Topology preserving thinning for cell complexes
TTOPOLOGY PRESERVING THINNING OF CELLCOMPLEXES
PAWE(cid:32)L D(cid:32)LOTKO, AND RUBEN SPECOGNA
Abstract.
A topology preserving skeleton is a synthetic representationof an object that retains its topology and many of its significant mor-phological properties. The process of obtaining the skeleton, referred toas skeletonization or thinning, is a very active research area. It playsa central role in reducing the amount of information to be processedduring image analysis and visualization, computer-aided diagnosis or bypattern recognition algorithms.This paper introduces a novel topology preserving thinning algo-rithm which removes simple cells —a generalization of simple points—ofa given cell complex. The test for simple cells is based on acyclicitytables automatically produced in advance with homology computations.Using acyclicity tables render the implementation of thinning algorithmsstraightforward. Moreover, the fact that tables are automatically filledfor all possible configurations allows to rigorously prove the generalityof the algorithm and to obtain fool-proof implementations. The novelapproach enables, for the first time, according to our knowledge, to thina general unstructured simplicial complex. Acyclicity tables for cubi-cal and simplicial complexes and an open source implementation of thethinning algorithm are provided as additional material to allow theirimmediate use in the vast number of practical applications arising inmedical imaging and beyond.
Keywords: skeleton, skeletonization, iterative thinning, topology preservation, al-gebraic topology, homology, topological image analysis Introduction
Thinning (or skeletonization) is the process of reducing an object to itsskeleton. The topology preserving skeleton may be informally defined as athinned subset of the object that retains the same topology of the originalobject and often many of its significant morphological properties. Thinningis a very active research area thanks to its ability of reducing the amount a r X i v : . [ c s . C V ] F e b PAWE(cid:32)L D(cid:32)LOTKO, AND RUBEN SPECOGNA of information to be processed for example in medical image analysis andvisualization as well as simplifying the development of pattern recognitionor computer-aided diagnosis algorithms. Hence, it is not surprising thatthinning gained a pivotal role in a wide range of applications. An exhaustivereview of the literature is beyond the scope of this work. Two dimensionalskeletons have been used for digital image analysis and processing, opticalcharacter and fingerprint recognition, pattern recognition and matching, andbinary image compression since a long time ago, see for example the surveypaper [1]. More recently, three dimensional skeletons have been widely usedin computer vision and shape analysis [2], in computer graphics for meshanimation [3] and in computer aided design (CAD) for model analysis andsimplification [4], [5] and for topology repair [6].There is also a vast literature of applications of skeletons in medical imag-ing. They have been used for route planning in virtual endoscopic naviga-tion [7], for example in virtual colonoscopy [8]-[12] or bronchoscopy [13].Skeletons have also been an important part of clinical image analysis byproviding centerlines of tubular structures. In particular, there is a largebody of literature showing applications of skeletons to blood veins centerlineextraction from angiographic images [12], [14]-[20], and intrathoracic airwaytrees classification for the evaluation of the bronchial tree structure [21].Also protein backbone models can be produced with techniques based onskeletons [22]. Furthermore, many computer-aided diagnostic tools rely onskeletons. For example, skeletons have been used to identify blood vesselsstenoses [23]-[25], tracheal stenoses [26], polyps and cancer in colon [27] andleft atrium fibrosis [28].There are medical applications of skeletons where topology preservationis essential. Non invasively determine the three-dimensional topological net-work of the trabecular bone [29] is a good example. Indeed, many studiesdemonstrate that the elastic modulus and strength of the bones is deter-mined by the topological interconnections of the bone structure rather thanthe bone volume fraction [30], [31]. Therefore, topological analysis plays afundamental role in computer-aided diagnostic tools for osteoporosis [30].Topology preserving thinning is non trivial and a vast literature, brieflysurveyed in Sec. 1.1, has been dedicated to this topic. In particular, thin-ning by iteratively removing simple points [32] is a widely used and effectivetechnique. It works locally and for this reason is efficient and easy to imple-ment.While reading the literature one may notice that thinning algorithms areclaimed to be “topology preserving,” even though in most cases a precisestatement of what that means is left unaddressed. This paper uses homologytheory [33] to rigorously define what the virtue of being topology preservingactually consists of. This theory is less intuitive than the concepts usedso far, including simple homotopy type [34], but exhibits some importanttheoretical and practical advantages that will be highlighted later in thepaper. We remark that a homological definition of simple points has already
