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On the empty space function of some germ-grain models
Pattern Recognition 32 (1999) 1587}1600
On the empty space function of some germ-grain models G. Last*, M. Holtmann Technical University of Braunschweig, Germany Accepted for publication 25 January 1999
Abstract We derive and discuss formulas for the density and the hazard rate of the empty space function of a germ-grain model $ in 1B generated by a stationary point process ' and i.i.d. convex primary grains $ , n3-, that are independent of '. L Our formulas are based on the Palm probability of the germ process and the mean generalized curvature measure of the grain. Particular attention is paid to cluster models, where the grains form a Poisson cluster process. Our discussion of speci"c Gauss}Poisson models with spherical grains provides some motivation for the use of the failure rate of F to detect clustering e!ects. In the general case we propose a family of functions comparing the behaviour in the neighbourhood of a typical germ with the neighbourhood of an arbitrary point in space. These characteristics can be used to measure e!ects of clustering and spatial interactions between the locations of the individual grains. 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Germ-grain model; Empty space function; Spherical distance; Point process; Poisson cluster process; Hazard rate; Stochastic geometry; Simulation
1. Introduction In this paper, we deal with certain characteristics of a random germ-grain model $" 8 ($ #m )" 8 +x#m : x3$ , L L L L L L in the d-dimensional Euclidean space 1B, where the random points m , n3-, represent the locations of the germs L and the primary grains $ , n3-, are assumed to be nonL empty convex and random compact subsets of 1B (i.e. random convex bodies). We assume that the point process '" : +m : n3-, is stationary (i.e. spatially homogeneous). L That is to say that the distribution of the shifted point process '#x " : +m #x : n3-, is the same for all L x31B. We also make the simplifying assumption that ' is
* Corresponding author. Tel.: 0049-531-391-7572. E-mail address: [email protected] (G. Last)
independent of all the primary grains $ and that the L $ , n3-, are independent and identically distributed. L Our "nal, somewhat technical, assumption is that each bounded subset of 1B is hit by only a "nite number of the grains $ #m . This ensures that $ is closed, and in fact, L L $ is a random closed set in the sense of Matheron [1]. Our assumptions ensure that $ is also stationary, i.e. the distribution of $#x is the same for all x31B. Germgrain models of this form are a rather #exible class of random closed sets and may be used to describe a great variety of (random) patterns ocurring, e.g. in stochastic geometry, stereology, geology and material sciences. An important special case is the Boolean model, where the germs form a homogenous Poisson process. We refer to Stoyan et al. [2] for a detailed introduction into the subject containing a rather detailed list of references. The subject of the present paper is the empty space function (e.s.f.) F of $, de"ned by F(r)"P($5B(x, r)O), r*0, where B(x, r) is the closed ball with center x and radius r. Stationarity ensures that this does not depend on x. Hence F(r) is just the probability that $ hits the ball B(x, r). The closely related spherical contact distribution
0031-3203/99/$20.00 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 1 - 3 2 0 3 ( 9 9 ) 0 0 0 2 2 - 9
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
function H of $ is de"ned by the conditional probabilitQ ies H (r)"P($5B(x, r)O " x,$), r*0. The empty Q space function is a basic characteristic of a random closed set. It can be easily estimated from an observation of $ and can therefore be used for statistical purposes. A comparison of the e.s.f. of $ with that of a Boolean model is a common method for con"rming a Boolean hypothesis. Recently, Baddeley and Gill [3] proposed to use the density f or the hazard rate h : "f/(1!F) of F rather than F itself. If $ is a Boolean model with convex grains, then the empty space hazard function h is a polynomial. For more general models, however, exact formulas for F are rather rare. In Section 2 we recall some basic de"nitions and facts concerning germ-grain models and their empty space function. In Section 3 we will demonstrate the analytic complexity of the e.s.f. by computing it for a model with germs forming a Gauss}Poisson process and spherical grains. This is a simple example of a Boolean model with non-convex grains. Starting with a general result in [4] we derive and discuss in Section 4 a formula for the density of F which is based on the Palm probability of the germ-process and the mean generalized curvature measure of a typical grain. The formula uses a family of distribution functions which are similar to the nearestneighbour distance distribution function of a point process. Section 5 deals with Poisson cluster models. The hazard rate of the e.s.f. then takes a particularly nice form which is amenable to both computation and simulation. In particular, we simulated the hazard rate of speci"c Gauss}Poisson models with spherical grains. Our discussion shows that the hazard rate can be a useful instrument for detecting clustering e!ects. Following an idea in [5], we "nally introduce in Section 6 a family of nonparametric characteristics which compare, in a sense the behaviour of $ in `typicala boundary points of $ with the behaviour around a typical point in space. These characteristics can be used to measure e!ects of clustering and spatial interactions between the locations of the individual grains.
2. Some general facts We consider a stationary germ}grain model $" 8 ($ #m ), L L L as introduced in the previous section. Note that we do not consider a general stationary germ-grain model: "rst we have assumed that '"+m : n3-, and the primary L grains are independent and second we have assumed that the grains are convex. Formally, all random variables occurring in this paper are de"ned on a probability space (), F, P), consisting of a sample space ), a system F of
subsets of ) representing the events and a probability measure P assigning to each event A3F its probability P(A). The system F is assumed to be a p-"eld, i.e. it contains ) and is closed with respect to di!erences as well as with respect to countable unions and intersections. To de"ne a point process ' in a more formal way, we introduce the space N of all in"nite countable subsets u"+x : n3-,L1B such that each bounded set in L 1B contains only a "nite number of the points x . It is L then convenient to denote by u(B) the number of points of u in the Borel set BL1B and to de"ne N as the smallest p-"eld of subsets of N containing the sets +u : u(B)"n, for all Borel sets B and all n"0, 1,2 The point process ' is then a measurable mapping from ) to N. The theory of point processes is well developed and has been proven useful in many applications. We may again refer to Stoyan et al. [2], a book that also contains a short introduction into the basic notions of measure and probability theory. According to the de"nitions we may regard ' as a random measure assigning to each Borel set BL1B the cardinality '(B) of the set +n3- : m 3B,. In fact, we even identify ' with the ranL dom measure d L, where d is the Dirac measure V L K located at x31B. This is a simple random counting measure, where each point has at most mass 1. Therefore ' is also called a simple point process. If f : 1BP1 is a measurable function, then the integral f (x) '(dx) is just another notation for the sum f (m ). L L The intensity measure " of ' is the measure on 1B de"ned by
"(B) " : E'(B)" '(B) dP, where BL1B is a Borel set and the symbol E denotes expectation with respect to the underlying probability measure P. As said in the introduction we assume that ' is stationary, i.e. distributional invariant under translations. Hence we have for all x31B and all A3N that P('#x3A)"P('3A), where '#x"+m #x : n3N,. L A "rst basic consequence of stationarity is that " must be a multiple of Lebesgue measure: "(dx)"j HB(dx), ECP where the intensity j of ' is de"ned as the mean ECP number of points falling in a set of volume 1, i.e.
j "E'([0, 1]B)" '([0, 1]B) dP. ECP Throughout the paper we assume that j (R. The ECP most important instance of a stationary point process is a Poisson process distributing points `completely randoma in space. In this case, '(B ),2,'(B ) are stochasti I cally independent random variables, whenever B ,2, B I
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
are pairwise disjoint Borel sets. Moreover, we have jK P('(B)"m)" ECP exp[!j HB(B)], m"0, 1, 2, ECP m! for all Borel sets B, i.e. '(B) is Poisson distributed with parameter j HB(B). ECP The grains $ as well as the germ-grain model $ itself L are measurable mappings from ) into the set F of all closed subsets of 1B. Measurability refers here to the smallest p-"eld of subsets of F containing the sets +F : F5KO, for all compact sets KL1B, see [1,2]. It is convenient to denote by $ a typical grain having the distribution of $ , n*1. L Let ""x"" denote the Euclidean norm of a vector x31B and let d(x, B) " : inf+""x!y"" : y3B, denote the distance of x31B from a set BL1B. We then recall that the empty space function F of $ is given by F(r)"P(d(0, $))r), r*0, where 0 denotes the zero vector if there is no risk of ambiguity. The number p" : P(03$)"F(0) is the volume fraction of $. The Minkowski addition of two sets A, BL1B is de"ned by A B " : +x#y : x3A, y3B,. Since d(0, $))r if and only if 03$ B(r), F(r) is just the volume fraction of the stationary random closed set $ B(r), where B(r) is a shorthand notation for B(0, r). Hence F(r)"(HB(B))\EHB(($ B(r))5B), r*0, provided that the Lebesgue measure HB(B) of the Borel set B is positive. The independence from the chosen B is implied by the stationarity. The set $ B(r) is the parallel set of $ at distance e. Using geometric measure theory, Baddeley and Gill [3] showed that the empty space function of any random closed set is absolutely continuous on (0, R) with density f (r)"(HB(B))\EHB\(*($ B(r))5B),
