Inhomogeneous K-function for germ-grain models
IInhomogeneous K-function for germ-grainmodels
M. ´Angeles Gallego (1) , M. Victoria Ib´a˜nez (2) and Amelia Sim´o (2) . (1) Departament de Matem`atiques. Universitat Jaume I. Spain.(2) Departament de Matem`atiques-IMAC. Universitat Jaume I. Spain. April 23, 2018
Abstract
In this paper, we propose a generalization to germ-grain models ofthe inhomogeneous K-function of Point Processes. We apply them toa sample of images of peripheral blood smears obtained from patientswith Sickle Cell Disease, in order to decide whether the sample belongsto the thin, thick or morphological region. keyword
Germ-grain models; K-function; Binary images; SickleCell Disease
Nowadays, digital images of different phenomena of interest are commonlyobserved in almost every experimental fields, however, their subsequent post-processing and use tend to be very simple. The information from each imageis frequently resumed in just a few numbers, and simple statistical proceduressuch as hypothesis testing or parameter estimation procedures are appliedto them. In this paper we will work with binary images of random clumps.Stochastic-geometric models and in particular random closed sets Matheron[1975], Stoyan et al. [1995], Molchanov [1997, 2005] have been broadly usedto model irregular random patterns in various fields, such as communicationnetworks [Baccelli et al., 1997], materials science [Ohser and M¨ucklich, 2000]or physics [Mecke, 1998] among others. They have also been useful in model-ing many medical and biological problems, especially those which require the1 a r X i v : . [ s t a t . O T ] J a n xtraction of information from microscopic images. Thus, Boolean modelsin particular and germ-grain models in general have been extensively usedto analyze binary images of random clumps in many scientific fields [Stoyanet al., 1995, Serra, 1982, Plaza, 1991, Margalef, 1974, Lyman, 1972]. Let F be the class of closed subsets in the Euclidean space IR d and σ f = σ ( F K , K compact subset of IR d ) where F K = { F ∈ F : F ∩ K (cid:54) = ∅ } orequivalently the Borel σ -algebra generated by the the Fell topology on F [Fell, 1962]. If P denotes a probability measure in ( F , σ f ) then, accordingwith the original definition given by Matheron [1975], ( F , σ f , P ) is a randomclosed set.The germ-grain model [Hanisch, 1981] is a particular class of randomclosed set model that has proved to be suitable for working with imagesformed by random clumps. Its mathematical definition is as follows: Definition 1
Let
Ψ = { [ x n ; Ξ n ] } be a marked point process, where x n arepoints of IR d and Ξ n are compact subsets of IR d . A germ-grain model, Ξ , isdefined as: Ξ = (cid:91) n (Ξ n + x n ) . The points x n are called germs and the sets Ξ n are known as grains [Hanisch,1981]. The most widely known and used germ-grain model is the Boolean model[Molchanov, 1997]. The Boolean model is obtained when the germ process, φ λ = { x , x , · · ·} , is a Poisson process with intensity function λ ( x ) and thegrains, Ξ , Ξ , Ξ , · · · , are i.i.d. and independent of the germs.Under homogeneity, the intensity of the Poisson germs process is assumedto be constant, and it is denoted by λ . Additionally, if Ξ is stationary, i.e. ifΞ and the translated sets Ξ x = Ξ + x have the same distribution ∀ x ∈ IR d ,the typical grain , Ξ , can be defined as a closed random set with the samedistribution as the sets { Ξ n } but independent, both of them and of the germs.So, the distribution of the typical grain is the distribution of the marks ofthe marked point process Ψ = { ( x n , Ξ n ) : n ∈ N } . This is a marked pointprocess on IR d , with the marks Ξ n , n ∈ N , being random convex bodies in IR d . It is a distribution on the space K of compacts sets.2he probability distribution of a random closed set Ξ is given by its capacity functional , T Ξ , defined as: T Ξ ( K ) = P (Ξ ∩ K (cid:54) = ∅ ) ∀ K ⊂ IR d . Useful summary descriptions of the probability distribution of the randomclosed set are the coverage function, p ( x ), defined as the mean area of Ξ in theunit square centered in { x } , i.e. p ( x ) = T Ξ ( { x } ), and the covariance function C ( x, x + h ) = T Ξ ( { x, x + h } ). These functions describe the first-order andthe second-order