High resolution image data require a huge
amount of computational resources. Image pyramids
have shown high performance and flexibility to reduce
the amount of data while preserving the most relevant
pieces of information, and still allowing fast access to
those data that have been considered less important before.
They are able to preserve an existing topological structure
(Euler number, homology generators) when the spatial
partitioning of the data is known at the time of construction.
In order to focus on the topological aspects let us call this
class of pyramids “topological pyramids”. We consider
here four open problems, under the topological pyramids
context: The minimality problem of volumes representation,
the “contact”-relation representation, the orientation of
gravity and time dimensions and the integration of different
modalities as different topologies.Austrian Science Fund P20134-N13Junta de Andalucía FQM–296Junta de Andalucía PO6-TIC-0226

Kropatsch, Walter G.

Molina Abril, Helena

English

idUS. Depósito de Investigación Universidad de Sevilla

Computer Vision Winter Workshop 2009, Adrian Ion and Walter G. Kropatsch (eds.)Eibiswald, Austria, February 4–6Publisher: PRIP, Vienna University of Technology, AustriaOpen Issues and Chances for Topological Pyramids ∗Walter G. Kropatsch 1 and Helena Molina-Abril 1,21 Vienna University of Technology, Faculty of Informatics,Pattern Recognition and Image Processing Group.2University of Sevilla, Applied Math Department I, Spain.krw@prip.tuwien.ac.at, habril@prip.tuwien.ac.atAbstract High resolution image data require a hugeamount of computational resources. Image pyramidshave shown high performance and flexibility to reducethe amount of data while preserving the most relevantpieces of information, and still allowing fast access tothose data that have been considered less important before.They are able to preserve an existing topological structure(Euler number, homology generators) when the spatialpartitioning of the data is known at the time of construction.In order to focus on the topological aspects let us call thisclass of pyramids “topological pyramids”. We considerhere four open problems, under the topological pyramidscontext: The minimality problem of volumes representation,the “contact”-relation representation, the orientation ofgravity and time dimensions and the integration of differentmodalities as different topologies.1 IntroductionVisual data are characterized with a large quantity of infor-mation and high redundancy. These data require a hugeamount of computational resources (see Fig. 1). What isstill true in 2D becomes even more demanding in 3D (e.g.CT and MR images in medicine; and image and video se-quences in surveillance applications). The combination of3D and time brings us quickly into a four dimensional dis-crete space where online performance is not yet achieved.Figure 1: A 2D, 3D and 4D visual data, and the storage capacityneeded for each one of them.∗Partially supported by the Austrian Science Fund under grant P20134-N13, the “Computational Topology and Applied Mathematics” PAICYT re-search project FQM–296 and the “Andalusian research project” PO6-TIC-02268Image pyramids are a stack of images with decreasingresolutions [4]. Such pyramids present the following inter-esting properties within the Image Processing and Analysisframework ([2]):• Reducing the influence of noise by eliminating less im-portant details in lower-resolution versions of the image.• Making the processing independent of the resolution ofthe regions of interest in the image.• Converting global features to local ones.• Reducing the computational cost using the divide-and-conquer principle.• Finding regions of interest for plan-guided analysis at lowcost in low-resolution images, ignoring irrelevant details.• Visual inspection of large images.• Increasing speed and reliability of image matching tech-niques by applying coarse-to-fine strategy.Figure 2: Pyramid structureTopology aims at studying properties of objects whichare independent of geometrical transform. Some of thesetopological properties are useful in many applications, forexample in matching and indexation of structured objects.A topological pyramid is an image pyramid where eachlevel is a topologically equivalent representation of the ini-tial data.The construction of the pyramid hierarchy follows thephilosophy to reduce the data amount at each higher level121Open Issues and Chances for Topological Pyramids 1of the hierarchy by a reduction factor λ > 1 while preserv-ing important topological properties like connectivity andinclusion.There are topological representations for structured ob-jects that can be used in the hierarchical