New Algorithms Based on the Voronoi Diagram Applied in a Pilot Study on Normal Mucosa and Carcinomas

An adequate reproducibility in the description of tissue architecture is still a challenge to diagnostic pathology, sometimes with unfortunate prognostic implications. To assess a possible diagnostic and prognostic value of quantitiative tissue architecture analysis, structural features based on the Voronoi Diagram (VD) and its subgraphs were developed and tested. A series of 27 structural features were developed and tested in a pilot study of 30 cases of prostate cancer, 10 cases of cervical carcinomas, 8 cases of tongue cancer and 8 cases of normal oral mucosa. Grey level images were acquired from hematoxyline‐eosine (HE) stained sections by a charge coupled device (CCD) camera mounted on a microscope connected to a personal computer (PC) with an image array processor. From the grey level images obtained, cell nuclei were automatically segmented and the geometrical centres of cell nuclei were computed. The resulting 2‐dimensional (2D) swarm of pointlike seeds distributed in a flat plane was the basis for construction of the VD and its subgraphs. From the polygons, triangulations and arborizations thus obtained, 27 structural features were computed as numerical values. Comparison of groups (normal vs. cancerous oral mucosa, cervical and prostate carcinomas with good and poor prognosis) with regard to distribution in the values of the structural features was performed with Student's t‐test. We demonstrate that some of the structural features developed are able to distinguish structurally between normal and cancerous oral mucosa (P=0.001), and between good and poor outcome groups in prostatic (P=0.001) and cervical carcinomas (P=0.001). We present results confirming previous findings that graph theory based algorithms are useful tools for describing tis‐ sue architecture (e.g., normal versus malignant). The present study also indicates that these methods have a potential for prognostication in malignant epithelial lesions.

New algorithms based on the Voronoi Diagram applied in a pilot study on normal mucosa and carcinomas J. Sudbø a, * , R. Marcelpoil b and A. Reith a a Section for Digital Pathology, The Norwegian Radium Hospital, Oslo, Norway b Université Joseph Fourier, Grenoble, France Accepted 30 October 2000 An adequate reproducibility in the description of tissue architecture is still a challenge to diagnostic pathology, sometimes with unfortunate prognostic implications. To assess a possible diagnostic and prognostic value of quantitiative tissue architecture analysis, structural features based on the Voronoi Diagram (VD) and its subgraphs were developed and tested.
A series of 27 structural features were developed and tested in a pilot study of 30 cases of prostate cancer, 10 cases of cervical carcinomas, 8 cases of tongue cancer and 8 cases of normal oral mucosa. Grey level images were acquired from hematoxyline-eosine (HE) stained sections by a charge coupled device (CCD) camera mounted on a microscope connected to a personal computer (PC) with an image array processor. From the grey level images obtained, cell nuclei were automatically segmented and the geometrical centres of cell nuclei were computed. The resulting 2dimensional (2D) swarm of pointlike seeds distributed in a flat plane was the basis for construction of the VD and its subgraphs. From the polygons, triangulations and arborizations thus obtained, 27 structural features were computed as numerical values. Comparison of groups (normal vs. cancerous oral mucosa, cervical and prostate carcinomas with good and poor prognosis) with regard to distribution in the values of the structural features was performed with Student's t-test.
We demonstrate that some of the structural features developed are able to distinguish structurally between normal and cancerous oral mucosa (P = 0.001), and between good and poor outcome groups in prostatic (P = 0.001) and cervical carcinomas (P = 0.001).
We present results confirming previous findings that graph theory based algorithms are useful tools for describing tis-
In vitro transformation studies have demonstrated that the addition of carcinogens to contact inhibited ordered fibroblast monolayer cultures results in loss of contact inhibition, with cells displaying striking criss-crossing growth patterns, where the degree of criss-crossing pattern may reflect the extent of oncogenic transformation [16][17][18][19]. These in vitro findings are an indication that the biological status of cells also is expressed in the tis-sue architecture. Hence, it is biologically meaningful to extract structural features from tissues for diagnostic and prognostic purposes, and to do this in a quantitative manner might improve the prognostic value in some tissues [20][21][22][23][24][25][26][27][28][29][30][31][32]. Previous findings in transitional carcinomas of the bladder indicate that graph theory based methods are useful tools in grading of malignant lesions [29], but a prognostic value was not demonstrated.
We have undertaken the present study in order to develop tools for fast, strictly quantitiative and reproducible tissue architecture analysis in epithelial tissues (squamous cell carcinomas from the prostate, cervix and oral cavity and normal oral mucosa) and to evaluate the diagnostic and prognostic potential of these methods in such tissues. By employing graphs such as the Voronoi Diagram (VD [Figs 1-4]) [33] and its subgraphs, the Delaunay Triangulation (DT [ Fig. 2 [39] and the Gabriel Graph (GG [ Fig. 7]) [40], the structural manifestations of cellular interactions in tissues may be quantified [41][42][43][44][45][46][47]. A total of 27 structural features were developed, taking into consideration the shape of individual structural entities (polygons, triangulations, arborizations), particularly derived from the VD; clusterings, particularly from the GG, and studying the order or randomness in the distribution of pointlike seeds, particularly derived from the UT and MST.

