1. Introduction
In many 2D and 3D images, such as medical images, there appears a configuration of objects, and the analysis of objects in the image benefits from modeling the interplay of the different objects’ shapes and their relative positions. First steps for such an approach for medical images was begun by the MIDAG group at UNC led by Pizer, see e.g. [16], [18], [15], [14], and [2]. These results use a modification of the classical Blum medial axis to model the individual regions together with user chosen, somewhat ad hoc, approaches to relating neighboring objects. Results for the Blum medial axis of an individual region, introduced by BlumNagel [1], have concerned its generic structure using a number of different approaches (see, e.g., Mather [19], Yomdin [27], Kimia et al [17], Giblin and Kimia [11], [12]), and its computation (see, e.g., for “grassfire flow” Siddiqi et al. [25], the surveys by Pizer et al. [22] and [24] including Voronoi methods, and for bsplines, Musuvathy et al. [21]). The modification uses methods from “skeletal structure”models for objects as single regions with smooth boundaries (for 2D and 3D see [5] or [7] and more generally [3], [4]). These results add considerable flexibility and stability to the classical Blum medial axis.
In [9] we introduced for configurations of objects in or medial/skeletal linking structures which capture both the shapes of the individual objects and their relative positions in the configuration. In this paper we develop an approach to the “positional geometry” of a configuration using mathematical tools defined in terms of the linking structure, which build upon the methods already developed for skeletal structures for single regions. Moreover, we will see that certain constructions and operators defined for skeletal structures and used for determining the geometry of individual objects can be extended to give simultaneously the positional geometric properties of the entire configuration. As such this provides a natural progression from individual objects to configurations of objects.
Given a collection of configurations, we may ask what are the statistically meaningful shared geometric properties of the collection of configurations, and how the geometric properties of a particular configuration differ from those for the collection. To provide quantitative measures for these properties, we will directly associate geometric invariants to a configuration. Such invariants may be globally defined depending on the entire configuration or locally defined invariants depending on local subconfigurations associated to each object.
For example, if we view the union of the objects as a topological space, then we can measure the GromovHausdorff distance between two such configurations. We may also use the geodesic distance between the two configurations measured in a group of global diffeomorphisms mapping one configuration to another. Such invariants give a single numerical global measure of differences between two configurations. Instead, we will use skeletal linking structures associated to the configurations to directly associate both global and local geometric invariants which can be used to measure the differences between a number of different features of configurations in a variety of different ways.
In introducing these invariants, we will be guided by several key considerations. The first involves distinguishing between the differences in the shapes of individual objects versus their positional differences and how each of these contributes to the differences in the configurations, as illustrated in Figure 1. A first question for objects that do not touch is when they should be considered “neighbors”and what should be the criterion? Second, in measuring the relative positions of neighboring objects, more than just the minimum distance between their boundaries is required; we also wish to measure how much of the regions are close, see, e.g., Figure 2. A goal is then to define numerical measures of closeness of objects which takes into account both aspects.
Third, we seek a measure to distinguish how significant are objects within the configuration and to identify those that are mainly outliers. This would provide for a configuration a
hierarchical structure for the objects, indicating which objects are most central to the configuration and which are less positionally significant. For example, in Figure 3, the position of object makes it more important for the overall configuration in b) than in a), where it is more of an “outlier.” A small movement of in a) would be less noticeable and have a smaller effect to the overall configuration than in b). By having a smaller effect we mean that the deformed configuration could be mapped to the original by a diffeomorphism which has smaller local distortions near the configuration in the case of a) versus b). Finally, there is the question of whether there are numerical invariants which can be used to determine when there are identifiable subconfigurations. An example of this is seen in Figure 4.In Sections 2 and 3 we consider the “linking flow”associated to the linking structure. The nonsingularity of the linking flow is guaranteed in §2 by “linking curvature conditions” on the linking functions, given in a form which extends that given in [3, Thm 2.5] for the radial flow. Next, in §3, we use this linking flow to identify both the internal regions of neighboring objects and external regions shared by them, which are the “linking neighborhoods” between objects. Then, in §4, we show how to compute integrals over the boundaries of the objects or over general regions inside or outside the objects as “medial and skeletal linking integrals,”which are integrals defined on the skeletal sets in the interiors of the objects. Lastly, in §5, we introduce and compute several “volumetric–based” numerical invariants that include measures of the relative closeness of neighboring objects and relative significance of the individual objects. We furthermore show how they may be computed from the medial/skeletal linking structure via skeletal linking integrals.