(2.1)
where HB\ denotes (d!1)-dimensional Hausdor! measure and *A denotes the boundary of a set A. (If SL1B is a smooth (d!1)-dimensional surface, then HB\(S) is the surface content of S.) Independently, Last and Schassberger [4] derived for general stationary germ-grain models with convex grains a more speci"c formula for f that will be used later in this paper. The above interpretation of the e.s.f. as the volume fraction of a parallel set suggests rather intimate relationships to concepts and tools of convex geometry. Schneider [11] gives an extensive introduction to that subject. Here we con"ne ourselves with recalling some simple notions. Some slightly more advanced tools will
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be used in Section 4. For any compact and convex set KL1B we let < (K), i"0,2, d, denote the ith intrinsic G volume of K. We have < (K)"1, while < (K) is the B volume (i.e. the Lebesgue measure) of K. If the dimension of the a$ne hull of K does not equal d!1, then 2< (K) is the (d!1)-dimensional surface area of K. B\ (Otherwise this is true without the factor 2.) Let b " : < (B(0, 1)) denote the volume of the unit ball in 1B. B B Then 2(d!1)b /b < (K) is the mean width of K. The B\ B properties of the intrinsic volumes which are most important for our purposes are based on Minkowski addition. The set K B(e) is the outer parallel body of K at distance e and we have the following Steiner formula: B < (K B(e))" b eB\G< (B), (2.2) B B\G G G where b " : 1. For the typical grain we now make the natural assumption that the means
(2.3)
are "nite. The number jM " : j
(2.4)
!ln(1!¹ (K))"j EHB($ K[ ), (2.5) ECP where K[ " : +!x : x3K,. This, in fact, holds for general compact primary grains satisfying EHB($ B(r))(R, r*0. (2.6) The latter condition guarantees that each compact set is indeed hit by only a "nite number of the grains, see Heinrich [8]. In fact, (2.6) is even necessary for this property. Under our assumption of convexity, one can use the Steiner formula for the volume of $ B(r), to derive from (2.5) the following well-known formula for the e.s.f. of a Boolean model:
B rB\Gb
3. Poisson cluster process models In this section we deal with the special case, where the germs form a Poisson cluster process. Hence '" 8 Z #x, V VZN?P
(3.1)
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
where ' is a Poisson process (of parents) with positive N?P and "nite intensity j and the family +Z : x3' , N?P V N?P consists of independent random point processes on 1B having all the same distribution. We let Z denote a typical cluster, i.e. a point process with the same distribution as the Z , x3' , and assume that the mean V N?P number j of cluster points AJ
j " : EZ(1B)" Z(1B) dP, AJ is "nite and positive. The intensity j of ' is then given ECP by j j . The generating functional G of ' (see [2]) is N?P AJ given by the formula G ( f )"G (fH), N?P where f : 1BP[0, 1] is measurable, (1!f ) is non-negative and 0 outside a bounded set, and the function f H is de"ned by
f H(x) " : E f (y#x) WZ8 (this is the generating functional at f of the cluster Z#x), and G denotes the generating functional of the Poisson N?P process ' , i.e. N?P
and all random elements are independent. Then we have ln G ( f )"j N?P
Example 3.1. If ' is a Neyman}Scott process, then the > all have the same distribution <, say, and (3.5) G L implies
ln G ( f )"j N?P
> "8 $ #x, (3.3) V VZ8 where the $ , x3Z, are independent and distributed like V $ . Hence one can use (2.5) with $ replaced by > to obtain an expression for F. Rataj and Saxl [9] used this formula to derive expressions for the e.s.f. of the point process '. (Note that ' can be interpreted as a germgrain model with $ "+0,.) In view of (2.6) we have to ensure that EHB(> B(r))(R, r*0, which, in fact, is an easy consequence of (2.6) and j (R. In the following we consider the special case AJ if g"0, Z" (3.4) +> : i"1,2, n, if g"n*1, G L where the random cluster size g is a random element of 9 ,> , n3N, i"1,2, n, are random elements of 1B, > G L
g
f(x#y)<(dy) !1 dx,
(3.6)
where g is the generating function of g. Example 3.2. Assume for all n3- that > "0 and for L n*2 that > ,2,> have all the same distribution < , L L L L say. In that case,
ln G ( f )"j N?P
(3.2) F(r)"1!G (1!¹ (B(r)! ) )), where ¹ is the capacity functional of the typical grain, de"ned by (2.4) with $ replaced by $ . We can interpret $ as a Boolean model where the grain itself is a ("nite) germ-grain model
where p " : P(g"n). L
p !1
# p f (x) L L
G ( f )"exp j ( f (x)!1) dx , N?P N?P where dx always refers to integration with respect to Lebesgue measure. From the de"nition of F we obtain rather directly (see e.g. [8]), that
L p !1# p % Ef(x#> ) dx, L G L L G (3.5)
f(x#y)< (dy) L
L\
dx. (3.7)
If p "0, 1!p "p "p, for 0)p)1,> "> "0, then ' is a Gauss}Poisson process. In this case a cluster Z#x associated with a parent point x say, contains x and, with probability 1!p, also a second point x#> . The random point > is called secondary point. Based on (3.2) and (3.7) we next compute F for a special Gauss}Poisson process with spherical grains. For AL1B we denote by x C R+x3A, the indicator function of A, which is 1 for x3A and 0, otherwise. A similar notation will be used for other spaces as well. Theorem 3.3. Assume that ' is a Gauss}Poisson process as described in Example 3.2, where the secondary point > is uniformly distributed in B(e) for some e'0. Assume further that the primary grain $ is a ball B(R ), with the dis tribution of the random radius R being exponential with parameter k'0. Then the empty space function F of the germ-grain model $ satisxes !ln(1!F(r))
B d "j b m!rB\Kk\K!j pb\e\B ECP B N?P B m K
; 1+""x""'r,exp(!k(""x""!r))a(x, r) dx
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
where the radius R and the secondary point > are inde pendent. Since B(R )5B(!x, r)O), i! x3B(R #r), the right-hand side of (3.10) equals
!j pb\eB 1+""x"")r,a(x, r) dx N?P B
!j pb\e\B 1+""x""'r,exp(!k(""x""!r)) N?P B
B d j b E(R #r)B"j b m!rB\Kk\K, N?P B N?P B m K
;HB(B(e)5B(x, r)) dx
where we used the moments of the exponential distribution. Similarly, (3.11) equals
!j pb\e\B 1+""x""*r,HB(B(e)5B(x, r)) dx, N?P B (3.8)
where
a(x, r) " : 1+""y!x"")e ,""y""'r,exp(!k(""y""!r)) dy. If d"3 then, in particular,
4n 6 6 3 3 3 "j #r #r #r #r !j p ECP 3 k N?P 4ne k k k 3 ; 1+""x""'r,(exp(!k(""x""!r))a(x, r) dx!j p N?P 4ne
Using the distributions of R and >, and distinguishing the four possible cases for the values of (x, >), we obtain that (3.12) equals
(3.9)
where e "(1, 0, 0) is a specixc unit vector in 1.
#p f (x)Ef (x#>), dx, where > is the secondary point. By (3.2) and some simple algebra,
!ln(1!F(r))"j P(B(R )5B(!x, r)O) dx (3.10) N?P (3.11)
;exp(!k(""y""!r)) dx dy
(3.14)
#j pb\e\B N?P B
1+""x""'r, ""y"")e, ""y!x"")r,
;exp(!k(""x""!r)) dx dy
(3.15)
1+""x"")r, ""y"")e, ""y!x"")r,
;dx dy.
(3.16)
3 P>C j p u . exp(!k(. !r))H(B(0, e)B(. e , r)) d. , N?P e P (3.17)
3 P j p . H(B(0, e)5B(. e , r)) d. . N?P e
(3.18)
This proves the second statement of the theorem. 䊐 Heinrich and Muche [10] give a formula to calculate the volume
!j p P(B(R )5B(!x, r)O)P(B(R ) N?P 5B(!x!>, r)O) dx,
1+""x"")r, ""y!x"",)e, ""y""'r,
Putting things together and noting that j "j (1#p), ECP N?P we obtain the "rst assertion. We now assume that d"3. It is convenient to introduce spherical coordinates. Then (3.15) and (3.16) transform to
ln G ( f )"j +!1#(1!p ) f (x) N?P
;exp(!k(""x""!r))exp(!k(""y""!r)) dx dy (3.13)
#j pb\e\B N?P B
Proof. According to (3.5), the generating functional of ' is given as
#j p P(B(R )5B(!x!>, r)O) dx N?P
1+""x""'r, ""y!x"")e, ""y""'r,
P>C . exp(!k(. !r))H(B(0, e)5B(. e , r)) d. P
3 P !j p . H(B(0, e)5B(. e , r)) d., N?P e
"j pb\e\B N?P B
#j pb\e\B N?P B
3 ; 1+""x"")r,a(x, r) dx!j p N?P e ;
B d j pb m!rB\Kk\K. N?P B m K
j p P(R '""x""!r)P(R '""x#>""!r) dx N?P
!ln(1!F(r))
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(3.12)
H(. , e, r) " : H(B(0, e)5B(. e , r))
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
. q . q(q!e) ; . q! ! dq d. 2 2
of the intersection of two balls in 1B ocurring in (3.17) and (3.18). We have to distinguish three cases:
. )e#r and !. )e!r). : H(B(0, e)5B(. e , r)) n 6 " 16(e#r)!12. (e#r)! (e!r)#2. , 24 .
. )e#r and . )r!e:
H(. , e, r)"H(B(0, e)),
. )e#r and . )e!r:
H(. , e, r)"H(B(. e , r)).
The intersection is empty if . 'e#r. In order to calculate the integrals (3.13), (3.14), (3.17) and (3.18) we have to distinguish three cases for the values of r, e, and . . The right column below speci"es the corresponding intersection volumes, taking into account the relative size of e!r to r.
(3.19)
exp(!k(. !r)) C>P . >C . q . q(q!e) ; dq d. exp(!k(q!r)) . q! ! 2 2 . \C (3.20) #K8n
H(. , e, r)"H(B(. e ,r)), e 0). )e!r, case 1 0) r ( 2 e!r(. )e#r, formula in (3.19),
0). )e!r; e )r(e 2 e!r(. )e#r;
case 2
case 3 r*e
H(. , e, r)"H(B(. e , r)), , formula in (3.19).
0). )r!e;
H(. , e, r)"H(B(0, e)),
r!e(. )e#r;
formula in (3.19).
We will consider the "rst case only, as the others are treated analogously. With K " : j p(3/4ne) expression ECP (3.13) transforms to
;
P
;
C\P exp(!k(. !r)). P
C\.
qexp(!k(q!r)) dq d. #K8n
. C\P >C exp(!k(. !r)) exp(!k(q!r)) P C\.
. q . q(q!e) ; . q! ! dq d. 2 2
#K8n
;
.
P
C>P exp(!k(. !r)) C\P
>C exp(!k(q!r))
P
.
