an area « « lines,a width « « planes.

• The more often, stereology intervenes between specific measures, hence in a stationary framework. Denote by VV(X) the volume of X by unit volume of the space, by SV(X) the specific surface, by NA and NL the specific EPC in !2 and !1 (or, if so, their rotation averages). The above relationships become :

VV = AA = LL SV = 4 NL = (4/ππππ) UA MV = 2ππππNA


Minkowski Functionals in R3 Minkowski Functionals in R3

• In !3, the four functionals are the volume V, the area S , the mean width or norm M , and the EPC νννν3333. We have the two expressions

π π π π S(X) = ∫4π4π4π4π dω ω ω ω ∫R νννν1111[ X ∩∩∩∩ ∆ ∆ ∆ ∆ (x , ωωωω) ] dx

2 2 2 2 M (X) = ∫4π4π4π4π dω ω ω ω ∫R νννν2222[ X ∩∩∩∩ Π Π Π Π (x , ωωωω) ] dx .• In particular, when set X is convex, its area and its norm has a meaning in

terms of projections XΠ ω and X∆ ω on the planes Πωand lines ∆ω :

π π π π S(X) = ∫4π4π4π4π A ( X ΠΠΠΠωωωω ) dωωωω

2 2 2 2 M (X) = ∫4π4π4π4π L ( X ∆∆∆∆ωωωω ) dωωωω = ( 2/π π π π )∫4π4π4π4π U ( XΠΠΠΠωωωω ) dωωωω ....

• Finally, if ∂X admits curvatures C1 and C2 everywhere, then2 2 2 2 M (X) = ∫∫∫∫∂∂∂∂X ( ( ( ( C1 + C2 ) ds .

• In !3, the four functionals are the volume V, the area S , the mean width or norm M , and the EPC νννν3333. We have the two expressions

π π π π S(X) = ∫4π4π4π4π dω ω ω ω ∫R νννν1111[ X ∩∩∩∩ ∆ ∆ ∆ ∆ (x , ωωωω) ] dx

2 2 2 2 M (X) = ∫4π4π4π4π dω ω ω ω ∫R νννν2222[ X ∩∩∩∩ Π Π Π Π (x , ωωωω) ] dx .• In particular, when set X is convex, its area and its norm has a meaning in

terms of projections XΠ ω and X∆ ω on the planes Πωand lines ∆ω :

π π π π S(X) = ∫4π4π4π4π A ( X ΠΠΠΠωωωω ) dωωωω

2 2 2 2 M (X) = ∫4π4π4π4π L ( X ∆∆∆∆ωωωω ) dωωωω = ( 2/π π π π )∫4π4π4π4π U ( XΠΠΠΠωωωω ) dωωωω ....

• Finally, if ∂X admits curvatures C1 and C2 everywhere, then2 2 2 2 M (X) = ∫∫∫∫∂∂∂∂X ( ( ( ( C1 + C2 ) ds .


Steiner Formulae for Convex SetsSteiner Formulae for Convex Sets

When sets X and B are compact convex, the functionals of X ⊕ B derive from those of X and of B. The relation has already been given for the norm; as for the volume, we have

in !2,

A( X ⊕ B ) = A( X ) + U( X ). U( B ) / 2 π + A( B )in !3,

V( X ⊕ B ) = V( X ) + [M( X ). S( B ) + S( X ). M( B )] / 4 π + V( B )

where A and V stand for the rotation averages as B takes all possible orientations with respect to set X .These formulae, established by Steiner in 1839 when B is a ball, are useful for calculating random sets models (see ch. XIV and XVI)

When sets X and B are compact convex, the functionals of X ⊕ B derive from those of X and of B. The relation has already been given for the norm; as for the volume, we have

in !2,

A( X ⊕ B ) = A( X ) + U( X ). U( B ) / 2 π + A( B )in !3,

V( X ⊕ B ) = V( X ) + [M( X ). S( B ) + S( X ). M( B )] / 4 π + V( B )

where A and V stand for the rotation averages as B takes all possible orientations with respect to set X .These formulae, established by Steiner in 1839 when B is a ball, are useful for calculating random sets models (see ch. XIV and XVI)


