Chapter XXII The Covariance

Transcription

1 Chapter XXII The Covariance Geometrical Covariogram Random Version : -Sets -Functions - Properties of the covariance Intrinsic Theory : - Variogram J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 1

2 Set Covariogram Let h be be a vector of of origin 0, 0, module h and direction α in in! n n.. Take for for structuring element B the the point doublet { 0, 0, h } and consider erosion X!B = X X -h n -h = { x! n,, x X X -h } -h.. Set X!B is is made of of all all those points common to to X and to to its its translate by by vector h.. Definition :: The set set covariogram of of X is is the the Lebesgue measure of of X X -h ; -h ; it it is is denoted by: K(h) = K α (h) (h) = Mes [X X -h ]= -h ]= Mes [X X h ] ] with K α (0) (0) = Mes X and K(h) dh dh = [Mes X ]] 22 The integral of of the the covariogram equals the the square of of the the area (volume) of of X J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 2

3 An Example X K( h ) "c:\wmmorph\set.dat" 2500 h = 0 α = h = 20 covariogramme h = 40 h = h h J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 3

4 Anisotropies of the covariogram "c:\wmmorph\hset.dat" "c:\wmmorph\set.dat" α 1 = α 2 = 60 covariogramme α 1 α h The tangent at at the the origin -K -K α (0) = --[ [ K α (h) (h)/ h ]] h h = 00 is is nothing but the the total variation of of X in in direction α, α, and in in the the convex case its itsapparent contour, as as indicated by by Crofton s formula :: - K dα (in 2 ) ; dα (in 3 α (0) dα = 2U(X) (in! 2 ) ; - K α (0) dα = π S(X) (in! 3 )) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 4

5 Random Version Associate with set set X its its indicator function f. f. We have :: K α (h) (h) = Mes{X X h } = f(x). f(x+h) dx dx where the the sum is is extended to to the the whole space. If If now f f is is no no longer an an indicator, but an an arbitrary measurable numerical function, the the the the above integral defines its its covariogram. It It may happen that the the object represented by by f f be be reproduced as as one likes (e.g. a crystal of of quartz) or, or, even if if it it is is unique, reproduces itself indefinitely through the the space. It It is is then interpreted as as a realization of of Random Function. In In the the first case, for for ff 22 a.s. integrable, the the covariogram becomes K α (h) (h) = E[ E[ Mes{X X h } ]] K α (h) (h) = E[ f(x). f(x+h) dx dx ]] (sets) (functions )) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 5

6 Stationary Approach When the the object under study is is large with respect to to the the working mask, it it is is preferable to to interpret it it in in terms of of stationary random close sets, or or functions. Covariogram K α (h) (h) is is then replaced by by the the covariance C(h,α) = E [( [( f(x) --m).(f(x+h) --m )] )] = E [[ f(x). f(x+h) ]]-- m 22.. where m = E [[ f(x) ]] is is the the mathematical expectation of of the the process. For h = 0, 0, we we have C(0,α) = E [( [( f(x) --m)] 22 = σ 22 The value at at the the origin of of the the covariance is is equal to to the the variance of of the process. In In the the set set case of of a volume proportion m = p we we find C(0,α) = p (1 (1--p) p) For h =, the the correlation between x and x+h vanishes, so so that C(h,α) = E [( [( f(x) --m)]. E[(f(x+h) --m )] )] = 0 For the the sake of of clarity, we we treat below the the set set case first, and then the the function case J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 6

7 Set Covariance When the the object under study is is modeled by by a stationary random close set setx, the the notion that corresponds directly to to the the covariogram K α (h) (h) is is the the non centered covariance C 1 (h,α) 1 = C (h,α) + p 22 = Prob { x X X h }. }. Experimentally, the the non centered set set covariance is is estimated in in a zone Z as as the the ratio of of the the favorable locations of of the the doublet B N = A [(X Z )! )! B] B] over all all its its possible locations in in Z D = A [Z! B] B] i. i. e. e. C 1 (h,α)* 1 = N/D.. X Z An example of objet X whose dimensions exceed those of the field Z. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 7

8 Properties of the Set Covariance We find again the the properties of of K(h), enriched with stochastic interpretations. Near the the origin o, o, we we have C 1 (0, 1 (0, α) α) = A A (X) = p (for all all α) α) ;; As As h is is very large, the the two events x X and x X h become h independent, hence lim C 1 (h, 1 (h, α) α) = p 22 = [A [A A (X)] 22.. The directional average of of the the tangent at at the the origin equals the the specific perimeter (area) - C 1 (0,α) 1 dα dα = (( 2/ 2/ π )U )U L (X) = S V (X) // 4 (in! 22 )) (( in in! 33 )) The range a = 1/p(1-p) [C 1 (h) 1 (h)--p 22 ]dh has an an ergodic meaning (see below XII-21) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 8