framework of topo-logical pyramids. These representations are plane graphs,combinatorial maps and generalized maps [12].The paper is structured as follows. Basic notions on ir-regular graph pyramids, combinatorial and generalized-mappyramids, and topological pyramids are recalled in Sections1.1, 1.2 and 1.3. The four proposed open problems arepresented in the following sections. The minimality prob-lem of volumes representation is presented in Section 2, the“contact”-relation representation in Section 3, the orienta-tion of gravity and time dimensions in Section 4, and theintegration of different modalities as different topologies inSection 5. A summary is found in Section 6.1.1 Irregular graph PyramidsIrregular graph pyramids are defined as a stack of succes-sively reduced graphs [11]. In irregular pyramids, each levelrepresents an arbitrary partition of the pixel set into cells, i.e.connected subsets of pixels.The construction of an irregular pyramid is iteratively lo-cal [15][8]. This means that we use only local properties tobuild the hierarchy of the pyramid.On the base level (level 0) of an irregular image pyramidthe cells represent single pixels and the neighborhood of thecells is defined by the connectivity of the pixels. A cell onlevel k+1 (parent) is a union of neighboring cells on level k(children). This union is controlled by so called contractionkernels (decimation parameters [13]).Every parent computes its values independently of othercells on the same level. This implies that an image pyramidis built in O[log(d)] parallel steps being d the image diame-ter.A level of a dual graph pyramid consists of a pair(Gk, Gk) of plane graphs (see Fig. 3), Gk and its geometricdual Gk, in order to correctly represent the embedding ofthe graph in the image plane [7].Figure 3: A plane graph G and its dual GkThe vertices of Gk represent the cells on level k andthe edges of Gk represent the neighborhood relations of thecells. The edges of Gk represent the borders of the cells onlevel k, including so called pseudo edges needed to representneighborhood relations to a cell completely enclosed by an-other cell. Finally, the vertices of Gk, represent junctions ofborder segments of Gk.1.2 Combinatorial and Generalized-map PyramidsCombinatorial maps and generalized maps, define a generalframework which allows to encode any subdivision of nDtopological spaces orientable or non-orientable with or with-out boundaries.Combinatorial maps were introduced in [6], at first as aplanar graph representation model, and extended in [14] indimension n to represent orientable or not-orientable quasi-manifolds. In dimension n a combinatorial map is a (n +1)-tuple M = (D,β1, β2, ..., βn) such that D is the set ofabstract elements called darts, β1 is a permutation on D andthe other βi are involutions on D (see Fig. 4). An involutionis a permutation whose cycle has the length of two or less.The differences between combinatorial and generalizedmaps is that in the case of combinatorial maps, for each di-mension, there is more than one way of attributing the per-mutations, but the number of permutations used for a certaindimension and how many of them are involutions is fixed.Figure 4: Combinatorial map where D = (1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7), β1(d) = −d ∀d ∈ D, β2 =(2, 1)(−1,−4, 3)(−2, 6, 5, 4)(−5, 7,−3)(−6,−7)A Combinatorial pyramid is a hierarchical stack of com-binatorial maps. The definition is analogous for generalizedpyramid and generalized maps [3].1.3 Topological PyramidsAs mentioned before, a topological pyramid is a stack oftopological encodings (graphs or maps).The basic operations to construct these hierarchies areedge contraction and edge removal. Some restrictions havebeen defined in order to preserve topology while applyingthese operations [9]. In that way connectivity, holes, Eulernumber and Betti number are preserved along the hierarchy.122Walter G. Kropatsch and Helena Molina-AbrilThis preservation allows topological pyramids to be auseful tool in many analysis and image processing appli-cations, where this topological information is crucial. Theyare a useful i.e. to distinguish between different parts of anobject, between solid objects and objects that enclose otherobjects, etc. As is shown in [16] topological informationlike homology generators is computed efficiently on the toplevel of a topological pyramid, since the number of cells issmall.There are some open problems, in which analyzing andprocessing large amounts of data is necessary, and topolog-ical information need to be preserved. Organization and ag-gregation principles are needed in order to cope with thecomputational complexity. Topological pyramids