Then, in §6 we will combine the invariants which measure these geometric features in two different ways. One is to construct a “proximity matrix”which captures the closeness of all objects in the configuration and to which the PerronFrobenius theorem can be applied, yielding a unique set of “proximity weights”assigned to the objects, measuring their overall closeness in the configuration. The second is to construct a “tiered graph structure,” which is a graph with vertices representing the objects, edges between vertices of neighboring objects, and values of significance assigned to the vertices, and closeness assigned to the edges. Then, as thresholds for closeness and significance vary, the resulting subgraph satisfying the conditions will exhibit the central objects (and outliers) of the configuration, various subgroupings of objects and a hierarchical ordering of the relations between objects. The skeletal linking structure also allows for the comparison and statistical analysis of collections of objects in and , extending the analyses given in earlier work for single objects.
2. Linking Flow and Curvature Conditions
We model a configuration of objects in 2D or 3D images by a collection of regions in either or , with each modeling one of the objects, whose boundaries may share common boundary regions (see e.g. [9, Fig. 1 and 2]) and along their edges there are singularities of generic type corresponding to whether the objects are flexible or rigid (see e.g. [9, Fig. 5 and 6]).
Medial/Skeletal Linking Structures
We recall from [9] the definition of a skeletal linking structure for the multiobject configuration . First of all, it consists of skeletal structures for each , where each is a stratified set in and
is a multivalued vector field on
whose vectors end at boundary points in (see[5] for 2D or 3D regions, or more generally [4]). By being a stratified set we mean for regions in it is a disjoint collection of smooth curve segments ending at branching or end points, and for , it is a disjoint collection of smooth surface regions, curve segments and points, with the surfaces ending at the curves and points. We may define a stratified set from by replacing points by pairs , where varies over the multiple values of at , with strata formed from strata of together with a choice of smoothly varying values for . The mapping sending sends the strata to strata of . This has the benefit of being able to consider multivalued objects on as singlevalued objects on .We express , where are multivalued unit vector fields on . In addition, the are multivalued “linking functions”defined on , and there are then defined the multivalued “linking vector fields” on each . These become single valued on .
There are additional conditions of [9, Def. 3.2] to be satisfied for to be the skeletal linking structure for . Conditions S1  S3 concern a refinement of the stratification of and the differentiability properties of on the strata of the refinement. Also, the conditions L1  L4 concern the relations between the linking vector fields from different objects and the nonsingularity of the “linking flows” generated by the linking vector fields. Regions defined by the linking flows are what we use to identify properties of the positional geometry of the configuration.
Nonsingularity of the Linking Flow
The nonsingularity of the radial flow, which occurs within the regions, was established in [3, §4] (and see [5, §2]) using the radial and edge shape operators and , which are multivalued operators defined on , with defined at all points except the edge points of and defined at these points. These capture the geometric properties of the radial vector field and play an important role in determining the local geometric properties of the boundary and the relative and global geometric properties of the region (see [4] and [5, §3, 4]
). Although their properties differ from those of the differential geometric shape operators appearing in differential geometry, their eigenvalues
, the principal radial curvatures and generalized eigenvalues , the principal edge curvatures, play an equally important role.We next explain how they appear in the sufficient conditions we give for nonsingularity of the linking flow. As well we give formulas for the evolution of the radial and edge shape operators under the linking flow.
Recall [9, (3.1)], the linking flow from is defined by
(2.1) 
where ranges over all possible values and
(2.2) 
For , this flow is the radial flow at twice the speed and it extends the radial flow to the exterior of the regions for and ends at the “linking axis”. We let for each and refer to as the linking mapping from strata of the refinement of to strata of the linking axis. We refer to the combined union of the for all by .
To establish the conditions for the nonsingularity of the linking flow for the skeletal linking structure , we introduce the following two conditions:

(Linking Curvature Condition ) For all points and all values ,
for all positive principal radial curvatures ;

(Linking Edge Condition ) For all points (the closure of ),
for all positive principal edge curvatures .
In these conditions, the values of and either or are at the same point and for the same value of .
The nonsingularity of the linking flow (and the radial flow) for a skeletal linking structure is given by the following.
Theorem 2.1 (Nonsingularity of the Linking Flow).
Let be a skeletal linking structure in or which satisfies: the Linking Curvature Conditions and Linking Edge Conditions on all of the strata of all . Then, the radial flow also satisfies the radial curvature and edge curvature conditions on each stratum. Hence, the following properties hold.

On each stratum of the refinement of , the linking flow is nonsingular and remains transverse to the radial lines.

Hence, the image of the linking map on is locally a smooth stratum of the same dimension and which may only have nonlocal intersections from distant points in . If there are no nonlocal intersections then is a smooth stratum.

The image of a stratum of under the radial map is a smooth stratum of of the same dimension.

For both flows, at points of the top dimensional strata, the backward projection along the lines of will locally map strata of , resp. , diffeomorphically onto the smooth part of .

Thus, if there are no nonlocal intersections, each will be a piecewise smooth embedded surface.
Evolution of the Shape Operators Under the Linking Flow
We may translate the vectors along the lines of each to the level sets of the linking flow. We may use these translated vectors as a radial vector field on the level set. Hence, for each we may define corresponding radial shape operators on the level sets (curves or surfaces) in a neighborhood of . We can best relate them to the radial or edge shape operators on in terms of their matrix reprsentations. For and each choice of , there is a smooth stratum of containing in its closure and which smoothly extends through and a smoothly varying value of defined in a neighborhood of extending . We denote the tangent space to this stratum at by . For we choose a basis for , which is either a single vector for configurations in , or for . We then let denote the image of under which is a basis for the tangent space to the level set at . At points for , instead a nonzero vector is completed to a basis using for the source and the unit normal vector to for the target. We denote the resulting matrix representation of by ; but for , it evolves to also become radial shape operators ; and we use a basis , which is the image of the basis of under with adjoined.
Remark 2.2.
In the 3D case and are matrices; while in the 2D case, is a matrix formed from the single radial curvature (see e.g. Examples 2.3 and 2.4 in [5]).
Then, the evolved radial shape operators under the linking flow are given by the following.
Proposition 2.3 (Evolution of the Shape Operators).
Suppose with value (with a smooth value of in a neighborhood of ). Provided the following conditions are satisfied, the linking flow is nonsingular and the evolved radial shape operator on the level surface in a neighborhood of is given by the following.