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 3 been used in the context of skeletonization in [16], [35], but only in the caseof cubical complexes. This paper generalizes this idea to cell complexesthat are more general than cubical complexes. There are many applicationsthat would benefit from an algorithm that deals with general unstructuredsimplicial complexes [p. 35][36]. In fact, the geometry of three-dimensionalobjects is frequently specified by a triangulated surface, obtained for exampleby using an isosurface algorithm as marching cubes [p. 539][36], [12] appliedon voxel data from computed tomography, magnetic resonance imaging orany other three-dimensional imaging technique. Another possibility is toobtain the triangulations from the convex hull of point clouds provided forexample by 3d laser scanners. Triangulated surfaces offer two potentialadvantages over voxel representation. They allow to adaptively simplify thesurface triangulation, see for example Fig. 16.20 in [p. 549][36]. They alsoallow to visualize and edit the object efficiently with off-the-shelf software(for example the many visualization and editing tools for stereo lithography)and without the starcase artifacts typical of voxel representation of objectswith curved boundary. One may even easily print the object with additivemanufacturing technology (i.e. 3d printers).Another issue that arises reading the literature is that many different def-initions of topology preserving skeleton exist. In some papers, the skeletonis obtained by removing simple pairs in the spirit of simple homotopy theoryby what is well known as collapsing in algebraic topology [33]. The resultingskeleton, if no other constraints are used, has a lower dimension with respectto the input complex. On the contrary, this paper assumes that the skeletonis always a solid object of the same dimension as the initial complex. Thedifference is highlighted in Fig. 1.In this paper the skeleton of a given complex K is defined as a subset S ⊂ K that is obtained from K after removing a sequence of top dimensionalcells. We require that the homology [33] of the initial complex K is preservedduring this process. In particular, a top dimensional cell can be safelyremoved if this does not change the homology of the complement of S . Fig. 2provides an intuitive explanation why the last requirement is desirable. Thisadditional requirement, to the best of our knowledge, is not documented inother papers. We call this cell a simple cell , which is a generalization of theidea of simple points [32] in digital topology. Clearly, in nontrivial cases,the skeleton S is not unique.Resorting to explicit homology computations to detect simple points asin [16], [35], [32] is quite computationally intensive, as the worst-case com-plexity of homology computations is cubical, see also the discussion in [32].In this paper, we introduce a much more efficient solution by exploiting theidea of tabulated configurations, i.e. acyclicity tables , that are described indetail in Sec. 3.Usually, a skeleton also requires to preserve the shape of the object. Inthis paper we show some very simple proof of concept idea how to preserveboth homology and shape. Of course, this is just an example to illustrate PAWE(cid:32)L D(cid:32)LOTKO, AND RUBEN SPECOGNA
Figure 1.
Let us consider, as an example, a 2-dimensionalsimplicial complex K representing an annulus. On the left,the thick cycle represents a 1-dimensional skeleton of K ob-tained by means of standard collapsing of K [33]. On theright, the gray triangles represent a 2-dimensional skeletonof K according to the definition used in this paper. Thiskind of skeleton is obtained after removing a sequence of topdimensional cells. Figure 2.
Suppose the iterative thinning Algorithm 1 isused to skeletonize a 2-dimensional simplicial complex K rep-resenting an annulus. Let the dark gray triangles belongto the skeleton. On the left, the result obtained by check-ing whether the removal of a cell changes the topology of K complement. On the right, the result obtained by check-ing whether the removal of a cell changes the topology of K .The numbers inside triangles indicate the iteration number ofthe while loop in the thinning algorithm when they were re-moved. Both skeletons preserve topology. However, in mostapplications, the skeleton on the left is preferred.how the idea of acyclicity tables can be used together with some additionaltechniques that guarantee shape preservation.The rest of the paper is organized as follows. In Section 1.1 the priorwork on thinning algorithms is surveyed. Section 1.2 analyzes the original OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 5 contributions of the present paper. In Section 2 the property of being ahomology preserving thinning is rigorously stated. In Section 3 the conceptof acyclicity tables is introduced, whereas, in Section 4, the topology pre-serving thinning algorithm is presented. Section 5 discusses the results ofthe thinning algorithm on a number of benchmarks and, finally, in Section 6the conclusions are drawn.1.1.
Prior work.
There are hundreds of papers about thinning. Most ofthem fall into two categories. On one hand, there are papers using morpho-logical operations like erosion and dilatations to obtains skeletons, see [37]and references therein. They do not guarantee topology preservation ingeneral. The others use the idea of removing the so called simple pointsfrom the given cell complex, see [16], [35], [32]. Without pretending to beexhaustive, in the following we resume previous results.1.1.1.
Most of the work on thinning regard findingskeletons of 2-dimensional images. A very comprehensive survey on thistopic may be found in [1]. This case is well covered in literature and generalsolution exists, see for example [38]-[43].1.1.2.