C\.
qexp(!k(q!r)) dq d. P P C>. #K8n exp (!k (q!r)) C\. . q . q(q!e) ; . q! ! dq d. . 2 2
K16n
K16n
With analogous treatment, (3.14) becomes
(3.21)
Finally, (3.17) and (3.18) become
C\P r . exp(!k(. !r))4n d. 3 P C>P #K4n . exp(!k(q!r))H(. , e, r) d. C\P and
K4n
P 4nr 16nr . d. "K , 3 9 respectively. K4n
(3.22)
(3.23)
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
Remark 3.4. Heinrich and Muche [10] have noticed that the intersection volume of two balls in 1B is only for odd dimensions a rational function of the involved variables. Further integration steps (e.g. in (3.17) and (3.18)) can therefore be performed explicitly in this case. For even dimensions the expression for the intersection volume contains arcus functions of rational expressions. Exact calculation seems to be di$cult if not impossible. So numerical evaluation is adequate or even needed in these cases. 4. A representation involving curvature measures In this section we return to the general setting of Section 2. Our aim is to derive and discuss a formula for the density f of the empty space function F. To prepare the formula we "rst have to introduce some general characteristics of the typical grain $ and of the point process ' of germs. The characteristics of $ are again based on inte gralgeometric notions, see [6,11]. Let KL1B be a nonempty, compact and convex set. For all x31B there exists a unique point p(K, x) in K nearest to x and we set d(K, x) " : ""x!p(K, x)"". For x, K we let u(K, x) " : (x!p(K, x))/d(K, x) and we give u (K, x) some "xed value in the (d!1)-sphere SB\ " : +x31B : ""x"""1, if x3K. There exist "nite measures C (K, ) ),2,C (K, ) ) B\ on 1B;SB\ satisfying, for all e'0 and all Borel sets BL1B;SB\, the local Steiner formula HB(+x31B : 0(d(K, x))e,(p(K, x), u(K, x))3B,) B\ " eB\Gb C (K, B). (4.1) B\G G G These generalized curvature measures of K are uniquely determined by (4.1). The measures C (K, ) ;SB\), G i"0,2, d!1, are the curvature measures of K and are concentrated on the boundary *K of K. In contrast to the intrinsic volumes determined by (2.2), the de"nition of the generalized curvature measures is based on the local outer parallel sets +x31B : 0(d(K, x))e,(p(K, x), u(K, x))3B, taking into account the location p(K, x) of the nearest point as well as the direction of the contact vector x!p(K, x). Of course we have < (K)"C (K, 1B;SB\), G G i"0,2, d!1. It is easy to prove that the normalized contact vector u(K, x) is for x , K the outer normal of a support hyperplane of K at the point x. Since we have assumed that the mean intrinsic volumes de"ned in (2.3) are "nite, the mean generalized curvature measures of a typical grain, de"ned by
CM ( ) ) " : C ( $ , ) ) dP, G G
i"1,2, d!1,
are "nite measures on 1B;SB\.
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The other important concept we have to introduce is the Palm probability P of P with respect to ' (see [2]). This is a probability measure on (X, F) satisfying P (03')"1. We can interpret P (A) as the conditional probability of the event A3F given that 0 is a `randomly chosen pointa of '. Sometimes this is symbolically written as P ( ) )"P( ) "03'). Let us de"ne $(x) " : $ if L x"m for some n and $(x) " : , otherwise. If P is the L ruling probability measure, then +(x, $(x)) : x3',is again an independently marked point process, i.e. the grains +$(x) : x3', are independent, independent of ' and have the same distribution as $ . If
G (r) " :
(4.2)
where $ " : 8 ($(x)#x), VZ!+, is the union of all grains except the grain of the typical germ 0. For convenience, we set G ,1 for
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where $L " : 8 ($ #m ). K K K$L Our aim is to apply Theorem 2.1 in [4]. To do so we recall that the skeleton $H of 1B!$ is de"ned as the set of all points x31B!$ having the same distance from at least two di!erent points of $. For (x, w)3N($) we denote the distance from x to $H in direction w by d(x, w) " : inf+r'0 : x#rw3$H,, where inf " : R. For (x, w) , N($) we set d(x, w) " : 0. By virtue of Theorem 2.1 and Remark 2.2 in [4] we have B\ f(r)" (d!i)b rB\G\ B\G G
E
g(y)1+B(y#rw, r)5$L
(4.5)
where we have used an invariance property of the generalized curvature measure C to obtain the last equality G (see [11]). To continue the above computation it is convenient to introduce the following notation. For any locally "nite and simple counting measure t" d L L V on 1B, we let C(t, ) ) be the distribution of the germ-grain model 6 ($ #x ). In fact, we consider only t with the L L L property
P(K5($ #x)O)t(dx)(R, KL1B compact,
see [8]. Using the kernel ! and the independence properties of our germ-grain model, we can rewrite (4.5) as
E L
g(y#m )1+B(y#m #rw, r)5A", L L
!('!d ,dA)CM (d(y, w)) KL G
g(y#x)1+B(y#rw, r) A",
!('!d , dA) CM (d(y, w))dx G
1+B(y#rw, r)5A",
!('!d , dA)CM (d(y, w)) . G From the properties of the Palm probability P it easily follows that
1+B(y#rw, r)5A",!('!d , dA)
Remark 4.2. We have assumed that the reduced second moment measure of ' is absolutely continuous w.r.t. Lebesgue measure. The above proof shows that Theorem 4.1 remains valid under the more general assumption (4.4).
g(y#m )1+B(y#m #rw, r)5$L L L
",C ($ , d(y, w)) , G L
j E ECP
"P (B(y#rw, r)5$") proving the result. 䊐
",C ($ #m , d(y, w)) G L L " E L
Making use of the re"ned Campbell theorem (see [2]) as well as of Fubini's theorem we obtain that the latter sum equals
E
g(y)1+d(y, w)'r,C ($, d(y, w)) G
" E L
!(('!m )!d , dA)CM (d(y, w)) . L G
g(y)1+d(y, w)'r,C ($, d (y, w)) , G
g(y#m )1+B(y#rw, r)5A", L
"j E ECP
where g : 1BP[0,R) is a measurable function satisfying g(x)dx"1. Let n3-. If (y, w) is a supporting element of $ #m , then d(y, w)'r and y , $Li! B(y#rw, r)5 L L $L". Hence we obtain from (4.4) that E
" E L
Remark 4.3. The function G can be interpreted as the G distribution function of a random variable m which can be constructed as follows. First one selects a point > of ' at random. Then one samples a random element (X, =) according to the distribution is not G G covered by 6 !+ ,($(x)#x), then m is the distance from VZ 7 >#X to the skeleton $H in direction =. Otherwise m"0. The distribution functions G are similar to the nearG est-neighbour distance distribution function of a point process: Example 4.4. Assume that $ "+0,, i.e. that '"$ is L a point process. Then
G (r)" P (B(rw, r)5$O);B\(dw), where ;B\ is the uniform distribution on the sphere SB\.
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
If $ is a Boolean model, i.e. if ' is a Poisson process, then Slivnyak's theorem (see e.g. [2]) implies that P (B(y#rw, r)5$")"1!F(r), (4.6) and the density of the empty space function becomes B\ f (r)" (d!i)b j
P($3 ) )"E
1+A3 ) ,!(', dA) , KL1B compact.
A similar formula holds for P . The kernel ! is com pletely determined by the distribution of $ and has the property
1+A6A 3 ) ,!(t, dA)!(t , dA ),
1+'#t o ) ,Q(dt) , 8
(4.9)
where
Q( ) ) " : j\E 8 AJ
1+Z!x3 ) ,Z(dx)
is the Palm distribution of the typical cluster Z. This shows in particular that a su$cient condition for the absolute continuity of the reduced second moment measure of ' is that ' is given as in (3.4) with the > having an absolutely continuous distribution for G L n*2 and i*2. Using (4.9) together with (4.8) gives P (B(y#rw, r)5$")
"E
1+B(y#rw, r)5A",
1!G8(r)"
1+B(y#rw, r)5A",
;!(t!d , dA)Q(dt)CM (d(y, w)). (4.11) 8 G Here and below we use the convention 0/0 " : 0. This is the de"nition of the function G with P replaced by Q. G 8 If, in particular, ' is a Poisson process, then G "F for G
f ('!d , x)'(dx) "j E V ECP
f(', x)j(', x)dx
for all measurable f : N;1BP1 . We refer to [13,14] > for the meaning of this condition. The point process ' is then called a Gibbs process with local energy function !log j(', x). For our purposes it is important to note that j E f ('!d )"E j (', 0) f (') ECP holds for all measurable functions f : NP1. Applying this to the function mapping ' to 1+B(y#rw, r)5$", we obtain that
1!G (r)"jM \ E[1+B(y#rw, r)5$", G G ;j(', 0)]CM (d(y, w)). G 5. The empty space hazard of Poisson cluster models The hazard rate of the empty space function is the function h de"ned by f (r) h(r) " : , r*0. 1!F(r)
;!('#t!d , dA)Q(dt) 8 "E
1!G (r)"(1!F(r))(1!G8(r)), (4.10) G G where the function G8 is de"ned via the equation G
E
Example 4.5. Assume that ' is a Poisson cluster process as described in the previous section. A fundamental formula for the Palm probability P (see e.g. [2]) says that
;!(t!d , dA )Q(dt). 8 Hence
(4.8)
provided that t and t have disjoint supports.
P ( ' o ) )"E
1+B(y#rw, r)5A ",
"(1!F(r))
Example 4.6. Assume that ' admits a Papangelou conditional intensity j : N;1BP1 , where N has been de> "ned at the beginning of Section 2. This is to say that
1+A3 ) ,!(t#t , dA)
"
1595
1+B(y#rw, r)5A",1+B(y#rw, r)5A
",!(', dA)!(t!d , d A )Q(dt) 8
This is also the hazard rate of the spherical contact distribution function. Following Baddeley and Gill [3] we might call it the empty space hazard (rate) of $. The hazard rate h of a distribution function F of a nonnegative random variable X is a common tool in reliability theory. Quite often X describes the lifetime of a
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
technical system such that, heuristically speaking, h(t)"P(X3(t, t#Dt]"X*t) is the conditional probability that the system fails in a small interval (t, t#Dt] given that the system has survived time t. In our context we might also introduce a random variable with distribution function F by assuming that the point 0 starts growing in all directions at unit speed. The space which is covered at time t is just the ball B(t). Taking X as the time it takes the ball to hit $, we obtain a random variable with distribution function F and hazard rate h. Given h, the empty space function can be obtained from the formula R 1!F(t)"(1!p)exp ! h(s) ds , t*0. If $ is a Boolean model, then h is a polynomial
B\ h(r)" (d!i)b j rB\G\. (5.1) B\G G G It is important to note that this simple formula is determined by the intensity j and the mean intrinsic volECP umes
h(r)"db j rB\ B ECP
1+(t!d )(B(rw, r))"0,
;Q(dt);\(dw). 8 If, for instance, ' is a Neyman}Scott process as in Example 3.1, then
Example 5.2. Assume that $ "B(R), where the random radius R has distribution M I and a "nite dth moment. Using the speci"c form of the generalized curvature measures of a ball, we obtain
uG1+B((u#r)w, r)5A",
1!G8(r)"m\ G G
;!(t!d , dA)Q(dt);B\(dw)M I (du), 8 where m is the ith moment of R. Due to our convention G 0/0 " : 0 this is also true for ER"0 in which case we are back in Example 5.1. Assume now, for instance, that ' is a Gauss}Poisson process (see Example 3.2) and let > be independent of R with the distribution of the secondary point. Assume also that > and !> have the same distribution. We have j "1#p and the de"nition of AJ Q easily implies that 8
1!GG (r)"m\ 8 G
(1#p)\
uG(1!p#2pc(u, w, r))
;;B\(dw)M I (du),
(5.3)
where c(u, w, r) " : P(B((u#r)w, r)5B(>, R)"). If ' is a Neyman}Scott process, then a similar calculation as in the preceding Example 5.1 shows that
1!G8(r)"m\ uG p # g (P(B(>I , R)5B((u#r)w, r) G G "));B\(dw) M I (du), where >I is independent of R and distributed like > !> . Our next example is Neyman}Scott processes of germs with general convex grains. Example 5.3. Let ' be a Neyman}Scott process. Similarly as above,
1+B(y#rw, r)5A",!(t!d , dA)Q(dt) 8
j 1+(t!d )(B(rw, r))"0,Q(dt) AJ 8 L "p # p P(+> !> : lOk,5B(rw, r)") L J L I L L I "p #g (1! !> and g is the derivative of the generating function of g. Hence
One natural generalization of the preceding example are spherical grains:
h(r)"db j rB\ p # g (1!