Steiner Formulae, particular casesSteiner Formulae, particular cases

• For B the unit segment, we have– in !2, A( X ⊕ r B ) = A( X ) + r U( X ) / ππππ– in !3, V( X ⊕ r B ) = V( X ) + r S( X ) / 4

• If B is now the unit disc, we find– in !2, A( X ⊕ r B ) = A( X ) + r U( X ) + ππππr2

– in !3, V( X ⊕ r B ) = V( X ) + ππππr S( X ) / 4 + r2M( X ) / 2

• and finally, in !3, when B is the unit ballV( X ⊕ r B ) = V( X ) + r S( X ) + r2 M( X ) + 4/3 ππππr3 .

In all cases, the measure of the dilate is a polynomial function of the size of the structuring element, and has for degree the dimension of the latter. One shows that, for r small, the first order terms extend to the convex ring.

• For B the unit segment, we have– in !2, A( X ⊕ r B ) = A( X ) + r U( X ) / ππππ– in !3, V( X ⊕ r B ) = V( X ) + r S( X ) / 4

• If B is now the unit disc, we find– in !2, A( X ⊕ r B ) = A( X ) + r U( X ) + ππππr2

– in !3, V( X ⊕ r B ) = V( X ) + ππππr S( X ) / 4 + r2M( X ) / 2

• and finally, in !3, when B is the unit ballV( X ⊕ r B ) = V( X ) + r S( X ) + r2 M( X ) + 4/3 ππππr3 .

In all cases, the measure of the dilate is a polynomial function of the size of the structuring element, and has for degree the dimension of the latter. One shows that, for r small, the first order terms extend to the convex ring.


Weights and Individuals (I) : VolumesWeights and Individuals (I) : Volumes

• Global Measurements or individual analysis?So far, set X was always considered as a whole. What additional

results can be found when X partitioned into individuals Y (grains, clusters of grains) which can be identified on sections ? How the sizes of the sectioned grains are they related to those of the space ones, for example ?

Denote by E[∗ ] the mean of ∗ over all objets, and by ∗ the mean of ∗for all sections of all objets in all directions.

• Inaccessibility of the mean volume In !3, the major stereological relationship is the following

E [V (Y)] = A (Y ∩ Π ) . L (Y ∆ ) = A (Y Π ) . L (Y ∩ ∆ )

A rather disappointing result, since the projection terms cannot be accessed from plane sections.

• Global Measurements or individual analysis?So far, set X was always considered as a whole. What additional

results can be found when X partitioned into individuals Y (grains, clusters of grains) which can be identified on sections ? How the sizes of the sectioned grains are they related to those of the space ones, for example ?

Denote by E[∗ ] the mean of ∗ over all objets, and by ∗ the mean of ∗for all sections of all objets in all directions.

• Inaccessibility of the mean volume In !3, the major stereological relationship is the following

E [V (Y)] = A (Y ∩ Π ) . L (Y ∆ ) = A (Y Π ) . L (Y ∩ ∆ )

A rather disappointing result, since the projection terms cannot be accessed from plane sections.


Weights and Individuals (2) : NumbersWeights and Individuals (2) : Numbers

Let I(∗ ) be the number of the individuals of ∗ , a quantity equal to the EPC of X when the sets are convex . We have

I ( X ∩∩∩∩ Π Π Π Π ) = I (X) . L ( X ∆ ∆ ∆ ∆ ) number = number . diameter (in !2 ) (in !3 ) ( mean)

and similarlyI ( X ∩∩∩∩ ∆ ∆ ∆ ∆ ) = I (X) . A ( X Π Π Π Π ).

For example, in fig.a) the average value of I ( X ∩ Π ) is 9

whereas in fig.b) where there are more spheres, but

smaller, I ( X ∩ Π ) = 7 .