For , if is not an eigenvalue of for , then

For , if is not a generalized eigenvalue of for , then
The derivation of these results can be found in [8, §7] and [10, Chap. 6] extending the results in [4], or see [5, §2].
Shape Operators on the Boundary and Linking Medial Axis
As a consequence of the corollaries, we can deduce the shape operator for the linking axis , and in a region where the partial Blum condition is satisfied, the differential geometric shape operator on the boundary. First, for the boundary, it is reached at . If for the radial vector field is orthogonal to at the point , then we say that the skeletal structure satisfies the partial Blum condition at . If it satisfies the partial Blum condition in a neighborhood of a smooth point of , then in Proposition 2.3 gives the differential geometric shape operator for at , and hence completely describes the local geometry of the boundary at . This result and its consequences follow from [4, §3] and also see [5, §3].
If instead we consider the image of under the linking flow, then to such a point there is the corresponding point . We then have a value of a radial vector field at , with , , and associated to the value . This vector at the point ends at . This defines a multivalued vector field on . Thus, we can view as a skeletal structure for the exterior region. We can determine the corresponding radial or edge shape operator at by the following (see [8, Cor. 8.7]).
Corollary 2.4.
If is as in the above discussion, then the radial shape operator for the skeletal structure at is given by either: if is a nonedge closure point, then with the notation of Proposition 2.3,
or if is an edge closure point, then
This follows because the associated unit vector field at is . Thus, the radial shape operator for at is the negative of that for the stratum of , viewed as a level set of the linking flow from . Hence, by Proposition 2.3 we obtain the result.
3. Positional Properties of Regions Defined Using the Linking Flow
We next consider how the medial/skeletal linking structure for a multiobject configuration in or allows us to introduce numerical measures capturing various aspects of the object’s positions within the configuration. There are two possibilities for this. One is to base the numerical quantities on geometric properties of the boundaries of the objects. The second is to use instead volumetric measures for subregions of the objects and identified regions in the external complement which capture positional information about the objects.
The problem with the first choice is that two regions may show the same boundary region to each other even though there are other pairs of regions showing the same boundaries to each other but which may have completely different shapes and volumes, as shown in Figure 5. The alternative is to keep track of the relation between all points of all boundaries of each pair of regions. However, this involves an enormous redundancy in the data structure. The skeletal linking structure avoids this redundancy, allowing additional geometric information to be computed directly from the linking structure. We also shall see in §4 that for general skeletal linking structures, using volumetric measures to capture positional information will allow us to express these numerical quantities as integrals over the internal skeletal sets of appropriate mathematical quantities derived from the linking structure .
Because finite volumetric measures will require bounded regions in the complement, we will first consider regions defined in the unbounded case and then introduce bounded versions.
Medial/Skeletal Linking Structures in the Unbounded Case
We begin by considering regions and modeling objects in the configuration that are linked via the linking structure and identifying regions using the linking flow on each . We recall in [9] that and are said to be “linked” if there are strata in and in which map to the same stratum in under the linking flow. We then introduce the following regions as illustrated in Figure 6.
Regions Defined by the Linking Flow:

will denote the union of the strata of which are linked to strata of , and we refer to it as the strata where is linked to (the strata being in indicate on which “side”of the linking occurs).

denotes the region of linked to .

denotes the linking neighborhood of linked to .

is the boundary region of linked to .

, is the total region for linked to .
Then, we make a few simple observations. First, will consist of the strata of the linking axis where the linking between and occurs; and second, the regions for a fixed but different may intersect on the images under the linking flow of strata where there is linking between and two or more other objects.
Next, strata of may involve “selflinking,” which means different strata of may be linked to each other (see e.g. [9, Fig. 9]). We will still use the notation , , etc. for the strata, regions etc. involving selflinking. Then, will intersect on strata where “partial linking” occurs.
Finally the remaining strata in lie in , which is introduced in [9] and consists of the union of strata which are unlinked. By property in [9, Def. 3.2], the radial flow from the union of the defines a diffeomorphic parametrization of the complement of the regions reached by the linking flow. We may flow at twice the radial flow speed to agree with the linking flow on the rest of . We still refer to this completed flow as the linking flow, and then denote the corresponding regions for by: , , and .
Then, we have the decompositions
(3.1) 
with
but the various and/or may have nonempty intersections, as explained above. There are analogous decompositions for and . Also, we denote the total linking neighborhood by . Then, is the total neighborhood of (in the complement of the configuration), whose interior consists of points external to the configuration which are closest to .
We let denote the portion of the boundary that is not shared with any other region. We recall that on the strata of corresponding to those in , the linking flow is constant in for . Hence, at these points only consists of boundary points of . Off of these points we can describe the structure of using the linking flow. We summarize the consequences of the properties of the linking flow (see [8, Cor. 9.2]).
Corollary 3.1.
For a multiobject configuration with skeletal linking structure , there are the following parametrizations for each region associated to by the portions of the level sets of the linking flow in those regions:

is parametrized by the level sets of the linking flow for ;

is parametrized by the level sets of the linking flow for ;

is parametrized by the level sets of the linking flow for ; and

is parametrized by the level sets of the radial flow for .
Medial/Skeletal Linking Structures for the Bounded Case
As we have mentioned, because the regions are often unbounded, there is no meaningful numerical measure of their sizes. We can overcome this problem for practical considerations by introducing a bounded region containing the configuration and such that its boundary is transverse both to the stratification of and to the linking vectors on . In [9], we describe a number of different possibilities for obtaining bounded regions including: a bounding box or bounding convex region, the convex hull of the configuration, a natural or intrinsic bounding region, or a region defined by a userspecified threshold for linking, see, e.g., Figure 8.
Then, we can either truncate the linking vector field, or define it on all of for all by defining it on and subsequently refining the stratification so that the linking vector field ends at on appropriate strata. This maintains the nonsingularity of the linking flow as we are merely either reducing or defining on .
Thus, we have the corresponding properties from Corollary 3.1, except that for properties 2), 3), and 4) the linking flow and corresponding regions , , and may only extend to . These regions, which are now compact, are obtained from the unbounded regions by intersecting them with . We refer to this as the bounded case, referring the reader [9, §3] for more details. Moreover, we still obtain analogous formulas for the evolution of the radial shape operators for those level sets of the linking flow while they remain within .
Relevance of the Linking Regions for Positional Geometry
The regions capture various aspects of the positional geometric properties of the configuration. Two regions and are neighbors if the regions , , , etc are nonempty, as are the corresponding , etc. Then, the and represent the boundary regions “between”these neighbors. Moreover, the and represent the internal portions of the regions which are “closest”to the neighbors. These can be compared to the linking neighborhoods and to see how close the neighbors are. The larger the linking neighborhoods are compared to the internal neighboring regions the further away are the neighboring regions. If one region has large linking neighborhoods relative to all of its neighbors, then it plays a less significant positional role for the configuration. This perspective will lead us in §5 to introduce volumetric invariants of these regions which capture this positional geometry. Before doing so, we next explain how numerical volumetric invariants can be obtained from the linking structures using “linking integrals”on the skeletal sets of the regions.
4. Global Geometry via Medial and Skeletal Linking Integrals
We now will use the associated regions we have introduced for a configuration via a skeletal linking structure to define quantitative invariants measuring positional geometry for the configuration. We will do so in terms of integrals which are defined on the internal skeletal sets. We begin by defining these integrals, and then we give a number of formulas for integrals of functions on the regions or boundaries of the configuration in terms of these integrals on the internal skeletal sets (see [8, §10] and for 2D and 3D single regions [7, §3.4]).
In practice, skeletal structures have been modeled discretely, as can be skeletal linking structures. For these the integrals then can be discretely approximated from the linking structure to compute the appropriate numerical invariants.
Defining Medial and Skeletal Linking Integrals
We begin by considering a medial or skeletal linking structure for the configuration in or . We again let denote the disjoint union of the for each region for . Each has its double , so we introduce the double for the configuration by . For each , there is a canonical finitetoone projection , mapping . The union of these defines a canonical projection , such that for each .
We will define the skeletal integral on for a multivalued function , by which we mean for any , may have a different value for each different value of at . Such a pullsback via to a welldefined map so that .
We recall that by Proposition 3.2 of [6], for each , there is a positive Borel measure on , which we call the medial measure. If we let , , denote the inverse images of under the canonical projection map , then each is a copy of , representing “one side of ” with the smoothly varying value of associated to the copy. For each copy we let . Here denotes the Riemannian length or area on , as , resp. . Also, where is a value of the unit vector field corresponding to the smooth value of for and is the normal unit vector pointing on the same side as .