"j\[p #g (1!¹ (B(y#rw, r))]. AJ Y Here ¹ denotes the capacity functional of the random Y closed set $ " : $ #>I , where $ and >I are indepen dent and >I is distributed like > !> . Consequently, B\ h(r)" (d!i)b j rB\G\ B\G N?P G
; [p #g (1!¹ (B(y#rw, r))]CM (d(y, w)). G
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
Gauss}Poisson processes admit an interesting property of stochastic monotonicity: Theorem 5.4. Fix a number j'0 and let 'N denote a Gauss}Poisson process as in Example 3.2 with intensity j "j and probability p for having a secondary point. ECP Assume that the secondary point > " : > has the same distribution as !>. ¸et hN(HN) denote the hazard rate Q (spherical contact distribution function) of a germ-grain model with germ process 'N and typical grain $ . Then hN(r) is for all r*0 decreasing in p. In particular, 1!HN(r) is increasing in p. Q Proof. A straightforward generalization of (5.3) is 1!G8(r)"
; (1!p#2pa(y, w, r))CM (d(y, w)), G where a (y, w, r) " : P (B (y#rw, r)5($ #>)"), and $ and > are independent. Function a(y, w, r) and the quermass densities jM are independent of p. By (5.2) it G su$ces to show that 1!p#2pa 1!p #2p a * , 1#p 1#p provided that p)p and 0)a)1. Simple algebra shows that this is correct. The second assertion follows from the equality
1!HN(t)"exp ! Q
R
hN(u)du ,
t*0.
䊐
Remark 5.5. The second statement of Theorem 5.4 says that HN is stochastically increasing in p. The parameter Q p is a measure for the clustering in 'N. Hence, an increase of clustering leads to an stochastically increasing spherical distance. This is in perfect coincidence with intuition. If $ "+0,, then HNis just the spherical con Q tact distribution function of the point process 'N itself. In many cases of interest there is no closed simple expression for the functions G8. Instead we propose to G use Monte}Carlo integration. To be more speci"c we assume that the typical cluster Z is given as in (3.4). Then we have G8(r)"j\E G AJ
E E 1+B(>#r=, r) 8 ($ I K I I$K
#> !> )O, , I L K L
(5.4)
1597
where (>, =) is a random element with distribution
1598
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
Fig. 3. In#uence of e on the hazard rate of a cluster model with one secondary point ( k"10, p"0.8)
Fig. 1. Comparision of the empty space function and the corresponding hazard rate for various cluster models (one secondary point, k"10, e"0.5)
Fig. 4. Hazard rates of a cluster model and of ann adapted Boolean Model.
Fig. 2. In#uence of e on the empty space function of a cluster model with one secondary point ( k"10, p"0.8).
Fig. 5. Hazard rates of various cluster models with two secondary points (k"10, p "0.2, p "0.8).
3 points. The cluster process is now a generalized Gauss}Poisson process as described in Example 3.2 with p "0, p "0, n*4, where the points > , > , > L are all uniformly distributed in the ball B(e) for some
e'0. The primary grains are again balls, where the radius follows an exponential distribution with parameter k'0. For p "0 we obtain the model discussed above. Fig. 5 shows the empty space hazard rates for
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
k"10, p "0.2, p "0.8 and two values of e. For e"0.5 the rate very much resembles that of a Boolean model, i.e. it is approximately a quadratic function of r. In this case e is apparently small enough to ensure that the typical grain Z (see (3.3)) can be approximated by a convex grain without neglecting too much empty space. For e"2 the situation changes drastically. The empty space `between the grainsa of a cluster cannot be neglected anymore and a quadratic function cannot satisfactorily be "tted to the rate. The ordering between the rates of the two cluster models on the interval [0,2.1] can be interpreted as above. It is interesting to note that, for larger values of r, the rate for e"2 behaves concave. Summarizing we can say that the empty space hazard rate might be an appropriate tool to reveal clustering phenomena and to explain distinctions between di!erent cluster models.
6. Measures of spatial interaction We consider the general model of Section 2, assume that (4.4) is satis"ed, and de"ne 1!G (r) G , i"1,2, d!1, J (r) " : G 1!F(r)
(6.1)
for all r*0 with F(r)(1. The idea of this de"nition is taken from van Lieshout and Baddeley [5], who de"ned their J-function for point processes using the nearestneighbour distance distribution function G instead of our G . The functions J compare the behaviour in the neighG G bourhood of a typical point of the germ process with the behaviour in the neighbourhood of a typical point in space in order to detect possible interactions and clustering e!ects. Our de"nition is also motivated by the discussion in Section 5 which gave some evidence that distinctions between the empty space hazard rates can be explained by the structure of clusters. If ' is a Poisson process, then J (r)"1 for
(6.2)
Hence the functions J are non-parametric measures for G the distance between the germ-grain model and a Boolean model with the same quermass densities. For statistical purposes it might be more convenient to consider jM J instead of J . In the following we show that our G G G functions J share many properties with the J-function of G van Lieshout and Baddeley [5].
Example 6.2. If ' is a Gibbsian point process as in Example 4.6, then
J (r)"jM \ E[j(',0)"B(y#rw, r)5$"]CM (d(y, w)). G G G Theorem 6.3. Assume that $"$6$ is the superposition of two germ-grain models $I" ($ #m ), L I L I L satisfying our assumptions and such that the marked point processes +(m , $ ) : n*1,, k"1, 2, are independent. L I L I For k"1, 2, let j denote the intensity of I ' " : +m : n*1, and J the J -function of $I. Then, I L I G I G for i"1,2, d!1, j j J (r)# J (r), J (r)" G G j #j j #j G provided that
"E
1+B(y#rw, r)5(A 6A )",
;!(' !d , dA )!(' , dA ) "(1!F (r))E
1+B(y#rw, r)5A",
;!(' !d , dA) "(1!F (r)) ;P (B(y#rw, r)5$"), where F is the empty space function of $I. The remainI der of the proof is the same as in the proof of Theorem 2 in [5]. ) We conclude the section by indicating how to estimate the functions jM (1!G ) and jM J using an observation of G G G G $. Because the point process ' is not directly observable we cannot use the de"nition. However, as outlined in [4] it follows for any measurable set BL1B of positive volume that jM (1!G (r)) G G
"(HB(B))\E Example 6.1. If ' is a Poisson cluster process as in Example 4.5, then J (t)"1!G8(t). G G
1599
1+x3B, d(x, w)'r,C ($, d(x, w)) , G (6.3)
1600
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
where d(x, w) has been de"ned in the proof of Theorem 4.1. Ignoring censoring e!ects due to bounded observation windows, this can be used in an obvious way to construct estimators for jM (1!G ) and jM J . We "nally G G G G note that j> " : jM (1!G (0)) G G G
"(HB(B))\E
1+x3B, (x, w)3N($),C ($, d(x, w)) G
is the intensity of the curvature measure C ($, ) ) reG stricted to the generalized normal bundle N($) de"ned in the proof of Theorem 4.1. Hug and Last [15] have recently shown that this restriction coincides with the non-negative curvature measure introduced by Matheron [1] and Schneider [6].
Acknowledgements The authors thank an anonymous referee for his very careful reading of this paper which has led to a considerable improvement of the exposition.
References [1] G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1975. [2] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications, second ed., Wiley, Chichester, 1995. [3] A. Baddeley, R.D. Gill, The empty space hazard of spatial pattern, preprint 845, Department of Mathematics, University Utrecht, 1994.
[4] G. Last, R. Schassberger, On the distribution of the spherical contact vector of stationary grain models, Adv. Appl. Proba. 30 (1998) 36}52. [5] M.N.M. van Lieshout, A. Baddeley, A nonparametric measure of spatial interaction in point patterns, Statist. Neerland. 50 (1996) 344}361. [6] R. Schneider, Parallelmengen mit Vielfachheit und Steiner-Formeln, Geom. Dedicata 9 (1980) 111}127. [7] W. Weil, J.A. Wieacker, Stochastic geometry, in: Handbook of Convex Geometry, North-Holland, Amsterdam, 1993, pp. 1391}1438. [8] L. Heinrich, On existence and mixing properties of germgrain models, Statistics 23 (1992) 271}286. [9] J. Rataj, I. Saxl, Boolean cluster models: mean cluster dilations and spherical distances, Math. Bohemica 122 (1995) 21}36. [10] L. Heinrich, L. Muche, On the pair correlation function of the point process of nodes in a Voronoi tesselation, preprint 94-07, TU Bergakademie Freiberg, 1994. [11] R. Schneider, Convex Bodies : the Brunn-Minkowski Theory, in: Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge, 1993. [12] L. Heinrich, I.S. Molchanov, Central limit theorem for a class of random measures associated with germ-grain models, CWI Report BS-R9518; Adv. Appl. Probab., 1999, in press. [13] X.X. Nguyen, H. Zessin, Integral and di!erential characterizations of the Gibbs process, Mathematische Nachrichten 88 (1979) 105}115. [14] O. Kallenberg, Random Measures, Akademie}Verlag, Berlin and Academic Press, London, 1983. [15] D. Hug, G. Last, On support measures in Minkowski spaces and contact distributions in stochastic geometry, preprint, UniversitaK t Freiburg, 1999, submitted for publication.
About the Author*GUG NTER LAST received the diploma degree in mathematics from Humboldt-University in Berlin in 1984 and Ph.D. also from the same university in 1986. He worked there as scienti"c assistant till 1992. Then he moved to Technical University in Braunschweig, where he received Dr. Sc. degree in 1995. His reserach interests include random point processes and measures and their application in reliability theory, queueing and stochastic geometry. About the Author*MARKUS HOLTMANN studied mathematics at the University of Braunschweig where he received his diploma in 1997. He is currently a Ph.D. student at the Freiberg University of Mining and Technology in the graduate school `Spatial Statisticsa. The main topic in his thesis covers aspects of experimental design of stochastic "elds involving derivative information.