Material
The biological material investigated consisted of 8 cases of normal oral mucosa obtained from surplus tissue after plastic surgery on the gingiva in relation to implant surgery, 8 cases of carcinomas of the tongue, 10 cases of cervical carcinomas and 30 cases of prostate carcinomas. HE stained sections were made from paraffin embedded tissue blocks fixed in 4% formaldehyde.

Data acquisition
Grey level images from 5-7 µm thick HE stained sections were digitised using a charged coupled device (CCD) camera (Philips LDH 0670/00 equipped with a Hamamatsu AC Adaptor, type A3472) mounted on a Zeiss Axioplan 2 microscope using a Plan-Neofluar 40×/0.75 lens in addition to a Prior HI52V2 microscope stage. The final magnification was 400× at a resolution of 876 nm (0.9 micrometers) per pixel.

Segmentation
Local segmentation was used, and developed from an algorithm based on the size of the elements to be detected and their contrast to the background. Thresholding was based on the pixel darkness measured as integrated optical density (IOD). Any pixel with an IOD within a given range is turned ON, otherwise OFF. The algorithms for construction of a continuous area of interest (e.g., cell nucleus) were further based on mathematical morphology [48]. From the nuclear profiles, the geometrical center of gravity was computed. The resulting data were stored as files of coordinates, where the coordinates represented a center of gravity. Further analysis of these raw data with computation of structural features was done on a Pentium based PC running Windows 98 . Among the software facilities developed was the possibility to define digitally a subset of pointlike seeds in order to run the analyses in a limited window of analysis (Fig. 4).

Building a composite picture
A composite picture consisting of up to 50 fields of view was constructed by aligning each field of view according to a simple algorithm developed by the authors. For composite pictures generated by manual movement of the microscope stage, an algorithm using the binary mask of segmented nuclei was employed. One field of vision is composed of a matrix of 512 × 512 pixels, i.e., 512 separate columns and rows. When moving to the left in the visual field, the 128 left columns of the binary image are copied from the left to the right margin of the screen. The microscope stage is then moved until the binary mask is congruent to the grey level image. The offset of the movement thus is 384 columns (75% of the field of view), giving an overlap from one field to another of 25%. No further correction of the image alignment was performed. For automatic movement of the microscope stage a predetermined pattern of movement (spiralling) using the same offset as default was employed. Hereby, the stage was first moved one offset (384 columns) to the left, then one offset (384 rows) up, then two offsets (768 columns) to the right. Thereafter two offsets down, then three offsets to the left, then three offsets up, three followed by four offsets to the right and so on. The typical number of fields of view to be included was 25, 36 or 49, which makes up a square composite picture. For the automatic image acquisition, alignment of visual fields relied on the mechanical accuracy of the microscope stage.

Space partitioning
Space partitioning in our context is based on the geometrical center of segmented cell nuclei. Computing the geometrical centers for each nucleus within a considered area creates a 2D swarm of pointlike seeds, from which the VD (Fig. 1) is constructed. All other graphs employed (DT, MST, GG and UT) are subgraphs of the main graph, the VD (Fig. 8). This graph was chosen as the principle tool for exploring the tissue structure, as it is considered to be the most informative [33]. The algorithms for generating these graphs have been presented elsewhere [9,33,39,49] and are only briefly commented here. The schematic relationships between the VD versus DT and VD versus GG are shown in Figs 2 and 5, respectively. Figure 3 shows relationship of the VD to tissue structures. In Panel A (HE stained section) an epithelial island (E) can be seen bordering onto the underlying stroma (S). Panel B is the corresponding grey level images. To the right the detail is shown in larger magnification, where the pointlike seeds are superimposed on the cell nuclei, and Voronoi polygons are constructed to make up the VD for the considered area. A program for eliminating border effects in marginal polygons ( Fig. 8) was also developed. The window of analysis was defined digitally, by defining a closed contour with a digitizing pad and storing the coordinates of the contour. Only pointlike seeds within the contour were included in the analysis. The coordinates of the part of the contour crossing a marginal polygon were defined as the new edge in the polygon.