Then, the integral of the multivalued function over is, by definition, the sum of the integrals of the corresponding values of over each copy with respect to the medial measure . It is shown in [6] that for continuous , this gives a welldefined integral and this extends to integrals of “Borel measurable functions” on , which include piecewise continuous functions. Then, these distinct medial measures on together define a medial measure on ; and the integral of a Borel measurable multivalued function on is defined to be
(4.1) 
where each integral on the RHS is the integral of over with respect to the measure , and it can be viewed as an integral of over “both sides of .”. If the integral is finite then we say the function is “integrable”.
We refer to the integrals in (4.1) as medial or skeletal linking integrals, depending on whether the linking structure is a Blum medial linking structure or a skeletal linking structure.
Computing Boundary Integrals via Medial Linking Integrals
We now show how, for a multiobject configuration with “full” Blum linking structure, we may express integrals of functions on the combined boundary as medial integrals. We emphasize that the full Blum linking structure allows the Blum medial axis to extend up to the edgecorner points of the boundaries. This does not alter the existence nor definition of the integrals, see [8, §10].
First, we consider a Borel measurable and integrable function which is multivalued in the sense that for any edgecorner point , may take distinct values for each region , , containing on its boundary. Thus, if denotes the shared boundary region of and , may take different values on and . For example, and may have different boundary properties such as densities measured by .
By the integral of such a multivalued function over we mean
where denotes the values of on for and denotes the Riemannian length (for 2D) or area (for 3D) for each .
Then, for the radial flow map , we define by , where the value on is the value associated to . Then, is a multivalued Borel measurable function on . We may compute the integral of over by the following result [8, Thm 10.1].
Theorem 4.1.
Let be a multiobject configuration with (full) Blum linking structure. If is a multivalued Borel measurable and integrable function, then
(4.2) 
where is the radius function of each .
In the case of a skeletal structure, there is a form of Theorem 4.1 which still applies. For each region , with , let denote a Borel measurable region of which under the radial flow maps to a Borel measurable region of . Let and . We suppose that the skeletal structure satisfies the “partial Blum condition” on , by which we mean: for each , the compatibility form vanishes on (recall this means that the radial vector at points of is orthogonal to at the point where it meets the boundary). Note that for a skeletal structure this forces to be contained in the complement of .
Then, there is the following analogue of Theorem 4.1.
Corollary 4.2.
Let be a multiobject configuration with skeletal linking structure which satisfies the partial Blum condition on the region , with image under . If is a multivalued Borel measurable and integrable function, then
(4.3) 
Remark 4.3.
One may compute the length (2D) or area (3D) of by choosing in Theorem 4.1. There is an analogous result for a region which is the image of a region under the radial flow. If the configuration is modeled by a skeletal linking structure which satisfies the partial Blum condition on , then the length, resp. area, of is given by .
Computing Integrals over Regions as Skeletal Linking Integrals
Next, we turn to the problem of computing integrals over regions which may be partially or completely in the external region of the configuration. Quite generally we consider a Borel measurable and integrable scalarvalued function defined on or , but only nonzero on a compact region. We shall see that we can compute the integral of as an integral of an appropriate related function on the internal skeletal sets.
Since we are in the unbounded case, we first modify the skeletal linking structure by defining on to be . The linking flow on is a diffeomorphism for , see [8, Prop. 14.11].
Next, we replace the linking flow by a simpler elementary linking flow defined by , for (or if ). The elementary linking flow is again along the radial lines determined by ; however, the rate differs from that for the usual linking flow. This means that the level surfaces will differ, although the images of strata under the elementary linking flow agree with that for the linking flow. In addition, as the linking flow is nonsingular, the linking curvature and edge conditions are satisfied. Then, viewing the linking vector field as a radial vector field, the radial curvature and edge curvature conditions are satisfied, and hence imply the nonsingularity of the elementary linking flow.