Let I(∗ ) be the number of the individuals of ∗ , a quantity equal to the EPC of X when the sets are convex . We have

I ( X ∩∩∩∩ Π Π Π Π ) = I (X) . L ( X ∆ ∆ ∆ ∆ ) number = number . diameter (in !2 ) (in !3 ) ( mean)

and similarlyI ( X ∩∩∩∩ ∆ ∆ ∆ ∆ ) = I (X) . A ( X Π Π Π Π ).

For example, in fig.a) the average value of I ( X ∩ Π ) is 9

whereas in fig.b) where there are more spheres, but

smaller, I ( X ∩ Π ) = 7 .

D = 0.3

D = 0.1

a)

b)


Generalization to Numerical FunctionsGeneralization to Numerical Functions

• A Model for numerical functionsIn order to generalize the above results, we can model gray tone images

by the class F of those functions f : !2 (or !n) → [-∞ , +∞] whose all sections

Xt (f) = {x : x ∈ ∈ ∈ ∈ R2 , f(x) ≥ ≥ ≥ ≥ t }are elements of the convex ring. The W become mappings from F into

[-∞ ,+∞] • Generalization of the prerequisites

– C - additivity becomesW(f∨∨∨∨g) + W(f∧∧∧∧g) = W(f) + W(g) f,g ∈ F

– displacement invariance holds only on horizontal translations,and on rotations of vertical axis;

– finally the condition increasingness extends directly.

• A Model for numerical functionsIn order to generalize the above results, we can model gray tone images

by the class F of those functions f : !2 (or !n) → [-∞ , +∞] whose all sections

Xt (f) = {x : x ∈ ∈ ∈ ∈ R2 , f(x) ≥ ≥ ≥ ≥ t }are elements of the convex ring. The W become mappings from F into

[-∞ ,+∞] • Generalization of the prerequisites

– C - additivity becomesW(f∨∨∨∨g) + W(f∧∧∧∧g) = W(f) + W(g) f,g ∈ F

– displacement invariance holds only on horizontal translations,and on rotations of vertical axis;

– finally the condition increasingness extends directly.


What about Homogeneity ?What about Homogeneity ?

• Magnifications or affinities ?Class F models gray tones images, where the vertical axis T ( gray intensity) is heterogeneous to the space. In such conditions, compatibility under magnification turns out to be cumbersome. What about affinities ?

• An example : length l of curvein the affinity of ratio 2, the flat zones

keep their lengths, although the sloping ones become longer

• Reason :

l = ∫∫∫∫[ 1 + (f ’(x))2 ] dxthe length is no longer a convenient measurement ( it still remains at our

disposal the integral under the curve !)

• Magnifications or affinities ?Class F models gray tones images, where the vertical axis T ( gray intensity) is heterogeneous to the space. In such conditions, compatibility under magnification turns out to be cumbersome. What about affinities ?

• An example : length l of curvein the affinity of ratio 2, the flat zones

keep their lengths, although the sloping ones become longer

• Reason :

l = ∫∫∫∫[ 1 + (f ’(x))2 ] dxthe length is no longer a convenient measurement ( it still remains at our

disposal the integral under the curve !)

x

f(x)T


DimensionalityDimensionality

• We must separate the space magnifications from the intensity ones. A functional W is said to be dimensional when

W [ λ.λ.λ.λ.f (µ.µ.µ.µ.x) ] = λλλλk . µ . µ . µ . µ p .... W [ f(x)] x ∈ !n , f ∈ Fwhere k and p, integers ≠ 0 , are the dimensions of measurement W.

Dimensionality preservation orients us towards products of planar operations by «vertical» ones ( i.e. where the neighborhood is reduced to a single point).

• Each numerical Minkowski functional is obtained by summing up the set corresponding one over of T axis. In !2, we draw from the areas of the sections the (possibly infinite) volume V(f) and the cumulative histogramGf (t) :

V(f) = ∫R2 f(x) dx = ∫T A [Xt (f)] dt

Gf (t) = ∫ t

−−−− ∞∞∞∞ A [Xt (f)] dt / V(f)

• We must separate the space magnifications from the intensity ones. A functional W is said to be dimensional when

W [ λ.λ.λ.λ.f (µ.µ.µ.µ.x) ] = λλλλk . µ . µ . µ . µ p .... W [ f(x)] x ∈ !n , f ∈ Fwhere k and p, integers ≠ 0 , are the dimensions of measurement W.