VD:
The VD for a set of random distributed pointlike seeds is shown in Fig. 1. When two points, p x and p y are in a plane π, a half-plane, denoted H(p x p y ), is defined by the perpendicular bisector of p x p y . The locus of points closer to p x than to any other point is the intersector of N − 1 perpendicularly oriented halfplanes, where N is the number of points in the considered space. Hence, Fig. 1).
A single Voronoi polygon is defined by the intersection (∩) of N −1 halfplanes in the considered space, i.e., the center point and its surrounding pointlike seeds (Panel A, Fig. 1). When applied to every point in the considered area, this rules gives the VD (Panel B, Fig. 1).

DT:
The DT represents the dual of the VD and is constructed by drawing the lines between the pointlike  seeds in adjacent Voronoi polygons (Fig. 2). The completed construction is a triangular network that covers the considered area. A Delaunay network in two dimensions consists of non-overlapping triangles where no pointlike seeds in the network are enclosed by the circumscribing circles of any neighbouring triangle.
MST: Considering n distinct points in a d-dimensional space allows for (n − 1)!/2 closed paths (or tours) through the space. Determining L(n, d), the minimum tour length is possible by defining the smallest constant Additionally, β(d) given by Fig. 6. The Ulam Tree represents a mathematical object growing in space and time according to specified rules [39]. The UT is generated from the VD, in such a manner that the "branches" of the tree only traverses polygons that are not traversed by any other branch of the tree. The structural feature ELH_av (average Edge Length Heterogeneity) is derived from the UT.
applies to almost all optimal tours in the considered ddimensional space. The above limit fails only for a negligible subset of tours [37]. Furthermore, the above approximation applies to any dimension [50]. The solution to the problem can be reached by several different computations. For our purpose, the MST was derived by a decimation of the DT [51].  Accordingly, p x and p y (and not p k ) are neighbours (Panel A, Fig. 7). A graph similar to the DT can be constructed. However, it differs from the DT in that it contains polygons in addition to triangles (Panel B, Fig. 7). Cases are matched by comparing the lengths and orientations of the edges associated with each pointlike seed in one graph with those of every pointlike seed in a second graph. The Gabriel graph is particularly sensitive to subtle differences in the number or relative positions of pointlike seeds, making it a suitable tool for detecting changes in cellular organization within tissues.
UT: The Ulam Tree represents a mathematical object growing in space and time according to specified rules [39]. The UT is generated from the VD, in such a manner that the "branches" of the tree only traverses polygons that are not traversed by any other branch of the tree (Fig. 6).

Topographical analysis
Applying the 27 algorithms we have developed on the polygons obtained when constructing the VD and its subgraphs (space partitioning) we have performed topographical analysis on tissue specimens from normal and cancerous epithelium. The number of epithelial cells included in the analysis varied from 1500-5000. For a more detailed description of the VD-based algorithms, see Appendix.

Border effects
Marginal seeds represent a source of error as they yield polygons with a morphology that deviates from the population as a whole (Fig. 8). Structural features derived from marginal polygons are irrelevant, as they represent a non-representative population, and thus are a source of errors. Accordingly, we developed software to eliminate these aberrations (Fig. 4).

Temporal aspects
Scanning 30-50 fields of view is done in 8-12 minutes, depending on whether it is done manually or automatically. Segmentation requires another 5-10 minutes, depending on the number of cells in the specimen. For computation of structural features, 30-90 seconds are required, giving a total time expenditure of approximately 14-25 minutes.

Statistical analysis
Student's t-test was used for comparison of groups. All P -values were two-tailed, and values less than 0.05 were considered to indicate statistical significance.