In case one wants to skeletonizethree (or higher) dimensional images, there are much less papers availablein literature. Most of them rely on case study, see [44]-[57]. The problem isthat it is hard to prove that a rule-based algorithm is general, i.e. it removesa cell if and only if its removal does not change topology. In 3d there aremore than 134 millions possible configurations for a cube neighborhood andonly treating correctly all of them gives a correct thinning algorithm. Ref-erences [16], [35], [32] use explicit homology computations to detect simplepoints.There are a number of papers presenting thinning algorithms for 3-dimensionalimages in which Euler characteristic is used to guarantee topology preserva-tion, see for example [43], [54] and references therein. The problem is thatEuler characteristic is a rather raw measure of topology and it is not suf-ficient to preserve topology in general for 3-dimensional cubical complexes.For three dimensional images one needs to use both Euler characteristic andconnectivity information to preserve topology, but this is not sufficient forfour dimensional images.1.1.3.
All the strate-gies presented so far are applicable only to cubical grids (pixels, voxels, ...).To our best knowledge, there are just a few papers dealing with 2d gridsthat are not cubical, and they are restricted to 2d binary images modeled bya quadratic, triangular, or hexagonal cell complex, see [58]-[61]. The mainreason for the lack of results on general 2d simplicial complexes may be theabsence of regularity in unstructured simplicial grids that makes case-studyalgorithms very hard to devise and to implement. This gap in the literatureis covered by the present paper.
PAWE(cid:32)L D(cid:32)LOTKO, AND RUBEN SPECOGNA
To the best ofour knowledge, we are not aware of algorithms that deal with unstructured3d simplicial complexes or more general cell complexes. There are onlysome papers that find the 1-dimensional skeleton by using the well knowncollapsing in algebraic topology [62]-[65]. Again, this gap in the literatureis covered by the present paper.1.2.
Summary of paper contributions.
In this Section the main novel-ties presented in this paper are summarized:(1) The claim “topology preserving thinning” is rigorously defined, forany cell complex, by means of homology theory.(2) A novel topology preserving thinning algorithm that removes simplecells is introduced. Conceptually this algorithm falls into the cate-gory of thinning algorithms based on simple points and generalizesall previous papers. In fact, the acyclicity tables introduced in thispaper give a classification of all possible simple points that can occurin a given cell complex. Therefore, no rules are needed since all ofthem are encoded into the acyclicity tables.(3) The most important advantage of the novel approach is that acyclic-ity tables are automatically filled in advance, for any cellular decom-position, with homology computations performed by a computer.Therefore, once the tables are available, the implementation of athinning algorithm is straightforward since identifying simple cellsrequires just queering the acyclicity table. No other topological pro-cessing is needed.(4) The fact that acyclicity tables are filled automatically and correctly ,for all possible configurations, provides a rigorous computer-assistedmathematical proof that the homology-based thinning algorithmpreserves topology. It is also verified, simply by checking all acyclicconfigurations, that using Euler characteristic is not enough to en-sure preservation of topology in 3-dimensional or higher dimensionalcubical and simplicial complexes. However, when one checks bothEuler characteristic and that the number of connected componentbefore and after cell removal remains one, then topology is preserved.Checking Euler characteristic together with connectivity does notsuffice to preserve topology in 4d.(5) The acyclicity tables for simplicial complexes of dimension 2, 3 and4 and for cubical complexes of dimension 2 and 3, that can be freelyused in any implementation of the proposed algorithm, are providedas supplemental material at [66]. This way, we dispense readers toimplement homology computations to produce the acyclicity tables.(6) The thinning algorithm, unlike the standard collapsing of algebraictopology [33], does not require the whole cell complex data structurebut it uses only the top dimensional elements of the complex, withobvious memory saving.
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 7 (7) As a proof of concept, an open source C++ implementation thatworks for 3-dimensional simplicial complexes is provided to the readeras supplemental material at [66]. We remark that the code is opti-mized for readability and memory usage and not for speed.2.