On the empty space function of some germ-grain models G. Last*, M. Holtmann Technical University of Braunschweig, Germany Accepted for publication 25 January 1999
Abstract We derive and discuss formulas for the density and the hazard rate of the empty space function of a germ-grain model $ in 1B generated by a stationary point process ' and i.i.d. convex primary grains $ , n3-, that are independent of '. L Our formulas are based on the Palm probability of the germ process and the mean generalized curvature measure of the grain. Particular attention is paid to cluster models, where the grains form a Poisson cluster process. Our discussion of speci"c Gauss}Poisson models with spherical grains provides some motivation for the use of the failure rate of F to detect clustering e!ects. In the general case we propose a family of functions comparing the behaviour in the neighbourhood of a typical germ with the neighbourhood of an arbitrary point in space. These characteristics can be used to measure e!ects of clustering and spatial interactions between the locations of the individual grains. 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Germ-grain model; Empty space function; Spherical distance; Point process; Poisson cluster process; Hazard rate; Stochastic geometry; Simulation
1. Introduction In this paper, we deal with certain characteristics of a random germ-grain model $" 8 ($ #m )" 8 +x#m : x3$ , L L L L L L in the d-dimensional Euclidean space 1B, where the random points m , n3-, represent the locations of the germs L and the primary grains $ , n3-, are assumed to be nonL empty convex and random compact subsets of 1B (i.e. random convex bodies). We assume that the point process '" : +m : n3-, is stationary (i.e. spatially homogeneous). L That is to say that the distribution of the shifted point process '#x " : +m #x : n3-, is the same for all L x31B. We also make the simplifying assumption that ' is
* Corresponding author. Tel.: 0049-531-391-7572. E-mail address: [email protected] (G. Last)
independent of all the primary grains $ and that the L $ , n3-, are independent and identically distributed. L Our "nal, somewhat technical, assumption is that each bounded subset of 1B is hit by only a "nite number of the grains $ #m . This ensures that $ is closed, and in fact, L L $ is a random closed set in the sense of Matheron [1]. Our assumptions ensure that $ is also stationary, i.e. the distribution of $#x is the same for all x31B. Germgrain models of this form are a rather #exible class of random closed sets and may be used to describe a great variety of (random) patterns ocurring, e.g. in stochastic geometry, stereology, geology and material sciences. An important special case is the Boolean model, where the germs form a homogenous Poisson process. We refer to Stoyan et al. [2] for a detailed introduction into the subject containing a rather detailed list of references. The subject of the present paper is the empty space function (e.s.f.) F of $, de"ned by F(r)"P($5B(x, r)O), r*0, where B(x, r) is the closed ball with center x and radius r. Stationarity ensures that this does not depend on x. Hence F(r) is just the probability that $ hits the ball B(x, r). The closely related spherical contact distribution
0031-3203/99/$20.00 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 1 - 3 2 0 3 ( 9 9 ) 0 0 0 2 2 - 9
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
function H of $ is de"ned by the conditional probabilitQ ies H (r)"P($5B(x, r)O " x,$), r*0. The empty Q space function is a basic characteristic of a random closed set. It can be easily estimated from an observation of $ and can therefore be used for statistical purposes. A comparison of the e.s.f. of $ with that of a Boolean model is a common method for con"rming a Boolean hypothesis. Recently, Baddeley and Gill [3] proposed to use the density f or the hazard rate h : "f/(1!F) of F rather than F itself. If $ is a Boolean model with convex grains, then the empty space hazard function h is a polynomial. For more general models, however, exact formulas for F are rather rare. In Section 2 we recall some basic de"nitions and facts concerning germ-grain models and their empty space function. In Section 3 we will demonstrate the analytic complexity of the e.s.f. by computing it for a model with germs forming a Gauss}Poisson process and spherical grains. This is a simple example of a Boolean model with non-convex grains. Starting with a general result in [4] we derive and discuss in Section 4 a formula for the density of F which is based on the Palm probability of the germ-process and the mean generalized curvature measure of a typical grain. The formula uses a family of distribution functions which are similar to the nearestneighbour distance distribution function of a point process. Section 5 deals with Poisson cluster models. The hazard rate of the e.s.f. then takes a particularly nice form which is amenable to both computation and simulation. In particular, we simulated the hazard rate of speci"c Gauss}Poisson models with spherical grains. Our discussion shows that the hazard rate can be a useful instrument for detecting clustering e!ects. Following an idea in [5], we "nally introduce in Section 6 a family of nonparametric characteristics which compare, in a sense the behaviour of $ in `typicala boundary points of $ with the behaviour around a typical point in space. These characteristics can be used to measure e!ects of clustering and spatial interactions between the locations of the individual grains.
2. Some general facts We consider a stationary germ}grain model $" 8 ($ #m ), L L L as introduced in the previous section. Note that we do not consider a general stationary germ-grain model: "rst we have assumed that '"+m : n3-, and the primary L grains are independent and second we have assumed that the grains are convex. Formally, all random variables occurring in this paper are de"ned on a probability space (), F, P), consisting of a sample space ), a system F of
subsets of ) representing the events and a probability measure P assigning to each event A3F its probability P(A). The system F is assumed to be a p-"eld, i.e. it contains ) and is closed with respect to di!erences as well as with respect to countable unions and intersections. To de"ne a point process ' in a more formal way, we introduce the space N of all in"nite countable subsets u"+x : n3-,L1B such that each bounded set in L 1B contains only a "nite number of the points x . It is L then convenient to denote by u(B) the number of points of u in the Borel set BL1B and to de"ne N as the smallest p-"eld of subsets of N containing the sets +u : u(B)"n, for all Borel sets B and all n"0, 1,2 The point process ' is then a measurable mapping from ) to N. The theory of point processes is well developed and has been proven useful in many applications. We may again refer to Stoyan et al. [2], a book that also contains a short introduction into the basic notions of measure and probability theory. According to the de"nitions we may regard ' as a random measure assigning to each Borel set BL1B the cardinality '(B) of the set +n3- : m 3B,. In fact, we even identify ' with the ranL dom measure d L, where d is the Dirac measure V L K located at x31B. This is a simple random counting measure, where each point has at most mass 1. Therefore ' is also called a simple point process. If f : 1BP1 is a measurable function, then the integral f (x) '(dx) is just another notation for the sum f (m ). L L The intensity measure " of ' is the measure on 1B de"ned by
"(B) " : E'(B)" '(B) dP, where BL1B is a Borel set and the symbol E denotes expectation with respect to the underlying probability measure P. As said in the introduction we assume that ' is stationary, i.e. distributional invariant under translations. Hence we have for all x31B and all A3N that P('#x3A)"P('3A), where '#x"+m #x : n3N,. L A "rst basic consequence of stationarity is that " must be a multiple of Lebesgue measure: "(dx)"j HB(dx), ECP where the intensity j of ' is de"ned as the mean ECP number of points falling in a set of volume 1, i.e.
j "E'([0, 1]B)" '([0, 1]B) dP. ECP Throughout the paper we assume that j (R. The ECP most important instance of a stationary point process is a Poisson process distributing points `completely randoma in space. In this case, '(B ),2,'(B ) are stochasti I cally independent random variables, whenever B ,2, B I
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
are pairwise disjoint Borel sets. Moreover, we have jK P('(B)"m)" ECP exp[!j HB(B)], m"0, 1, 2, ECP m! for all Borel sets B, i.e. '(B) is Poisson distributed with parameter j HB(B). ECP The grains $ as well as the germ-grain model $ itself L are measurable mappings from ) into the set F of all closed subsets of 1B. Measurability refers here to the smallest p-"eld of subsets of F containing the sets +F : F5KO, for all compact sets KL1B, see [1,2]. It is convenient to denote by $ a typical grain having the distribution of $ , n*1. L Let ""x"" denote the Euclidean norm of a vector x31B and let d(x, B) " : inf+""x!y"" : y3B, denote the distance of x31B from a set BL1B. We then recall that the empty space function F of $ is given by F(r)"P(d(0, $))r), r*0, where 0 denotes the zero vector if there is no risk of ambiguity. The number p" : P(03$)"F(0) is the volume fraction of $. The Minkowski addition of two sets A, BL1B is de"ned by A B " : +x#y : x3A, y3B,. Since d(0, $))r if and only if 03$ B(r), F(r) is just the volume fraction of the stationary random closed set $ B(r), where B(r) is a shorthand notation for B(0, r). Hence F(r)"(HB(B))\EHB(($ B(r))5B), r*0, provided that the Lebesgue measure HB(B) of the Borel set B is positive. The independence from the chosen B is implied by the stationarity. The set $ B(r) is the parallel set of $ at distance e. Using geometric measure theory, Baddeley and Gill [3] showed that the empty space function of any random closed set is absolutely continuous on (0, R) with density f (r)"(HB(B))\EHB\(*($ B(r))5B),
