integrating digital topology in image-processing libraries

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  • computer methods and programs in b iomed ic ine 8 5 ( 2 0 0 7 ) 5158

    journa l homepage: www. int l .e lsev ierhea l th .com/ journa ls /cmpb

    Integr -

    Julien LaIrcad, 1, Pla

    a r t i c

    Article histor

    Received 14

    Received in

    28 August 2

    Accepted 28


    Digital topo



    Code generi



    , for

    st i

    e sa


    s as

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    1. Introduction

    Digital imathough sigtal images,account thaanymore buimages.

    This dislevel, as moon an imaglogical levebase of topcontinuoustopologicalwhen appli

    Algorithous, from c[1] or loop rical imagelibraries, ITduring the

    Tel.: +33E-mail a

    causes of course a loss of exibility, but also a loss of coher-

    0169-2607/$doi:10.1016/ge processing is by nature a discrete process:nal processing algorithms can be applied to digi-they have to be adapted in some way to take intot the image they process is not a continuous signalt a set of pixels, or of voxels for three-dimensional

    crete nature causes few problems at the geometricst of the geometric operations that can be appliede are independent of the underlying grid. At a topo-l, this is however different. The notion at the veryology, the neighborhood, is radically different fromspaces to discrete spaces. This leads to continuousalgorithms not respecting topological constraintsed on a digital based on topological information are numer-onnected component labeling to skeletonizationemoval, all of which are used in the eld of med-processing [24]. In two major image-processingK [5] and Vigra [6], the type of neighborhood usedalgorithm is hard-coded in that algorithm. This

    388 119 079; fax: +33 388 119 099.ddress:

    ence when processing both the foreground and the back-ground of a digital image, as it is a well-known fact thattwo different connectivities must be used for the backgroundand the foreground [7]. If this condition is not respected, theJordan theorem [8] will not hold and situations will arisewhere a closed curve does not partition the space in tworegions.

    This rigid characteristic of those both major libraries isdue to the lack of pertinent data structures to correctly rep-resent the topology of digital images. In this paper, we showhow to integrate the digital topology information in an image-processing library in a way that is:

    generic, with respect to both the image dimension and thedifferent types of neighborhood;

    automated, in the sense that as few special case code aspossible has to be written;

    fast, by pre-computing as many things as possible.

    The language used in this paper is C++, as this work wasrealized within the ITK framework. However, the concepts

    see front matter 2006 Elsevier Ireland Ltd. All rights reserved.j.cmpb.2006.08.006ating digital topology in image


    ce de lHopital, 67091 Strasbourg Cedex, France

    l e i n f o


    February 2006

    revised form


    August 2006


    essing library


    a b s t r a c t

    This paper describes a method

    processing libraries. This addition

    respecting topological constraints

    digital topology is absent from mo

    fullled. We describe and give cod

    tion, and show a use case in the fo

    lter can be up to a hundred time

    dimension. This paper mainly dea

    minor modications to other imaprocessing libraries

    integrate digital topology informations in image-

    information allows a library user to write algorithms

    example, a seed ll or a skeletonization algorithm. As

    mage-processing libraries, such constraints cannot be

    mples for all the structures necessary for this integra-

    of a homotopic thinning lter inside ITK. The obtained

    fast as ITKs thinning lter and works for any image

    f integration within ITK, but can be adapted with only

    rocessing libraries.

    2006 Elsevier Ireland Ltd. All rights reserved.

  • 52 computer methods and programs in b iomed ic ine 8 5 ( 2 0 0 7 ) 5158

    explained here could easily be adapted to other languages,and other image-processing libraries.

    We willprocessingtopology, apresent thein an imageuse these stion algoritgiving lead

    2. Pre

    In this sectlack the neogy in the i

    In ITK, talgorithmsneighborhoin two-dimit is not poground and

    In Vigrathe Neighblimited toeight-connis not an etreated ascode everyby the num

    The lackimage-proceven impoconstraints

    3. Di

    We will nothe notatioframeworkof the imagdecomposi

    In this rrepresenterelations, ror voxels (c

    3.1. Cel

    Let us condened as

    Ii is eithe k of the n

    are of th

    The dimz1 + 1[ ]z2,the image,

    Cells in R2, with underlying digital grid andnates. Green: 0-cells (vertices); yellow: 1-cells (edges);-cells (faces). (For interpretation of the references toin this gure legend, the reader is referred to the

    ersion of the article.)

