COMPUTATIONAL TOPOLOGY y z Tamal K Dey Herb ert Edelsbrunner and Sumanta Guha Abstract connectivity continuity on space and on maps This do es not mean that the more geometric notions have The authors of this article b elieve there is or should b e to b e absent but rather that their role is deemphasized a research area appropriately referred to as computa b eneting the fo cused attention on top ological prop er tional top ology Its agenda includes the identication ties The authors b elieve in a synergy b etween geom and formalization of top ological questions in computer etry and top ology and in the value of intuition and applications and the study of algorithms for top olog visualization as a general path to understanding They ical problems It is hop ed this article can contribute also b elieve in the general scientic metho d of study to the creation of a computational branch of top ology ing asp ects in delib erate isolation In other words it with a unifying inuence on computing and computer is worthwhile to o ccasionally fo cus on top ological prob applications lems The connection to geometry can b e forgotten temp orarily but needs to b e remembered eventually Keywords Survey top ology geometry algorithms com Many computer application areas deal with geometric puter applications data and problems and there we nd a rich collection of questions with top ological avor Typicallythese questions are not well dened and it is part of the task INTRODUCTION to nd the most meaningful formalization in the given The title of this article combines computation with context Once a concrete problem has b een formulated topology suggesting a general research activitythat we can study its computational complexity It is im studies the computational asp ects of problems with p ortant to notice this is an oversimplication of the top ological avor What wehave in mind is distinctly actual situation the problem formalization cannot b e dierent from studying the topology of computing or the indep endent of studying the computational complexity computer animation of topology Computational stud or else we are likely to generate many computationally ies of top ological questions can b e found in the math e need to b e convinced there infeasible problems First w ematics and the computer science literature but no really are essentially top ological questions in computer concerted eort is apparent The authors hop e that application areas Part I of this article provides su together with the likeminded survey pap er byVegter cient evidence for their existence by discussing problems this article can contribute to the general aware in image pro cessing cartography computer graphics ness of the p ervasive presence of top ological notions in solid mo deling mesh generation and molecular mo d computer applications eling Second we need the necessary background in What is it ab out a problem that makes it top ologi top ologyPart I I surveys top ological metho ds catego cal The standard answer mentions a typ e of geometry rized under decomp ositions xedp oints surfaces em devoid of concrete spatial notions such as straightness b edding threemanifolds and homology computation convexity distance and the like The emphasis lies on It has b een observed that a go o d fraction of the lit erature in top ology is not or only barely intelligible to Department of Computer Science and Engineering Indian Institute of Technology Kharagpur India readers with interest but without prop er top ological ed y Department of Computer Science University of Illinois at ucation The rst barrier is the o ccasionally complex UrbanaChampaign Urbana Illinois USA notation and the usual assumption of accumulated con z Department of Electrical Engineering and Computer Science cepts and denitions Indeed theory building is the University of Wisconsin at Milwaukee Milwaukee Wisconsin USA main purp ose of top ology and as a consequence def initions are more imp ortant than in some other areas dierent notions of connectivity for the curve and the of mathematics The authors attempt to pro duce a comp onents do cument accessible to nonsp ecialists by collecting for a simple closed connected curve partitions an im mal and standard denitions in an app endix The main age into the path and two connected comp onents parts of the pap er replace technical detail byintuitive see gure Symmetrically if the closed curveis examples and arguments However it seems imp ossible o comp onents are connected A connected then the tw to avoid technical denitions altogether and wehope necessary assumption for b oth results is that the curve the app endix will b e helpful in recalling their meaning touches the image b oundary at most once I APPLICATION PROBLEMS We consider problems in six applications areas of com puter science image pro cessing cartography computer graphics solid mo deling mesh generation and molec ular mo deling Image Pro cessing Figure A simple closed connected curve with two A pixel is a unit square and an image is a rectangle connected comp onents in the complement decomp osed into pixels Images are the most common means for representing pictorial information in a com The image version of the Jordan curve theorem can b e puter which explains the enormous literature in the generalized to and higherdimensional arrays of unit area of image pro cessing The pap ers collected in cub es and hyp ercub es see Herman This should b e cover a fairly wide range of the topics The related area compared to the situation in Euclidean space Already of mathematical morphology is treated in Serra R p ermits a counterexample to the straightforward ex We restrict our attention to binary images where each tension of the Jordan curve theorem there is a wild pixel is either black or white The foreground consists sphere in R so that the b ounded comp onentofR of all black pixels and the background consists of all minus the sphere is not homeomorphic to the op en white pixels ball see eg chapter Suchcounterexamples do not exist for tame emb eddings and the Jordan curve theorem generalizes as exp ected Connectivity Each pixel shares an edge each with other pixels and it shares a vertex each with addi tional pixels The numb ers are smaller for pixels at the Skeletonizing A common op eration in image pro image b oundary Questions of proximity and connec cessing replaces the foreground by its skeleton There tivity can b e based on the two corresp onding notions of is no commonly accepted denition of what exactly this adjacency pixels p q are adjacent if they share an is One exp ects a thin subset that reects the connec edge and they are adjacent if they share an edge or tivity of the foreground Often the skeleton is stored avertex adjacency implies adjacencyApath is a together with additional information that p ermits the sequence of pixels so that anytwocontiguous ones are exact or approximate reconstruction of the foreground adjacent It is simple if two pixels are adjacentonlyif Algorithms for constructing skeletons are abundant they are contiguous and it is a closed curve if the rst in the image pro cessing literature One class of algo is adjacent to the last pixel Dep ending on the notion rithms is based on the medial axis transform For each of adjacency wegetconnected and connected paths foreground pixel p consider a shortest path that con and curves For i two pixels are iconnected if nects p to a background pixel q The length of this they b elong to a common iconnected path path is p Boundary pixel are characterized by A basic result in plane top ology is the Jordan curve The medial axis consists of all pixels p with at least two theorem which states that every simple closed curvein shortest paths to the background that intersect in no R decomp oses the plane into two disjoint op en con pixel other than p nected comp onents Neither notion of connectivity Another class of algorithms constructs a skeleton by among pixels supp orts an analoguous result for images rep eatedly removing pixels of the b oundary com The following version of the Jordan curve theorem uses pare this with the discussion on collapsing in section that agrees with b at all p oints of P is p opularly known The foreground is ero ded layer bylayer and pixels are as rubber sheeting removed as long as the connectivity of the foreground see for example do es not change The skeleton is the collection of fore Supp ose K and L are simplicial complexes whose sim ground pixels that remain Although this collection re plices cover M and N M jKj j and N jLj j Suppose tains the connectivity of the foreground it often do es also that the p oints in P and Q are vertices of K and L not preserve its overall shap e P Vert K and Q Vert L and that there is a vertex map v Vert KVert L Inner and Outer Boundaries Consider a collec tion of disjoint connected simple closed curves that that agrees with b at all p oints of P The extension of b ound the foreground of a binary image Eachcurve C v to a simplicial map f M N is a simplicial homeo in this collection denes two connected comp onents morphism eectively solving the rubb er sheet problem an outer and an inner region C is an outer boundary if Several variations of the construction of suchcom the foreground comp onent b ounded by C is part of its plexes K and L havebeentackled in the recent inner region Otherwise C is an inner boundaryHow past Aronov Seidel and Souvaine consider simply can we distinguish outer from inner b oundaries connected p olygons M and N with n vertices each