Dimensionality preservation orients us towards products of planar operations by «vertical» ones ( i.e. where the neighborhood is reduced to a single point).

• Each numerical Minkowski functional is obtained by summing up the set corresponding one over of T axis. In !2, we draw from the areas of the sections the (possibly infinite) volume V(f) and the cumulative histogramGf (t) :

V(f) = ∫R2 f(x) dx = ∫T A [Xt (f)] dt

Gf (t) = ∫ t

−−−− ∞∞∞∞ A [Xt (f)] dt / V(f)


Perimeter and GradientPerimeter and Gradient

• From Steiner formula (extended to the convex ring), the perimeter is equal to the derivative at the origin of the dilation by a disc

r → 0→ 0→ 0→ 0 ⇒ ⇒ ⇒ ⇒ [ A(Xt ⊕ r B ) - A( Xt )] →→→→ r U(Xt ) • When Xt is finite union of convex Yi (t), this limit is upper bounded by

the finite sum Σ [ A(Yi (t) ⊕ r B) - A(Yi (t) ] ; therefore we can write

∫T U [Xt (f)] dt = lim r → 0 → 0 → 0 → 0 [ ((((V (f ⊕⊕⊕⊕ r B ) - V( f )) / r ]a relationship the provide Beucher gradient by dilation g+ with an Euclidean meaning (and proves its existence on the convex ring), hence

∫T U [Xt (f)] dt = ∫R2 g+(x) dx N.B. Beucher gradient can be still defined even when the usual one is not

• From Steiner formula (extended to the convex ring), the perimeter is equal to the derivative at the origin of the dilation by a disc

r → 0→ 0→ 0→ 0 ⇒ ⇒ ⇒ ⇒ [ A(Xt ⊕ r B ) - A( Xt )] →→→→ r U(Xt ) • When Xt is finite union of convex Yi (t), this limit is upper bounded by

the finite sum Σ [ A(Yi (t) ⊕ r B) - A(Yi (t) ] ; therefore we can write

∫T U [Xt (f)] dt = lim r → 0 → 0 → 0 → 0 [ ((((V (f ⊕⊕⊕⊕ r B ) - V( f )) / r ]a relationship the provide Beucher gradient by dilation g+ with an Euclidean meaning (and proves its existence on the convex ring), hence

∫T U [Xt (f)] dt = ∫R2 g+(x) dx N.B. Beucher gradient can be still defined even when the usual one is not

.x

xf(x) g+(x)


The sum of the ECP taken on the sections of function f measures the differences in height of f . In the example of the figure, we have

∫T νννν [Xt (f)] dt = D1 + D2 - D3

The sum of the ECP taken on the sections of function f measures the differences in height of f . In the example of the figure, we have

∫T νννν [Xt (f)] dt = D1 + D2 - D3

Connectivity Number for Gray tone functionsConnectivity Number for Gray tone functions

D1D2 D3


Others measurementsOthers measurements

• Other measurements are introduced in the next chapters. When they involve derivatives of erosions, the underlying model is still the convex ring : – E.g. Roughness ( ch.XIII), Rose of directions

If they involve only areas of dilates, it suffices to suppose set X locallycompact :..– E.g. Star (ch.XIII), Range (ch.XII), boolean counting (ch.XIV)

• Although they often admit a stereological interpretation, these functionals no longer satisfy all hadwigerian conditions (c-additivity, in particular).

• We will conclude the chapter by two non hadwigerian measurements, the first one in the convex ring (convexity number), and the second in the more general class of the compact sets (fractal dimension).

