Persistent topology for natural data analysis - A survey
PPersistent topology for natural data analysis —A survey
Massimo Ferri
Dip. di Matematica and ARCES, Univ. di Bologna, Italy [email protected] , Abstract.
Natural data offer a hard challenge to data analysis. One setof tools is being developed by several teams to face this difficult task:Persistent topology. After a brief introduction to this theory, some ap-plications to the analysis and classification of cells, liver and skin lesions,music pieces, gait, oil and gas reservoirs, cyclones, galaxies, bones, brainconnections, languages, handwritten and gestured letters are shown.
Keywords:
Homology; Betti numbers; size functions; filtering function; classi-fication; retrieval.
What is the particular challenge offered by natural data, which could suggestthe need of topology, and in particular of persistence? Simply said, it’s qualityinstead of quantity. This is especially evident with images.If one has to analyze, classify, retrieve images of mechanical pieces, vehicles,rigid objects, then geometry fulfills all needs. On the images themselves, matrixtheory provides the transformations for superimposing a picture to a template.More often, pictures are represented by feature vectors, whose components aregeometric measures ( shape descriptors ). Then recognition, defect detection, re-trieval etc. can be performed on the feature vectors.The scene changes if the depicted objects are of natural origin: the rigidity ofgeometry becomes an obstacle. Recognizing the resemblance between a sittingand a standing man is difficult. The challenge is even harder when it comes tobiomedical data and when the context is essential for the understanding of data[34,51].It’s here that topology comes into play: the standing and sitting men are homeomorphic , i.e. there is a topological transformation which superimposes oneto the other, whereas no matrix will ever be able to do that. It is generally difficultto discover whether two objects are homeomorphic; then algebraic topology turnshelpful: It associates invariants — e.g. Betti numbers — to topological spaces,such that objects which are homeomorphic have identical invariants (the conversedoes not hold, unfortunately).(Algebraic) topology seems then to be the right environment for formalizingqualitative aspects in a computable way, as is nicely expressed in [35, Sect. a r X i v : . [ m a t h . A T ] A ug .1]. There is a problem: if geometry is too rigid, topology is too free. This isthe reason why persistent topology can offer new topological descriptors (e.g.Persistent Betti Numbers, Persistence Diagrams) which preserve some selectedgeometric features through filtering functions . Classical references on persistenceare [50,12,8,22].Persistent topology has been experimented in the image context, particularlyin the biomedical domain, but also in fields where data are not pictures, e.g. ingeology, music and linguistics, as will be shown in this survey. It is out of the scope of this survey to give a working introduction to homologyand persistence; we limit ourselves to an intuitive description of the concepts,and recommend to profit of the technical references, without which a real un-derstanding of the results is impossible. An essential (and avoidable) technicaldescription of a particular homology is reported in Section 2.1.
Homology
There is a well-structured way (technically a set of functors) toassociate homology vector spaces (more generally modules) H k ( X ) to a simplicialcomplex or to a topological space X , and linear transformations to maps [33,Ch. 2] [23, Ch. 4]. Betti numbers
The k -th Betti number β k ( X ) is the dimension of the k -thhomology vector space H k ( X ), i.e. the number of independent generators (ho-mology classes of k -cycles ) of this space. Intuitively, β ( X ) counts the number ofpath-connected components (i.e. the separate pieces) of which X is composed; β ( X ) counts the holes of the type of a circle (like the one of a doughnut); β ( X )counts the 2-dimensional voids (like the ones of gruyere or of an air chamber). Homeomorphism
Given topological spaces X and Y , a homeomorphism from X to Y is a continuous map with continuous inverse. If one exists, the two spacesare said to be homeomorphic . This is the typical equivalence relation betweentopological spaces. Homology vector spaces and Betti numbers are invariantunder homeomorphisms. Remark 1.
As hinted in the Introduction, geometry is too rigid, but topologyis too free. In particular, homeomorphic spaces can be very different from anintuitive viewpoint: the joke by which “for a topologist a mug and a doughnutare the same” is actually true; the two objects are homeomorphic!