structure of the set respectively.Under stationarity the coverage function is constant, p ( x ) = p (0) = p ,and is called volume fraction and C ( x, x + h ) = C ( h ) ∀ x ∈ IR d . In this case,the K-function [Daley and Vere-Jones, 1988, Jensen et al., 1990] provides amore intuitive and practical way to describe the second order structure. Itis defined as: K ( t ) = 1 p E [ ν (Ξ ∩ B (0 , t )] , where B (0 , t ) is the ball centered at the origin with radius t ; ν is the Lebesguemeasure and E denotes the expectation with respect to the Palm distributionof Ξ. When { } , the origin of IR d is chosen as the typical point of Ξ, pK ( t )becomes the average measure of the intersection of Ξ with a ball of radius t centered at { } .We have the following relation between the covariance and the K -function: K ( t ) = 1 p (cid:90) B (0 ,t ) C ( h ) dh. The K -function, also called the Ripley K -function, has been extensivelyused in the point processes literature [Ripley, 1977, Diggle, 1983] mainly toanalyze the strength of the interaction between points in the point process.However, it has also been used to analyze isotropic Boolean models [Ayalaand Sim´o, 1995, 1998]. Additionally, Ayala and Sim´o [1993] proposed anapproximation for the K-function in overlapping Boolean models based onan approximation of the covariogram of the primary grain. Originally, theK-function was defined to characterize stationary point processes, but thedefinition was later extended to inhomogeneous point processes [Baddeleyet al., 2000, Diggle et al., 2007]. As far as we know, it has not yet beenextended to non-stationary germ-grain models. This is our objective in thispaper. 3he assumptions of stationarity and isotropy facilitate the estimation ofthe parameters of the germ-grain model. However, the hypothesis of spatialhomogeneity frequently fails when real data sets are analysed. An impor-tante example of non stationary germ-grain model is the non-homogeneousBoolean model, i.e. the Boolean model obtained when the germ process isa Poisson process with intensity function λ ( x ). Non-homogeneous Booleanmodels have been used to model functionally graded materials [Hahn et al.,1999, Quintanilla and Torquato, 1997], distributions of galaxies [Bond et al.,1995] and complex fluids [Brodatzki and Mecke, 2001]. Methods to esti-mate parameters of non-homogeneous Boolean models have been studied byMolchanov and Chiu [2000] and by Schmitt [1996].From now on we will restrict our work to the case d = 2.To define the inhomogeneous K-function for point patterns, Baddeleyet al. [2000] considered a point process Y in IR , with first-order intensityfunction λ ( s ) , s ∈ IR . Given B the class of bounded Borel sets in IR , andassuming that the function M ( A, B ) = E (cid:88) y i ∈ Y (cid:84) A (cid:88) y j ∈ Y (cid:84) B λ ( y i ) 1 λ ( y j ) (1)is finite for all A, B ∈ B , they defined Y as a second-order intensityreweighted stationary point process if M ( A, B ) = M ( A + x, B + x ), be-ing A + x the translation of A by the vector x. From this point, they definedthe inhomogeneous K-function for point processes as: Definition 2 (Inhomogeneous K-function for point processes)
Let Y be a second-order intensity reweighted stationary point process. Then, the in-homogeneous K-function of Y is defined as: K inhom ( t ) = 1 ν ( W ) E (cid:88) y i ∈ Y ∩ W (cid:88) y j ∈ Y \{ y i } (cid:107) y i − y j (cid:107) ≤ t ) λ ( y i ) λ ( y j ) , t ≥
0; (2) for any W ∈ B , the class of bounded Borel sets in IR , where · ) denotesthe indicator function, ν ( W ) is the area (Lebesgue measure) of W , , and a/ for a ≥ . This expression does not depend on the choice of W . Given a realization of Y in an observation window W , its correspondingsample estimator [Baddeley et al., 2000] becomes:4 K inhom ( t ) = 1 ν ( W ) (cid:88) y i ∈ Y ∩ W (cid:88) y j ∈ Y \{ y i } w y i ,y j (cid:107) y i − y j (cid:107) ≤ t )ˆ λ ( y i )ˆ λ ( y j ) , ≤ t ≤ t ∗ , (3)where w y i ,y j is an edge corrector function and t ∗ = sup { r ≥ ν ( { s ∈ W : ∂B ( s, r ) ∩ W (cid:54) = ∅} ) > } , where ∂B ( s, r ) denotes the boundary of B ( s, r ). As stated below, the inhomogeneous K-function has been defined to workwith inhomogeneous point processes [Baddeley et al., 2000, Diggle et al.,2007], and our aim is to extended it for non-stationary germ-grain models.We have to note that there is an important difference between point processesand germ grain models. The probability distribution of a point process ischaracterized by its random count measure, but the random coverage measureof a random closed set does not characterize its probability distribution.