Extreme elements and Granulometries in the estimation problem.

Ivan R. Terol Villalobos. Centro de Inv. y Desarrollo Tec. en Electroquimica. S.C.

Sanfandila Pedro Escobedo. Qureretaro Mexico.

Abstract - In this obtain size and shape information of random structures.

paper, we use the extreme elements to

I.- INTRODIJCTION. In textural analysis, Mathematical Morphology M.M. [ lluse the probabilistic models and the Granulometry to obtain size and shape information. On the other hand, fractal objets are characterized by applying morphological dilations. Here, we proppose an approach working with the extreme elements of a Granulometry and of the Eroded.

II.-MORPHC)LC)GICAL TRANSFORMATIONS In M.M.the basic morphological transfonnations, the Dilation and Erosion by B (structuring element), are given by, 6 B ( x ) = x&= U {x. : b 4 E g o ( ) = x&= n [ X b : b E B ] ; B= { -x : x E B }

v v

and the morphological closing and opening are defined by,

Definition 1 . . A Grantilometry is a family Y't, for P O , such that Y't is antiextensive, increasing for all t and for all s,DO,

The opening ym , with B a compact convex set satisfies these axioms and two functions are associated; the probability distribution function and its derivate:

q B ( X ) = E; 6B(X) YB = 8; EB(X).

Yt(YS(x>) = Ys(Yt(x)) = ys!Jp(s , t ) (m

where p is the Lebesgue measure (area in this case). For the h parameter we associate the critical element h=hn for a given set X. hn=sup{h: yhn(X) # 0). In the same way for the erosion case,hn=sup{h: ~hp, (X) # 0).

111.- GRANIJLOMETRY O F CRITICAL ELEMENTS. Let be r,(h,X) =ry(h) = X - y h ~ ( X ) the residue operator of X after application of Y ~ B and hi: the critical element of X. Invariably we use y k ~ = ykand r y ( h ~ l ) =A' for kn+l>kn. hi a

recursive way, we have ry(hn) =ry(hn+l) - ym(ry(hn+i)) and

where hk is the critical element of ry(hk+l). In other words,

We associate two functions, the probability distribution function of critical elements and its derivate:

for a give11 k, ry(hk,ry(hk+l)) =ry(hk+l) -yhk(ry(hk+l)),

h, = sup{h : YhB(ry(hl+l)) # 0 1

We define FC(h,X) = Fc(hk,X), b?. E [hk, hk+l)

gc(h,X) = d(Fc(h,X))/dh , IJsing a linear stnicturing element, we have g(h,X)-gc(h,X) and F(h,X)=Fc(h,X).

To test this approach, we realize a random geometrical characterization by using a deterministic approach. In [2] it is showed, that the deterministic Sierpinski Gasket objet S.G. has a similar behavior, in the percolation studies, than the random S.G. We use this physical assumption. In fact, the Fc and gc functions calculated on the complement of deterministic S.G. are similar than the random case. In this case we have Fcrj(hk) = 1 - (c:=k P(rx~(ry(h,+l)))ll*(M)) = ((4-P)/4lk (3) where M is the mask or the frame and P is the probability filling to create a random S.G. From (3) we obtain a family of straight lines with slope log((4-P)/4),

By calculating the slope we estimate the filling factor and the fractal dimension. We realized experiments to estimate the fractal dimension of the union of two S.G. objet with the same fractal dimension. We obtain the same fractal dimension.?his approach is now used on other fractal objets.

log (Fcri(hk)) = k log ((4 - P)/4)

1v.- DEAD LEAVES MODEL An appropriate model for grains overlaps when the contour of the grains is apparent, is the Dead Leaves Model [3] . A Dead Leaves simulation X, is constructed by implanting independent realizations of primary grains X' at random poins of a Poisson point process (density 8) using a masking law.The probability for a connected set B to be included in a grain is given by, P(B, f) = P(B c X(t ) ) = F(X' 0 B)/F(X' @ B)[ 1 - Q(B, t)] wliere Q(B, t) = exp (-Otp(X' @ B)) Let X i a random disk (radius Ri ) with fi unknow frecuency (dicrete case) for "n classes" and B(r) a ball of radius r. Then,

H(r) = x CRi,r - r l2

LOG(Q(B(r))) P(B(r),t) where

H( r)= - Q(B(~NI

H(r) can be estimated from the images. Initially, we estimate the value f, by calculating the size "r" (R"., -r=O)of the extreme element of the class 11-1 (primary grain). Next, a similar procedure is used to estimate fn., by calculating the extreme element of the classe n-2. We realize the same operation until all the f , are estimated. The number of classes (limits of application) is four or five.

[ 11 Serra J. "Mathematical Morpholgy and Image Analysis". Academic Press 1988 Vol. II. [2] Clerc et all "The electrical conductibity of binary desordered systems, percolation clusters, fractal and related models".Advances in Physics 1990,Vo1.39,No3. [3] Jeulin D., Terol I. "Application of the Dead Leaves Modelto Powders Morphological Analysis". Acta Stereo1 11, Suppl. 1,

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