• Other measurements are introduced in the next chapters. When they involve derivatives of erosions, the underlying model is still the convex ring : – E.g. Roughness ( ch.XIII), Rose of directions

If they involve only areas of dilates, it suffices to suppose set X locallycompact :..– E.g. Star (ch.XIII), Range (ch.XII), boolean counting (ch.XIV)

• Although they often admit a stereological interpretation, these functionals no longer satisfy all hadwigerian conditions (c-additivity, in particular).

• We will conclude the chapter by two non hadwigerian measurements, the first one in the convex ring (convexity number), and the second in the more general class of the compact sets (fractal dimension).


Convexity Number Convexity Number

Consider, in !2 the test line ∆∆∆∆αααα of normal α ± dα/2 , and let N+(X,α) be

the number of its first contacts with set X when ∆α sweeps the plane. We have

N+(X) = ∫2π2π2π2π N+(X,α) dα = ∫R> 0 du / R ,( if ∂X admits everywhere a radius of curvature R). Similarly

N −−−−(X) = ∫2π2π2π2π N −−−−(X,α) dα = ∫R< 0 du / R .These two convexity numbers are linked with the EPC the relationship

ν(X) = N+(X)-N −(X)

Consider, in !2 the test line ∆∆∆∆αααα of normal α ± dα/2 , and let N+(X,α) be

the number of its first contacts with set X when ∆α sweeps the plane. We have

N+(X) = ∫2π2π2π2π N+(X,α) dα = ∫R> 0 du / R ,( if ∂X admits everywhere a radius of curvature R). Similarly

N −−−−(X) = ∫2π2π2π2π N −−−−(X,α) dα = ∫R< 0 du / R .These two convexity numbers are linked with the EPC the relationship

ν(X) = N+(X)-N −(X)

∆∆∆∆αααα

N+ =1 N −−−−= 0

N+ = 1,1N −−−−= 0,1

N+ =1,7 N −−−−= 0,7

N+ = 2 N −−−−= 0

N+ =1 N −−−−= 1


Fractal Dimension (I)Fractal Dimension (I)

• DefinitionConsider a compact set X in !2. When set X is dilated, or eroded, by a small

disc r B, whose radius tends towards zero, it may happen that, under the development of finer and finer details, the area increment divided by r does not tend towards the perimeter, finite, of Steiner formula, but towards the infinity. In such a case, there exists a smaller value α such that

A( X ⊕ r B ) - A (X ) r B ) = kr αααα + 0 (r αααα ) 1 ≤ α ≤ ≤ α ≤ ≤ α ≤ ≤ α ≤ 2(k constant). The value d = 2 - α defines the Minkowski Dimension of X.

• Examples : for all sets of the convex ring , d = 1for a brownian trajectory, d = 1,5 for a Peano curve, d = 2.

• Fractal SetsThe model of a fractal set due to B. Mandelbrot brings into play the dimension

d , a local notion, and also self-similarity, i.e. a notion which is global .

• DefinitionConsider a compact set X in !2. When set X is dilated, or eroded, by a small

disc r B, whose radius tends towards zero, it may happen that, under the development of finer and finer details, the area increment divided by r does not tend towards the perimeter, finite, of Steiner formula, but towards the infinity. In such a case, there exists a smaller value α such that

A( X ⊕ r B ) - A (X ) r B ) = kr αααα + 0 (r αααα ) 1 ≤ α ≤ ≤ α ≤ ≤ α ≤ ≤ α ≤ 2(k constant). The value d = 2 - α defines the Minkowski Dimension of X.

• Examples : for all sets of the convex ring , d = 1for a brownian trajectory, d = 1,5 for a Peano curve, d = 2.

• Fractal SetsThe model of a fractal set due to B. Mandelbrot brings into play the dimension

d , a local notion, and also self-similarity, i.e. a notion which is global .


Fractal Dimension (II)Fractal Dimension (II)

• Stereology Fractal dimension extends to !n , but is not a stereological notion. In

practice, in the isotropic case, ones passes from !2 to !3 by adding 1 to d. • ExperimentsThe object under study is accessed via a series of magnifications θ, whose

limit of resolution r decreases. For a given θ, the perimeter of X isUX (r) = [A( X ⊕ r B ) - A (X ) r B ) ] / 2 r = k/2 r d −−−− 1111 ....