( ? ), establish a sufficient condition to guarantee that the random coveragemeasure determine the probability distribution of the random closed set.Previously to define the inhomogeneous K-function, the following funda-mental concept must be introduced.Assuming that Ξ has a strictly positive coverage function, i.e. p ( x ) > ∀ x , given A , B bounded Borel sets in R , the measure M is defined in R as: M ( A, B ) = E (cid:90) Ξ ∩ A (cid:90) Ξ ∩ B p ( x ) p ( y ) dxdy. Definition 3 (Second-order intensity-reweighted stationary)
The germ-grain model Ξ is ”second-order intensity-reweighted stationary” if M ( A, B ) = M ( A + x, B + x ) for all x ∈ R A second-order stationary germ-grain model is also second-order intensity-reweighted stationary. A non-homogeneous Boolean model is second-orderintensity-reweighted stationary because of the second-order intensity-reweightedstationary property of its germ process and the independence of the grains.5 efinition 4 (Inhomogeneous K-function for germ-grain models)
Let Ξ be a second-order intensity-reweighted stationary germ-grain model in anobservation window W . The inhomogeneous K-function of Ξ is defined as: K inhom ( t ) = 1 ν ( B ) E ( (cid:90) Ξ ∩ B (cid:90) Ξ ∩ B ( y,t ) p ( x ) p ( y ) dydx ) (4) for any B ∈ σ f . Property 1
Definition 4 does not depend on B . Dem.
Let A t = { ( x, y ) : x ∈ B, y ∈ B ( x, t ) } , M ( A t ) = E ( (cid:90) Ξ ∩ B (cid:90) Ξ ∩ B ( y,t ) p ( x ) p ( y ) ) dydx. Because of the second-order intensity-reweighted stationary property, M ( A t ) = M ( A t + ( z, z )) ∀ z ∈ IR , and as a result1 ν ( B ) E ( (cid:90) Ξ ∩ B (cid:90) Ξ ∩ B ( y,t ) p ( x ) p ( y ) dydx ) = 1 ν ( B + z ) E ( (cid:90) Ξ ∩ B + z (cid:90) Ξ ∩ B ( y,t ) p ( x ) p ( y ) dydx )c.q.d. K inhom ( t ) has an interpretation as a Palm expectation, similar to that forthe stationary case: Property 2 K inhom ( t ) = E s ( (cid:90) Ξ ∩ B ( s,t ) p ( x ) dx ) ∀ s ∈ IR . where E s denotes the expectation with respect to the Palm distribution P s of Ξ at s that can be interpreted as the conditional distribution of Ξ giventhat s ∈ Ξ . Dem.
Consider B ∈ σ f . 6 ν ( B ) (cid:90) B E s ( (cid:90) Ξ ∩ B ( s,t ) p ( x ) dx ) ds = 1 ν ( B ) (cid:90) B p ( s ) p ( s ) E s ( (cid:90) Ξ ∩ B ( s,t ) p ( x ) dx ) ds =1 ν ( B ) (cid:90) IR p ( s ) E s (1 B ( s ) (cid:90) Ξ ∩ B ( s,t ) p ( s ) p ( x ) dx ) ds. Applying the Campbell-Mecke formula [Jensen et al., 1990]:1 ν ( B ) (cid:90) IR p ( s ) E s (1 B ( s ) (cid:90) Ξ ∩ B ( s,t ) p ( x ) dx ) ds p ( s ) p ( x ) dx ) ds = 1 ν ( B ) E ( (cid:90) Ξ ∩ B (cid:90) Ξ ∩ B ( s,t ) p ( s ) p ( x ) dxds From the result of the Prop. 1, we know that the second term of the equalitydoes not depend on B . Then:1 ν ( B ) (cid:90) B E s ( (cid:90) Ξ ∩ B ( s,t ) p ( x ) dx ) ds = ν ( B ) ν ( B ) E s ( (cid:90) Ξ ∩ B ( s,t ) p ( x ) dx ) . And so: E s ( (cid:90) Ξ ∩ B ( s,t ) p ( x ) dx ) = 1 ν ( B ) E ( (cid:90) Ξ ∩ B (cid:90) Ξ ∩ B ( s,t ) p ( s ) p ( x ) dxds )c.q.d. Given a realization of Ξ in an observation window W its corresponding sampleestimator isˆ K inhom ( t ) = 1 ν ( W ) (cid:90) Ξ ∩ W (cid:90) Ξ ∩ W w y i ,y j (cid:107) y i − y j (cid:107) ≤ t )ˆ p ( y i )ˆ p ( y j ) dy i dy j ∼ = (5) ∼ = 1 ν ( W ) (cid:88) y i ∈ Ξ ∩ W (cid:88) y j ∈ Ξ ∩ W w y i ,y j (cid:107) y i − y j (cid:107) ≤ t )ˆ p ( y i )ˆ p ( y j ) , ≤ t ≤ t ∗ , with w y i ,y j an edge corrector function, and t ∗ = sup { r ≥ | { s ∈ W : ∂B ( s, r ) ∩ W (cid:54) = ∅} | > } , as above, ˆ p is a kernel estimator of the coveragefunction, and ∼ = denotes the numerical approximation.7igure 1 shows realizations of three different germ-grain models and themean of the estimated inhomogeneous K-function computed from a sampleof them. It can be seen that for t ≈ <
60, ˆ K inhom ( t ) is greater for the clustermodel than for the Boolean model, which is due to the effect of the clusteringof germs. For t ≈ <
30 it also increases much faster but for larger t -valuesthe increase in ˆ K inhom ( t ) for the Cluster model is slower than for the Booleanmodel. Finally, for t ≈ >