Topology preserving thinning by preserving homology
When one claims that an algorithm ”preserves topology,” in order to givea precise meaning to this statement, one needs to specify which topologicalinvariant is preserved. In the literature, the invariant is assumed to be, inmost cases implicitly, the so called homotopy type [33]. The problem ofthis choice is that this strong topological invariant in general is not com-putable according to Markov [67]. This is the reason why in this paper wepropose to use homology theory which is computable in place of homotopytheory, even if it is weaker than the former. Indeed, homology seems tobe the strongest topological invariant that can be rigorously and efficientlycomputed. Therefore, every time we claim that topology is not changed,implicitly we mean that the homology is not changed.Homology groups may be used to measure and locate holes in a givenspace. Zero dimensional holes are the connected components. One dimen-sional holes are handles of a given space, whereas two dimensional holes arevoids totally surrounded by the considered space (i.e. cavities). One canlook at a n -dimensional hole as something bounded by a deformed n -sphere.A space is homologically trivial (or acyclic ) if it has one connected compo-nent and no holes of higher dimensions. A rigorous definition of homologygroups is not presented in this paper due to the availability of rigorous math-ematical introductions in any textbook of algebraic topology as [33] and thelack of space. For a more intuitive presentation for non mathematicians oneone may consult [68], [69].In this paper, we consider in particular two standard ways of represent-ing spaces, namely the simplicial and cubical complexes. A n-simplex isthe convex hull of n + 1 points in general position (point, edge, triangle,tetrahedron, 4-dimensional tetrahedron). A simplex spanned with vertices x , . . . , x n is denoted by [ x , . . . , x n ]. By a face of a n -simplex A we meanthe simplices spanned by a proper subset of vertices spanning A . A simpli-cial complex S is a set of simplices such that for every simplex A ∈ S andevery face B of simplex A , B ∈ S . Pixels (2-cubes) and voxels (3-cubes) arewidely used in image analysis. They form a Cartesian grid, that is a specialcase of grid where cells are unit squares or unit cubes and the vertices haveinteger coordinates. Even though we assume to deal with a Cartesian grid,the results presented in this paper hold also for more general grids such asa rectilinear grid, that is a tessellation of the space by rectangles or paral-lelepipeds that are not, in general, all congruent to each other. Therefore,we define a cubical complex K as a set of cubes such that for every cube A ∈ K and for every B being face of we have A , B ∈ K . We want to stress PAWE(cid:32)L D(cid:32)LOTKO, AND RUBEN SPECOGNA here that we assume every cell to be closed, i.e. if a cell is present in acomplex, so do are its faces.In the iterative thinning algorithm presented in this paper, the top dimen-sional cells (simplices, voxels) are iteratively removed from the object. Ho-mology theory is used to ensure that removing of a given cell (simplex/cube)does not change the topology of the object. If removal of a cell does notchange the topology, the cell is said to be simple . Due to efficiency reasons,the homology cannot be recomputed after removing every single element. Infact, one may compute the homology of a cell complex for instance with [70]software, but the worst case computational complexity is cubical. Therefore,the main idea is to rely on the so called Mayer–Vietoris sequence [33]. Letus express the considered space X as X = X (cid:48) ∪ V , where V is a single topdimensional simplex or voxel. The sequence states that once the intersection X (cid:48) ∩ V is homologically trivial, then the homology of X (cid:48) is the same as thehomology of X . This important result is the key of the method presentedin this paper. In fact, it implies that, in order to preserve the homology of X with respect to X (cid:48) , one should check the homology of the intersection X (cid:48) ∩ V . In practice, this may be easily performed with the [70] software.The main novelty of this paper is to present a different idea to speed upthe computations. Let V be a simplex or voxel. By bd V we denote theboundary of V , i.e. all lower dimensional cells which are entirely containedin the closure of V . The idea is based on the observation that in bd V thereare not too many elements, namely(1) 6 in case of triangle (2-dimensional simplex);(2) 14 in case of tetrahedron (3-dimensional simplex);(3) 30 in case of 4-dimensional simplex (i.e. the convex hull of 5 pointsin R in general position);(4) 8 in case of pixel (2-dimensional cube);(5) 26 in case of voxel (3-dimensional cube).By configuration we mean any subset of bd V . When looking for simplecells, the configuration characterizes the way V intersects the complementof the set that we aim to thin. A configuration is acyclic if its homology—computed as the homology of the corresponding chain complex—is trivial.Since the number of all possible configurations in bd V is 2 i , where i is thenumber of boundary elements of V , one may pre-compute the homology of allthe configurations and store them in a lookup table. In this case, homologycomputations are done only in a pre-processing stage and once and for all.After creating such an acyclicity table, one may instantly (i.e. in O (1) time)get the answer whether the intersection X (cid:48) ∩ V is homologically trivial ornot. This is the strategy that we aim to use in the thinning algorithm.Next section shows how to use the acyclicity tables and how to obtain themautomatically. OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 9 Use of acyclicity tables and their generation
Let us consider a generic cell V of a cell complex. Let us fix an order ofall boundary elements { b , . . . , b n } of V . We consider all subsets of the set { b , . . . , b n } and enumerate them in the following way. For J ⊂ { , . . . , n } the number of a subset { b i } i ∈ J is (cid:80) i ∈ J i . The acyclicity table of V isan array of size 2 n having, at position j = (cid:80) i ∈ J i , the value true if theconfiguration { b i } i ∈ J is acyclic and false otherwise.Let us describe how the acyclicity table is constructed and used startingby considering a tetrahedron (3-dimensional simplex), see Fig. 3a. Let usenumerate vertices, edges and faces of tetrahedron as in Fig. 3a. We now (a) (b) (c) Figure 3.