Experimentally, we have to check whether Log UX (r), as a function of Log r fits a straight line. If so, its slope provides an estimate of d - 1.

• Numerical FunctionsWhen the object under study is a numerical function, by applying the previous approach to the sections Xt , we obtain the following algorithm

log grad r (f ) = (d - 1) log r + k’

• Stereology Fractal dimension extends to !n , but is not a stereological notion. In

practice, in the isotropic case, ones passes from !2 to !3 by adding 1 to d. • ExperimentsThe object under study is accessed via a series of magnifications θ, whose

limit of resolution r decreases. For a given θ, the perimeter of X isUX (r) = [A( X ⊕ r B ) - A (X ) r B ) ] / 2 r = k/2 r d −−−− 1111 ....

Experimentally, we have to check whether Log UX (r), as a function of Log r fits a straight line. If so, its slope provides an estimate of d - 1.

• Numerical FunctionsWhen the object under study is a numerical function, by applying the previous approach to the sections Xt , we obtain the following algorithm

log grad r (f ) = (d - 1) log r + k’


Fractal Dimension : an ExampleFractal Dimension : an Example

• Specimen of clay seen in scanning electron microscopy, at magnifications

102, 3.102, 103, 3.103 et 104.• The apparition of new details at each step suggests the

fractal model. Indeed, the increments of the gradient yield the estimate d* = 1.8

• Specimen of clay seen in scanning electron microscopy, at magnifications

102, 3.102, 103, 3.103 et 104.• The apparition of new details at each step suggests the

fractal model. Indeed, the increments of the gradient yield the estimate d* = 1.8

100

1000

10000

100 1000 1000

"c:\wmmorph\argile.dat"

Clay :G×××× 1000

Clay : G×××× 300


Basic References on the measurementsBasic References on the measurements

On integal Geometry and Minkowski Functionals • H. Minkowski, Math. Ann., 57 (1903) 447. • H. Hadwiger (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer

Verlag, Berlin.• G. Matheron (1975) Random Sets and Integral Geometry, Wiley, N.Y.• L.A. Santalo (1976) Integral Geometry and Geometrical Probability, Encyclopedia of

Mathematics and its Applications, Addison Wesley, Reading, Mass., USA.

On stereology • E.E. Underwood (1970) Quantitative Stereology, Addison Wesley.• E. R. Weibel (1981) Stereological Methods,Vol 1 and 2, Ac. Press, London.

• J. Serra(1982) Image Analysis and Mathematical Morphology, Academic Press, London

• M. Coster, J.L. Chermant (1989) Précis d'Analyse d'Images, Les Presses du CNRS (1985); 2nd Edition, Les Editions du CNRS.

On fractal Sets• B. Mandelbrot (1977) Fractals: form, chance, dimension . W.H. Freeman San Francisco

On integal Geometry and Minkowski Functionals • H. Minkowski, Math. Ann., 57 (1903) 447. • H. Hadwiger (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer

Verlag, Berlin.• G. Matheron (1975) Random Sets and Integral Geometry, Wiley, N.Y.• L.A. Santalo (1976) Integral Geometry and Geometrical Probability, Encyclopedia of

Mathematics and its Applications, Addison Wesley, Reading, Mass., USA.

On stereology • E.E. Underwood (1970) Quantitative Stereology, Addison Wesley.• E. R. Weibel (1981) Stereological Methods,Vol 1 and 2, Ac. Press, London.

• J. Serra(1982) Image Analysis and Mathematical Morphology, Academic Press, London

• M. Coster, J.L. Chermant (1989) Précis d'Analyse d'Images, Les Presses du CNRS (1985); 2nd Edition, Les Editions du CNRS.

On fractal Sets• B. Mandelbrot (1977) Fractals: form, chance, dimension . W.H. Freeman San Francisco

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