60, ˆ K inhom ( t ) is lower for the cluster model thanfor the Boolean model.As Baddeley et al. [2000] state, in practice it is difficult to make a dis-tinction between large-scale variation given by p ( x ) and variation due tointeractions. In this case it is of particular importance the choice of thebandwidth parameter in the kernel estimator of the coverage function (Eq.5). In our experiments, we choose it comparing the value of the analyticalexpression of the volume fraction of an homogeneous Boolean model, withthe empirical one for different values of h . On the other hand, taking intoaccount that the theoretical volume fraction in the Cluster model is constantwe choose the greatest h value among those that provided a good approxi-mation. This choice of h allows to distinguish clearly the ˆ K inhom ( t ) for thenon-homogeneous Boolean model and the Cluster model, as can be seen infigure 1 (d). It is due to the fact that the estimation of the coverage functionin the ”accumulation” area of the non-homogeneous Boolean model is quitegreater than the corresponding to the Cluster model.In contrast to what happens in point processes context it does not existan exact expression for the K-function for an homogeneous Boolean model.As said above, Ayala and Sim´o [1993] gave an approximate expression for itbased on an approximation of the covariogram of the primary grain. Thisapproach is valid for values of t close to zero. In Fig. 2 we compare the meanof the estimated inhomogeneous K-function corresponding to 10 realizationsof a Boolean model with the approach given by Ayala and Sim´o [1993]. As an example of application we use the inhomogeneous K-function to per-form unsupervised classification when the sample information are digital im-ages of peripheral blood smears.Examination of peripheral blood smears is an essential component inthe evaluation of all patients with hematologic disorders. In particular it is8a) (b) (c) (d)Figure 1: Realization of different germ-grain models: (a) HomogeneousBoolean model, (b) non-homogeneous Boolean model, (c) Cluster model,and (d) their Sample Inhomogeneous K-function9
Figure 2: Mean of the estimated inhomogeneous K-function correspondingto 10 realizations of a Boolean model and the approach given by Ayala andSim´o [1993].used in the diagnosis and monitoring of an important genetic disease calledSickle Cell Disease (SCD). SCD causes the hardening or polymerization ofthe hemoglobin that contains the erythrocytes. The cells are deformed andtend to block blood flow in the blood vessels of the limbs and organs. Blockedblood flow can cause pain and organ damage. This disease has been recog-nized as a major public health problem by international agencies such as theWorld Health Organization (WHO) and the United Nations Educational,Scientific and Cultural Organization (UNESCO). Depending on the state ofthe disease, i.e., depending on the quantity of deformed cells in their bloodflow, patients are classified into three groups: those with a benignant form,without pain crises, those with a moderate form who only have one crisisper year, and those who are seriously ill with two or more crises per year.The quantitative analysis of digital images of peripheral blood smears offersuseful results in the clinical diagnosis of this illness, and guides the specialistin allocating the most suitable treatment.A peripheral blood smear (peripheral blood film) is a glass microscopeslide coated on one side with a thin layer of venous blood. The slide is stainedwith a dye and examined under a microscope. A blood-smear preparationrequires dropping the blood sample, spreading the sample, and staining.Sample spreading is done by pulling a wedge to spread a sample drop ofblood on the slide. A well-made peripheral smear is thick at the frostedend and becomes progressively thinner toward the opposite end. In thethicker region, most of the cells are clumped, which increases the difficulty inidentifying and analyzing blood components. At the thinner region the cells10re unevenly distributed, and grainy streaks, troughs, ridges, waves, or holesmay be present. This portion of the smear has insufficient useful informationfor analysis. The ”zone of morphology” (area