The model of the (a) 3-dimensional sim-plex (tetrahedron of edges and faces ), the (b) 2-dimensional cube (pixel ofedges ) the (c) 3-dimensional cube (voxel ofedges and faces ).introduce an ordering on boundary elements of the 3-dimensional simplex(bold numbers are the indexes of elements in the given order):(3.1)
12 13 23 012 013 023 123 . Let l , . . . , l k be the indexes of elements in the considered configuration (i.e.bold numbers corresponding to elements that are present in the configura-tion). The index of the configuration in the acyclicity table is computedwith index := k (cid:88) i =1 l i . The acyclicity table is automatically generated in advance as follows. Allpossible configurations of the elements are automatically generated and thehomology group of each of these configurations is computed with the [70]software. If a configuration turns out to be acyclic, then a true is set tothe place in the array corresponding to the examined configuration, false otherwise.
Example 3.1.
Suppose the 3-dimensional simplex [4 , , ,
20] is given asinput. Let [4 , ,
19] and [20] be the maximal elements in the configuration(i.e. the configuration consists of those elements and the vertices [4] , [5] , [19],edges [4 , , ,
19] that are the faces of [4 , , → , 5 → , 19 → , 20 → . It is naturally extended to the mapping on simplices.Namely, the triangle [4 , ,
19] is mapped to [ , , ] in the 3-dimensional sim-plex model, whereas vertex [20] is mapped to vertex [ ]. Therefore, theelements in this configuration are:(1) Vertices: [ ], [ ], [ ], [ ] (indices 1, 2, 3, 4);(2) Edges: [ , ], [ , ], [ , ] (indices 5, 6, 8);(3) Face: [ , , ] (index 11).Consequently, the index of this configuration is index = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 2430 . One may check, at this position of the provided acyclicity table for 3-dimensional simplices, that this configuration is not acyclic .In the same spirit, we introduce an ordering for the 2-dimensional simplex(3.2) , and for the 4-dimensional simplex(3.3)
02 03 04 12 13 14
13 14 15 16 17 18
23 24 34 012 013 014
19 20 21 22 23 24
023 024 034 123 124 134
25 26 27 28 29 30
234 0123 0124 0134 0234 1234 . In the case of cubes, unlike the case of simplices, the model cube is ex-pressly needed to specify the location of vertices in the cube . The model for Clearly it cannot be, since it has two connected components. This happens because in a cube not all vertices are connected with edges as in case ofsimplices. Therefore, the model cube is needed to point out the incidences of the vertices.
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 11
2- and 3-dimensional cubes is represented in Fig.s 3b and 3c. The orderingfor the 2-dimensional cube is(3.4) , whereas, for the 3-dimensional cube (voxel) is(3.5)
13 14 15 16 17 18
15 23 26 37 45 47
19 20 21 22 23 24
56 67 0123 0145 0347 1256
25 26 . Of course, in order to compute the index in the acyclicity table, exactly thesame procedure as the one described for the 3-dimensional simplex is used.Historically, the acyclicitiy tables for cubes [71] and simplices [72] wereintroduced in order to speed up homology computations. In this paper weprovide an even stronger result. Not only the homology of the initial set andits skeleton is the same, but one can construct a retraction from the initialset to its skeleton. The existence of retraction implies the isomorphismin homology, but the existence of retraction is a stronger property thanhomology preservation. We demonstrated the existence of a retraction by abrute-force computer assisted proof, i.e. checking all acyclic configurations.Thus, the following lemma holds.
Lemma 3.2.
For every acyclic configuration C in the boundary of 2-, 3- or4-dimensional simplices and 2- or 3-dimensional cubes (denoted as bd ( K ) )there exist a simple homotopy retraction from bd ( K ) \ C to C . At the end of this section, let us define more rigorously a simple cell . Definition 3.3.
A cell T in a complex K is simple if ( bd T \ ( T ∩ ( K \ T )))is acyclic.In the supplemental material, we already provide the acyclicity tables for2 , , , . All configurations for 4-dimensional cube require almost 10 PB. All configurationsfor 5-dimensional simplex require 4096 PB. On the contrary, the acyclicity table for the3-dimensional simplices provided as supplemental material requires no more than 32kB. Topology preserving thinning algorithm
In this section we propose a simple thinning technique that iterativelyremoves simple cells. The algorithm is valid both for cubes and simplicesprovided that the corresponding acyclicity table is used. We want to pointout that the algorithm works on top dimensional cells (cubes, simplices).Therefore—unlike the case of homological algorithms or collapsing—thereis no need to generate the whole lower dimensional cell complex data struc-ture . The input of the algorithm consists of a list K of top dimensional cellsin the considered set. The output is a subset of K being its skeleton.At the beginning, we present a first version of the algorithm that preservesonly the topology of K . At the beginning, one searches the list K to find allthe cells K , . . . , K n that are simple and store them into a queue L . Then,the queue L is processed as long as it is not void. In each iteration, anelement K is removed from the queue L . Then, with the acyclicity table,one has to check if K is simple in the set S ( K ). We want to point out thatelements already removed form the considered set S ( K ) in previous iterationsare treated as the exterior of S ( K ) at a given iteration. If K is simple in S ( K ), then it is removed from the set S ( K ). In this case, all neighbors of K that are still in S ( K ) are added to the queue L . The details of thepresented procedure are formalized in Alg. 1. We want to stress that Alg. 1 Algorithm 1
Topology preserving thinning.