Persistenttopology then tries to overcome this difficulty by studying not just topologicalspaces but pairs, once called size pairs , (
X, f ) where f is generally a continuousfunction, called measuring or filtering function , from X to R (to R n in multi-dimensional persistence ) which conveys the idea of shape, the viewpoint of theobserver. Shape similarity is actually very much dependent on the context. TheBetti numbers of the sublevel sets then make it possible to distinguish the twoobjects although they are homeomorphic: see Figure 1. ig. 1. Sublevel sets of mug and doughnut.
Sublevel sets
Given a pair (
X, f ), with f : X → R continuous, given u ∈ R ,the sublevel set under u is the set X u = { x ∈ X | f ( x ) ≤ u } . Persistent Betti Numbers
For all u, v ∈ R , u < v , the inclusion map ι u,v : X u → X v is continuous and induces, at each degree k , a linear transformation ι u,v ∗ : H k ( X u ) → H k ( X v ). The k -Persistent Betti Number ( k -PBN) function assigns to the pair ( u, v ) the number dim Im ι u,v ∗ , i.e. the number of classesof k -cycles of H k ( X u ) which “survive” in H k ( X v ). See Figure 2 (left) for the1-PBN functions of mug and doughnut. Note that a pitcher, and more generallyany open container with a handle, will have very similar PBNs to the ones of themug; this is precisely what we want for a functional search and not for a strictlygeometrical one. Fig. 2.
From left to right: 1-PBN functions of mug and of doughnut, 1-PDs of mugand of doughnut.
Persistence Diagrams . The k -PBN functions are wholly determined by the po-sition of some discontinuity points and lines, called cornerpoints and cornerlines (or cornerpoints at infinity ) The coordinates ( u, v ) of a cornerpoint representthe levels of “birth” and “death” respectively of a generator; the abscissa of acornerline is the level of birth of a generator which never dies. The persistence ofa cornerpoint is the difference v − u of its coordinates. Cornerpoints and corner-lines form the k -Persistence Diagram ( k - PD ). Figure 2 (right) depicts the 1-PDsof mug and of doughnut. For the sake of simplicity, we are here neglecting thefact that cornerpoints and cornerlines may have multiplicities. emark 2. Sometimes it is important to distinguish even objects for which thereexists a rigid movement superimposing one to the other — so also geometricallyequivalent — as in the case of some letters: context may be essential! See Figure3, where ordinate plays the role of filtering function.
Fig. 3.
Above: the objects “M” and “W”. Below, from left to right: 0-PBN functionsof M and of W, 0-PDs of M and of W.
Matching distance
Given the k -PDs D X,f , D Y,g of two pairs (
X, f ) , ( Y, g ),match the cornerpoints of D X,f either with cornerpoints of D Y,g or with theirown projections on the diagonal u = v ; the weight of this matching is the supof the L ∞ -distances of matching points. The matching distance (or bottleneckdistance ) of D X,f and D Y,g is the inf of such weights among all possible suchmatchings.
Natural pseudodistance
Given two pairs (
X, f ) , ( Y, g ), with
X, Y homeomor-phic, the weight of a given homeomorphism ϕ : X → Y is sup x ∈ X | g ( ϕ ( x )) − f ( x ) | .The natural pseudodistance of ( X, f ) and (
Y, g ) is the inf of these weights amongall possible homeomorphisms. If we are given the k -PDs of the two pairs, theirmatching distance is a lower bound for the natural pseudodistance of the twopairs, and it is the best possible obtainable from the two k -PDs. Much is knownon this dissimilarity measure [20,19,21]. There are several homologies. The classical and most descriptive one, at least forcompact spaces, is singular homology with coefficients in Z ; we refer to [33, Ch. 2]for a thorough exposition of it. Anyway, the homology used in most applicationss the simplicial one, of which (with coefficients in Z ) we now give a very shortintroduction following [23, Ch. 4]. Simplices A p -simplex σ is the convex hull, in a Euclidean space, of a set of p + 1 points, called vertices of the simplex, not contained in a Euclidean ( p − generated by its vertices. A face of a simplex σ is the simplex generated by a nonempty set of vertices of σ . Simplicial complexes
A finite collection K of simplices of a given Euclideanspace is a simplicial complex if 1) for any σ ∈ K , all faces of σ belong to K , 2) theintersection of two simplices of K is either empty or a common face. The space of the complex K is the topological subspace of Euclidean space | K | formed bythe union of all simplices of K . Simplicial homology with Z coefficients . Given a (finite) simplicial com-plex K , call p -chain any formal linear combination of p -simplices with coefficientsin Z (i.e. either 1 or 0, with 1 + 1 = 0). p -chains form a Z -vector space C p .Note that each p -chain actually identifies a set of p -simplices of K and that thesum of two p -chains is just the symmetric difference (Xor) of the correspond-ing sets. We now introduce a linear transformation ∂ p : C p → C p − (called boundary operator ) for any p ∈ Z . We just need to define it on generators, i.e.on p -simplices, and then extend by linearity. Writing σ = [ u , u , . . . , u p ], wedenote by [ u , . . . , ˆ u j , . . . , u p ] its face generated by all of its vertices except u j ( j = 0 , . . . , p ). Then we define ∂ p ( σ ) = n (cid:88) j =0 [ u , . . . , ˆ u j , . . . , u p ] Fig. 4.