Results
A total of 10 of the 27 structural features we investigated were able to distinguish between normal and malignantly altered tissue and/or were shown to have a possible prognostic value (Tables 1-3). (Table 1) Eight biopsies from assumptively normal oral mucosa (acquired during gingivoplastic procedures in relation to serial extraction of teeth) and 8 cases of oral carcinomas of the tongue were compared with regard to values of 10 structural features. Of these, 6 features made it possible to distinguish between normal oral mucosa and carcinoma of the tongue, usually situated at the lateral border of tongue, bordering onto the floor of the mouth. This pertains to the features RF (roundness factor, from the VD, P = 0.01), RF_dis (disorder of the roundness factor disorder derived from the VD, P = 0.05), DKNN_av (average distance to the K-nearest neighbours, P = 0.03), DEL_av (average Delaunay Edge Length, P = 0.001), ELH_av (average edge length heterogeneity of the Ulam Tree, P = 0.02) and RMPB (radius of the maximum percolating ball, percolating the Delaunay network, P = 0.03). The ability of these methods to discern normal and cancerous oral epithelium points to a diagnostic potential as they obviously detect structural differences between normal and malignantly changed mucosa. (11.6-15.1) * Two-tailed. † Measured in pixels. ‡ Structural features for which the differences between groups reaches statistical significance. ¶ See Appendix for an explanation of the separate structural features. Results from running 10 different form parameters on altogether 10 cases of carcinomas of the cervix, 4 with good (relapse-free survival more than 12 years) and 6 with poor (relapse-free survival less than 5 years) prognosis when 5000 cells are include in the analysis. The featured DEL_av and ELH_av display significant differences in the two prognosis groups in this test set. P -values for the best structural feature are 0.001 (DEL_av and ELH_av).

Carcinomas of the cervix (Table 2)
Altogether 10 biopsies; 4 with a good (relapse-free survival more than 12 years) and 6 cases with a poor (relapse-free survival less than 5 years) prognosis were examined. A total of 4 structural features made it possible to distinguish between the two outcome groups. These were A_dis (area disorder, from the VD, P = 0.02), ELH_av (P = 0.02), NNRR (number of nearest neighbours within a restricted radius of 75 pixels, P = 0.03) and RMPB (P = 0.04).

Carcinomas of the prostate (Table 3)
The values of the same 10 form features as above when applied to 30 cases of carcinomas of the prostate (15 cases with good and 15 cases with poor prognosis) are shown. Altogether 5 structural features made it possible to distinguish between carcinomas of the prostate with a good and poor prognosis. These structural features were RF_dis (P = 0.01), A_dis (P = 0.02), DEL_av (P = 0.001), ELH_av (P = 0.02), DKNN (average distance to the K nearest neighbour, from the DT, P = 0.01) and RMPB (P = 0.03).
Note in particular that DEL_av and/or ELH_av as significant descriptors are common to all three sets of tissues.