9 Pseudo-periodicities and Covariance "c:\wmmorph\eutect.dat" The maxima of of the the covariance, when their abcissae are are multiple of of each other (( here 30, 30, 60, 60, 90) indicate (pseudo) periodicities of of the the objet; Unlike, the the covariance is is blind to to connectivity :: the the covariances of of the the two above sets are are graphically identical J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 9

10 Clusters and Covariance "c:\wmmorph\part2.dat" Features of of different scales (here, the the particles and their cluster) add their covariances (e.g. the the associated tangents at at the the origin) Moreover, we we see in in the the present case an an oscillation due to to the the equal inter particle distance in in the the clusters. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 10

11 Noise and Covariance "c:\wmmorph\noise2.dat" As As a limit of of the the previous situation we we obtain the the Poisson noise.. Hence, in in digital case the the range of of such an an impulse noise is is equal to to 1, 1, and its its variance p 00 (1- (1-p 00 )) corresponds to to the the jump down of of the the ordinate near the the origin. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 11

12 Rectangle (or Cross) Covariance The division of of the the space into a set set and its its complement is is replaced sometimes by by a partition into n phases X 11,, X X n n.. The study of of their space relations is is then carried out by by means of of the the rectangle covariance. Three phases polished section Corresponding partition Definition : The rectangle (or cross) covariance is the probability that point x belongs to phase X i and point x+h to X j. One writes : C ij (h, α) = Prob { x X i, x+h X j }. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 12

13 Properties of the rectangle covariance When phases X et et X cc are are complementary C 10 (h, 10 (h, α) α) = A A (X) --C 11 (h, 11 (h, α) α) where C (h, (h, α) α) is is the the covariance of of phase X 11.. In In case of of a multiphased structure A A (X (X i ) i ) = C ij (h, ij (h, α) α) and i j i j C ij (0, ij (0, α) α) = 0.. For h large, the the two events x X i i and x+h X j j are are independent, hence lim C ij (h, ij (h, α) α) = [A [A A (X (X i )] i )][A A (X (X j )] j )] The directional average of of the the derivative at at the the origin is is related to to the the specifie perimeter U L (area S V ) ) between phases i i and j j :: - C ij (0,α) ij dα dα = (( 2/ 2/ π )U )U L (X (X i / i / X j ) j ) = S V (X (X i / i / X j )/ j )/ 4 J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 13

14 A Metallographical Example "c:\pore-fer.dat" "c:\fer-hem.dat" "c:\hem-pore.dat" Specimen of of iron ore sinter. Light gray: hematite ;; dark gray :: ferrite ;; black: pores The two ferrite covariances are are similar. That between hematite and pores --starts from more below :: smaller contact surface --presents a hole effect for for h =50 :: this indicates a halo of of ferrite around hematite. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 14

15 From Sets to Functions The transition binary numerical shows itself essentially in in two ways :: on on the behavior near the origin :: the the numerical functions may be be q. q. m. m. derivable,, the the sets cannot ;; on on the behavior at at infinity :: large though the the working field is, is, the the variance of of a EFA inside it it remains bounded by by sup { p(1-p),, p [0,1] } = 0,25 unlike, certain physical phenomena can exhibit a quasi infinite range of of fluctuations, i.e. i.e. a variance which increases without limit with the the size of of the the domain of of experiments (( ore deposits, rains...) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 15

16 Behavior near the origin (I) "c:\wmmorph\chrom.dat" The function is is derivable in in quadratic mean; therefore its itscovariance is is derivable at at the the origin, where, by by symmetry, the the derivative is is zero Remark that the the assumption of of stationarity, obviously false here, does not disturb in in the the study near the the origin. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 16

17 Behavior near the origin (II) "c:\wmmorph\chromk.dat" An An additional Poisson noise is is reflected on on the the covariance by by a jump down at at the the origin (called «nugget effect»),»), whose value is is nothing but the the variance of of the the noise. However, if if the the phenomenon and the the noise are are independent, the the rest of of the the covariance is is no no more affected :: a very useful robustness!! J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 17

18 Behavior near the origin (III) a) Healthy lung "c:\wmmorph\lung3.dat" "c:\wmmorph\lung4.dat" b) Lung of a smoker the the piecewise continuous functions (e.g. indicators) are are m.q. continous. Hence, their covariances are are not derivable at at the the origin (oblique tangent). In In addition, the the covariance sums up up the structures which are are independent (e.g. nugget effect). In In the the smoker radiograph, we we see the the superimposition of of rib rib structures (8µ), and of of the the dark alveoli of of the the lung (35µ). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 18

19 Random Functions of Dilution What sort of of link does exists between a basic shape (e.g. the the lung nodule) and the the stationary structure it it generates? Can we we build a stationary random function whose covariance be be a given covariogram? How to to justify the the super-impositions of of scales on on the the covariances? To To answer such questions, a very simple model consists in in starting from Poisson points {x {x i i,, i i I I } in in R n n,, of of variance λ,, putting, independently, at at each point x i i,, a realization ff ii of of a primary set set (or function) of of covariogram K α (h) (h),, and in in taking the the sum f f = f f i i of of all all these primary objects. The covariance of of the the resulting stationary random function f f is is then C(h,α) = λ K α (h) (h) and that of of the the sum f f = f f + + f f of of two such independent functions λ K α (h) (h) + λ K α (h) (h).. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 19