Cycles.
It is possible to prove that ∂ p ∂ p +1 = 0, so that B p = Im ∂ p +1 is contained in Z p = Ker ∂ p . Elements of B p are called p -boundaries ; elements of Z p are called p -cycles . The p -homology vector space is defined as the quotient H p ( K ) = Z p /B p .omology classes are represented by cycles which are not boundaries. Two cyclesare homologous is their difference is a boundary. In Figure 4, representing thesimplicial complex K formed by the shaded triangles and their faces, the bluechain b is a 1-cycle which is also a boundary; the red chain c and the green one c (cid:48) are 1-cycles which are not boundaries; c and c (cid:48) are homologous. The application of persistence to shape analysis and classification has a longstory, since it started in the 90’s when it still had the name of Size Theory [50].In the last few years it has taken various, very interesting forms. The constantaspect is always the presence of qualitative features which are difficult to captureand formalize within other frames of mind.
Leukocytes, or white blood cells, belong to five different classes: lymphocyte;monocyte; neutrophile, eosinophile, basophile granulocytes. Eosinophile and neu-trophile granulocytes are generally difficult to be distinguished, so they wereconsidered in a single classification class in an early research by the Bolognateam [26].
Fig. 5.
A radius alongwhich the three filteringfunctions are computed.
Fig. 6.
Persistent Betti Number functions relative to the sumof grey tones (different colors represent different values).
As a space, the boundary of the starlike hull of the cell is assumed. Theimages are converted to grey tones.hree filtering functions are put to work, all computed along radii from thecenter of mass of the cell (Figure 5): – Sum of grey tones – Maximum variation – Sum of variations pixel to pixel.Classification (with very good hit ratios for that time) is performed by mea-suring distance from the average PBN function of each class.
Again in Bologna we faced recognition of handwritten letters with time infor-mation; our goal was to recognize both the alphabet letter and the writer [25].The space on which the filtering functions are defined is the time interval ofthe writing. The filtering functions are computed in the 3D “plane-time”: – Distance of points from the letter axis – Speed – Curvature – Torsion – Distance from center of mass (in plane projection).
Fig. 7.
A monogram with its outline (above) and the directions along which the filteringfunctions are computed (below).
Classification comes from fuzzy characteristic functions, obtained from nor-malized inverse of distance. Cooperation of the characteristic functions comingfrom the single filtering functions is given by their rough arithmetic average.A later experiment, which was even repeated live at a conference, concernedthe recognition of monograms for personal identification, without time informa-tion [24].wo topological spaces are used. The first is the outline of the monogram andthe filtering function is the distance from the center of mass (see upper Figure7). The second space is a horizontal segment placed at the base of the monogramimage. Filtering functions: – Number of black pixels along segments (3 directions) (see lower Figure 7) – Number of pixel-pixel black-white jumps (3 directions).Classification is performed by a weighted average of fuzzy characteristic func-tions.