(2.1)
where HB\ denotes (d!1)-dimensional Hausdor! measure and *A denotes the boundary of a set A. (If SL1B is a smooth (d!1)-dimensional surface, then HB\(S) is the surface content of S.) Independently, Last and Schassberger [4] derived for general stationary germ-grain models with convex grains a more speci"c formula for f that will be used later in this paper. The above interpretation of the e.s.f. as the volume fraction of a parallel set suggests rather intimate relationships to concepts and tools of convex geometry. Schneider [11] gives an extensive introduction to that subject. Here we con"ne ourselves with recalling some simple notions. Some slightly more advanced tools will
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be used in Section 4. For any compact and convex set KL1B we let < (K), i"0,2, d, denote the ith intrinsic G volume of K. We have < (K)"1, while < (K) is the B volume (i.e. the Lebesgue measure) of K. If the dimension of the a$ne hull of K does not equal d!1, then 2< (K) is the (d!1)-dimensional surface area of K. B\ (Otherwise this is true without the factor 2.) Let b " : < (B(0, 1)) denote the volume of the unit ball in 1B. B B Then 2(d!1)b /b < (K) is the mean width of K. The B\ B properties of the intrinsic volumes which are most important for our purposes are based on Minkowski addition. The set K B(e) is the outer parallel body of K at distance e and we have the following Steiner formula: B < (K B(e))" b eB\G< (B), (2.2) B B\G G G where b " : 1. For the typical grain we now make the natural assumption that the means
(2.3)
are "nite. The number jM " : j
(2.4)
!ln(1!¹ (K))"j EHB($ K[ ), (2.5) ECP where K[ " : +!x : x3K,. This, in fact, holds for general compact primary grains satisfying EHB($ B(r))(R, r*0. (2.6) The latter condition guarantees that each compact set is indeed hit by only a "nite number of the grains, see Heinrich [8]. In fact, (2.6) is even necessary for this property. Under our assumption of convexity, one can use the Steiner formula for the volume of $ B(r), to derive from (2.5) the following well-known formula for the e.s.f. of a Boolean model:
B rB\Gb
3. Poisson cluster process models In this section we deal with the special case, where the germs form a Poisson cluster process. Hence '" 8 Z #x, V VZN?P
(3.1)
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
where ' is a Poisson process (of parents) with positive N?P and "nite intensity j and the family +Z : x3' , N?P V N?P consists of independent random point processes on 1B having all the same distribution. We let Z denote a typical cluster, i.e. a point process with the same distribution as the Z , x3' , and assume that the mean V N?P number j of cluster points AJ
j " : EZ(1B)" Z(1B) dP, AJ is "nite and positive. The intensity j of ' is then given ECP by j j . The generating functional G of ' (see [2]) is N?P AJ given by the formula G ( f )"G (fH), N?P where f : 1BP[0, 1] is measurable, (1!f ) is non-negative and 0 outside a bounded set, and the function f H is de"ned by
f H(x) " : E f (y#x) WZ8 (this is the generating functional at f of the cluster Z#x), and G denotes the generating functional of the Poisson N?P process ' , i.e. N?P
and all random elements are independent. Then we have ln G ( f )"j N?P
Example 3.1. If ' is a Neyman}Scott process, then the > all have the same distribution <, say, and (3.5) G L implies
ln G ( f )"j N?P
> "8 $ #x, (3.3) V VZ8 where the $ , x3Z, are independent and distributed like V $ . Hence one can use (2.5) with $ replaced by > to obtain an expression for F. Rataj and Saxl [9] used this formula to derive expressions for the e.s.f. of the point process '. (Note that ' can be interpreted as a germgrain model with $ "+0,.) In view of (2.6) we have to ensure that EHB(> B(r))(R, r*0, which, in fact, is an easy consequence of (2.6) and j (R. In the following we consider the special case AJ if g"0, Z" (3.4) +> : i"1,2, n, if g"n*1, G L where the random cluster size g is a random element of 9 ,> , n3N, i"1,2, n, are random elements of 1B, > G L
g
f(x#y)<(dy) !1 dx,
(3.6)
where g is the generating function of g. Example 3.2. Assume for all n3- that > "0 and for L n*2 that > ,2,> have all the same distribution < , L L L L say. In that case,
ln G ( f )"j N?P
(3.2) F(r)"1!G (1!¹ (B(r)! ) )), where ¹ is the capacity functional of the typical grain, de"ned by (2.4) with $ replaced by $ . We can interpret $ as a Boolean model where the grain itself is a ("nite) germ-grain model
where p " : P(g"n). L
p !1
# p f (x) L L
G ( f )"exp j ( f (x)!1) dx , N?P N?P where dx always refers to integration with respect to Lebesgue measure. From the de"nition of F we obtain rather directly (see e.g. [8]), that
L p !1# p % Ef(x#> ) dx, L G L L G (3.5)
f(x#y)< (dy) L
L\
dx. (3.7)
If p "0, 1!p "p "p, for 0)p)1,> "> "0, then ' is a Gauss}Poisson process. In this case a cluster Z#x associated with a parent point x say, contains x and, with probability 1!p, also a second point x#> . The random point > is called secondary point. Based on (3.2) and (3.7) we next compute F for a special Gauss}Poisson process with spherical grains. For AL1B we denote by x C R+x3A, the indicator function of A, which is 1 for x3A and 0, otherwise. A similar notation will be used for other spaces as well. Theorem 3.3. Assume that ' is a Gauss}Poisson process as described in Example 3.2, where the secondary point > is uniformly distributed in B(e) for some e'0. Assume further that the primary grain $ is a ball B(R ), with the dis tribution of the random radius R being exponential with parameter k'0. Then the empty space function F of the germ-grain model $ satisxes !ln(1!F(r))
B d "j b m!rB\Kk\K!j pb\e\B ECP B N?P B m K
; 1+""x""'r,exp(!k(""x""!r))a(x, r) dx
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
where the radius R and the secondary point > are inde pendent. Since B(R )5B(!x, r)O), i! x3B(R #r), the right-hand side of (3.10) equals
!j pb\eB 1+""x"")r,a(x, r) dx N?P B
!j pb\e\B 1+""x""'r,exp(!k(""x""!r)) N?P B
B d j b E(R #r)B"j b m!rB\Kk\K, N?P B N?P B m K
;HB(B(e)5B(x, r)) dx
where we used the moments of the exponential distribution. Similarly, (3.11) equals
!j pb\e\B 1+""x""*r,HB(B(e)5B(x, r)) dx, N?P B (3.8)
where
a(x, r) " : 1+""y!x"")e ,""y""'r,exp(!k(""y""!r)) dy. If d"3 then, in particular,
4n 6 6 3 3 3 "j #r #r #r #r !j p ECP 3 k N?P 4ne k k k 3 ; 1+""x""'r,(exp(!k(""x""!r))a(x, r) dx!j p N?P 4ne
Using the distributions of R and >, and distinguishing the four possible cases for the values of (x, >), we obtain that (3.12) equals
(3.9)
where e "(1, 0, 0) is a specixc unit vector in 1.
#p f (x)Ef (x#>), dx, where > is the secondary point. By (3.2) and some simple algebra,
!ln(1!F(r))"j P(B(R )5B(!x, r)O) dx (3.10) N?P (3.11)
;exp(!k(""y""!r)) dx dy
(3.14)
#j pb\e\B N?P B
1+""x""'r, ""y"")e, ""y!x"")r,
;exp(!k(""x""!r)) dx dy
(3.15)
1+""x"")r, ""y"")e, ""y!x"")r,
;dx dy.
(3.16)
3 P>C j p u . exp(!k(. !r))H(B(0, e)B(. e , r)) d. , N?P e P (3.17)
3 P j p . H(B(0, e)5B(. e , r)) d. . N?P e
(3.18)
This proves the second statement of the theorem. 䊐 Heinrich and Muche [10] give a formula to calculate the volume
!j p P(B(R )5B(!x, r)O)P(B(R ) N?P 5B(!x!>, r)O) dx,
1+""x"")r, ""y!x"",)e, ""y""'r,
Putting things together and noting that j "j (1#p), ECP N?P we obtain the "rst assertion. We now assume that d"3. It is convenient to introduce spherical coordinates. Then (3.15) and (3.16) transform to
ln G ( f )"j +!1#(1!p ) f (x) N?P
;exp(!k(""x""!r))exp(!k(""y""!r)) dx dy (3.13)
#j pb\e\B N?P B
Proof. According to (3.5), the generating functional of ' is given as
#j p P(B(R )5B(!x!>, r)O) dx N?P
1+""x""'r, ""y!x"")e, ""y""'r,
P>C . exp(!k(. !r))H(B(0, e)5B(. e , r)) d. P
3 P !j p . H(B(0, e)5B(. e , r)) d., N?P e
"j pb\e\B N?P B
#j pb\e\B N?P B
3 ; 1+""x"")r,a(x, r) dx!j p N?P e ;
B d j pb m!rB\Kk\K. N?P B m K
j p P(R '""x""!r)P(R '""x#>""!r) dx N?P
!ln(1!F(r))
1591
(3.12)
H(. , e, r) " : H(B(0, e)5B(. e , r))
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
. q . q(q!e) ; . q! ! dq d. 2 2
of the intersection of two balls in 1B ocurring in (3.17) and (3.18). We have to distinguish three cases:
. )e#r and !. )e!r). : H(B(0, e)5B(. e , r)) n 6 " 16(e#r)!12. (e#r)! (e!r)#2. , 24 .
. )e#r and . )r!e:
H(. , e, r)"H(B(0, e)),
. )e#r and . )e!r:
H(. , e, r)"H(B(. e , r)).
The intersection is empty if . 'e#r. In order to calculate the integrals (3.13), (3.14), (3.17) and (3.18) we have to distinguish three cases for the values of r, e, and . . The right column below speci"es the corresponding intersection volumes, taking into account the relative size of e!r to r.
(3.19)
exp(!k(. !r)) C>P . >C . q . q(q!e) ; dq d. exp(!k(q!r)) . q! ! 2 2 . \C (3.20) #K8n
H(. , e, r)"H(B(. e ,r)), e 0). )e!r, case 1 0) r ( 2 e!r(. )e#r, formula in (3.19),
0). )e!r; e )r(e 2 e!r(. )e#r;
case 2
case 3 r*e
H(. , e, r)"H(B(. e , r)), , formula in (3.19).
0). )r!e;
H(. , e, r)"H(B(0, e)),
r!e(. )e#r;
formula in (3.19).
We will consider the "rst case only, as the others are treated analogously. With K " : j p(3/4ne) expression ECP (3.13) transforms to
;
P
;
C\P exp(!k(. !r)). P
C\.
qexp(!k(q!r)) dq d. #K8n
. C\P >C exp(!k(. !r)) exp(!k(q!r)) P C\.
. q . q(q!e) ; . q! ! dq d. 2 2
#K8n
;
.
P
C>P exp(!k(. !r)) C\P
>C exp(!k(q!r))
P
.