    ilarlrtexraph. 1 anng the hav

    ll: vell: edlls: flls: s


    i if Iizi 1

    Cells in R3. Green: 0-cells (vertices); yellow: 1-cells); blue: 2-cells (faces); red: 3-cells (solids). (Forretation of the references to colour in this gure, the reader is referred to the web version of the.)rst present the limitations of two major image-libraries, ITK and Vigra, with respect to digital

    nd detail the theoretical background. We will thennecessary structures to integrate digital topology-processing library, and give an example on how totructures to implement a homotopic skeletoniza-hm. We conclude by summarizing our work and bys for future improvements.

    vious work

    ion, we show how both the ITK and Vigra librariescessary structures to correctly embed digital topol-mages and is no structure related to digital topology. Alluse a hard-coded neighborhood, usually the 18-od in three-dimensions, and the 4-neighborhoodensions. As these neighborhoods are hard-coded,ssible to use coherent connectivities for the back-the foreground.

    , there is a basic support for digital topology, usingorhoodCirculator class. This class is howevertwo-dimensional images and treats the four- andectivity in a non-generic way. In our sense, thisasily maintainable solution: each connectivity isa special case. And as the developer has to hand-case, this multiplies the possible sources of errorsber of cases.of digital topology information in those twomajoressing libraries shows that it will be difcult, orssible, to write algorithms respecting topological.

    gital topology basics

    w recall the basics of digital topology, and presentns used in the rest of the article. To ensure that ourwill be generic with respect both to the dimensione and the type of neighborhood, we will use a celltion representation [9].epresentation, a binary image of any dimension isd as a set of cells, with well-dened neighboringather that just a set of pixels (cells of dimension 2)ells of dimension 3).


    sider images of dimension n. A k-cell of Rn [9] isa subset of Rn of the form c= I1 In such that:

    r of the form ]zi, zi +1[ or the form {zi} (zi Z);sets Ii are of the form ]zi, zi +1[ and the other n k

    e form {zi}.

    ension of a k-cell is k. A n-cell has the form ]z1,z2 + 1[ ]zn, zn +1[, and is thus a pixel/voxel ofoccupying a unit area/volume.

    Fig. 1coordiblue: 2colourweb v

    Simis a ve

    A gin Figs

    Usithat w

    0-ce 1-ce 2-ce 3-ce

    Weused incell c iswhere

    Ji = I Ji = ]

    Fig. 2(edgesinterplegendarticley, a 0-cell has the form {z1} {z2} {zn}, andof a pixel/voxel, of coordinates (z1, z2, . . ., zn).ical denition of the cells of R2 and R3 is presentedd 2.e more usual vocabulary of modeling, we can seee the following equivalences, up to three-cells:


    now dene the open star of a cell, which will bedenition of connectivity. The open star St(c) of a k-smallest open set containing c, i.e. St(c) = J1 Jn

    = ]zi, zi +1[,, zi +1[ if Ii = {zi}.

  • computer methods and programs in b iomed ic ine 8 5 ( 2 0 0 7 ) 5158 53

    Fig. 3 Open stars of 0-cell (vertex, left) and 1-cell (edge,right) in R2.

    From this denition,we can see that the open star of a k-cellc contains the cell c as well as every -cell adjacent to c suchthat k R

    2 can be s

    3.2. Con

    This cell dedenition arespect toneighborho

    Let di becells (i.e. piexist a dj-ce

    As shoneighborho1)-neighborlarly, in threto the 26-nto the 18-nlent to the

    Using thent possiblcan also sedj coordinawhich diffedj)-neighbo




    This resultbors in a la

    Fig. 4 (2,0(right).

    Table 1 Equivalences between connectivities expressedwith usua

    Regular co

    4, in two di8, in two di6, in three18, in three26, in three


    m th

    1 andkdsamd in


    cont(i.e. non-overlapping) closed surface divides the spaceparts, the interior and the exterior. In a discrete space,ding on how the connectivities are chosen for the fore-and the background, this theorem can be invalidated.ple closed curved show in Fig. 5 intuitively divides the

    in two parts. However, if the curve is (2, 0)-connectede background is also (2, 0)-connected, the backgroundore divided in two parts. In the opposite case, i.e. the (2,nectivity is used for the background and the foreground,ckground has two components, but the curve enclosingt closed beenproven [10] that the (di, di 1)-connectivitymustd either for the foreground or the background. The (di,nectivity is usually chosen for the other connectivity.n. A graphical representation of the open stars ineen in Fig. 3.



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