Automatic recognition of the symbols expressed by the hands in the sign lan-guage is a task which was of interest for different teams. The first one was thegroup led by Alessandro Verri in Genova [49]. The signs were performed with awhite glove on a black background; translation into common letters was done inreal time in a live demo at a conference.The domain space is a horizontal segment; the filtering functions assign toeach point of the segment the maximum distance of a contour point within astrip of fixed width, with 24 different strip orientations.
Fig. 8.
Four filtering functions and the corresponding 0-Persistent Betti Number func-tions.
The choice of S. Wang in Sherbrooke, instead, is to use a part of the contour,determined by principal component analysis, as a domain and distance fromcenter of mass as filtering function [32].The team of D.Kelly in Maynooth uses the whole contour as domain, anddistances from four lines as filtering functions [36] (see Figure 8). ig. 9.
Four filtering functions on silhouette stacks for gait identification.
Personal identification and surveillance are the aim of a research by the Cubanteam of L. Lamar-Le´on, together with the Sevilla group of computational topol-ogy [37].Considering a stack of silhouettes as a 3D object, and using four different fil-tering functions, makes 0- and 1-degree persistent homology a tool for identifyingpeople through their gait (Figure 9).
S. Banerjee in Kolkata makes use of persistence on sequences of satellite imagesof cloud systems (Figure 10), in order to evaluate risk and intensity of forminghurricanes [2].
Fig. 10.
Time evolution of cyclones.
Time interval is the domain of two filtering functions which are commoncharacteristic measures of cyclones: – Central Feature portion – Outer Banding Feature .6 Galaxies
Again S. Banerjee [3] applies similar methods to another type of spirals: galaxies.Various filtering functions are used. One is defined as a function of distancefrom galaxy center, and is the ratio between major and minor axis of the corre-sponding isophote. Another one is a “pitch” parameter defined by Ringermacherand Mead [45]. A third filtering function is a compound based on color.The classification results agree with the literature.
In [48] a powerful construction (the Persistent Homology Transform) is intro-duced. It consists in gathering the “height” filtering functions according to allpossible directions. The paper shows that the transform is injective for objectshomeomorphic to spheres. By using the transform it is possible to define aneffective distance between surfaces. An application is shown by classifying heelbones of different species; the comparison with the ground truth produced byusing placement of landmarks on the surfaces is very good.
A very important part of natural shape analysis is the detection of malignantcells and lesions, since there generally are no templates for them. As far as weknow, the first attempt through persistence (called size theory at that time) isthe ADAM EU Project, by the Bologna team together with CINECA and withI. Stanganelli, a dermatologist of the Romagna Oncology Institute [17,47,27].The analysis is mainly based on asymmetry of boundary, masses and color dis-tribution: the lesion is split into two halves by 45 equally spaced lines, and thedifference between the two halves is measured by the matching distance of thecorresponding Persistence Diagrams.
Fig. 11.
One of the 45 splittings of a melanocytic lesion, and the whole A -curve cor-responding to the filtering function luminance. he three functions ( A -curves) relating these distances to the splitting lineangles give parameters which are then fed into a Support Vector Machine clas-sifier.The same team is presently involved with a biomedical firm in the realizationof a machine for smart retrieval of dermatological images [28]. A morphological classification of normal and tumor cells of the epithelial tissueof the mouth is proposed in [40,41]: the filtering function is distance from thecenter of mass; the discrimination is statistically based on the distribution ofcornerpoints (see Fig. 12).
Fig. 12.
Distribution of cornerpoints in the diagrams of normal and tumor mouth cells.
The advantages of a multidimensional range for the filtering functions are shownin [1], where several classification experiments are performed on the images ofhepatic cells (see Figure 13). The domain space is the part of image occupied bythe lesion; the two components of the filtering function are the greyscale of eachpixel and the distance from the lesion boundary. ig. 13.
Various types of hepatic lesions.
So far we have seen applications of persistence to images of natural origin. Butthe modularity of the method opens the possibility to deal with data of verydifferent nature. A first example is given by [43], where persistence is used onthe Vietoris-Rips complex in a space where points are complex phenotypes re-lated together by the
Jaccard distance . This made it possible to find systematicassociations among metabolic syndrome variates that show distinctive geneticassociation profiles.