20 Covariance and Spectral Analysis We will pass to to the the behavior at at the the infinity by by quoting first a famous theorem Theorem (Wiener-Khinchine) :: The covariance C(h) of of a stationary random function f f is is the the Fourier transform of of its its energy spectrum φ (ν) (ν),, hence of of a non negative measure. Corollary 1 :: This last condition amounts to to saying that C(h) is is definite positive, i.e. i.e. such that we we have, for for all all positive weights λ i i and all all points x ii λ i λ i j C(x j i -x i -x j ) j ) Corollary 2 :: Since Fourier transformation exchanges the the behaviors at at the the origin and at at the the infinity, we we obtain, in in particular :: φ φ (ν) (ν) dν dν = C(0) = E [( [( f(x) -m)] 22 = σ The sum of of the the energies associated with all all frequencies is is thus equal to to the the point variance σ J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 20

21 Ranges and Variances Conversely, the the range a of of the the covariance corresponds to to frequency zero a C(0) = C(h) dh dh = φ (0) (0).. Let then Z be be a large working zone. If If ff Z (x) (x) = Z f(x-y)dy,, we we have Z σ 22 Ζ = E [(f [(f Z (x) (x)-m)] 22 [a [a// Mes Z] Z] C(0) With respect to to the the variance σ 22 Ζ of of a sample Z, Z, the the range turns out to to be, be, asymptotically, the the size unit of of the the phenomenon under study. More generally, whatever large Z is, is, if if σ / / Ζ stands for for the the point variance in in Z, Z, it it can be be shown that σ // Z = σ σ 22 Z ; ; This remarkable identity, known as as Krige s formula,, constitutes the the starting point for for Gostatistics. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 21

22 Intrinsic Theory, Variogram When the the phenomenon under study is is extremely large, it it may happen that the the estimated variance [σ [σ // Z ] ] ** of of a point inside Z increases indefinitely with Z, Z, at at any scale of of analysis. From Krige s formula, this means that the the stationary model is is just inadequate :: σ 22 0 = 0 C(0) being infinite, the the covariance no no longer exist. Intrinsic Model (G.Matheron) :: However, the the increments of of f f may still exist and have a meaning. By By assuming them stationary, we we can study their variances, also called «variogram» γ (h,α) = E [[ f(x) --f(x+h) ]] 22 Properties :: We have γ (0,α) = 0,, and σ 22 0 < 0,, γ (h,α) = 2[C(0) --C(h)] When σ 22 0 = 0,, the the notion of of a range vanishes, and new behaviors at at the the infinity appear. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 22

23 An Example : Poisson Steps (I) One may argue :: anyway all all numerical data are are finite!! Let us us examine the the point more accurately, in in the the framework of of a model that lends itself to to easy calculations. Definition :: Poisson steps are are generated from a Poisson point process in in! 11,, of of variance λ,, by by placing a jump of of amplitude δ at at each Poisson point. The random value δ follows a law of of mean 0 and of of variance σ 22.. Properties :: In In this model, the the increments only are are defined, their average is is equal to to zero and their variogram to to γ (h) (h) = λ σ 22 h J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 23

24 An Example : Poisson Steps (II) The estimated variogram 2γ 2γ ** (h) (h) = 1/(L-h) L-h L-h 0 [f [f(x+h) --f(x)]2 f(x)]2 dx dx admits for for expectation E [2γ [2γ ** (h) (h)]] = γ (h) (h) = λ σ 22 h.. But it it is is always possible to to put ff L = 1/L L f(x) dx, and then Cov*(h) = 1/(L-h) L-h L-h 0 (f (f(x+h) --fl fl )( )( f(x) --ff L )dx.. Such an an experimental «covariance» admits for for expectation E [[ Cov*(h) ]] = L/3 -- 4/3. h + 2/3. h 22 /L /L (( 0 h L )) where the the actual structure (behaviors at at the the origin and at at the the infinity) is is completely falsified "c:\wmmorph\steps6.dat" "c:\wmmorph\steps66.dat" Simulation of Poisson Steps, its «covariance», its variogram J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 24

25 An Example of Complex Structure (I) Thin section of beech taken normally to the axis of the trunk. The horizontal direction is that of the the radii that start from the center of the tree; the vertical one is that of the year rings. Dimensions : pixels, with 1 pixel = 1 micron. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 25

26 An Example of Complex Structure (II) "c:\wmmorph\boisvv.dat" "c:\wmmorph\boishh.dat" "c:\wmmorph\boisvd.dat" "c:\wmmorph\boishd.dat" ( Blue) vertical and (red) horizontal variograms from steps 1 to 200, and beginings of the same curves when multiplying the x-axis by 10 Interpret J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology XXII. 26