Input:
List of maximal cells K ; Output:
List of maximal cells S ( K ) that belong to the skeleton of K ; Queue L ; S ( K ) = K ; for Every element T ∈ S ( K ) do if bd T \ ( T ∩ ( S ( K ) \ T )) is acyclic (check with acyclic table) then L.enqueue ( T ); while L (cid:54) = ∅ do T = L.dequeue (); if T (cid:54)∈ S ( K ) then Continue; if ( bd T \ ( T ∩ ( S ( K ) \ T ))) is acyclic then S ( K ) = S ( K ) \ T ; Put all neighbor cells of T in S ( K ) to the queue L ; return S ( K ); is just an illustration. It may be turned into an efficient implementationby using more efficient data structures (for instance removing from the list S ( K ) can be replaced by a suitable marking the considered element.) Alsosearching for intersection of T with current S ( K ) should be performed by This structure have to be generated only locally for the boundary of a cell T whenchecking if T is simple. A neighbor of cell/simplex K is any cell/simplex K ∈ K such that K ∩ K (cid:54) = ∅ . OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 13 using hash tables that, for the sake of clarity, are not used explicitly inAlg. 1. Let us now discuss the complexity of the algorithm. Clearly the for loop requires O ( (cid:107)K(cid:107) ) operations. We assume that one can set and check aflag of every cell in a constant time. This flag indicates if a cell is removedfrom S ( K ) or not. Every cell T ∈ K appears in the while loop only k times, where k is maximal number of neighbors of a top dimensional cell inthe complex. Therefore, the while loop performs at most k (cid:107)K(cid:107) iterationsbefore its termination. The time complexity of every iteration is O ( k ), whichmeans that the overall complexity of the procedure is O ( k (cid:107)K(cid:107) ). Typicallythe number k is a dimension dependent constant and, in this case, thecomplexity of the algorithm is O ( (cid:107)K(cid:107) ). The same complexity analysis isvalid for Alg. 2.We now present in Alg. 2 a simple idea that enables to preserve the shapeof the object in addition to its topology. We stress that the aim of this secondalgorithm is just to show how to couple topology and shape preservation.In Alg. 2 there is one basic difference with respect to Alg. 1. In Alg. 2, Algorithm 2
Shape and topology preserving thinning.
Input:
List of maximal cells K ; Output:
List of maximal cells S ( K ) that belong to the skeleton of K ; Queue L ; S ( K ) = K ; for Every element T ∈ S ( K ) do if bd T \ ( T ∩ ( S ( K ) \ T )) is acyclic (check with acyclic table) then L.enqueue ( T ); Queue K ; while L (cid:54) = ∅ do T = L.dequeue (); if T ∈ S ( K ) then if ( bd T \ ( T ∩ ( S ( K ) \ T ))) is acyclic then S ( K ) = S ( K ) \ T ; Put all the neighbor cells of T in S ( K ) to the queue K ; if L = ∅ then L = K ; K = ∅ ; if all the cells in S ( K ) have a top dimensional face in external boundary then Break ; return S ( K ); after removing a single external layer of cells, a check is made at line 16 todetermine whether all cells that remain in S ( K ) are already in the boundaryof S ( K ). Once they are, the thinning process terminates. The topology isstill preserved due to line 10. The additional constraint used at line 16 ofAlg. 2 is very simple and it gives acceptable results in practice. It may be easily coupled with other techniques to preserve shape already described inliterature.Finally, we discuss the situation when one wants to keep the skeletonattached to some pieces of the external boundary B of the mesh. In thiscase, when testing whether a top dimensional cell T is simple, one shouldconsider B ∩ bd T as elements in S ( K ). In other words, elements from B are not considered as an interface between the object to skeletonize and itsexterior.4.1. Proofs.
Now we are ready to give a formal definition of skeleton.
Definition 4.1.
Let us have a simplicial or cubical complex K . A skeleton of K , denoted by S ( K ), is a set of top dimensional simplices or cubes suchthat:(1) S ( K ) is obtained from K by iteratively removing top dimensionalelements T , . . . , T n , provided that the intersection of T i with K \ (cid:83) i − j =1 T j complement is acyclic. Consequently, homology groups of S ( K ) and K are isomorphic;(2) There is no top dimensional element T ∈ S ( K ) that has an acyclicintersection with S ( K ) complement (i.e. the process of removingsuch elements has been run as long as possible.)We want to point out that sometimes, due to some deep phenomenaarising in simple homotopy theory, some skeleton may be redundant. Forinstance it is possible to have a skeleton of a 3-dimensional ball that is aBing’s house [73] instead being a single top dimensional element. In gen-eral it is impossible to avoid this issue due to some intractable problems intopology.In the follwing, we formally show that the skeleton obtained from Alg. 1satisfies Def. 4.1. This fact is shown with a sequence of two simple lemmas. Lemma 4.2.