C\.
qexp(!k(q!r)) dq d. P P C>. #K8n exp (!k (q!r)) C\. . q . q(q!e) ; . q! ! dq d. . 2 2
K16n
K16n
With analogous treatment, (3.14) becomes
(3.21)
Finally, (3.17) and (3.18) become
C\P r . exp(!k(. !r))4n d. 3 P C>P #K4n . exp(!k(q!r))H(. , e, r) d. C\P and
K4n
P 4nr 16nr . d. "K , 3 9 respectively. K4n
(3.22)
(3.23)
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
Remark 3.4. Heinrich and Muche [10] have noticed that the intersection volume of two balls in 1B is only for odd dimensions a rational function of the involved variables. Further integration steps (e.g. in (3.17) and (3.18)) can therefore be performed explicitly in this case. For even dimensions the expression for the intersection volume contains arcus functions of rational expressions. Exact calculation seems to be di$cult if not impossible. So numerical evaluation is adequate or even needed in these cases. 4. A representation involving curvature measures In this section we return to the general setting of Section 2. Our aim is to derive and discuss a formula for the density f of the empty space function F. To prepare the formula we "rst have to introduce some general characteristics of the typical grain $ and of the point process ' of germs. The characteristics of $ are again based on inte gralgeometric notions, see [6,11]. Let KL1B be a nonempty, compact and convex set. For all x31B there exists a unique point p(K, x) in K nearest to x and we set d(K, x) " : ""x!p(K, x)"". For x, K we let u(K, x) " : (x!p(K, x))/d(K, x) and we give u (K, x) some "xed value in the (d!1)-sphere SB\ " : +x31B : ""x"""1, if x3K. There exist "nite measures C (K, ) ),2,C (K, ) ) B\ on 1B;SB\ satisfying, for all e'0 and all Borel sets BL1B;SB\, the local Steiner formula HB(+x31B : 0(d(K, x))e,(p(K, x), u(K, x))3B,) B\ " eB\Gb C (K, B). (4.1) B\G G G These generalized curvature measures of K are uniquely determined by (4.1). The measures C (K, ) ;SB\), G i"0,2, d!1, are the curvature measures of K and are concentrated on the boundary *K of K. In contrast to the intrinsic volumes determined by (2.2), the de"nition of the generalized curvature measures is based on the local outer parallel sets +x31B : 0(d(K, x))e,(p(K, x), u(K, x))3B, taking into account the location p(K, x) of the nearest point as well as the direction of the contact vector x!p(K, x). Of course we have < (K)"C (K, 1B;SB\), G G i"0,2, d!1. It is easy to prove that the normalized contact vector u(K, x) is for x , K the outer normal of a support hyperplane of K at the point x. Since we have assumed that the mean intrinsic volumes de"ned in (2.3) are "nite, the mean generalized curvature measures of a typical grain, de"ned by
CM ( ) ) " : C ( $ , ) ) dP, G G
i"1,2, d!1,
are "nite measures on 1B;SB\.
1593
The other important concept we have to introduce is the Palm probability P of P with respect to ' (see [2]). This is a probability measure on (X, F) satisfying P (03')"1. We can interpret P (A) as the conditional probability of the event A3F given that 0 is a `randomly chosen pointa of '. Sometimes this is symbolically written as P ( ) )"P( ) "03'). Let us de"ne $(x) " : $ if L x"m for some n and $(x) " : , otherwise. If P is the L ruling probability measure, then +(x, $(x)) : x3',is again an independently marked point process, i.e. the grains +$(x) : x3', are independent, independent of ' and have the same distribution as $ . If
G (r) " :
(4.2)
where $ " : 8 ($(x)#x), VZ!+, is the union of all grains except the grain of the typical germ 0. For convenience, we set G ,1 for
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
where $L " : 8 ($ #m ). K K K$L Our aim is to apply Theorem 2.1 in [4]. To do so we recall that the skeleton $H of 1B!$ is de"ned as the set of all points x31B!$ having the same distance from at least two di!erent points of $. For (x, w)3N($) we denote the distance from x to $H in direction w by d(x, w) " : inf+r'0 : x#rw3$H,, where inf " : R. For (x, w) , N($) we set d(x, w) " : 0. By virtue of Theorem 2.1 and Remark 2.2 in [4] we have B\ f(r)" (d!i)b rB\G\ B\G G
E
g(y)1+B(y#rw, r)5$L
(4.5)
where we have used an invariance property of the generalized curvature measure C to obtain the last equality G (see [11]). To continue the above computation it is convenient to introduce the following notation. For any locally "nite and simple counting measure t" d L L V on 1B, we let C(t, ) ) be the distribution of the germ-grain model 6 ($ #x ). In fact, we consider only t with the L L L property
P(K5($ #x)O)t(dx)(R, KL1B compact,
see [8]. Using the kernel ! and the independence properties of our germ-grain model, we can rewrite (4.5) as
E L
g(y#m )1+B(y#m #rw, r)5A", L L
!('!d ,dA)CM (d(y, w)) KL G
g(y#x)1+B(y#rw, r) A",
!('!d , dA) CM (d(y, w))dx G
1+B(y#rw, r)5A",
!('!d , dA)CM (d(y, w)) . G From the properties of the Palm probability P it easily follows that
1+B(y#rw, r)5A",!('!d , dA)
Remark 4.2. We have assumed that the reduced second moment measure of ' is absolutely continuous w.r.t. Lebesgue measure. The above proof shows that Theorem 4.1 remains valid under the more general assumption (4.4).
g(y#m )1+B(y#m #rw, r)5$L L L
",C ($ , d(y, w)) , G L
j E ECP
"P (B(y#rw, r)5$") proving the result. 䊐
",C ($ #m , d(y, w)) G L L " E L
Making use of the re"ned Campbell theorem (see [2]) as well as of Fubini's theorem we obtain that the latter sum equals
E
g(y)1+d(y, w)'r,C ($, d(y, w)) G
" E L
!(('!m )!d , dA)CM (d(y, w)) . L G
g(y)1+d(y, w)'r,C ($, d (y, w)) , G
g(y#m )1+B(y#rw, r)5A", L
"j E ECP
where g : 1BP[0,R) is a measurable function satisfying g(x)dx"1. Let n3-. If (y, w) is a supporting element of $ #m , then d(y, w)'r and y , $Li! B(y#rw, r)5 L L $L". Hence we obtain from (4.4) that E
" E L
Remark 4.3. The function G can be interpreted as the G distribution function of a random variable m which can be constructed as follows. First one selects a point > of ' at random. Then one samples a random element (X, =) according to the distribution
G (r)" P (B(rw, r)5$O);B\(dw), where ;B\ is the uniform distribution on the sphere SB\.
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
If $ is a Boolean model, i.e. if ' is a Poisson process, then Slivnyak's theorem (see e.g. [2]) implies that P (B(y#rw, r)5$")"1!F(r), (4.6) and the density of the empty space function becomes B\ f (r)" (d!i)b j
P($3 ) )"E
1+A3 ) ,!(', dA) , KL1B compact.
A similar formula holds for P . The kernel ! is com pletely determined by the distribution of $ and has the property
1+A6A 3 ) ,!(t, dA)!(t , dA ),
1+'#t o ) ,Q(dt) , 8
(4.9)
where
Q( ) ) " : j\E 8 AJ
1+Z!x3 ) ,Z(dx)
is the Palm distribution of the typical cluster Z. This shows in particular that a su$cient condition for the absolute continuity of the reduced second moment measure of ' is that ' is given as in (3.4) with the > having an absolutely continuous distribution for G L n*2 and i*2. Using (4.9) together with (4.8) gives P (B(y#rw, r)5$")
"E
1+B(y#rw, r)5A",
1!G8(r)"
1+B(y#rw, r)5A",
;!(t!d , dA)Q(dt)CM (d(y, w)). (4.11) 8 G Here and below we use the convention 0/0 " : 0. This is the de"nition of the function G with P replaced by Q. G 8 If, in particular, ' is a Poisson process, then G "F for G
f ('!d , x)'(dx) "j E V ECP
f(', x)j(', x)dx
for all measurable f : N;1BP1 . We refer to [13,14] > for the meaning of this condition. The point process ' is then called a Gibbs process with local energy function !log j(', x). For our purposes it is important to note that j E f ('!d )"E j (', 0) f (') ECP holds for all measurable functions f : NP1. Applying this to the function mapping ' to 1+B(y#rw, r)5$", we obtain that
1!G (r)"jM \ E[1+B(y#rw, r)5$", G G ;j(', 0)]CM (d(y, w)). G 5. The empty space hazard of Poisson cluster models The hazard rate of the empty space function is the function h de"ned by f (r) h(r) " : , r*0. 1!F(r)
;!('#t!d , dA)Q(dt) 8 "E
1!G (r)"(1!F(r))(1!G8(r)), (4.10) G G where the function G8 is de"ned via the equation G
E
Example 4.5. Assume that ' is a Poisson cluster process as described in the previous section. A fundamental formula for the Palm probability P (see e.g. [2]) says that
;!(t!d , dA )Q(dt). 8 Hence
(4.8)
provided that t and t have disjoint supports.
P ( ' o ) )"E
1+B(y#rw, r)5A ",
"(1!F(r))
Example 4.6. Assume that ' admits a Papangelou conditional intensity j : N;1BP1 , where N has been de> "ned at the beginning of Section 2. This is to say that
1+A3 ) ,!(t#t , dA)
"
1595
1+B(y#rw, r)5A",1+B(y#rw, r)5A
",!(', dA)!(t!d , d A )Q(dt) 8
This is also the hazard rate of the spherical contact distribution function. Following Baddeley and Gill [3] we might call it the empty space hazard (rate) of $. The hazard rate h of a distribution function F of a nonnegative random variable X is a common tool in reliability theory. Quite often X describes the lifetime of a
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G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
technical system such that, heuristically speaking, h(t)"P(X3(t, t#Dt]"X*t) is the conditional probability that the system fails in a small interval (t, t#Dt] given that the system has survived time t. In our context we might also introduce a random variable with distribution function F by assuming that the point 0 starts growing in all directions at unit speed. The space which is covered at time t is just the ball B(t). Taking X as the time it takes the ball to hit $, we obtain a random variable with distribution function F and hazard rate h. Given h, the empty space function can be obtained from the formula R 1!F(t)"(1!p)exp ! h(s) ds , t*0. If $ is a Boolean model, then h is a polynomial
B\ h(r)" (d!i)b j rB\G\. (5.1) B\G G G It is important to note that this simple formula is determined by the intensity j and the mean intrinsic volECP umes
h(r)"db j rB\ B ECP
1+(t!d )(B(rw, r))"0,
;Q(dt);\(dw). 8 If, for instance, ' is a Neyman}Scott process as in Example 3.1, then
Example 5.2. Assume that $ "B(R), where the random radius R has distribution M I and a "nite dth moment. Using the speci"c form of the generalized curvature measures of a ball, we obtain
uG1+B((u#r)w, r)5A",
1!G8(r)"m\ G G
;!(t!d , dA)Q(dt);B\(dw)M I (du), 8 where m is the ith moment of R. Due to our convention G 0/0 " : 0 this is also true for ER"0 in which case we are back in Example 5.1. Assume now, for instance, that ' is a Gauss}Poisson process (see Example 3.2) and let > be independent of R with the distribution of the secondary point. Assume also that > and !> have the same distribution. We have j "1#p and the de"nition of AJ Q easily implies that 8
1!GG (r)"m\ 8 G
(1#p)\
uG(1!p#2pc(u, w, r))
;;B\(dw)M I (du),
(5.3)
where c(u, w, r) " : P(B((u#r)w, r)5B(>, R)"). If ' is a Neyman}Scott process, then a similar calculation as in the preceding Example 5.1 shows that
1!G8(r)"m\ uG p # g (P(B(>I , R)5B((u#r)w, r) G G "));B\(dw) M I (du), where >I is independent of R and distributed like > !> . Our next example is Neyman}Scott processes of germs with general convex grains. Example 5.3. Let ' be a Neyman}Scott process. Similarly as above,
1+B(y#rw, r)5A",!(t!d , dA)Q(dt) 8
j 1+(t!d )(B(rw, r))"0,Q(dt) AJ 8 L "p # p P(+> !> : lOk,5B(rw, r)") L J L I L L I "p #g (1! !> and g is the derivative of the generating function of g. Hence
One natural generalization of the preceding example are spherical grains:
h(r)"db j rB\ p # g (1!