Discussion
We present data from several tissues that demonstrate the possible diagnostic and prognostic value of  15 with good (relapse-free survival more than 10 years) and 15 with poor (relapse-free survival less than 3 years) prognosis.5000 cells were included in the analysis. P -value for the best structural feature is 0.02 (A_dis and ELH_av). tissue architecture analysis by VD-based algorithms. However, it should be kept in mind that we have tested out a large number of putatively informative structural features (27) on a limited number of objects (56). Testing out such a large number of structural features on a limited number of objects statistically is not without pitfalls [52]. The interpretations from this preliminary study must therefore be cautious, and a limited number (2-4) of structural features (e.g., DEL_av and ELH_av) should be applied to an independent test set with a number of objects considerably larger than the number of features explored [52]. Furthermore, these methods have to be tested out in a series of different tissues, to avoid the serious problem of over-fitting, where selected features characterise well the samples in specific training data, but not the general classes (e.g., prostatic carcinomas, but not all carcinomas). Over-fitting typically occurs when the number of features analyzed is high in relation to the number of samples considered [52].
The main graph in our context (VD) encompasses a number of subgraphs, such as the DT, MST, UT and GG, and is generally regarded as the most informative graph [33]. The VD is generated from the pointlike seeds representing the centres of gravity within cell nuclei of the considered tissues. The Voronoi polygon represents the region of influence of each seed. A priori, this does not have any direct biological correlate. However, in epithelial tissues, with only sparse intercellular substance, it roughly corresponds to the somata of the epithelial cells. The shape and size of cells is a structural feature of considerable interest in tradi-tional histological assessment of pathologically altered tissue, e.g., carcinomas. This relationship breaks down when we consider the stromal tissue, with abundance of intercellular substance (Panel C, Fig. 3). Thus, the algorithms we have developed directly derived from the VD we believe are best suited for tissues with a minimum of intercellular substance, as in epithelial tissues, although graph theory based methods could be applied to any tissue. Also, there is no reason that other features based on, e.g., the MST or UT should have such limitations.
The segmentation algorithms we developed performed with an acceptable level of precision (Panel C, Fig. 3). The algorithms represent a compromise of speed and precision. For the calculations, a minimum number of 1000-1500 cells were included. We have chosen such a fairly large number of objects to be included in the analysis, as preliminary runnings of computations indicated that the values of the structural features did not stabilise until at least 1000-1500 objects were included, the exact number depending on which tissue was analysed. The imaginary flat plane we consider in fact represents a 3-dimensional entity. Depending on the thickness of the sections considered some cell nuclei might be below or above the focal plane and therefore missed in the segmentation. However, for the sections thickness we have employed (5-7 µm), this has not been a major problem (Fig. 3). However, several observer note that in the most aggressive carcinoma of the oral cavity, the epithelial cells bordering onto the underlying stroma show a distinct blurring of their structural features, with a typically glossy appearance of the somata and nuclei (M. Bryne, personal communication). It is conceivable that because of this, a considerable number of nuclei in the area of interest could escape segmentation, with a resultant error in the estimation of the structural features. If this is abundant, a distinct prognostic group of lesions might be missed, at least with the HE staining procedures. In our study, however, this was not a prominent feature, and the precision in the segmentation is acceptable (Fig. 3).
A particular point of interest when investigating the invasive front of carcinomas, are the border effects, that tend to have a dominant effect when the epithelial islands become small and numerous (Fig. 8). In particularly aggressive lesions, the gross structures of the invasive front of carcinomas tend to disintegrate, with multiple small fingerlike projections into the underlying stroma. In sections, these projections will present as small epithelial islands, consisting of a very limited number of cells. This poses a possible problem with re-gard to border effects, as these will become dominant in small cluster of cells, perhaps eliminating them entirely. Again, these cases may be of particular prognostic interest.
Only structural features of the epithelial tissue have been investigated in this study. Most likely, the tumorhost response will result in structural alterations of diagnostic and prognostic value also in the underlying stroma. Such features could be the amount of inflammatory response, which can easily be assessed by our methods. However, current algorithms do not detect the specific nature of subepithelial lymphocyte infiltration. New methods for segmentation of immunohistochemically stained cell nuclei [53,54] might contribute to shedding more light on biological information contained in the pattern of inflammatory response.
Twenty-nine structural features based on algorithms derived from the Voronoi Diagram or its subgraphs on different sets of epithelial tissues, 10 of which were demonstrated to have a diagnostic or prognsotic potential. The ultimate test for these methods will be to employ a limited number of structural features (e.g., DEL_av and ELH_av) on an independent test set [52]. Polygon_nb denotes the number of polygons.

5) DEP_av
Average Delaunay Edge Probability. One edge of the DT belongs to two triangles, each being associated with an overlapping circle.
Here, d1 denotes the distance of the first vertex to its nearest neighbour. d2 denotes the distance of the second vertex to its nearest neighbour. r1 is the radius of the first circle associated with the two triangles sharing the considered edge d1. r2 is the radius of the first circle associated with the two triangles sharing the considered edge d2 (Panel B, Fig. 2). √ 3 is a normalization factor with respect to the triangular lattice. 6) DEP_dis Delaunay Edge Probability disorder. This feature denotes the disorder of the abovementioned Delaunay edge probability, and is given by the following equation:

7) DFRAC_av
This feature denotes theaverage fractal dimension of the Ulam Trees, more precisely an application of the Hausdorff fractal dimension which express the properties of topological defects in the tree structure, e.g., as related to a highly regular tree. Consider a closed contour (e.g., Ulam Tree) within a 3-dimensional space, on which two points, x and y, are placed. The Ulam Tree may be viewed as projected onto a 2dimensional flat plane with a unit of length correponding to the size of a pixel. The mean value of the area A covered by traversing in N steps from x to y along branches of the tree is given by A ∼ r 2 where r is given by the equation The quantity r effectively expresses the radius of the circle or square that has the same area as the projection of the traversed part of the tree projected onto the 2D plane where ][ denotes the mean values for all the possible contours within the considered space.
x and y are expressed as vectors.
The Hausdorff dimension of the Ulam Tree is given by the general equation A value of ∆ = 1 indicates that the contour is fractal, i.e., how fuzzy the contour is. The algorithm for computing ∆ is fairly simple. From the projection of the tree on a 2D plane (with pixels as the unit of length), one defines the upper, lower, left and right margins of the projected tree, which yields a rectangle with sides lx, ly. The r accordingly is defined as N can be derived from the number of nearest neighbouring distances required to build the Ulam Tree within a square where p x and p y lies on two opposing margins of the square. 8) DKNN_av Average distance to the K nearest neighbours. K was set to 16 after preliminary simulations showed that normal oral mucosa was statistically different from carcinomas of the tongue with regard to this feature (  Hence, the DRT is the sum of the weighted absolute differences between all the elements of the actual tree matrix (M p ) and a theoretical tree reference (M ref ) [51]. 11) ELH_av This is the average edge length heterogeneity of the UT's. ELH is representative of the intrinsic node to node distance variations in the current tree [39]. 12) HA_av Average area of holes within the considered area. Concidering the empty circles associated with each DT, a triangle is set to be a hole if a ball of the current radius is able to go through at least one edge of the triangle (i.e., the radius of the percolating ball is equal to or less than at least one of the edges of the triangle). The number of DTs that can be percolated by the ball is the average number of DT's belonging to the defined holes within the architecture. Considering a Voronoi Edge (VE, denoted ls) and the Delaunay edge that it bisects (lg), let r1 and r2 be the radius of the circles associated with the vertices of VE. If the length of ls is smaller than or equal to both r1 and r2, then the two DT's associated with the vertices of VE and accordingly belong to the same hole. Considering a circle of which radius is set to 75 pixels around one seed, NNRR is the number of other seeds (neighbours) lying within the circle. NNRR_dis is the disorder in the number of seeds over all the population located at least at 75 pixels from the border of the analysis window. 19) PTS probability of topological stability. The center of gravity of the nuclei are stored as coordinates. For a small error tolerance (one pixel in diameter), we move at random all the pointlike seeds and rebuild the graph. As long as one seed keeps the same neighbourhood as without any disorder, the local PTS is computed to be 1, otherwise it is computed to be zero. At each step in the simulation (with one pointlike seed moved one pixel in any direction) the PTS represents the number of seeds that kept (after disorder is introduced) the same neighbourhood as initially. The more the PTS is close to 1, the more stable the graph is to a random alteration. The closer the PTS is to zero the less valid local topology is, only statistics can then be computed about local topology. The marginal polygons are not considered. Considering the empty circles associated with each DT, a triangle is set to be a hole if the radius of its associated empty circle is below the current threshold. If a ball of the current radius is able to go through one edge of the DT, then the neighbouring triangle can be associated to the same hole as the considered triangle. The RMBP is the value of the radius of the biggest ball that is able to percolate the Delaunay network. In other terms, it is the maximum allowed radius to obtain a hole that is larger in area than 50% of the area of all the holes. 23) RF_av Average Roundness Factor, given as RF av = 4π perimeter 2 .

20) PTS_av
This equals the roundness factor of one polygon average RF is the average over the entire population except for the marginal polygons. 24) RF_dis Roundness Factor disorder.
This features acquires the value of 1 if all the RF's are the same and tends towards zero otherwise. The entire population except the marginal polygons were considered.

25) WGC Weighted global compacity. Consider one
Voronoi vertex and its associated three Delaunay seeds (Fig. 2). To each of these seeds corresponds a set of nearest neighbours located at distance d1, d2, d3 etc. If r is defined as the radius of the considered Delaunay circle, the compacity is defined as follows: The weighted compacity (WC) is equal to C multiplied by the area of the DT. By doing this, one takes into account the possibility that two different DT's can have exactly the same proportions, but different size and accordingly represent two differents structural entities. Only considering the unweighted compacity does not take this possibility into consideration. Only considering the WGC is the average WC over all nonmarginal polygons. 26) WGC_av Average Compacity.

27) WGC_dis
This represents the disorder of the Compacity, and is given by the equation