The homology of K and S ( K ) are isomorphic.Proof. The proof of this lemma is a direct consequence of the Mayer–Vietorissequence [33]. Let T , . . . , T n be the elements removed during the course ofthe algorithm (enumeration is given by the order they were removed bythe algorithm.) Let us show that, for every i ∈ { , . . . , n } , homology of K \ (cid:83) i − j =1 T j and homology of K \ (cid:83) ij =1 T j are isomorphic. Let us writethe Mayer–Vietoris sequence in reduced homology for K \ (cid:83) i − j =1 T j = ( K \ (cid:83) ij =1 T j ) ∪ T i : . . . → H n (( K \ i (cid:91) j =1 T j ) ∩ T i ) → The difference used in the formulas in this proof is not a set theoretic difference. Allthe objects are assumed to contain all their faces.
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 15 → H n ( K \ i (cid:91) j =1 T j ) ⊕ H n ( T i ) → H n ( K \ i − (cid:91) j =1 T j ) → . . . . The intersection (
K \ (cid:83) ij =1 T j ) ∩ T i is acyclic. This is because the intersectionof T i with the set complement is checked in the acyclicity tables to be acyclic.Once it is, also ( K\ (cid:83) ij =1 T j ) ∩ T i is acyclic. Therefore, H n (( K\ (cid:83) ij =1 T j ) ∩ T i )is trivial. Also, since T i is a simplex or cube, it is acyclic. This provides H n ( T i ) being trivial (we are considering reduced homology.) Consequentlyfrom the exactness of the presented sequence we have the desired isomor-phism between H n ( K\ (cid:83) ij =1 T j ) and H n ( K\ (cid:83) i − j =1 T j ). The conclusion followsfrom a simple induction. (cid:3) Lemma 4.3.
After termination of the algorithm there is no element T ∈ S ( K ) that has an acyclic intersection with the S ( K ) complement.Proof. Let T , . . . , T n be the elements removed during the course of the al-gorithm (enumeration is given by the order they were removed by the algo-rithm.) Suppose, by contrary, that a T ∈ S ( K ) exists such that it has anacyclic intersection with the S ( K ) complement. Let i ∈ { , . . . , n } denotesthe index of last element among T , . . . , T n that has nonempty intersectionwith T . If i = 0, then T would be put to the queue L in the line 5 of Alg. 1and removed from S ( K ) in the line 11 of the algorithm, since no changeto its intersection with S ( K ) complements is made by removing T , . . . , T n .If i >
0, then after removing T i the intersection of T with S ( K ) comple-ment does not change. Therefore, it is acyclic after removing T j for j ≤ i .When Alg. 1 removes T i in the line 12, T is added to the list L and it isgoing to be removed in the line 11, since removing T j for j > i does notaffect the acyclicity of the intersection of T with the S ( K ) complement. Inboth cases we showed that T is removed from S ( K ) by Alg. 1. Therefore, acontradiction is obtained. (cid:3) Experimental results
Assessing aortic coarctation and aneurism.
Skeletons can be usedin computer-aided diagnostic tools for coarctation and aneurism, by evalu-ating the transverse areas of any vessel structure, see for example [23].5.1.1.
Aortic coarctation.
Aorta coarctation is a congenital heart defect con-sisting of a narrowing of a section of the aorta. Surgical or catheter-basedtreatments seek to alleviate the blood pressure gradient through the coarc-tation in order to reduce the workload on the heart. The pressure gradientis dependent on the anatomic severity of the coarctation, which can bedetermined from patient data. Gadolinium-enhanced magnetic resonanceangiography (MRA) has been used in a 8 year old female patient to image amoderate thoracic aortic coarctation, see Fig. 9a. Fig. 9b shows a render-ing of the 3D triangulated surface, obtained by segmenting the MRA data,which models the ascending aorta, arch, descending aorta, and upper branch
Thoracic coarctationDescending aortaLeft subclavian arteryLeft common carotid arteryArchAscending aortaRight subclavianartery Right commoncarotid artery (b) (c)(a)
Figure 4. (a) Magnetic resonance angiography (MRA) im-age of a moderate thoracic aortic coarctation. (b) Renderingof the 3D triangulated surface (20922 triangles) that repre-sents the patient-specific thoracic aortic coarctation anatomyobtained by segmenting MRA data and (c) the skeleton ex-tracted with Alg. 2.vessels. The interior of the surface has been covered with 94756 tetrahedra.The skeleton of this vessel structure, obtained with Alg. 2, is shown in Fig.9c.5.1.2.