promiseto be very useful in order to solve this mega-data problems,in an efficient way.2 Non-unique minimal configurationsThe extension of the pyramidal concept from 2D to 3D is adifficult issue dealing with dual graphs. On the other hand,the combinatorial map formalism has been defined in anydimensions. That is why the 3D pyramidal extension hasbeen treated using this topological representations [9].But although this mathematical model allows the con-struction of a 3D pyramid, the problem of non-unique mini-mal configurations arises.Combinatorial maps encode space subdivisions and allincidence relations [5][1]. The underlying representation ofcellular complexes is based on so-called bounding relations.Each i-cell with i > 0 is bounded by at least one (i-1)-cell; a volume is bounded by surfaces, a surface is boundedby curves and a curve is bounded by points. Consequentlythe smallest volumetric description consists of at least onevolume (|V | ≥ 1), one surface (|S| ≥ 1), one curve (|C| ≥1) and one point (|P | ≥ 1).A combinatorial map will be considered minimal if it hasthe minimal possible number of i-cells. The top level of acombinatorial pyramid is the minimal topological equivalentrepresentation of the initial data.To ensure that applying an operation on an i-cell will pro-duce consistent combinatorial map topologically equivalent,some conditions need to be imposed [9].Let us consider the minimal representation of the sim-plest 2D object (Fig. 5). It exists only one possible minimalconfiguration, as is shown in Fig. 5. This configuration con-tains a single face and one cell of each lower dimension [9].Figure 5: 2D minimal configurationThe simplest object in 3D, a filled sphere must satisfythe boundary representational constraint but also the Eu-ler number (number of points minus number of curves plusnumber of surfaces minus number of vertices) must be one:|P | − |C|+ |S| − |V | = 1.There are two possible solutions that also resulted as top-level solution in 3D combinatorial pyramids [9]: either thenumber of points is two (|P | = 2, |C| = 1, |S| = 1, |V | =1) or the number of surfaces is two (|P | = 1, |C| = 1, |S| =2, |V | = 1), and all the other cells appear only once. Thesetwo configurations are shown in Fig. 6.Figure 6: 3D minimal configurationsDifferent representations at the top of the pyramid occurin the simplest case of a 3D object, but also in many other3D and 4D configurations.This non-uniqueness of the top of a combinatorial pyra-mid restricts the potential uses of combinatorial pyramids inpattern recognition, classification or identification applica-tions.How can we deal with this two different descriptions ofthe same object?3 Contact versus connectivityIn most current representations connectivity is the main re-lation forming the structure of the composite objects. Tointroduce the concept of connectivity within the image con-text, we first need to introduce what is considered as a digitalbinary picture.In digital topology, following the terminology given in[10], a 3D digital binary-valued picture space (or, briefly,DPS) is a triple (V, β, ω). V is the set of grid points in a 3Dgrid and the set β (resp. the set ω) determines the neighbor-hood relations between black points (resp. white points) inthe grid.A 3D digital binary-valued picture is a quadruple I =(V, β, ω,B), where (V, β, ω) is a DPS and B (the set ofblack points) is a finite subset of V.Given a digital binary picture P , a black path (resp.white) in P is a sequence p1, ..., pn of n ≥ 1 black points inP , in which each pi is β(ω) -adjacent to pi−1(1 < i ≤ n).Two black points p, q ∈ P are connected if there exist ablack path in P from p to q.Under this path connectivity concept, representation ofcellular complexes is based on bounding relations; boundingsurfaces separate volumes, bounding curves separate sur-faces, and bounding points separate curves.In several real situations there is a variant of connectivitywhere cells are in contact but no real connectivity is estab-lished. If two people shake hands parts of the surface oftheir hands are in close contact without creating a connec-123Open Issues and Chances for Topological Pyramids 2tivity between their bodies. Geometrically the two touchingsurfaces are aligned but their surface structure (e.g. texture)is not. It occurs also, when for example we put some clothesone over the other. Both surfaces are in contact, but not con-nected.In terms of cell complexes this corresponds to a surfacebeing a neighbor of another surface contradicting the re-quirement that a surface always separates two volumes.Is it possible to model these two realities without anycontradiction?4 Oriented versus non-oriented dimensionsThere are