Titre : TDA and SHAPE RECOGNITION Armel MAGANGA MIHINDOU Work in progress Joint with

1. Titre : TDA and SHAPE RECOGNITION Armel MAGANGA MIHINDOU Work in progress Joint with DRISS BENNIS ProfessorHabilited My Ismail MAMOUNI : ProfessorHabilited

2. Plan Introduction 1. Problem - Cognitiv dysfonction undercosmic radiation 2. ImageryApproach - Shape recognition - Segmentation 3. Mathematicaltools - Persistent Homology - graphe theory 4. Computational 4.1 Algorithm Conclusion

3. Introduction: • The environnement in witchwemouve , gives a large diversity of situations fromwitch man afterobservedthem, isinvited to suggest the links witch support and explainconsistencies. • Afterthat, weexpect the man’s skills to show the cartesian relations betweennaturalsphenomens. Thus, this man canquantify, qualify, predict and estimate the situations convertibles in numerallogic. • In thisnumerallogic, wealso have to need how to convertthe partition of a picture in the graph?

4. 1. Problem The cognitiv dysfonction isremarcably the aimtask of severalsectors. Because the cognitivattempt of humanbeingis for long time copied by artificial intelligence to run the powerfull computer. To do this, ingeniersneed to represent the neuronal network and for less part theyneed to know how works the neuronal network. In the medecin, manydeseasestouch the brain and in this case, doctors and otherssearch to catch the picture of the brain and analyse it. Withthe aerospace innovation, weexpect to travel and land in manyplaneteslike mars, mercure and so far… But the trip should not beeasy, because all spaceiscovered by cosmic radiation comesfrommany source and principalyfrom supernova( choc betwenntwo stars). This radiation canmake a cognitiv dysfonction of astraunot. In United State of America, since a manyyears, a NASA team has been constitued to study how the radiation can cause damage and how to protect the astronaut. To this, theymakemousesin laboratory and expose them to manycentigray of radiation of oxygen and titaniumisotop. Aftertowelve and fortyfourweeks, theynoticed a modification of the mindbehaviors, theygetpictures of neuronal networks before and after test. Our workconcernsspecialy the damage issuedfrom the cosmic radiation. and ourpulposeis to compare picturesbefore and after the cosmique radiation.

5. Spines and dendrites. • Neuralsreceivedsignalsfrom dendrites and send information throwtheirextend legs.

6. Neural network before test Image originale 1.

7. 48 Ti 5 cGy Image after 2. The mouses have been exposedunder 5 centigray of 48Ti aftertwolveweeks, we have observedcognitivfeatures changement. contacts betweentwoneurons, from axone to a dendrit to spine. The dendrits are in green. The spine are in red.

8. Image after 3. The mouses have been exposedunder 30 centigray of 48Ti aftertwolveweeks, we have observedcognitivfeatures changement.

9. Statisticalsresutls. In thishistogramm, weseethatunder 0 centigray of none isotopwe have a highlevel of spinedensity. And under 5 and 30 centigray of Titanium radiation, we have a lowlevel of spinedensity. Wewonder, if the densityis not more the sameafter test, wecanobserved thatwe have loosesomespine. And then the graph after segmentation can not be the same. Wewouldlike to developpe a new approachwitchcangive us a new method to compare the result by comparing the images issuedfrombefore and after test. This methodbegins by segmentation and goes to take a graph issued by segmentation and developpepersitenthomology of this graph. Finalywe compare the persistent diagramme of all images betweenthem in the objectiv to renforece the results and get a persistent signature as difference of test. This technique isbased on imageryapproach and shape recognition.

10. 2. ImageryApproach Comparaison To this, wenaturalywonderourselveswithwitchtoolswecan do thistask? Naturaly, this comparaison can not bedonemanuely. Becauseittakes time, and each time to compare, wealways call a template. This templateis original pictureproducedbefore test. Withcomputers, this comparaison wouldbe more easy and precise. Thus, we use to compare with computers.