"j\[p #g (1!¹ (B(y#rw, r))]. AJ Y Here ¹ denotes the capacity functional of the random Y closed set $ " : $ #>I , where $ and >I are indepen dent and >I is distributed like > !> . Consequently, B\ h(r)" (d!i)b j rB\G\ B\G N?P G
; [p #g (1!¹ (B(y#rw, r))]CM (d(y, w)). G
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
Gauss}Poisson processes admit an interesting property of stochastic monotonicity: Theorem 5.4. Fix a number j'0 and let 'N denote a Gauss}Poisson process as in Example 3.2 with intensity j "j and probability p for having a secondary point. ECP Assume that the secondary point > " : > has the same distribution as !>. ¸et hN(HN) denote the hazard rate Q (spherical contact distribution function) of a germ-grain model with germ process 'N and typical grain $ . Then hN(r) is for all r*0 decreasing in p. In particular, 1!HN(r) is increasing in p. Q Proof. A straightforward generalization of (5.3) is 1!G8(r)"
; (1!p#2pa(y, w, r))CM (d(y, w)), G where a (y, w, r) " : P (B (y#rw, r)5($ #>)"), and $ and > are independent. Function a(y, w, r) and the quermass densities jM are independent of p. By (5.2) it G su$ces to show that 1!p#2pa 1!p #2p a * , 1#p 1#p provided that p)p and 0)a)1. Simple algebra shows that this is correct. The second assertion follows from the equality
1!HN(t)"exp ! Q
R
hN(u)du ,
t*0.
䊐
Remark 5.5. The second statement of Theorem 5.4 says that HN is stochastically increasing in p. The parameter Q p is a measure for the clustering in 'N. Hence, an increase of clustering leads to an stochastically increasing spherical distance. This is in perfect coincidence with intuition. If $ "+0,, then HNis just the spherical con Q tact distribution function of the point process 'N itself. In many cases of interest there is no closed simple expression for the functions G8. Instead we propose to G use Monte}Carlo integration. To be more speci"c we assume that the typical cluster Z is given as in (3.4). Then we have G8(r)"j\E G AJ
E E 1+B(>#r=, r) 8 ($ I K I I$K
#> !> )O, , I L K L
(5.4)
1597
where (>, =) is a random element with distribution
1598
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
Fig. 3. In#uence of e on the hazard rate of a cluster model with one secondary point ( k"10, p"0.8)
Fig. 1. Comparision of the empty space function and the corresponding hazard rate for various cluster models (one secondary point, k"10, e"0.5)
Fig. 4. Hazard rates of a cluster model and of ann adapted Boolean Model.
Fig. 2. In#uence of e on the empty space function of a cluster model with one secondary point ( k"10, p"0.8).
Fig. 5. Hazard rates of various cluster models with two secondary points (k"10, p "0.2, p "0.8).
3 points. The cluster process is now a generalized Gauss}Poisson process as described in Example 3.2 with p "0, p "0, n*4, where the points > , > , > L are all uniformly distributed in the ball B(e) for some
e'0. The primary grains are again balls, where the radius follows an exponential distribution with parameter k'0. For p "0 we obtain the model discussed above. Fig. 5 shows the empty space hazard rates for
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
k"10, p "0.2, p "0.8 and two values of e. For e"0.5 the rate very much resembles that of a Boolean model, i.e. it is approximately a quadratic function of r. In this case e is apparently small enough to ensure that the typical grain Z (see (3.3)) can be approximated by a convex grain without neglecting too much empty space. For e"2 the situation changes drastically. The empty space `between the grainsa of a cluster cannot be neglected anymore and a quadratic function cannot satisfactorily be "tted to the rate. The ordering between the rates of the two cluster models on the interval [0,2.1] can be interpreted as above. It is interesting to note that, for larger values of r, the rate for e"2 behaves concave. Summarizing we can say that the empty space hazard rate might be an appropriate tool to reveal clustering phenomena and to explain distinctions between di!erent cluster models.
6. Measures of spatial interaction We consider the general model of Section 2, assume that (4.4) is satis"ed, and de"ne 1!G (r) G , i"1,2, d!1, J (r) " : G 1!F(r)
(6.1)
for all r*0 with F(r)(1. The idea of this de"nition is taken from van Lieshout and Baddeley [5], who de"ned their J-function for point processes using the nearestneighbour distance distribution function G instead of our G . The functions J compare the behaviour in the neighG G bourhood of a typical point of the germ process with the behaviour in the neighbourhood of a typical point in space in order to detect possible interactions and clustering e!ects. Our de"nition is also motivated by the discussion in Section 5 which gave some evidence that distinctions between the empty space hazard rates can be explained by the structure of clusters. If ' is a Poisson process, then J (r)"1 for
(6.2)
Hence the functions J are non-parametric measures for G the distance between the germ-grain model and a Boolean model with the same quermass densities. For statistical purposes it might be more convenient to consider jM J instead of J . In the following we show that our G G G functions J share many properties with the J-function of G van Lieshout and Baddeley [5].
Example 6.2. If ' is a Gibbsian point process as in Example 4.6, then
J (r)"jM \ E[j(',0)"B(y#rw, r)5$"]CM (d(y, w)). G G G Theorem 6.3. Assume that $"$6$ is the superposition of two germ-grain models $I" ($ #m ), L I L I L satisfying our assumptions and such that the marked point processes +(m , $ ) : n*1,, k"1, 2, are independent. L I L I For k"1, 2, let j denote the intensity of I ' " : +m : n*1, and J the J -function of $I. Then, I L I G I G for i"1,2, d!1, j j J (r)# J (r), J (r)" G G j #j j #j G provided that
"E
1+B(y#rw, r)5(A 6A )",
;!(' !d , dA )!(' , dA ) "(1!F (r))E
1+B(y#rw, r)5A",
;!(' !d , dA) "(1!F (r)) ;P (B(y#rw, r)5$"), where F is the empty space function of $I. The remainI der of the proof is the same as in the proof of Theorem 2 in [5]. ) We conclude the section by indicating how to estimate the functions jM (1!G ) and jM J using an observation of G G G G $. Because the point process ' is not directly observable we cannot use the de"nition. However, as outlined in [4] it follows for any measurable set BL1B of positive volume that jM (1!G (r)) G G
"(HB(B))\E Example 6.1. If ' is a Poisson cluster process as in Example 4.5, then J (t)"1!G8(t). G G
1599
1+x3B, d(x, w)'r,C ($, d(x, w)) , G (6.3)
1600
G. Last, M. Holtmann / Pattern Recognition 32 (1999) 1587}1600
where d(x, w) has been de"ned in the proof of Theorem 4.1. Ignoring censoring e!ects due to bounded observation windows, this can be used in an obvious way to construct estimators for jM (1!G ) and jM J . We "nally G G G G note that j> " : jM (1!G (0)) G G G
"(HB(B))\E
1+x3B, (x, w)3N($),C ($, d(x, w)) G
is the intensity of the curvature measure C ($, ) ) reG stricted to the generalized normal bundle N($) de"ned in the proof of Theorem 4.1. Hug and Last [15] have recently shown that this restriction coincides with the non-negative curvature measure introduced by Matheron [1] and Schneider [6].
Acknowledgements The authors thank an anonymous referee for his very careful reading of this paper which has led to a considerable improvement of the exposition.
References [1] G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1975. [2] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications, second ed., Wiley, Chichester, 1995. [3] A. Baddeley, R.D. Gill, The empty space hazard of spatial pattern, preprint 845, Department of Mathematics, University Utrecht, 1994.
[4] G. Last, R. Schassberger, On the distribution of the spherical contact vector of stationary grain models, Adv. Appl. Proba. 30 (1998) 36}52. [5] M.N.M. van Lieshout, A. Baddeley, A nonparametric measure of spatial interaction in point patterns, Statist. Neerland. 50 (1996) 344}361. [6] R. Schneider, Parallelmengen mit Vielfachheit und Steiner-Formeln, Geom. Dedicata 9 (1980) 111}127. [7] W. Weil, J.A. Wieacker, Stochastic geometry, in: Handbook of Convex Geometry, North-Holland, Amsterdam, 1993, pp. 1391}1438. [8] L. Heinrich, On existence and mixing properties of germgrain models, Statistics 23 (1992) 271}286. [9] J. Rataj, I. Saxl, Boolean cluster models: mean cluster dilations and spherical distances, Math. Bohemica 122 (1995) 21}36. [10] L. Heinrich, L. Muche, On the pair correlation function of the point process of nodes in a Voronoi tesselation, preprint 94-07, TU Bergakademie Freiberg, 1994. [11] R. Schneider, Convex Bodies : the Brunn-Minkowski Theory, in: Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge, 1993. [12] L. Heinrich, I.S. Molchanov, Central limit theorem for a class of random measures associated with germ-grain models, CWI Report BS-R9518; Adv. Appl. Probab., 1999, in press. [13] X.X. Nguyen, H. Zessin, Integral and di!erential characterizations of the Gibbs process, Mathematische Nachrichten 88 (1979) 105}115. [14] O. Kallenberg, Random Measures, Akademie}Verlag, Berlin and Academic Press, London, 1983. [15] D. Hug, G. Last, On support measures in Minkowski spaces and contact distributions in stochastic geometry, preprint, UniversitaK t Freiburg, 1999, submitted for publication.
About the Author*GUG NTER LAST received the diploma degree in mathematics from Humboldt-University in Berlin in 1984 and Ph.D. also from the same university in 1986. He worked there as scienti"c assistant till 1992. Then he moved to Technical University in Braunschweig, where he received Dr. Sc. degree in 1995. His reserach interests include random point processes and measures and their application in reliability theory, queueing and stochastic geometry. About the Author*MARKUS HOLTMANN studied mathematics at the University of Braunschweig where he received his diploma in 1997. He is currently a Ph.D. student at the Freiberg University of Mining and Technology in the graduate school `Spatial Statisticsa. The main topic in his thesis covers aspects of experimental design of stochastic "elds involving derivative information.