of optimal thickness for lightmicroscopic examination) occupies the central area of the slide. Figure 3shows typical images captured on the same peripheral blood smear.Figure 3: Different images of different zones of the same peripheral bloodsmearThe thickness of the smear is influenced by the angle of the spreader (thewedge), the size of the drop of blood, and the speed of spreading. In cur-rent laboratory practice, skilled users manually identify the zone of morphol-ogy and acquire images of it. Due to the aforementioned reasons, this zonevaries in its morphology and offers specific appearances on different slides.Such manual identification is tedious, inconsistent, and prone to error, andis also biased in terms of statistics and user subjectivity. Advances in high-throughput microscopy have enabled the rapid acquisition of many images11ithout human intervention. Depending on the sample size on the slide, onecould easily acquire more than ten thousand images in a sample. An auto-matic detection of the ”zone of morphology” can increase consistency, reducelabor, and achieve better accuracy. Some papers in the literature about au-tomatic classification of this area are [Mutschler and Warner, 1987, Anguloand Flandrin, 2003].Our dataset consists of three peripheral blood smears that were obtainedfrom patients with Sickle Cell Disease. Thirty digital images were takenacross each smear (some of them are shown in Fig. 3). Our aim is to use theinhomogeneous K -function to classify these images in three groups, whichwould correspond to the thick, thin and morphological zones respectively.Prior to proceeding with the clustering, the original images are segmentedin order to convert them into binary images. Since there exists a good con-trast between cells and background, a good segmentation is obtained usingvery simple image processing techniques: thresholding followed by morpho-logical filtering. This binarization was performed using Matlab. In Figure 4we can see the binary images corresponding to the images shown in Figure 3.These binary images are considered realizations of three different germ-grainprocesses, so the inhomogeneous K-function can be used to define homoge-neous classes of images. In order to do that, the inhomogeneous K-function isestimated from each image (Eq. 5), and a partitioning method called PAM(Partition Around Medoids) [Kaufman and Rousseeuw, 1990] will be usedfor clustering, defining the distance between each couple of images as theEuclidean distance between their inhomogeneous K-functions.Fig. 5 shows the images corresponding to the medoids obtained for thethree clusters.After obtaining the clusters, an expert hematologist helped us to reviewthe results and confirm the goodness of the classification obtained by lookingat the medoids and the other images of each group. It is very importantto remark that this study should be carried out with a really large imagedatabase in order to obtain any valid clinical conclusions, but in any caseour results are very promising. In this paper we introduce a generalization of the inhomogeneous K-functionthat allows its application to non-homogeneous germ-grain models. We have12igure 4: Pre-processed images of different zones of the same peripheralblood smearFigure 5: Medoids of the three clusters obtained using the Euclidean dis-tance between the K-functions 13hown its capacity to discriminate between realizations of different modelsand we have applied it to a clinical application: a sample of images of pe-ripheral blood smears obtained from patients with SCD. Different groups ofimages corresponding with different patterns were found. The study shouldbe repeated with a really large image database in order to achieve valid med-ical conclusions. From our point of view, we believe our methodology can beused without modifications. Other future applications of the inhomogeneousK-function can include Monte Carlo goodness of fit test [Diggle, 1983] orparameter estimation.
We would like to thank Silena Herold from the Computation Faculty of theUniversidad de Oriente, Santiago de Cuba, for introducing us in this inter-esting problem and providing us the images.This work has been partially supported by the UJI projects P B −
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