Researchers in Ufa and Novosibirsk need to get a reliable geological and hy-drodynamical model of gas and oil reservoirs out of noisy data; the model hasto be robust under small perturbations. The authors have found an answer inpersistent 0-, 1- and 2-cycles. The domain space is the 3D reservoir bed, and thefiltering function is permeability, obtained as a decreasing function of radioac-tivity [4] (Russian; translated and completed in this same volume). .13 Brain connections
A complex research on brain connections and their modification under the as-sumption of a psychoactive substance (psilocybine) is performed in [42] andextended in [39]. The construction starts with a complete graph whose verticesare cortical or subcortical regions; these, and their functional connectivity (ex-pressed as weights on the edges) come from an elaborate processing of functionalMRI data. Then the simplicial complex is built, whose simplices are the cliques(complete subgraphs) of the graph.The filtering function on each simplex is minus the highest weight of itsbuilding edges. A difference between treated and control subjects already appearsin the comparison of the 1-Persistence Diagrams (see Figure 14). Then moreinformation is obtained from secondary graphs (called homological scaffolds ),whose vertices are the homology generators weighted by their persistence.
Fig. 14.
Probability densities for H generators: placebo (left) and psilocybin (right)treated. There are other applications of persistence to brain research: evaluation ofcortical thickness in autism [16]; study of unexpected connections between sub-cortex, frontal cortex and parietal cortex in the form of 1- and 2-dimensionalpersistent cycles [31,46].
Among other mathematical applications to music, M.G. Bergomi in Lisbon col-laborates with various researchers in exploring musical genres by persistence [6].As a space they adopt a modified version of Euler’s
Tonnetz [9]. The filteringunction is the total duration of each note in a given track. Classification canbe performed at different detail levels: experimentation is reported on tonal andatonal classical music of several authors (an example is in Figure 15), on popmusic and on different interpretation of the same jazz piece.
Fig. 15.
0- and 1-persistence diagrams for three classical pieces.
A blend of persistence and deep learning is the central idea of a research bythe team of I.-H. Yang in Taiwan [38]. They input audio signals to a Convo-lutional Neural Network (CNN); after a first convolution layer, a middle layerprocesses the output of the first in two different complementary ways: one isa classical CNN; the other computes the persistence landscape (an informationpiece derivable from the persistence diagram [10]) of the same output. Whereasthe persistence layer by itself does not perform any better than the normal CNN,their combination gives very good results in terms of music tagging.
An interdisciplinary team at Caltech investigates the metric spaces built by 79Indo-European and 49 Niger-Congo languages [44]. These appear as points in aEuclidean space of syntactic parameters; on them a Vietoris-Rips complex [23,Sect. III.2] is built and Euclidean distance is assumed as filtering function. TheIndo-European family reveals one 1-dimensional and two 0-dimensonal persistentcycles, the Niger-Congo respectively none and one. The interpretation of thesedifferences and of the link with phylogenetic and historical facts is still underway.
Open problems
There is a number of open problems in persistence, whose solution will affectapplications to natural data analysis, and to which only partial answers havebeen given so far: – Optimal choice of the foliations along which to perform the 1D reduction ofmultidimensional persistence [13] – Study of the discontinuities in multidimensional persistence [11,15] – Understanding the monodromy around multiple cornerpoints [14] – Restricting the group of homeomorphisms of interest by considering the in-variance required by the observer [29] – Modulation of the impact of different filtering functions for search engineswith relevance feedback [30] – Use of advanced tools of algebraic topology [5] – Use of persistence in the wider context of concrete categories, not necessarilypassing through homology of complexes or of topological spaces [7].
There are at least two ways in which persistence will interact with machinelearning, and this is likely to enormously boost the qualitative processing ofnatural data [18]: – Feeding a neural network with Persistence Diagrams instead of raw data willconvey the needs and viewpoints of the user – Deep learning might yield a quantum leap in persistence, by automaticallyfinding the best filtering functions for a given problem.
Acknowledgments
Article written within the activity of INdAM-GNSAGA.
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