11. 3. Comparaison tools • But before to use the computer: • Weneed to recall a certain nomber of imageryapproch. • - Shape recognition • - Segmentation

12. In computer science, itshouldbeinnapropriate to do anything but not matchedtwoshapes. With people behaviors front of securityrequirements, weneed to catch some part of humanbeing and donesomegeometricoperations in the objectiv to get a numeralnumberwitchidentify the human. For most time, these parts are eye, fingerprincipaly. Thuswefindthatthese part are geometricalformthatwe call shape. All toolswitchallowed us to recognise the shape constitue a new approach in computer science based on mathematicaltheory. The shape recognition isnow new methodwitchcanoperate to compare twopictures. o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o Unique discretisationbecause the ellipse of eyeis unique for human

13. Briefhistory The shape recognition theorywasbuiltsince the begening of the last centurymanycontributorsfromseveraluniversitiesaroundword have allowed us to getadvantage front of this challenge. Features The shape recognition throw the computer requires a mathematical pattern. Techniques Many techniques are used to implementshape recognition. For instance, we have choose to use computer with python modules implementedthrowsimplicials complexes and filtredcomplexlike alpha complex.

14. segmentation Definition: The segmentation is a technique used to separate image in many part witch constitute a partition of image. Motivation We can be motivated by the knowledge of distribution a pixel gray val throw the image. We can also be motivated by the knowledge of regions created by a group of pixels witch have the same gray val. Or we can want how a singel gray val goes from one geometrical point to another.

15. Segmentation To this, many techniques are employed. But principaly three techniques are generaly used. We have: - Segmentation based by threshold - segmentation based on region growing - segmentation based on edges . For our case, and corresponding to our requirement, we have choosen the segmentation by region growing. Because, to recall it, we need to know the region connected to another, in the objectiv to create a adjancy matrix. With adjancy matrix we will build other mathematical aspect witch called simplicial complex and get from it a topological identification that they call bare core from persistent homology.

16. segmentation • Target:

17. Mathematicaltools introduction To compare picture, required to makeoperations in picturesuchthat: - distance betweenmost importants parts of picture. - number of parts - geometrical position of each part Wenote thatpicture comparaison, ismathematicalorientedfeatures. And whenwe note distance between parts and geometricalpostion, we call an mathematical aspect calledtopology. Wenwe note comparaison, we note part growing to math them. Thus, wealso call an mathematical aspect calledalgebraictopology.

18. Mathematicaltools. Wecanaskourselvesthatwhywe are obliged to use mathematical aspects before to use computer. • Because, weneed to compute the sequence of comparaison. Likewhenwebuild an algorithm. Weneed to know how theory, how rational methodinvolve the task. • Now, we know thatweneedalgebraictopology to computepictures comparaison.

19. Mathematicaltools • persistent homology persistent homologycanbeused to measure the scale or resolution of topologicalfeature. There are twoingredients, one geometric, defining a function on a topologicalspace, and the otheralgebraic, turning the functionintomeasurements. The measurementsmakesenseonly if the functiondoes. For our case, the topologyspacewillbe an important cloud of points issuedfrom a discretisation of our image to getstep by stepsimplicial complexe.

20. Simplicial complexe: • Definition: A simplicial complex is a collection K of non-empty subsets of a set K0 such that τ ⊂ σ and σ ∈ K guarantees that τ ∈ K and {v} ∈ K for all v ∈ K0. The elements of K0 are called vertices of K, and the elements of K are called simplices. Additionally, we say that a simplex has dimension p or is a p-simplex if it has a cardinality of p + 1.

21. In the same time wedefine a chain complexe by a abelian group of simplicies and boundary application ∂ suchthat∂no∂n+1=0 The construction of thischain complexe gives us an algebraicbehaviorthatwe call filtration and throwthis filtration wegetmany techniques to buildpersistencehomology. Amongthese techniques we have matrixreductionwitch use boundarymatrix.

22. Persistent homology • For our case, wechoose alpha complexbecausethisking of complexe isrelated to voronoiregion and our python module works on voronoialreadyimplemented. • Voronoiregion • We subdivide the space Rd into regions of points that are closest to any of the points in S. precisely, for any s ∈ S, we define :

23. Alpha complex • We continue to assume that S is a finite set of points in Rd. Using the Voronoi decomposition, one can define a simplicial complex. Let ᵋ> 0, and let Sᵋ denote the . For every s ∈ S, consider the intersection Vs∩B(s, ᵋ).