Then, using the elementary linking flow, we can compute the integral of as a skeletal linking integral. We define a multivalued function on as follows: for with associated smooth value and linking vector in the same direction as (so ),
(4.4) 
provided the integral is defined. Then, we have the following formula for the integral of as a skeletal linking integral [8, Thm 10.6].
Theorem 4.4.
Let be a multiobject configuration in or with a skeletal linking structure. If is a Borel measurable and integrable function which is zero off a compact region, then is defined for almost all , it is integrable on , and
(4.5) 
Remark 4.5.
If we compare this formula with that given for a single region in Theorem 6.1 of [6], we notice they have a slightly different form in that a factor of appears to be missing . However, as noted in Remark 6.2 of that paper, it is possible to use a change of coordinates to rewrite
(4.6) 
so that the form of (4.6) agrees with the form given in [6]. The apparent difference in form will also appear in all of the following formulas compared with the corresponding ones in [6].
Reducing to Integrals for Bounded Skeletal Linking Structures
We may replace the unbounded skeletal structure by a bounded one and replace the integrals by integrals over bounded regions. If is zero off the compact region , then we may find a compact convex region with smooth boundary containing both the configuration and . Then, we can modify the linking structure by reducing the so it is truncated where it meets , the boundary of , and defining on as the extensions of the radial vectors to where they meet . Because the extended radial lines are transverse to , the new smaller values of remain differentiable on the strata of each . Still letting the denote the new smaller values, the corresponding truncated vector fields will still be denoted by . Then the formula for the integral of is still given by (4.5). We shall assume we have chosen a bounded linking structure for the remainder of this section.
To simplify the statements for the remainder of this section, we shall use the notation for a region or : for or for .
For each , we let
(4.7) 
We can view as a weighted dimensional measure of the length of the intersection of with the linking line from determined by . Then, applying Theorem 4.4 in the special case where , we obtain an analogue of Crofton’s formula giving the area, resp. volume, of as a skeletal integral of using [8, Cor. 10.9].
Corollary 4.6 (Crofton Type Formula).
Let be a multiobject configuration with skeletal linking structure in for , or . Suppose is a compact subset. Then
(4.8) 
Decomposition of a Global Integral using the Linking Flow
We next decompose the integral on the RHS of (4.5) into internal and external parts using the alternative integral representation of using the linking flow. We do so by applying the change of variables formula to relate the elementary linking flow with the linking flow , both of which flow along the linking lines but at different linear rates.
We define
(4.9) 
Then, we may decompose as follows, [8, Cor. 10.10].
Corollary 4.7.
Let be a multiobject configuration in or with a skeletal linking structure. If is a Borel measurable and integrable function for , resp. , which equals off a compact set, then and are defined for almost all , they are integrable on , and
(4.10) 
where
(4.11) 
with an analogous formula with replaced by everywhere in (4.11).
The first integral on the RHS of (4.10) is the “interior integral” of within the configuration using the radial flow, and the second integral is the“external integral” computed using the linking flow outside of the configuration. Then we may further decompose each of these integrals using (4.11) into integrals over the distinct linking regions as illustrated in Figure 7.
Skeletal Linking Integral Formulas for Global Invariants
We now express the areas (2D) or volumes (3D) of regions associated to the linking structure, which we introduced in §3, as skeletal linking integrals. We may apply the same reasoning as in Corollary 4.6, using Corollary 4.7 to compute the volume of a compact 2D or 3D region as a sum of internal and external integrals.
For these calculations we will use the polynomial expression in
(4.12) 
In the 2D case, ; and in the 3D case, , where and , for the principal radial curvatures.
Areas and Volumes of Linking Regions as Skeletal Integrals
Then, we can express the areas or volumes of various linking regions as integrals of for various choices of .