Cerebrovascular aneurism.
Cerebrovascular aneurysms are abnormaldilatations of an artery that supplies blood to the brain. Magnetic resonanceimaging (MRI) has been used to image the cerebral circulation in a 47year old female patient, see Fig. 5a. Fig. 5b shows a rendering of the3D triangulated surface, obtained from the segmentation of the MRI data.The interior of the surface has been covered with 390,081 tetrahedra. Theskeleton of this vessel structure, obtained with Alg. 2, is shown in Fig. 5c.
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 17
Analysis of pulmonary airway trees.
Pulmonary arteries connectblood flow from the heart to the lungs in order to oxygenate blood beforebeing pumped through the body. Skeletons have been used for quantitativeanalysis of intrathoracic airway trees in [21]. A 3D triangulated surface,shown in Fig. 6b, represents the 3D model of pulmonary airway trees ofa 16 year old male patient obtained by segmenting data from computedtomography (CT) images, see Fig. 6a. The interior of the surface is coveredwith 236,433 tetrahedra. The topology preserving skeleton obtained by Alg.2 is shown in Fig. 6c.5.3.
Extracting centerline for virtual colonoscopy.
A 3D triangulatedsurface that represents the 3D model of a colon is obtained by segmentingdata from computed tomography (CT) images, see Fig. 7. The interior ofthe surface is covered with 2,108,424 tetrahedra. The topology preservingskeleton obtained by Alg. 2, which may be used as a colon centerline toguide a virtual colonoscopy, is shown in black in Fig. 7.5.4.
Computing topological interconnections of bone structure.
A3D model of a human bone belonging to a 61 year old male patient has beenobtained from a stack of thresholded 2D images acquired by X-ray MicroCTscanning [74]. In particular, a region of interest (ROI) of size 4 mm × ×
200 pixels, pixel size 19 . µ m) is selected in the trabecular region. Astack of 195 2D images has been considered, resulting in a volume of interest(VOI) of approximately 4mm × × Some other non medical examples.
The results of Algorithm 1 onsome benchmarks are visible in Fig.s 9-16, whereas the results obtained withAlgorithm 2 are shown in Fig.s 17-20.6.
Conclusion
This paper introduces a topology preserving thinning algorithm for cellcomplexes based on iteratively culling simple cells. Simple cells, that may beseen as a generalization of simple points in digital topology, are characterizedwith homology theory. Despite homotopy, homology theory has the virtue ofbeing computable. It means that, instead of resorting to complicated rule-based approaches, one can detect simple cells with homology computations.The main idea of this paper is to give a classification of all possible simplecells that can occur in a cell complex with acyclicity tables. These tables arefilled in advance automatically by means of homology computations for allpossible configurations. Once the acyclicity tables are available, implement-ing a thinning algorithm does not require any prior knowledge of homologytheory or being able to compute homology. The fact that acyclicity tablesare filled automatically and correctly for all possible configurations provides a rigorous computer-assisted mathematical proof that the homology-basedthinning algorithm preserves topology. We believe that such rigorous topo-logical tools simplify the study of thinning algorithms and provide a clearand safe way of obtaining skeletons.
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OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 23 (a) (b)(c)
Figure 6. (a) Computed tomography (CT) image of thepulmonary airway trees. (b) Rendering of the 3D trian-gulated surface (71926 triangles) representing the patient-specific pulmonary arteries and (c) the skeleton extractedwith Alg. 2.
Figure 7.
Rendering of the 3D triangulated surface(506,188 triangles) representing the patient-specific colonanatomy obtained by segmenting CT colonography data and,in black, the skeleton extracted with Alg. 2.
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 25 (a) (b)
Figure 8. (a) Rendering of the 3D triangulated surface(285,346 triangles) representing the anatomy of a humanbone in a region of interest of the trabecular region obtainedby segmenting MicroCT data and (b) the skeleton extractedwith Alg. 1.
Figure 9.
A flange and its topology-preserving skeleton.
Figure 10.
A mechanical part and its topology-preserving skeleton.
Figure 11.
A racing rim and its topology-preserving skeleton.
Figure 12.
A mechanical support and its topology-preserving skeleton.
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 27
Figure 13.
A net and its topology-preserving skeleton.
Figure 14.
A geometric figure and its topology-preserving skeleton.
Figure 15.
A mechanical part and its topology-preserving skeleton.
Figure 16.
OPOLOGY PRESERVING THINNING OF CELL COMPLEXES 29
Figure 17.
A rope and its shape-preserving skeleton.
Figure 18.
Three twisted wires and their shape-preserving skeleton.
Figure 19.
A male body and its shape-preserving skeleton.