24. The collection of these sets forms a cover of Sᵋ, and the nerve complex of this cover is called the alpha (α) complex of S at scale ᵋ and is denoted by Aᵋ(S). The Nerve Theorem applies, and it therefore follows that Aᵋ(S) has the same homology as Sᵋ.

25. Persistent homology

26. Persistent homology

27. Graph theory Definition A graphe is a topologicalspace in dimension one noted by G = (S,A) where S is collection of vertices, and A collection of edges.

28. Graph theory • Topologicaly, a graphe is a simple structure. It canbeplanar or not. • Vocabulary Weoftenconsiderverticeswithnaturalnomber i ∈ {1...n} uniques. Wedistinguishseveralforms of cycles : *Cycle (generaly) : A set of edges {e1, ..., en} suchthat ∀i, ∃ u v w, suchthatei = (u,v) et ei+1 = (v,w) otherwiseedgesconstitute a walk. In addition, we have a closedwalkwith, e1 = (u,v) en = (w,x) → u = x *Elementary cycle : is a cycle in witch the walkdoesn’t have twice the samevertice. *(hole) : For couple of vertices of cycle non adjacent in cycle no edge relies them.

29. Graph theory

30. Graph theory If for examplewesearch to determinewhoisrelated to otheramongworkers and weprecisethatweneed to know who are workingtogether in the sametask. The graph as we know simplycan not saysomething. It can shows the workersrelated and not whoworkwithwho in the sametask. In fact, a problematiclikethisisformaly a description of a logical proposition witchdepend on twiceunknown. The resultisnaturaly a mathématical set as wesee in sqldatabase. From graph, to drawthis set, wewillbeobliged to considersubset of graph as result. Fromthisconsideration, weextend the notion of graph and build a new perspectiv of graph thatwe call hypergrapwitchalsoberegardedthrowdiscretisation and createotherapproach : simplicialcomplex.

31. Graph theory Clique

32. Computational To compute let taketools to build graph from image First : language : Python vers: 2.7.14 Second: Modules : Matplotlib(py2.py3), NUMPY(1.9), SCIPY(0.11), CV2(2.4.9) 32bits/win32 Editor: PYZO Three : OS Windows 10 64bits

33. Computational • get image #call the image • img_filename = "user1.jpg" • im = Image.open(img_filename) • build a pixels matrix #get size of image • i,j=im.size #get pixels matrix • for u in range(i): • for v in range(j): • (rouge,vert,bleu) = im.getpixel((u,v)) • #grey_val=0.299*rouge+0.587*vert+0.114*bleu • grey_val=rouge+vert+bleu • matrice_pixels[u][v]=grey_val

34. Computational - weightmatrix(matrix of differences) - define a treshold - build graph • i volontaryremovealgorithm of thisstepsbecause, wewillseethat in cv2 module of python, we have a vonoroi module witchautomaticalyproduce a graphe of image after segmentation. And this segmentation isalsodone by voronoiregions.

35. Conclusion Noticedthatthrow all speech, we mention the choice of filtration withoutcubicalcomplextheorywitchisadapted to get persistent homologyfrom images. Wewant to saythat the filtration by voronoiregionis a choicegiven by a python module used to do segmentation of image. As fromthis segmentation wegetdirectly a regionalisationunder a graph, thenwecannaturely use simplicials complexes to obtain the persistent homology. The article DOI: 10.1515/amcs-2016-0030, published by THANH THE VAN and THANH MANH LE of University of Sciences/Hue University of Vietnam promotes the graph signature witch allow them to store data from segmentation of image. The method they use is not persistent homology. We would like to use persistent homology to proove that it is possible to store an image in database by this persistent signature witch gives a new measure of difference between two aspects of networks issued from neural networks before and after test. In addition,from a part of grah to another, we would like to navigate under clique graph and this consideration promote what we call : deeper navigation.