Corollary 4.8.
Let be a multiobject configuration with a bounded skeletal linking structure in for , resp. . Then, the areas, resp. volumes, of linking regions in are given by the following:
(4.13) 
As well as these formulas, we can compute the volumetric invariants of other linking regions such as , , etc. using skeletal linking integrals of the polynomials or . For example,
(4.14) 
with an analogous formula for .
As a consequence, we obtain generalizations of the classical formula of Weyl for “volumes of tubes”and Steiner’s formula for volumes of “annular regions”.
Corollary 4.9 (Generalized Weyl’s Formula).
Let , for or , be a multiobject configuration with a bounded skeletal linking structure. Then,
(4.15) 
The sense in which this generalizes Weyl’s formula is explained for the case of a single region with smooth boundary in [6, §6, 7]. For Steiner’s formula, we note that as explained in §3, represents the total neighborhood of , which is the region about extending along the linking lines. This is a generalization of an “annular neighborhood”about a region which depends on the specific type of bounding region (see Figure 8).
Corollary 4.10 (Generalized Steiner’s Formula).
Let , for or be a multiobject configuration with a bounded skeletal linking structure. Then,
(4.16) 
Remark 4.11.
The regions in both generalizations of Weyl’s formula and Steiner’s formula for different and will only intersect in lower dimensional regions. Thus, in both cases we can sum the integrals on the RHS for multiple to obtain formulas for a union of .
5. Positional Properties of MultiObject Configurations
In this section we define positional geometric invariants of configurations in terms of volumetric measures of associated regions defined by the linking structure. We emphasize that the volumetric measures versus boundary measures of positional geometry have two advantages: 1) they are computable from the skeletal linking structure, and unlike surface measures, they do not require the partial Blum condition to compute the invariants; and 2) as in Figure 5, the volumetric invariants capture the total geometric structure of regions better than boundary measures.
We proceed as follows. We first use the linking structure to determine which of the objects should be regarded as neighboring objects. Then, we use the regions associated to the linking structure to define invariants which measure the closeness of such neighboring objects. Second, we further introduce numerical invariants measuring the positional significance of objects for the configuration. These allow us to identify which objects are central to the configuration and which ones are peripheral.
In §6 we will use the closeness measures to construct a “proximity matrix”which yields proximity weights for the objects based on their closeness to other objects. We also will use both types of invariants to construct a tiered linking graph, with vertices representing the objects, and edges between vertices representing neighboring objects, with the closeness and significance values assigned to the edges, resp. vertices. By applying threshold values to this structure we can exhibit the subconfigurations within the given thresholds.
Neighboring Objects and Measures of Closeness
We consider a configuration , with a bounded skeletal linking structure. We use linking between objects and as a criterion for their being neighbors, so that objects which are not linked are not considered neighbors. The simplest measure of closeness between neighboring objects is the minimum distance between the objects. However, this ignores the size of the objects and how big a portion of each object is close to the other object, as illustrated in Figure 2 where is close to for a small region but is close to over a larger region. Moreover, if we choose a more global definition of closeness involving all neighboring boundary points, then as in Figure 5, this will not measure the portions of the objects which are close. We do overcome both of these issues by using volumetric measures of appropriate regions defined using the linking structures.
For a configuration with a skeletal linking structure, we introduced in §3 regions and which capture the neighbor relations between and . Since and share a common boundary region in , they are both empty if one is, and then both and are empty. In that case and are not linked. Otherwise, we may introduce a measure of closeness.
There are two different ways to do this, each having a probabilistic interpretation. First, we let
Then,
is the probability that a point chosen at random in
will lie in (see Figure 9); so is the probability that a pair of points, one each in and both lie in the corresponding regions and