important applications in which temporal behav-ior is critical. Treating time representation just as another di-mension, we allow multiple configurations that are not plau-sible in real situations (see Fig. 7). For example, the age ofobjects is constantly increasing, and a consistent representa-tion of this reality should not allow time decreasing config-urations. A similar situation to time dimension occurs withgravity. The water in a glass of water remains there onlyif the glass is placed on a horizontal surface. Any rotationwould change this state.Figure 7: A plausible and a not plausible situationIf our representation allows this configurations that cannot occur in practise, we need highly complex processes torecognized them, in order to distinguish between these con-figurations and all possible ones.To what extent can topological pyramids take thus ori-ented dimensions into account such that impossible config-urations are excluded from being represented?5 Different modalities, different topologiesA window of a room physically separates the inside of theroom from the outside. However it lets light go through cre-ating pleasant and un-wanted visual artefacts on the surfacesof the inside objects while allowing the inside observer tosee a part of the surrounding of the house. In terms of topol-ogy the window can be a solid part separating the inside ofthe room from the outside or it can be seen as a tunnel (a 1Dhole) allowing light to enter the room.There are many more modalities (e.g. CT, MR, US inmedicine; other frequency bands of the electro-magneticspectrum) that can produce measurements that have differ-ent topologies but describe the same (part of) reality (seeFig. 8).Figure 8: Different pictures describing the same realityModeling and integrating data from these different mea-surements will allow us to better represent the reality.How can we deal with this integration and modelization?Is there a “multi-modal” topology of which the differentviews are just a type of projection?Can we learn more about the properties of real world bycombining and integrating the individual views?6 SummaryFour open problems are presented here: The minimalityproblem of volumes representation, the “contact”-relationrepresentation, the orientation of gravity and time-dimensions, and the integration of different modalities asdifferent topologies.We propose topological pyramids, as a representationwhich has the chance to efficiently cope with the mega-dataproblem.In a future work, we plan to study possible solutionsto these problems and how would topological pyramids beused to address them.References[1] Y. Bertrand, G. Damiand, and C. Fiorio. Topologicalencoding of 3d segmented images. In InternationalConference on Discrete Geometry for Computer Imagery, LectureNotes in Computer Science, volume 1953, pages 311–324.Springer-Verlag, 2000.[2] M. Bisterand, J. Cornelis, and A. Rosenfeld. Acritical view of pyramid segmentation algorithms. InPRL, pages 605–617, 1990.[3] L. Brun and W.G. Kropatsch. Introduction tocombinatorial pyramids. In Digital and Image Geometry,Lecture Notes in Computer Science, volume 2243, pages108–128. Springer-Verlag, 2001.[4] P. Burt, T-H. Hong, and A. Rosenfeld. Segmentationand estimation of image region properties throughcooperative hierarchial computation. In IEEETransactions on Systems, Man and Cybernetics, pages802–809. IEEE, December 1981.[5] G. Damiand and P. Resch. Topological map basedalgorithms for 3d image segmentation. In DGCI 2002Lecture Notes in Computer Science, volume 2301, pages220–231. Springer-Verlag, 2002.124Walter G. 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In TechnicalReport PRIP-TR-42. Institute f. Automation 183/2, Dept.for Pattern Recognition and Image Processing, TUWien, Austria, 1995.[12] W.G. Kropatsch, Y. Haxhimusa, and P. Lienhardt.Hierarchies relating topology and geometry. InCognitive Vision Systems, pages 199–220. Springer,October 2003.[13] W. G. Kropatsch. Building irregular pyramids by dualgraph contraction. In IEE-Proc. Vision, Image and SignalProcessing, pages 336–374, December 1995.[14] P. Lienhardt. Subdivisions of n -dimensional spacesand n-dimensional generalized maps. In Symposium onComputational Geometry, pages 228–236, 1989.[15] P. Meer. Stochastic image pyramids. In Computer Vision,Graphics, and Image Processing, pages 269–294, March1989.[16] S. Peltier, A. Ion, W.G. Kropatsch, G. Damiand, andY. Haxhimusa. Directly computing the generators ofimage homology using graph pyramids. Image andVision Computing, December 2008.125

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Open Issues and Chances for Topological Pyramids

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